Citation
Tracking and disturbance rejection for nonlinear systems with applications to robotic manipulators

Material Information

Title:
Tracking and disturbance rejection for nonlinear systems with applications to robotic manipulators
Creator:
Whitehead, Michael L., 1958- ( Dissertant )
Kamen, Edward W. ( Thesis advisor )
Lasiecka, Irena ( Reviewer )
Peebles, Peyton Z. ( Reviewer )
Sandor, George N. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1986
Language:
English
Physical Description:
v, 157 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Differential equations ( jstor )
Dynamic modeling ( jstor )
Eigenvalues ( jstor )
Matrices ( jstor )
Robotics ( jstor )
Servomechanisms ( jstor )
Signals ( jstor )
Simulations ( jstor )
Sine waves ( jstor )
Trajectories ( jstor )
Feedback control systems ( lcsh )
Manipulators (Mechanism) ( lcsh )
Nonlinear theories ( lcsh )
Servomechanisms ( lcsh )
System analysis ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
The servomechanism problem (i.e., output tracking in the presence of disturbances) is considered for a class of nonlinear systems. Conditions are given which guarantee the existence of a solution to the problem. The resulting controller requires and internal model system in the feedback loop; however, due to the non-linearity of the system, the internal model must contain dynamics other than those found in the reference and disturbance signals. A robotic manipulator system has been considered as one possible application for the proposed control scheme and various hypotheses are tested with respect to this system. simulations are provided which demonstrated the performance of the control scheme when applied to a 2-link manipulator.
Thesis:
Thesis (Ph.D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 154-156.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Michael L. Whitehead.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Michael L. Whitehead. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
030303409 ( ALEPH )
16568635 ( OCLC )
AEP9804 ( NOTIS )

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Full Text










TRACKING AND DISTURBANCE REJECTION FOR NONLINEAR SYSTEMS
WITH APPLICATIONS TO ROBOTIC MANIPULATORS







By

MICHAEL L. WHITEHEAD


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA 1986














ACKNOWLEDGEMENTS


I would like to express my deep appreciation to Dr. Edward W. Kamen for the suggestions and encouragement he provided over the duration of my graduate studies. It is safe to say that without his continuous support, this research would not have been completed. I would also like to thank Dr. Thomas E. Bullock for his valuable comments concerning this work. These comments were needed, and the time he spent reviewing the paper is greatly appreciated. Special thanks also go to Dr. Irena

Lasiecka, Dr. Peyton Z. Peebles, and Dr. George N. Sandor for serving on my Supervisory Committee; all three are professors under whom I have had the pleasure of taking courses.















TABLE OF CONTENTS


ACKNOWLEDGEMENTS . ii

ABSTRACT . v

CHAPTERS

ONE INTRODUCTION .1

TWO TRACKING AND DISTURBANCE REJECTION FOR
NONLINEAR SYSTEMS . 8

Notation . 8 Main Results for the Nonlinear Servomechanism Problem. 9
Stability of the Closed-Loop Transient System . 26
The Relation Between the Dimension of the Internal
Model System and the Input/Output Dimensions . 29
Summary . 34

THREE APPLICATION TO LINEAR SYSTEMS . 37

Review of Linear Servomechanism Results . 37
Solution to the Linear Problem via the
Nonlinear Formulation . 39

FOUR FEEDBACK CONTROL . 47

Stabilization Using the Linearized Equation . 47 Optimal Feedback for the Linear Servomechanism Problem. 66
Increased Degree of Stability Using the Optimal
Control Approach . 71


FIVE PRACTICAL CONSIDERATIONS . 74

Controller Based on the Nominal Trajectories . 74 Feedback Gain Selection . 80
Robustness with Respect to Generation of the
Nominal Signals . 84
Digital Implementation . 86









SIX APPLICATION TO THE ROBOTIC MANIPULATOR . 94

Manipulator Dynamics. 94 Actuator Driving Torques . 96
Feedback Control System for Tracking and Disturbance
Rejection. 97
Determining the Dynamics of the Internal Model System. 98
Feedback Gain Calculation . 101
Compensation for Flexibilities in the Manipulator's
Links . 106

SEVEN SIMULATION RESULTS . 109

The Simulated System . 109 Control about a Stationary Configuration . 112 Control over a Time-varying Nominal Trajectory . 124 Correcting for Flexibilities . 147

EIGHT CONCLUSIONS AND OPEN PROBLEMS . 153

REFERENCES . 155

BIOGRAPHICAL SKETCH . 158













Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

TRACKING AND DISTURBANCE REJECTION FOR NONLINEAR SYSTEMS
WITH APPLICATIONS TO ROBOTIC MANIPULATORS

By

MICHAEL L. WHITEHEAD

December 1986

Chairman: Dr. Edward W. Kamen Major Department: Electrical Engineering

The servomechanism problem (i.e., output tracking in the presence of disturbances) is considered for a class of nonlinear systems. Conditions are given which guarantee the existence of a solution to the problem. The resulting controller requires an internal model system in the feedback loop; however, due to the nonlinearity of the system, the internal model must contain dynamics other than those found in the reference and disturbance signals.

A robotic manipulator system has been considered as one possible application for the proposed control scheme and various hypotheses are

tested with respect to this system. Simulations are provided which

demonstrate the performance of the control scheme when applied to a 2link manipulator.














CHAPTER ONE
INTRODUCTION

One of the most important problems in applications of feedback control is to provide output tracking in the presence of external disturbances. This is commonly referred to as the servomechanism problem. More precisely, given a certain system, the servomechanism problem involves the design of a controller which enables the output to asymptotically track a reference signal r(t), in the' presence of a

disturbance w(t), where r(t) and w(t) belong to a certain class of functions. The class of functions might be, for example, combinations of step, ramp and sinusoidal signals. The frequency of the signals is usually assumed to be known. Typically, enough freedom is allowed,

however, so that it is not necessary to have apriori knowledge of the amplitude or phase of either the disturbance or the reference.

The assumption of known frequency but unknown amplitude and phase

provides a realistic model for many reference and disturbance signals encountered in practice. For example, an imbalance in a piece of

rotating machinery might cause a sinusoidal disturbance force to act on a certain system. Although the frequency of this force might be easy to

predict, it is doubtful that the exact amplitude could be determined. Even if the amplitude was known exactly, modeling errors in the plant would make such schemes as open-loop compensation unreliable. This

leads to an important feature of a controller design to solve the

servomechanism problem. Namely, there should be a certain amount of robustness with respect to plant variations and with respect to

variations in signal level.








The servomechanism problem has been successfully dealt with for linear, time-invariant systems. Many results are available [1-11] and an excellent summary is provided by Desoer and Wang [2]. A more

abstract discussion is give by Wonham [11]. It has been shown that an essential ingredient in a controller designed to solve the servomechanism problem is an internal model system. This internal model

system is a system which replicates the dynamics of the exogenous signals (i.e., reference and disturbance) in the feedback loop.

Because any real system is seldom linear, it is important to consider the servomechanism problem for nonlinear systems. Some results

exist for the nonlinear problem [12-15]; but for the most part, the results apply only when the reference and disturbance signals are constant.

Desoer and Wang [12] have approached the problem using input-output techniques. They have considered a linear system with nonlinearities

both preceeding it (input channel nonlinearities) and following it (output channel nonlinearities). They first treat the case of input channel nonlinearities (such as a sensor nonlinearity). Conditions are given as to when tracking (disturbance rejection is not considered) will occur. Although conditions are given, no method is provided which will enable one to construct a suitable controller nor is discussion given as

to ways of testing the conditions. The main results derived by Desoer and Wang, however, are for memoryless nonlinearities (both input and output channel). These results are valid only for reference and disturbance signals which tend to constants. The conditions given for a solution to the problem are precise, however, it appears that the

algorithm recommeded for selecting the control law is useful only for








single-input single-output systems. As expected, the resulting

controller requires integrators in the feedback loop.

Solomon and Davison [13] have used state-space techniques to treat

the servomechanism problem for a certain class on nonlinear systems. They too have considered only constant reference and disturbance signals. In addition, the nature of the disturbance is such that it affects the output directly without affecting the dynamics of the nonlinear system. It can be shown that such a disturbance can be regarded as simply a change in the level of the reference signal. Various assumptions are made and conditions are given stating when it is

possible to solve this servomechanism problem. The resulting control

law employs integrators in the feedback loop and nonlinear feedback is used to give global stability. Although global stability is obtained, the range in amplitude of the reference and disturbance signals which can be applied is limited.

Some appealing results, again only for the case of constant reference and disturbance signals, are derived in Desoer ad Lin [14] and in Anantharam and Desoer [15]. Desoer and Lin have shown that if the nonlinear plant has been prestabilized so that it is exponentially

stable and if the stabilized plant has a strictly increasing, dc, steady-state, input-output map then the servomechanism can be solved with a simple proportional plus integral controller. Using such a

control scheme, it is necessary that the gains of the integrators be sufficiently small and that the proportional gain be chosen appropriately. Anantharam and Desoer have derived results virtually identical to those found in Desoer and Lin. In their paper, however,

the proof is somewhat different and a nonlinear discrete-time system is








treated where proportional plus sum (discrete-time integral) control is employed.

Other than the case of constant reference and disturbance signals,

it appears that there were no satisfactory results for the nonlinear servomechanism problem. One could attempt to linearize the nonlinear system and design a controller based on linear servomechanism theory. However, such an approach will usually lead to steady-state tracking

error.

In this dissertation, the servomechanism problem is solved for a class of multi-input, multi-output, nonlinear systems. Here the results are valid for reference and disturbance signals which belong to a much wider class of signals than simply those which tend to constants.

The major contributions of this research are:

1. Conditions are given for a solution to the nonlinear servomechanism

problem. When these conditions are not satisfied exactly, employing the type of controller developed here still makes

intuitive sense.

2. The problem is solved in the time-domain using a completely new

approach. A time-domain approach is necessary because standard

techniques (i.e., frequency domain analysis) used for solving the

linear problem are not applicable to nonlinear systems.

3. It becomes apparent that the idea of an internal model system which

contains the dynamics of only the reference and disturbance signals

is not complete. It is shown that actually, the internal model system should include the dynamics found in both the input and in

the state which must be present during successful tracking.








4. The controller is rather simple to implement. The internal model

system is linear and stabilization is accomplished by the use of

well known linearization techniques.

The main results of this paper are contained in Chapter Two. Here the servomechanism problem is solved for a nonlinear system having the same number of inputs as outputs. Later, a method is introduced so that the results can be extended to a nonlinear system having more inputs than outputs.

The assumptions needed in the derivations are that a solution to the problem does indeed exist and that when tracking does occur, both the state and input will satisfy a linear differential equation.

Although the latter assumption is restrictive, when it does not hold, a design based on such an approximation may still result in very small tracking error.

After the assumptions are stated, an internal model system is introduced. This internal model system replicates the dynamics found in the state and input signals which are necessary to achieve tracking. The concept of including the dynamics of the state and input rather than the common practice of including the dynamics of the reference and disturbance is believed to be new.

The next step in the design involves the use of constant gain feedback with the internal model systm incorporated into the feedback loop. It is shown that observability of the internal model system, through its associated feedback gain, insures that zero tracking error will occur for all time provided the initial state of the combined plant and controller has the correct value. Since such an initial state is unlikely to occur in practice, it is next shown that certain stability








conditions will allow the true state trajectory to asymptotically converge to the trajectory which gives zero tracking error. These

stability conditions are easily checked using Liapunov's indirect method. It is noted, however, that with Liapunov's approach, the tracking error may only asymptotically converge to zero for a limited range of initial states. Roughly speaking, this can be considered

equivalent to requiring that the disturbance and reference signals remain small.

In Chapter Three, using the approach developed for the nonlinear problem, we rederive the well known conditions imposed for a solution to the linear servomechanism problem.

In Chapter Four, selection of locally stabilizing feedback based on linearization techniques is discussed in detail. Due to the complexity of the stability problem, the control law derived here is for timeinvariant systems which are acted upon by small reference and disturbance signals. Simulations of a nonlinear system are provided which verify the design technique. Also discussed in Chapter Four is the interpretation of using optimal control techniques to arrive at the feedback law required for the linear servomechanism problem. In a

nonlinear system, however, a certain degree of stability is often desired. Consequently, in order to achieve this stability using optimal control theory, a well known technique due to Anderson and Moore [16] is presented.

In Chapter Five, we develop a controller designed to solve the nonlinear servomechanism problem when a nominal input and state trajectory are supplied as open-loop commands. Here essentially no new theory is needed since the control problem can actually be treated using









previously developed ideas. Designing the controller about a nominal

trajectory is a standard technique often used in dealing with nonlinear systems. This technique usually reduces the stability requirements of

the feedback law since it is assumed that the state trajectory will never deviate far from a prescribed nominal trajectory.

Another topic discussed in Chapter Five is that of using a discrete-time control law to approximate the already developed continuous-time control law.

In Chapter Six we consider the robotic manipulator as a system for which to apply the results derived in this paper. The dynamic equations

modeling the manipulator are analyzed in view of the servomechanism problem. It is shown that the requirements needed for a solution to the servomechanism problem are satisfied for the robotic system. Also

considered for the manipulator is an application of tracking where the signals being tracked are used to compensate for errors in end-effector location due to link flexibilities.

Chapter Seven shows results, obtained using simulations, of applying the control scheme to a 2-link manipulator. These results

verify that the method will improve, or sometimes eliminate, steadystate tracking error.














CHAPTER TWO
TRACKING AND DISTURBANCE REJECTION FOR NONLINEAR SYSTEMS

In this chapter we derive a method to achieve tracking and disturbance rejection for certain nonlinear multi-input, multi-output systems. Conditions are given which reveal when the problem can be solved. An internal model system is used as a basis for the design,

however, unlike the case of the linear system, the internal model contains dynamics which may not appear in either the reference or disturbance signals.


Notati on


Given a positive integer n, let Rn denote the set of n-dimensional

vectors with elements in the reals and let Rmxn be the set of matrices of dimension mxn with elements in the reals. The symbol ii � Ii shall denote the Euclidian norm of a given vector. For a matrix A, the symbol ifAni will be the induced norm defined as
i


IIAIIi : sup oAxi = Exmax (AA)] (2-1)


The symbol := will mean equality by definition and the notation A' signifies the transpose of the matrix A.

When referring to square matrices, the notation A > 0, A > 0 A < 0 will mean that A is positive definite, positive semidefinite, and negative definite respectively.









Usually, capital letters (e.g., A, B, F) will denote matrices, while lower case letters (e.g., x, y, z) shall denote vectors. Both

standard lower case letters and Greek lower case letters shall indicate

a scalar. Any deviations made from the notation for matrices, vectors, and scalars will be clear from context.


Main Results for the Nonlinear Servomechanism Problem



Consider the nonlinear system


N: ;(t) = f(x(t), u(t), w(t))

y(t) = Hx(t)

e(t) = r(t) - y(t) (2-2)



where x(t) e Rn is the state of the system, u(t) e RP is the control input to the system, w(t) c Rd is a disturbance signal, y(t) e RP is the output of the system, and e(t) e RP is the error which occurs when

tracking the reference signal r(t) e RP. Note that the output is

assumed to be a linear function of the state and the dimension of the input is the same as that of the output.

It is our objective to design a closed-loop controller which will asymptotically regulate against disturbances and also asymptotically

track a reference signal. In particular, we desire e(t) = r(t) - y(t) + 0 as t + - where r(t) is a specified output chosen from a given class of functions.

The primary concern here will be tracking and disturbance rejection when the disturbance w(t) and the reference r(t) are comprised of









components which are either constant or sinusoidal in nature. To

simplify the development, we shall consider a particular disturbance signal, say w*(t), and a particular reference signal, say r*(t), to be represenative signals from a given class of functions. Once the
controller is derived with respect to these signals, the results can be generalized to cover a class of functions for which r*(t) and w*(t) are assumed to belong.

We now make the following assumptions:


(A.1) For some chosen reference signal r*(t) and a particular

disturbance w*(t) there exists an open-loop control u*(t) and an

initial state x*(O) = x* such that


x *t) = f(x*(t), u*(t), w*(t))

y*(t) = Hx*(t) = r*(t) (2-3)

e(t) = r*(t) - y*(t) = 0 for all t > 0


(A.2) The elements of both x*(t) and u*(t) satisfy the scalar, linear

differential equation

(.)(r) + ar-l(.)(r-1) + . + a1(.)(1) + a = 0 (2-4)




where the characteristic roots of (2-4) are all in the closed right half-plane.
The first assumption is merely a way of stating that it is possible

to provide output tracking. A typical example where (A.1) would not hold is for a system having more outputs than inputs. This particular








problem is avoided here, however, since we consider systems which have the same number of inputs as outputs.
The second assumption is perhaps the most restrictive. It is

different from the assumption commonly made in the linear servomechanism problem; namely, that the disturbance w*(t) and the reference r*(t) both satisfy a linear differential equation of the form given by (2-4). Here we are concerned with this class of disturbance and reference signals;

however, in the nonlinear case it is important to work also with the acutal state and input trajectories which arise during tracking.

If we assume that r*(t) satisfies a differential equation of the form given by (2-4), it is actually not unreasonable to assume that x*(t) will satisfy the same equation. This is because the output y*(t), which must be identically equal to r*(t) during tracking, is taken to be a linear combination of the state x*(t). Consequently, if all elements

of the state are reflected in the output, these elements must satisfy (2-4). The assumption on the input signal u*(t) is then the assumption which needs further discussion. In the nonlinear servomechanism

problem, it is often the case that u*(t) will contain terms (e.g., sinusoids) not present in either w*(t) or r*(t). To help clarify this point, consider the following proposition.


Proposition 2.1: Given the autonomous system


M(t) = f(x(t), u(t), w(t)) (2-5)


assume there exists an input u(t) such that x(t) = xp(t) is the solution to (2-5) with initial state xp(O) and with disturbance w(t) = wP(t).









Furthermore, assume that xp(t) and wp(t) are periodic with a common period of T. Under these conditions, for the same inital state xp(O) and disturbance wp(t), there exists a periodic input up(t) having a period of T which results in the state trajectory xp(t).


Proof: Make the definition


uT~tu {Mt


O

(2-6)


Then let


up(t) = E uT(t-nT)
n=O


Since xp(O) = xp(T) = xp(2T) = interval T, the result is obvious.


(2-7)


. and wp(t) repeats itself over every


Let xp(t), up(t), and wp(t) be periodic with a common period T and assume that the following differential equation is satisfied.


(2-8)


In Proposition 2.1 we have already asserted that a periodic up(t) will exist whenever xp(t) and wp(t) are periodic with a common period. Assuming up(t) is integrable over any period, let the Fourier series expansion of up(t) be

C0
Up(t) 0 o + E a kcos(kwt + k) (2-9)
k=1


Xp tM : f(x p(t), u p(t), W p(t))









where w is the fundamental frequency.

Given a positive integer K, let uK(t) denote the truncation


K
uK(t) � + E akcos(kwt + k) (2-10)
k= 1

The truncation UK(t) satisfies a differential equation of the form given by (2-4). Example 2.1 will show how to obtain the specific differential equation using Laplace transform theory.

Now let xK(t) denote the solution to


XK(t) = f(xK(t), uK(t), wp(t)) (2-11)


(assuming the solution xK(t) exists)

If lix p(t) - xK W)1 is suitably small for t > 0, the assumption that the input satisfies (2-4) is reasonable. Often, either by using

simulations or actual tests, it is possible to determine apriori how small lix p(t) - XK W11is for a given value of K. Also note that in practice there is always some error, so that demanding lix p(t) - xK(t)l = 0 is not reasonable.

We now mention an important practical point which was overlooked in the preceeding discussion. For ix p(t) - xK M11 to be suitably small, the nonlinear system N must be stable in the sense that bounded inputs give bounded outputs. If this is not the case, it would be necessary to

use a pre-stabilizing feedback so that the unstable portion of xK(t) could be eliminated. This allows one to make the most meaningful

assessment of how "good" the input uK(t) acutally is. The use of such a

stabilizing feedback would be needed only in simulations and testing








since later, a stabilizing feedback law will be developed for the actual implementation.
The following example shows how a differential equation of the form given by (2-4) is derived from a truncated Fourier series


Example 2.1

Suppose


uK(t) M o + k cos(kwt + k)
k=1k


(2-12)


Taking Laplace transforms of both sides we get


U K(s) =


N(s)
S[ I (S2 + K2w2


(2-13)


where N(s) is a polynomial in s. Equation (2-13) can be expressed as


K 2 22
s[ II (s + k W )]UK(s) = N(s)
k=1


(2-14)


Next, by writing


K 2 22
sE H (s2+ kw)] w as , J = 2K + 1
k=1 j=O



J ajsj] UK(S) = N(s)
j=O


(2-15) (2-17)


Now taking inverse Laplace transforms and noting that since N(s) is a polynomial in s and hence has an inverse Laplace transform consisting of


we have









impulse type terms we have


S di
. a. JuK(t) = 0 t > 0 (2-18)
j=O j dt


This is exactly of the form given by (2-4).


We now give further motivation for the assumption that u*(t) satisfies equation (2-4) by showing an example of a nonlinear system where this is indeed the case.


Example 2.2

Consider the system


k(t) = 3x(t) + x 2(t) + (2x 2(t) + 4)w(t) + u(t)

y(t) = x(t) (2-19)


where we desire y*(t) = r*(t) = 1 sinwt and the disturbance is

constant. Thus


x (t) = c1sinwt

w*(t) = a2 (2-20)

where a 1 and a2 are constants. Substituting (2-20) into (2-19) yields the following

1 2 2
tlwCOSot = 3oisinwt + l1-cos2 t) + [ I (l-cos2ot) + 4]a2 + u*(t)

(2-21)









Solving for u*(t) gives


u*(t) = lwCOSwt - 3aisinwt + a,22 )cos2wt - 12(+ -42

(2-22)
The characteristic polynomial for U*(s) is


(s2 + 4w2)(s2+w2)(s) = s' + 5w2s3 + 4w4s (2-23)



Hence, both u*(t) and x*(t) satisfy


d5 2 d3 4 d
- (.) + 5w - (.) + 4w a- (.) : 0 (2-24)
dt dt



Now that assumptions (A.1) and (A.2) have been justified, we proceed by introducing an internal model system. In the literature, an internal model system is usually taken as a system which replicates the dynamics of the reference and disturbance signals. Here it will take on a slightly different meaning which is made more precise by the following definitions. Let C . Rrxr and T e R r be defined as follows


0 I 0 . . . 0 0
0 0 1 . . . 0 0

C := . . . � T := . (2-25)

0 0 0 . 1 0
-a0 -a, -a2 . . . -ar 1


with the coefficients aj, i = 0,1, ., r-1 defined by (2-4).









Definition: Given the system N described by (2-2), suppose that for a particular r*(t) and a particular w*(t) assumptions (A.1) and (A.2) both hold. Then an internal model system of r*(t) and w*(t) with respect to N is a system of the following form:



*(t) = An(t) + Be(t)

e(t) = r (t) - y(t) = H[x (t) - x(t)] (2-26)



where

A = T-1 block diag. CC, C, ., C] T (2-27)

p blocks



B = T-1 block diag. [T, T, ., T] (2-28)

p blocks


where the state n(t) E Rpr, T is an arbitrary nonsingular matrix, B 6 Rprxp, and the pair (A,B) is completely controllable. In practice, T is usually taken as the identity matrix.

Roughly speaking, the above internal model system is seen to contain p copies of the dynamics of the state and input signals which must occur during tracking. This is different from the internal model system used in the linear servomechanism problem where only the dynamics of the reference and disturbance signals are included. The dynamics of the reference and disturbance will inevitably be included in (2-26); however, the nonlinear structure of N may necessitate the introduction of additional dynamics.









The closed-loop control scheme proposed to solve the nonlinear servomechanism problem has, incorporated into the feedback, the internal model system of the disturbance and reference signals with respect to the nonlinear system N. The implementation of this closed-loop controller is shown in Figure 2-1. The equations modeling the closedloop system are the following:



NC: (t) = f(x(t), u(t), w (t))

A(t) = An(t) + BH[x*(t) -x(t)]
(2-29)
u(t) = -Kx(t) - K2n(t)

e(t) = H[x*(t) -x(t)]


where K1 e Rpxn and K2 Rpxpr are constant feedback matrices. It is assumed that the state x(t) is available for feedback.

We will show in Theorem 2.1 that there exists an initial state for the closed-loop system NC such that tracking occurs with e(t) = 0 for t > 0. The following proposition shall be required in the proof of this theorem.



Proposition 2.2 Let z(t) e RP be any vector with elements satisfying the linear differential equation

(.)(r) + a rl(.)(r-1) + . + a,(.)(1) + ao(.) = 0 (2-30)



and let C e Rrxr be a matrix whose eigenvalues, including multiplicities, exactly match the characteristic roots of (2-30). If in

addition, the pair (A, K2) is observable with the constant matrix K2 . Rpxpr and A . Rprxpr defined by















w (t)


Figure 2-1. Closed-loop control system NC









A = T-1 block diag. [C, C, ., C] T (2-31)

p blocks

then for some no � Rpr, z(t) can be generated as follows


z(t) = K2n (t)


A*(t) An*(t) , n (0) = no (2-32)

*0

Furthermore, the initial state no is unique for any given z(t).


Proof: By the conditions given in Proposition 2.2 each of the p components of z(t) must satisfy (2-30) and hence the entire vector z(t) can be expressed uniquely in terms of a pr dimensional initial condition vector. By assumption, this vector can lie anywhere in pr dimensional space. From the definition of the A matrix, each element of the vector K2n (t) must also satisfy (2-30). Hence, if we show the initial

condition vector representing K2n (t) can be made to lie anywhere in pr dimensional space by appropriate choice of the initial state no,

the proof will be complete. Such an initial state can be shown to exist by noting that any state n0 can be observed through the output

K2n (t). Consequently, a linearly independent set of initial states must result in a linearly independent set of outputs K2n*(t). Thus, since n0 spans pr dimensional space, the initial condition vector defining K2n (t) will span pr dimensional space.
1
To prove uniqueness, let n0 be another initial state such that 1(t 1 1 1 1
z(t) = K2n , 1 (t) = An (t) , n (0) = no (2-33)








This gives
1 ,* Atr 1 *
K2[n (t) - n (t)] = K2e [no - no] = 0 (2-34)
1 2


and the vector [n - n] is not observable which is a contradiction.
0 0


Theorem 2.1 Given the p-input, p-output system N suppose that for a particular reference r*(t) and a particular disturbance w (t)

assumptions (A.1) and (A.2) both hold. In addition, suppose that the pair (A,B) defines an internal model system of r*(t) and w*(t) with respect to N. Furthermore, assume K2 of the system NC is such that the pair (A,K2) is observable and let K1 be arbitrary. Under these

conditions, there exist initial states x(O) = x0 and n(O) = n0 such that in the closed-loop system NC, e(t) = Er*(t) - y(t)] = 0 for all t > 0 when the exogenous signals r*(t) and w*(t) are present.


Proof: To prove Theorem 2.1 it is necessary to show that there exists an initial state* for the system NC such that perfect tracking occurs. Let [xo, no] denote this initial state and let [x*(t), n*(t)] be the
00
corresponding state trajectory. The following relationship must then hold for the system NC


x (t) = f(x*(t), u*(t), w*(t))

, (t) = An*(t) (2-35)

u (t) = -Klx*(t) - K2n*(t)

e(t) = r*(t) - Hx*(t) = 0

Henceforth, the initial state of the combined plant and controller will be grouped in a pair as [x n]. The state trajectory which results from this initial state will e grouped as [x(t), n(t)].









In order to verify (2-35) we first note that by assumption (A.1) there is an initial state x*(O) = xo and an input u *(t) such that e(t) = 0 for all t > 0. Hence, it must be shown that for some initial state n*(O) = no of the internal model system, the input u *(t) can be produced by feedback of the form


u*(t) = -K1x*(t) - K2n*(t) (2-36)


From assumption (A.2) we know that the elements of u*(t) and x*(t) will satisfy the differential equation (2-4) (or equivalently, equation (2-30)). Also observe that because e(t) = 0 in (2-35), the internal model system is completely decoupled from the original system. This

decoupling allows us to apply Proposition 2.2. Specifically, we may

verify (2-36) by letting z(t) = -u* (t) - Klx*(t) in Proposition 2.2. This completes the proof.


We have shown that if certain conditions have been met, then when the exogenous signals r*(t) and w*(t) are acting on the closed-loop system NC, there exists an initial state [x , no] such that perfect tracking occurs. However, if the initial state [x(O), n(O)] differs from [xo, no , the resulting state trajectory [x(t), n(t)] may not converge to [x (t), n (t)] as t + . To achieve (asymptotic)

tracking, we want [x(t), n(t)] to converge to [x*(t), n*(t)] for some range of initial states [x(O), n(O)]. This leads to the following

notation.








Definition: If the state trajectory for the closed-loop system NC converges to [x (t), n*(t)] for a set of initial states in the neighborhood of [Xo no] then we say there is local tracking of r (t) with disturbance w* (t). If this convergence occurs for all initial

states then we say there is global tracking of r*(t) with disturbance

w*(t).


To give conditions under which global or local tracking will occur, we first define a new set of state vectors as follows


(t) = x(t) - x*(t)
=(t) = n(t) - n*(t) (2-37)


where [x(t), n(t)] is the state trajectory of NC resulting from an arbitrary initial state and Ex (t), n (t)] is the trajectory which gives e(t) = 0, t > 0 and results from the initial state [x0, no].

Since it is our goal to have the trajectory [x(t), n(t)] converge, eventually, to the trajectory Ex (t), n*(t)] we may think of [(t), "(t)] as the transient trajectory. Using (2-29) and (2-35), it is then possible to write a dynamic equation modeling the transient
response of the closed-loop system. This will be referred to as the closed-loop transient system NCT. The system NCT is given by

NCT:


x(t) = f(x*(t)+'(t), u*(t)+u(t), w*(t)) - f(x*(t), u*(t), w*(t)) fl(t) = An(t) - BH'(t) (2-38)
"U(t) = -Kl X(t) - K2"(t)









It is seen that the system NCT has an equilibrium point at x(t) = 'n(t) = 0



Theorem 2.2: Suppose that the hypotheses of Theorem 2.1 are satisfied and that for some choice of K, and K2 the system NCT is locally asymptotically stable. Then, with the control scheme defined by system NC, local tracking of r*(t) with disturbance w*(t) will occur. If in

addition, system NCT is globally asymptotically stable then global tracking of r*(t) with disturbance w*(t) will occur.




Proof: Obvious since 'X(t) + 0 as t + - and e(t) = r*(t) - Hx(t) H[x*(t) - x(t)] = -H (t).


Since global stability is often difficult to obtain in many

practical systems using constant-gain feedback, the local stability result of Theorem 2.2 will most often apply. Consequently, success of the control scheme will depend on the initial state of the original system and of the internal model system. This will usually mean that tracking and disturbance rejection can be achieved only if the reference signal and the disturbance signal are not excessively large.

Stability of the system NCT will be a major topic of the next section as well as subsequent chapters. At this point, however, it is appropriate to generalize the results obtained so far. This is

important since previous results have been developed with the assumption that only one particular reference signal r*(t) and one particular disturbance signal w*(t) will be applied to the system. The









generalization is rather obvious. If tracking and disturbance rejection

is to hold for a certain class of signals r(t) and w(t), two conditions are required: 1) the internal model system must contain the necessary dynamics to cover the entire class of signals, and 2) the closed-loop tranasient system NCT must remain locally (or globally) asymptotically stable over this class of signals.

Often in practice, the precise reference and disturbance signals acting on the system are not known in advance and hence neither are x*(t) and u*(t). In order to determine the dynamics which must be included in the internal model system it is necessary to have some apriori knowledge of the state and input signals which will occur during tracking. Usually, knowledge of the frequencies of the anticipated disturbance and reference signals is available. Generally, the

frequencies of the reference r*(t) and the disturbance w*(t) will directly affect the frequencies of the corresponding state x*(t) and input u*(t). Assuming this to be true, the mathematical model

describing the nonlinear system can be used to determine x*(t) and u*(t) for various combinations of r*(t) and w*(t). Fourier analysis can then be used to determine the dominant frequencies in the signals comprising the various x*(t) and u*(t) and the internal model system can be designed accordingly. Even when the mathematical is not used, an educated guess or perhaps trial and error can enable one to design an internal model system with the appropriate dynamics. For example, if

sinusoidal disturbance and reference signals are expected, it might be advisable to design the internal model system to accomodate for various harmonics and subharmonics of the anticipated sinusoidal signals.









Stability of the Closed-Loop Transient System

In this section we investigate the stability of the system NCT. First, the previously defined condition of controllability for the pair

(A,B) and the required observability of the pair (A, K2) are related to the stability of NCT. Next, a method for checking local stability of NCT using Liapunov's indirect method is presented.

Now consider controllability of the pair (A,B) which has already been insured by the chosen structure for the internal model system. Suppose, for the sake of example, that the pair (A,B) is not controllable. This implies that the pair (A, BH) is not controllable. Consequently, there exists a linear transformation matrix P such that


p_lAP 1 A2 A4(-9
P : , P-1BH = (2-39)



Since the eigenvalues of A are in the closed right half-plane, the eigenvalues of A3 are in the closed right half-plane. It is apparent

that the modes* associated with A3 are not affected by any control law. Thus, we can conclude that when (A,B) is not controllable, the

system NCT can not be made asymptotically stable. This points out one

of the reasons behind the structure chosen for the internal model system.

Now consider the situation which arises when the pair (A, K2) is

not observable. Since the eigenvalues of A are in the closed right

* Modes are components of the form tke t which appear in the solutions to linear differential equations. For example, given the
system x(t) = Fx(t), x(O) = xo with solution x(t) = 0(t,O)xo; the elements of the state transition matrix cD(t,O) are made up of modes of the form tK e X. Here x, which is generally complex valued, represents an eigenvalue of the matrix F.









half-plane, all modes associated with A must be forced to zero or NCT will not be asymptotically stable. From (2-38) we see that ' (t) is the only signal which can accomplish this task. Consequently, X(t) should be a function of the modes in 'n(t) induced by the eigenvalues of A. Since '(t) depends on 'n(t) only through the feedback coupling from the

gain K2 and (A, K2) is not observable, it is impossible for 'X(t) to depend on the unobservable modes. Thus, we can conclude that

observability of the pair (A, K2) is necessary for asymptotic stability of the system NCT. We therefore have the following result.



Proposition 2.3: Controllability of (A, B) and observability of (A, K2) are necessary conditions for asymptotic stability (either local or

global) of NCT.



Although the above result is important, it is even more important that a practical method is available which allows one to ascertain directly whether or not NCT is asymptotically stable. It has already

been indicated that the local stability results of Theorem 2.2 will most often apply. One convenient method for showing local stability is Liapunov's indirect method which can be found in standard texts on nonlinear systems (e.g., see [17]). The required linearization of the closed-loop transient system NCT about the equilibrium point (t) = 0,

( 0 is



LF t -BH -G (t)K2j [ (2-40)
n BH A rl Mt









where



F* (t) f(x,u,w) F (x = x (t) u u*()(2-41) w w (t)



and



G*(t) = af(x,u,w) tu x = x*(t)
u *(t) (2-42)
w =w t)




Notice that the Jacobian matrices F*(t) and G*(t) are evaluated along the trajectory which gives tracking of r*(t) with disturbance w*(t).

Often this trajectory is not known in advance, however, we shall defer a more detailed discussion of this problem until a later chapter.

Let us make the following definitions:



XA(t) - �t) (2-43)


, FF*(t) -G* (t) K1 -G* (t) K21
F AMt :=L -B (2-44)


f(x *(t)+x, u* (t)+u, w*(t)) - f(x*(t), u*(t), w*(t) fA(t, xA) AB (2-45)











fA(t, XA) := fA(t, XA) - FA(t)xA (2-46)



We now present a theorem based on Liapunov's indirect method which can be used to show local tracking. In order to apply Liapunov's indirect method, the following two technical conditions are required


lim rsup 1fifA(t' XA)11
11 XsA 1+0 tO x ) = 0 (2-47)




F is bounded (2-48)



It is mentioned that the above conditions are almost always satisfied in practical systems.


Theorem 2.3: Suppose that the hypotheses of Theorem 2.1 are satisfied and also assume that conditions (2-47) and (2-48) hold true. If in

addition, the system (2-40) is asymptotically stable then, with the control scheme defined by system NC, local tracking of r*(t) with disturbance w (t) will occur.


Extensive use will be made of Theorem 2.3 in later chapters.


The Relation Between the Dimension of the

Internal Model System and the Input/Output Dimensions

In the previous sections, the servomechanism problem was treated where it was assumed that the number of inputs to the plant was the same









as the number of outputs. In this section, further insight into this assumption is presented by showing its relation to the chosen controller structure. In addition, sufficient conditions will be given to allow one to consider a system with more inputs than outputs. The case where the input dimension is less than that of the output shall not be considered since, in this circumstance, a solution to the servomechansism problem does not generally exist. The intuitive reason for

this is that it requires at least p independent inputs to control p degrees of freedom independently.

Let us now consider a nonlinear system with input u(t) e Rm and output y(t) E RP, where m > p. Assume that the controller is implemented in essentially the same way as the previously discussed controller except now consider changing the dimension of the internal model system. It is assumed that the matrix A of the internal model

system has q blocks on the diagonal rather than p blocks as before. Consequently, we now have A : Rqrxqr, B E Rqrxp and K2 . Rmxqr. The

exact change in the A matrix is shown by the following equation


A = T-1 block diag. [C, C, ., C] T (2-49)

q blocks



where C is again defined by (2-25). The corresponding change in the B matrix does not need to be considered in this analysis.



Proposition 2.4: Given the triple (A, B, K2) with A . Rqrxqr defined by

(2-49), B . Rqrxp arbitrary, and K2 . Rmxqr arbitrary. The following properties are true.









If p < q the pair (A, B) is not controllable If m < q the pair (A, K2) is not observable


Proof: The proofs to properties (1) and (2) are similar so we only prove (1). This will be accomplished by showing the existence of a row vector v' such that


v'EXI - A , B] = 0


(2-50)


for some x which is an eigenvalue of

From the structure of A, it is C and hence there is a row vector w'


A.

apparent that x is an eigenvalue of such that


w'[XI - C] = 0


Now define a matrix Q . Rqxqr to be the following


Q= 0


* o . Wm]


It can be readily seen that


Q [xI - A] = 0


Now let D . RqxP be the matrix product of Q and B. That is


(2-51)


(2-52)


(2-53)









D = QB (2-54)



By the assumption p < q there exists a row vector z' E Rq which is orthogonal to the range D. Hence


z'D = 0 (2-55)

Consequently, by letting v' = z'Q it is evident that (2-50) holds.


In the previous sections, it was shown that stability of the closed-loop transient system NCT is a key requirement in the solution of the nonlinear servomechansim problem. To achieve such stability, the conditions that the pair (A, B) be controllable and the pair (A, K2) be observable were shown to be crucial. Hence, from Proposition 2.4 we can conclude that the number of blocks q in the internal model system must be such that q < min(m,p).

Now recall a major earlier assumption; namely, the open-loop input u*(t) which forces the nonlinear system N to track r*(t) satisfies the differential equation (2-4). Ultimately, such an input is generated by

the internal model system as can be seen from Theorem 2.1 or equation (2-35). If u*(t) E Rm is arbitrary (aside from satisfying (2-4)) then it is not difficult to see that the internal model system must have at least m blocks. In otherwords, we must have the condition q > m.

Because of earlier condition that q < min(m,p) and because of the

assumption that m > p we are forced to consider systems with m = p.

*The modes of the various elements of u*(t) have a one to one corresondence with the eigenvalues of the C matrices which mak up the block diagonal A matrix. Since we assume all m elements of u (t) are independent of one another then m separate blocks will be needed in A to insure this independence. For further insight, see the proof of
Proposition 2.2.








The above restriction is actually somewhat misleading.

Specifically, consider a system having more inputs than outputs (i.e., m > p). It is quite possible that when only p out of the m inputs are used, all conditions for solving the servomechanism problem will be satisfied. In this case, we can define a new system NP which is nothing

more than the original system N operating with only p inputs. This is shown by the following equations:
NP:

<(t) = f(x(t), u(t), w(t)) = fp(x(t), Up(t), w(t))

y(t) = Hx(t) (2-56)

u(t) = MuP(t)


where M . RmxP is of full rank. If there exists an M such that the

system NP with input up(t) meets all conditions given in the previous sections, then the problem can be solved. Working with the input up(t),

let the feedback law which stabilizes the closed-loop transient system

be


Up(t) = -Kpl x(t) - Kp,2n(t) (2-57)



In terms of the original system N, the feedback law will then be


u(t) = -K1x(t) - K2n(t) (2-58)

where


Ki = MKp,1 , K2 = MKp,2


(2-59)








Summary

In this chapter, a nonlinear system was considered with the number of inputs equal to the number of outputs and with the output taken as a linear combination of the system's state. In the last part of the

chapter, conditions were given so that a nonlinear system with more inputs than outputs could be treated.

Prior to developing the theory for the nonlinear servomechanism problem, two major assumptions were made. The first of these, assumption (A.1), was absolutely necessary since without it, the servomechanism problem could not be solved under any circumstance. This being the case, the primary attention was focused on the second assumption, assumption (A.2). Here, the requirement was made that the input and state trajectories which occured during tracking were to satisfy a linear differential equation. It was noted that in practice, such an assumption may only be approximate, however, a design based on the approximation could be perfectly adequate. Typically, truncated Fourier series expansions approximating the true signals would be used for design purposes.

In the first part of the controller design we dealt with the development of an internal model system. It was indicated that this

internal model system would have to contain dynamics which matched the dynamics of the state and input which are necesary for tracking. The

importance of such an internal model system becomes evident when it is compared to a standard alternative. A typical approach to solving the nonlinear servomechanism problem is to first linearize the nonlinear system and then design a controller using linear servomechanism theory. This leads to an internal model system containing dynamics








corresponding only to the modes present in the reference and disturbance signals. From the results derived here, it is obvious that this type of approach may not be adequate. In fact, tracking error will always occur

when modes which are required to be present in the input for tracking are not incorporated into the dynamics of the internal model system. Hence, our idea is to incorporate enough modes into the internal model system's dynamics to insure that the tracking error is indeed small. These modes, if sinusoidal, could actually be sinusoids at frequencies

which are harmonics or subharmonics of the frequencies found in the reference and disturbance signals.

Once the formulation for the internal model system was complete, the controller design was given. In this design, state feedback was

used and the internal model system was incorporated into the feedback loop. It was shown that a necessary condition for a solution to the servomechanism problem (for arbitrary K1) was observability of the internal model system's state through its feedback gain matrix. This

was a key requirement which has not been postulated for the linear servomechamism problem, but was needed here due to the different

approach used in solving the nonlinear servomechansim problem. Later it

was indicated that the observability condition would also be required for stability of the closed-loop system. Consequently, it is enough to consider only the stability problem since the observability condition is satisfied automatically whenever stability is achieved.

The stability requirement for the nonlinear servomechanism problem was imposed upon a dynamic system which modeled the difference between

the actual state trajectory and the desired state trajectory of the closed-loop system. Thus, the dynamical model was referred to as the








closed-loop transient system. Again, the approach taken here is seen to be drastically different from the approach taken in the well known linear servomechanism problem. Asymptotic stability of the closed-loop transient system was allowed to be either global or local; however, with local stability it was indicated that tracking and disturbance rejection would occur only for certain initial states. These initial states were

restricted to the neighborhood of the particular initial state which defined the equilibrium point of the closed-loop transient system.

The dynamic equations modeling the closed-loop transient system were seen to be nonlinear and somewhat complicated. In order to apply Liaponov's indirect method, a much simpler dynamic system was derived

through linearizations. Although the linearized model is much more suitable for feedback gain selection, stability of the linearized model only insures local stability of the true system.

In this chapter, no discussion was given as to possible means of determining the stabilizing feedback gains. This topic is the subject of Chapters Four and Five.














CHAPTER THREE
APPLICATION TO LINEAR SYSTEMS


In this chapter the linear servomechanism problem is considered.

Since the linear problem can be regarded as a special case of the nonlinear problem, the methods developed in the previous chapter apply here as well.
When using the methods of the previous chapter, specific conditions must be met in order to guarantee a solution to the servomechanism problem. Here, it will be shown that these conditions are satisfied whenever the well established conditions [1-7] imposed for the linear problem are satisfied. Of course this is obvious; however, the insight obtained by approaching the problem from a different point of view will prove beneficial. In fact, many of the results obtained in this chapter will be used in subsequent chapters where feedback gains are selected for the nonlinear problem via a linearized model.






Review of Linear Servomechanism Results

In this section we give a brief review of the well known results for the linear servomechanism problem. A more general discussion of this topic can be found in [2] or [3].









Consider the linear time-invariant system



L: x(t) = Fx(t) + Gu(t) + Ew(t)

y(t) = Hx(t) (3-1)

e(t) = r(t) - y(t)


where x(t) e Rn is the state, u(t) E Rm is the input, w(t) E Rd is a disturbance, y(t) e RP is the output, and e(t) e RP is the error which arises in tracking the reference signal r(t) e RP. Conditions shall be given as to when it is possible to design a controller such that e(t) + 0 as t + -. It is assumed that the elements of the reference r(t) as well as the disturbance w(t) satisfy the linear differential equation


(.)(r) + yrl(.)(r-1) + . + y1(.)(1) + yO(.) = 0 (3-2)



where the characteristic roots of (3-2) are assumed to be in the closed right half-plane. We shall let Xi, i = 1, 2, ., F denote the

distinct characteristic roots of (3-2) where F 4 r due to multiplicities. The following well known result gives conditions under which the linear servomechanism problem can be solved.


Theorem 3.1: Assume the state x(t) is available for feedback. A

necessary and sufficient condition that there exists a linear timeinvariant controller for (3-1) such that e(t) + 0 as t +- for all

r(t) and w(t) with elements satisfying (3-2) is that the following two conditions both hold.









(B.1) (F, G) is stabilizable


(B.2) rank F n + p ,
-H0


Conditions (B.1) and (B.2) are essential for a solution to the linear servomechanism problem. Therefore, when the linear problem is solved using the framework developed for the nonlinear problem, conditions (B.1) and (B.2) should play important roles.


Solution to the Linear Problem via the Nonlinear Formulation

We now proceed to show that when conditions (B.1) and (B.2) are satisfied, the conditions given in Chapter Two for the nonlinear

formulation are also satisfied.

First consider assumptions (A.1) and (A.2) when applied to a linear system. The following theorem will relate these assumptions to conditions (B.1) and (B.2).


Theorem 3.2: Consider the linear system L and assume both the reference r (t) and the disturbance w*(t) satisfy the linear differential equation (3-2). Then, if conditions (B.1) and (B.2) are both satisfied, there exists an input u*(t) and an initial state x*(O) = xo such that

tracking occurs. Furthermore, u*(t) and x� can be chosen so that the

resulting state trajectory x*(t) and the input u*(t) satisfy the linear differential equation (3-2) (i.e., assumptions (A.1) and (A.2) are satisfied).









Proof: We shall show the existence of u*(t) and x*(t).

Let
u*(t) = -Kx*(t) + U(t) (3-3)


where K e Rmxn and 5(t) . Rm are still to be defined. Also let the Laplace transforms of r*(t), w*(t), x*(t), and U(t) be denoted as
R*(s), W*(s), X*(s), and U(s) respectively. Then, if tracking is to
occur, it can be verified using (3-1) that the following relationships must hold.
1*I1

X*(s) = [sI-F] x� + [slI-F]-G(s) + [sl-F]- EW*(s) (3-4)

and
R*(s) = H[sI-F]1 x + H[sI-FI 1GU(s) + H[sI-]-EW*(s) (3-5)
0


where
F:= F - GK (3-6)


Now consider the following conditions


(1) m > p
(2) rank [ x.I-F] = n for all xi , i : 1,2, .,
(3) rank HEXi I-FIG = p for all Xi ' i = 1,2, .,


where xi, i = 1,2, ., r are the characteristic roots of the linear differential equation (3-2).
When (1), (2), and (3) are satisfied then both (3-4) and (3-5) will

hold true (i.e., tracking will occur). Furthermore, when these three








conditions are met, x*(t), i(t) and hence u*(t) can be chosen to

satisfy the differential equation (3-2).

The proof to the above statement is quite tedious and will be omitted; however, an example shall be given later which will make the statement obvious.
Now it is only necessary to show that when (B.1) and (B.2) hold, conditions (1), (2), and (3) are satisfied. It immediately follows from condition (B.2) that (1) must be true. Also, by condition (B.1) we can select K so that X.i i = 1,2, ., F is not an eigenvalue of [F - GK] and hence (2) holds. Assuming that such a K has been chosen, it is easy to show that



XiI - F Gr Xi - F + GK G
rank [] = rank L(3-7) G]1G
1 H 0 0 H[Xil - F + GK]1G


This is accomplished by premultiplying and postmultiplying the left-hand side of (3-7) by


Inxn 0 an byixn 0

H[XiI - F + GK]-1 I pxpb y I Imxm

respectively. Here Inxn denotes the identity matrix of dimension nxn. From condition (B.2) and equation (3-7) the matrix H[Xil - F + GK]G must have rank p for all Xi, i = 1,2, ., j. This gives (3) and the proof is complete.


The following example helps to verify the statement given in

Theorem 3.2.








Example 3.1:

Let us take

1( a + b + c (3-8)
(S-X1 (s1


where a, b, c e RP are constant vectors. Also, for simplicity of this
example let


W*(s) = 0 (3-9)


We now show that when conditions (1), (2), and (3) used in Theorem 3.2 hold true, tracking will occur for some initial state x0 and input

U(s). The input U(s) can be chosen as 1S w + 2 v + r (3-10)
(s-X T

where w, v, r Rm are vectors given by


v = {H[X1 I-T- G}- b

w = {H[X I-F]-1G}'I {a + HEX1I-F]'2Gv} (3-11)

r = {H[X2 I-]-IG}- 1c


The initial state x0 should be taken as


x -{ [F-xl]1 'Gw + [T-Xl11-2Gv + [F-X21] 1Gr } (3-12)
012









Note that for the matrix inverses given in (3-11) to exist, conditions

(1), (2), and (3) are needed. In addition, if HEXI - F]-G is not
square, any right inverse can be used.
We shall make use of the following formula which can be obtained using a partial fraction expansion
N

N { X (-l)j-'II.l-]j.
(s-X) j=1 (S N+-j

+ { [sI-F]'- [F-XI]-N (3-13)


where it is assumed that x is not an eigenvalue of F. Using (3-13) and (3-10) it is then possible to write (3-5) as


R*(s) = H[sI-F]1 { x* + [F-Xll]'IGw + [T-XlI]-2Gv + [F-X21] -1 Gr


+ HEX IF]-IGw 1 + HEX I-I s1 HEX I-F]-2Gv 1
1 ~ (S-X) +Hx1-]G 1 (s-Xx12)
HEX I-F]- 1Gr 1 (3-14)
L2IF (s-X 2)


When (3-11) and (3-12) are substituted into (3-14) we obtain the desired
R (s).

It also follows from (3-4), (3-10), and (3-12) that the X*(s) which occurs during tracking can be expressed as


X*(s) = [XlI-F-1Gws + [xII-1-IGv 1
(S-Xl)2'
1 +(3-15)

- [ExI-F]2Gv ( + X21I-F]'IGr1








Consequently, x*(t), i(t) and u*(t) all satisfy the differential

equation


(3) + [X22 (.2) + [2X + 2 () + [-'2 X 0

(3-16)


This completes the example. Although a disturbance was not considered, the inclusion of a disturbance would have led to similar results.

Next we consider another requirement which was needed to solve the

nonlinear servomechanism problem and relate it to the linear problem. This requirement is that the closed-loop transient system NCT given by (2-38) must be asymptotically stable. When the linear system is

considered, (2-38) takes the following form


LCT:



BH A Wt) 0](t)



FL(t) = -K 1(t) - K2-(t) (3-17)


Now if conditions (B.1) and (B.2) both hold then it is possible to select K1 and K2 such that (3-17) is asymptotically stable. This is a consequence of the following well known theorem (see [2] or [3]).


Theorem 3.3: If (B.1) and (B.2) both hold and the eigenvalues of A correspond exactly to xi' i = 1, 2, ., r given by condition (B.2)








then the pair
F 0 G
(3-18)
BH A [ )


is stabilizable. If in addition, the word "stabilizable" in (B.1) is replaced by "controllable" then the pair of (3-18) is controllable.


The next theorem shows that when K1 and K2 are selected to stabilize (3-17) the pair (A, K2) is observable. This is the precise

condition needed for Theorem 2.1 and the final condition required for our discussion.


Theorem 3.4: If all eigenvalues of A are in the closed right half-plane

and the system LCT described by (3-17) is asymptotically stable then the pair (A, K2) is observable.


Proof: We use contradiction. Suppose that the system LCT is asymptotically stable but (A, K2) is not observable. This implies that there exists a vector v such that


xI-A
V = 0 (3-19)
K2



for some X which is an eigenvalue of A. Consequently, we can write


XI - F + GK1 GK2 [ 0 (3-20)
BH XI - A] v









This means that x is also an eigenvalue of the matrix


F- GK 1 -GK21
LF-BH Aj



Since the system LCT can be written as



t = - GK1 -GK2 (t1 (3-21)




and X is, by assumption, in the closed right half-plane then LCT is not asymptotically stable. This is a contradiction to the assertion that LCT is asymptotically stable and the proof is complete.


Theorems 3.2, 3.3, and 3.4 show that when conditions (B.1) and (B.2) are true, then all requirements for a solution to the linear servomechanism problem are satisfied. Furthermore, this has been accomplished in the framework developed for the nonlinear servomechanism problem. In the next chapter, we shall apply some of these results to obtaining the stability required for the nonlinear servomechansim problem using the linearization approach.















CHAPTER FOUR
FEEDBACK CONTROL



In this chapter various concepts related to selecting the feedback gains for the servomechanism problem are discussed. The beginning of the chapter deals primarily with the nonlinear servomechanism problem while some results for the linear servomechanism problem are given in later sections.

Since it is often difficult to show global stability in nonlinear systems, here the emphasis is placed on achieving only local stability. Conditions are given showing when it is possible to obtain a time-invariant feedback control law which stabilizes the system NCT. These conditions apply mainly to the case when the reference and disturbance signals are small. An example is provided to demonstrate the method.

Also discussed in this chapter is the interpretation of using optimal control techniques in the selection of the feedback gains for the linear servomechanism problem. The optimal control approach can be applied to the nonlinear servomechanism problem when the linearized equations are used; however, the interpretation is less precise.





Stabilization Using the Linearized Equation

To solve the tracking and disturbance rejection problem it is necessary to have asymptotic stability of the system NCT. Furthermore,









only state feedback with constant gain matrices can be used to achieve this stability. The system NCT is given again here for convenience.


NCT:

X(t) = f(x*(t)+W(t), u (t)+w(t), w*(t)) - f(x*(t), u*(t), w*(t)) "'( t) = An(t) - BH"X(t)

'i'( t) -Kl (tM - K2 rt (4-1)



Two forms of asymptotic stability are actually considered in the

solution to the servomechanism problem: global stability and local

stability. It was indicated that local stability allows tracking and disturbance rejection to occur only for certain initial states whereas global stability allows tracking and disturbance rejection for all initial states. Global stability is thus the most desirable form of

stability, however, due to the diversity which can occur in the system NCT, we shall limit our concern to the local stability problem. In

addition, the original nonlinear plant will be taken as time-invariant (i.e., the function f(x,u,w) is independent of time). These

restrictions will allow us to obtain a time-invariant feedback law which

gives local tracking and disturbance rejection for small reference and disturbance signals.

For convenience, when discussing the system NCT, the following definition shall be used



t : (t
�A~t : L(t!(4-2)








Adhering to this more compact notation, we may write the linearized approximation to NCT as


A(t) FA(t)xA(t) (4-3)

where



F*t) LF*(t) - G*(t)K1 -G*(t)K2(
A(-BH A

and




* af(x,u,w) G * (t): *
F(t) ax x= x (t) au x= x (t)

u u (t) u u (t) (45)
w w(t) w w(t)




Note that (4-3) is the linearized system needed in conjunction with Liapunov's indirect method (Theorem 2.3) and was originally given as equation (2-40).
Although by applying Liapunov's indirect method we reduce the problem from stabilizing a nonlinear system to stabilizing a linear system, the time-dependency of this linear system can create complications. Equations (4-4) and (4-5) show how this time-dependency enters into the linearized system due to the time-varying signals x*(t), u*(t), and w*(t). To further complicate the stability problem, it is very

likely that the matrices F*(t) and G*(t) will not even be known. This is because both F*(t) and G*(t) are implicitly dependent on the









reference and disturbance signals and these signals (especially the disturbance) are usually not known in advance.

At this point we consider one method of dealing with the stability problem which will be applicable when the reference and disturbance signals are small. This assumption, although restrictive, is necessary to show stability of the system NCT when the feedback gains are kept constant. We first give what is known as the Poincare-Liapunov theorem [18].


Theorem 4.1: Consider the system


= Fl(t)x + fl(t,x) , x(to) = xo (4-6)

where

fl(t, 0) = 0 (4-7)


Assume that the following conditions are also satisfied


(1) Fl(t) is such that the system


: Fl(t)x (4-8)



is exponentially asymptotically stable for the equilibrium point x : 0. In otherwords, the state transition matrix D(t, to) associated with (4-8) is such that


S110(t, t 0 )I i< me- a(t-t0) (4-9)


for some positive constants m and a.









(2) fl(t, x) satisfies the criterion


sup lif1(t,x)ii < Ltixii , L > 0 tt
0


(4-10)


Then if we have L suitably small so that


(mL - a) < 0,


(4-11)


the system (4-6) is exponentially asymptotically stable for the equilibrium point x = 0.

Proof: Let (t,t ) be the state transition matrix for the system (4-8). Consequently, we can write the solution to (4-6) as


x(t) = t(t,t0)X0 + f t(t, T)f1(T, x(T))dT


(4-12)


By taking the norm of both sides of (4-12) and using (4-9) it easily follows that


Iix(t) ii < me-a(t-to)x11 + f mea(t)L Ix(T) ItdT


(4-13)


Multiplying through by eat gives


ato t
eatilx(t)ii 4 me tox0 + mL f eaT ilx(T)IdT to


(4-14)


We may now apply the Bellman-Gronwall inequality (see [17]) to obtain









eat ix (t) ii


4 me


n1x(t)ii < m i


ato mL(t-to)


(mL-a)(t-t o) xo lie0


If L is sufficiently small so that (mL-a) < 0


then we have the desired result.


Consider again the linearization of NCT given by (4-3). be written as


kt) F'xAt + F* t- F0
RA(t) A F~A(t) +FA(t) - A]XA(t)


(4-15) (4-16)


(4-17)


This can


(4-18)


The matrix


is defined as follows


where


Fo : af(x,u,w)
ax


x=0 u 0 w= 0


FA:=


F 0 G�K1

-BH


-G 0K2]


(4-19)


GO : = f(xuw)
au


x 0 u 0 w= 0


(4-20)









We see that F 0 is simply the matrix resulting from a linearization of
A
the system NCT about the origin. In addition, because the original
system is time-invariant F�, Go, and F are constant matrices.
A
Let us investigate the stability of NCT using the Poincare-Liapunov theorem. To do this we give the following result.


Theorem 4.2: Suppose that for a particular time-invariant system assumptions (A.1) and (A.2) are satisfied and that the conditions needed

to apply Liaponov's indirect method to the system NCT (see eq. (2-47) and (2-48)) hold true. In addition, assume that the following

conditions are satisfied.


(i) sup IIF*(t) - F0ni 6
t)O


(ii) sup iIG*(t) - Gi=2
t)O 1


where s 1 and c2 are positive constants.


(iii) The pair [FO, GO] is stabilizable



(iv) rank F0 = n + p
-H 0



for all xi which are characteristic roots of the differential

equation given in assumption (A.2).









Then, provided e1 and e2 are suitably small, there exist feedback gains K1 and K2 so that with the controller given as system NC, (eq. (2-29)) local tracking of r*(t) with disturbance w (t) will occur.


Proof: Our main concern here will be to show asymptotic stability of the system NCT and observability of the pair (A,K2). With these

conditions verified, the remainder of the proof is immediate from the results obtained in Chapter Two.

First let us compare (4-18) to (4-6) letting F 0 take the role of
A
Fl(t) and [FA(t) - F ]XA(t) take the role of fl(t,x). Condition (1) of the Poincare-Liapunov theorem then requires exponential stability of the system


=A = FAXA (4-21)


In order to meet this stability requirement, the pair



o] [Go])(4-22)




must be at least stabilizable. Since the matrices given in (4-22) are constant, Theorem 3.3 can be employed. More specifically, conditions (iii) and (iv) imply that the pair given in equation (4-22) is

stabilizable and hence, proper selection of the feedback K1 and K2 will give exponential stability to the system (4-21).

Assuming that suitable feedback gains have been selected, the state transition matrix @�(t,t ) associated with (4-21) will satisfy the inequality










llD� (tt0 )Ii < me a(tt) (4-23)


for some positive constants m and a. Hence, to show condition (2) of the Poincare-Liapunov theorem and also to conclude stability we must have



sup IIFA(t) -F A L (4-24)
t)O


where L is small enough to insure that



(mL - a) < 0 (4-25)


Equation (4-24) can be verified by using the definitions for F*(t) and FA to obtain the relationship A A
IIAt*Ii * F0 I. * 0IG

IIFA(t) - Foil < lIF (t) - Foil + IIG(t) - Golli.l[Ki, K2]1. (4-26)


Using (i) and (ii) then gives



sup IIFA(t) -oFl< e 2 + II[K1, K2]II. (4-27)
t)O


If 1 and e2 are small enough, then condition (2) of the PoincareLiapunov theorem is satisfied and the system NCT is asymptotically stable.
The final condition which needs to be verified for a solution to the servomechanism problem is the observability of the pair (A, K2). This condition presents no problem, however, since it immediately

follows from Theorem 3.4 that (A, K2) is observable whenever the system









given by (4-21) is exponentially stable. This completes the proof.



In order to interpret Theorem 4.2 we need certain continuity

conditions to hold. That is, F*(t) and G*(t) should be continuous functions of the reference r*(t) and the disturbance w*(t). Then

assumptions (i) and (ii) are realistic since both iiF (t) - Fi and

viG*(t) - Goo i will be small whenever i r*(t)o and iw (t)I are small. Furthermore, because F*(t) and G*(t) are often periodic due to the periodicity of x*(t), u*(t), and w*(t), C I and 62 will be,

respectively, the maximum values that iiF (t) - F011i and iiG*(t) - iii assume over one period.

As a final point, note that if in condition (iii) of Theorem 4.2 the word "stabilizable" is replaced by "controllable" then the feedback gains K1 and K2 can be selected to arbitrarily assign the eigenvalues of

the system (4-21). This may, in turn, make it possible to obtain a large ratio of a/m with suitably chosen feedback gains. Then, provided that 11K1, K211ii does not become too large, L will increase and hence

larger reference and disturbance signals will be allowed. Note also

that if the input enters into the nonlinear system by a linear timeinvariant mapping, E2 will be zero so that increasing the ratio a/m will

always increase L.

We now give a rather lengthy example which makes use of many of the results obtained so far for the nonlinear servomechanism problem. In

this example, simulated test results are provided to show the performance of the control algorithm. Also, simulations are provided which show the consequence of using a controller based on linear servomechanism theory.











Example 4.1

Consider again the system of Example 2.2


(t) = 3x(t) + x2(t) + (2x2(t) + 4)w(t) + u(t) y(t) = x(t)


(4-28)


Assume that the disturbance is a constant with a value of a2 and that the reference is a sinusoid with amplitude a, and frequency w

1. Thus


r (t) = alsint

w (t) = a2 (4-29)



It was found in Example 2.2 that the input u* (t) and state x* (t) satisfy the differential equation


d5 d3 d
(. 5 + 5 3 .) + 4 C.) 0 0
dt5 + dt3


(4-30)


The internal model system is thus


0
0 0
(t) : 0
0


0 0 0 0 I 0 TI(t)
0 1
-5 0


01
0
+ 0 e(t)
10
L' 1i


For the nonlinear system (4-28) we have


(4-31)










f(x,u,w) = 3x + x2 + (2x2 + 4)w + u


so that


F (t)=


X=alsi nt


* ( f(x,u,w) G*(t)= . au


= (3+2x+4wx)
X=alsint
w: a2


= 3+(2a 1+412)sint

(4-33)


X=alsi nt w= a2


(4-34)


The linearization of NCT may be written as


xA(t)


[F (t)-K1] [
0( 0C 0( 0( -1(


1
0
0
0
-4


* -K2
0 1 0 0 0


0
0
1
0
-5


x A(t)


(4-35)


X (t)
where xA(t) := (4-36)


It is not difficult to show that all conditions needed to apply Theorem 4.2 are satisfied. Evaluation of the linearized system about the origin gives


F� = 3 4 Go = 1


(4-32)


(4-37)









Using (4-33) and (4-34) it readily follows that



sup IF(t) - Foii = (2a 1 + 4a 1) = '1 (4-38)
t>O
and

sup IG (t) - G = 0 2 (4-39)
t )O




In order to show stability by the Poincare-Liapunov theorem we must then have



m - a < 0 (4-40)


where -a corresponds to the real part of the right-most eigenvalue of the matrix given in equation (4-35) when F*(t) is replaced by FO. The constant m depends on the eigenvectors of this matrix.

In order to meet the stability requirement, the feedback gains K1

and K2 are selected using a standard technique for eigenvalue assignment. (Note: it can be shown that the linearized system is controllable and hence arbitrary eigenvalue assignment can be made). The

closed-loop eigenvalues are chosen to be:


-4.0
-5.0
-5.0 � j3.0
-4.0 * j2.0


This gives a = 4, however, we shall not concern ourselves with the calculation of m. Just as an example, let us assume m = 1, 1 =1/2









and a2 = 1. Then, from (4-38), we get e1 = 3 so that slm-a = -1 < 0 and the system NCT is locally asymptotically stable.


Various simulations have been obtained using a Runge-Kutta algorithm [19] to numerically integrate the closed-loop nonlinear system. In some of these simulations, an internal model system has been used which does not contain dynamics corresponding to the second harmonic of the reference signal. Such a design results when the

internal model system is chosen in accordance with linear servomechanism theory. To give a fair comparison between the two control schemes, the closed-loop eigenvalues of the design based on linear theory are identical to those given above, except that the eigenvalues at -4 t j2 are no longer needed due to a reduction in system order.
Figure 4-1 shows the responses obtained for both design approaches when only a reference signal is applied. The proposed control design works well (see Fig. 4-1(a)) and the tracking error is completely eliminated in steady-state. The design based on linear servomechanism theory, however, results in a steady-state tracking error which is sinusoidal with a frequency twice that of the reference signal.

Figure 4-2 again shows the responses obtained using both design approaches. Here, however, a constant disturbance has been introduced in addition to the sinusoidal reference signal.

In Figure 4.3 the amplitude of the disturbance has been increased and as a result, the transient response in the system designed by linear theory is very poor. Note that in order to show the complete response, the scaling of the plot in Figure 4-3(b) is different from the scaling used in previous figures.






61


As a final test of the control scheme, a sinusoidal reference at a frequency slightly different from the intended frequency is applied to the system. Figure 4-4 depicts the resulting responses. Although we have not considered this particular situation from a theoretical

standpoint, it is seen that the controller which contains a second harmonic works considerably better than the controller designed according to linear servomechanism results.

























iIi I



J










Aii
* 0

4








?n nn no nn U M c n aMi n 1 il
0.



ni tnutif llIf ii I I'IU mII


TIME (SECONDS)


(a) Design based on nonlinear servomechanism theory


(b) Design based on linear servomechanism theory


Figure 4-1. Tracking error: reference = 2sin(t), no disturbance


.0


UI UW


NI V .l
I


i. VV
g


L o u U W


&It,.� .P l,,w


AL

















. . . . . - --.--; / . .: I











0.00 2.00 .00 500 800 LO.0 1i00 11100 i


TIME (SECONDS)


(a) Design based on nonlinear servomechanism theory


(b) Design


B.AD 8.00 LD. DO 12.00
TIME (SECONDS)



based on linear servomechanism theory


15.DO


Figure 4-2. Tracking error: reference = 2sin(t), disturbance = 1.0


LLJ I--


.00

















. .
(i:

8










% 1. . --------0


2. 00


tL. 00


6.DD 5.00
TIME (SECONDS)


LG. 00


12.00


*,00


(a) Design based on nonlinear servomechanism theory


(b) Design based on linear servomechanism theory


Tracking error: reference = 2sin(t), disturbance = 2.0


U. 0D


16.00


Figure 4-3.






































e. UU


41UU


(a) Design based


8I






. . .



. . .
0 " . . . . . . . . . . . . . . . . . . . . . .i. . . . . . . . . . . , . . . . . . . . . . . - . . . . . . . . . . . $ . . . . . . . . . . . . . . . . . . . . . . . . .
Cm !








C, ;.I .




It, - :,


5.uu B.M La.n 2.0 t
TIME (SECONDS)



on nonlinear servomechanism theory


(b) Design based on linear servomechanism theory






Figure 4-4. Tracking error: reference = 2sin(t), disturbance = 1.0


C) ,J




Q= I-


UU


q=~


15.O0









Optimal Feedback for the Linear Servomechanism Problem

In this section we discuss the consequences of using optimal control theory as a means of determining the stabilizing feedback gains for the linear servomechanism problem. Only the linear problem shall be treated since the interpretation of the results for the nonlinear problem is not clear. It is true, however, that the actual method of feedback gain selection discussed in both this and the succeeding section can be applied to the nonlinear problem when linearization techniques are used.

Now consider the well known linear optimal control problem [16], [20]. That is, given the linear time-invariant system


(t) = Fx(t) + Gu(t) (4-41)


select the control u(t) to minimize the quadratic performance index



J = f [x'(t)Qx(t) + u'(t)Ru(t)]dt (4-42)
0

Where Q > 0 and R > 0 are symmetric matrices of appropriate dimension. The optimal control law is found to be of the form


u(t) = -Kx(t) (4-43)

where K is a time-invariant feedback gain determined by solving an algebraic Riccati equation.

The question answered here concerns the interpretation of applying such a control to the linear servomechanism system. The reason that an interpretation is considered necessary is that the optimal control is









derived assuming u(t) is the only external input to the system and no consideration is made for uncontrollable inputs such as a disturbance or reference signal.

In the linear servomechanism problem, the dynamics of the plant and the controller can be modeled by the following equation LC:


: x(t) F+o
= + u(t) + [ E]ww(t) + r(t)

BH A n(t)0B

u(t) = -K1x(t) - K2n(t) (4-44)

y(t) = Hx(t)


where x(t) e Rn is the state of the plant, n(t) . Rpr is the state of the internal model system, u(t) E Rm is the input, y(t) E RP is the output, w(t) e Rd is a disturbance, and r(t) e RP is the reference. The system LC is essentially the linear version of the system NC given in equation (2-29) for the nonlinear servomechanism problem. Note that for

the linear system, it is not necessary for the dimension of the input and the dimension of the output to be the same.

As already discussed, if the internal model system is chosen appropriately and the feedback gives asymptotic stability to the system LC without the exogenous inputs w(t) and r(t); then tracking and disturbance rejection will occur when w(t) and r(t) are applied. The precise conditions as to when it is possible to obtain a stabilizing feedback are conditions (B.1) and (B.2).

Suppose that the conditions for a solution to the linear servomechanism problem have been satisfied and the closed-loop system LC has








been constructed in agreement with equation (4-44). Since asymptotic
tracking and disturbance rejection will occur, there must be a steadystate solution to (4-44). Consequently, if both w(t) and r(t) are

applied to the system at the time t = 0 then there exist initial states x(O) = xo and n(O) = no such that no transients appear in the state trajectory and the tracking error is zero. We denote this trajectory by the pair [x ss(t), nss (t)]. In terms of this notation, (4-44) becomes



ss ~ s (tF 0 SSW
= + U SS(t) + w(t) + r(t)
Sss(t)] BH A_]Lnsst) M o B



u ss(t) -Kx ss(t) -K nss(t) (4-45)


Ysst) = r(t) = HxSS(t)


If (4-45) is subtracted from (4-44) the following equation results:


F~t2 VF 0l F~t1 G1
= + I(t) (4-46)
T BH A [ (t0 0
where


X(t) = x(t) - xss(t)
n(t) = n(t) - nss(t) (4-47)
i(t) = u(t) - uss(t) = -Kl(t) - K2 (t)









Equation (4-46) is a linear dynamic equation modeling the transient part of the state trajectory of the system LC. Figure 4-5, shows a typical illustration of the actual, steady-state, and transient trajectories which might result when a sinusoidal signal is being tracked. It is

important to note that neither w(t) nor r(t) appear in (4-46) so that these exogenous signals play no role in the transient trajectory or, in otherwords, how fast tracking occurs. In addition, without the exogenous signals, (4-46) is of a form which makes possible the interpretation of using optimal control techniques for the selection of the feedback gains.

Now let us assume that the stabilizing gain K := [K1, K2] is found by solving the algebraic Riccati equation for optimal control. That is, the positive semidefinite P satisfying


Q + PFA + F P - PGAR- G P = 0 (4-48)



is obtained and K is selected as


K = R1G'P (4-49)
A


Here, Q > 0 is a symmetric matrix of dimension n+pr x n+pr, R > 0 is a symmetric matrix of dimension m x m, and


F 0 G
FA [ H , GA [ (4-50)


If the reference signal r(t) and the disturbance w(t) are applied at time t=O then the following quadratic performance index is minimized
























x(t)


Figure 4-5. Actual and steady-state trajectories









00
J f [ EA(t)QXA(t) + '(t)R'f(t)]dt (4-51)
0
where
S=x(t) (4-52)
and xA(t) : n(t) nss(t(



U(t) = [u(t) - Uss(t)] = [u(t) + Klxss(t) + K2nss(t)] (4-53)



The above observation simply points out that the transients behave in some optimal fashion.

Now consider the state trajectory [xss(t), nss(t)] which is unique for a specific feedback gain K = [K1, K2]. By assumption, this gain has been selected by optimal control techniques. If this is not the case, it is quite possible that the steady-state trajectory [xss(t), nss(t)] as well as the input uss(t) might be improved in some respect. For

example, a different choice of gains might actually lessen the average power required to maintain tracking of a certain reference signal.



Increased Degree of Stability Using the Optimal Control Approach

In the previous section we discussed using the optimal control approach to obtain stabilizing feedback gains for the linear servomechanism problem. The method can also be employed in the nonlinear servomechanism problem when the linearized equations are used; however, the control can no longer be considered optimal. It is sometimes

advantageous to make a slight modification on the performance index to give a higher degree of stability to the linearized model. It was

previously indicated that when the linearization of NCT about the origin









is exponentially stable, tracking and disturbance rejection will occur for small reference and disturbance signals. It is also true that in certain cases, increasing the degree of stability of this system will allow for a larger range of reference and disturbance signals. To meet this condition using optimal control theory, a well known technique due to Anderson and Moore [16] can be used.

Consider the linear system



(t) = Fx(t) + Gu(t) (4-54)



where the pair (F,G) is completely controllable. Equation (4-54) could model the linearized servomechanism equations.

Consider also the exponentially weighted performance index



J = f e2,t[x'(t)Qx(t) + u'(t)Ru(t)]dt (4-55)
0


where Q > 0 and R > 0 are symmetric matrices with (F',Q) stabilizable. It can be shown that minimizing (4-55) with respect to the system (4-54) results in a feedback law u(t) = -Kx(t). Furthermore, the degree of stability of the closed-loop system is increased relative to using a performance index without exponential weighting.

The problem is actually easier to solve by defining a new system given as



x(t) = P(t) + O(t) (4-56)

where


F = F + al , G = G


(4-57)









A new performance index is taken as


00
J= f [x'(t)Q (t) + '(t)RU(t)]dt (4-58)
0

The standard algebraic Riccati equation may be solved yielding an optimal feedback gain K for the system (4-56). It just so happens that

if this same feedback gain is used for the original system then the integrand of (4-55) is minimized with respect to the original system. In addition, the eigenvalues of the closed-loop system will all have real parts less than -a. This is the desired result. A similar

technique can be used in discrete-time control [21].















CHAPTER FIVE
PRACTICAL CONSIDERATIONS



In this chapter practical aspects dealing with implementing the control algorithm for the nonlinear servomechanism problem are treated.

First discussed is a modification of the already proposed control algorithm which is specifically tailored for tracking and disturbance rejection with respect to a nominal trajectory. The modified design is based on deviations from the nominal trajectory and since these deviations are generally small, the demand on the stabilizing capabilities of the controller is minimized. Another benefit of the design is that

tracking of a wide class of reference signals is made possible by supplementing the closed-loop feedback with open-loop control inputs.

The final section of this chapter deals with discrete-time

techniques so that the control algorithm can be implemented using a digital computer.


Controller Based on the Nominal Trajectories

In this section we develop a controller for the servomechanism problem which operates relative to nominal reference and disturbance signals. The design presented here employs not only feedback control but also feedforward control to achieve the desired trajectory. Although feedforward control is introduced, the control scheme is

formulated in essentially the same way as was done in previous treatment of the nonlinear servomechanism problem.








Consider again the nonlinear system N


N: k(t) = f(x(t), u(t), w *(t))

y(t) = Hx(t)

e(t) = r*(t) - y(t) (5-1)


Here r*(t) and w*(t) indicate a particular reference and disturbance out of the class of signals r(t) and w(t).

Now assume that for a certain nominal reference r*(t) and a nominal (anticipated) disturbance w*(t) tracking can be achieved. In
otherwords, assumption (A.1) holds for the nominal signals so that the following solution for (5-1) exists


_x Mt = f( (t) M (tI U M(t))i
y*Wt= 1F(t) (5-2)
0 = F W*()



where x(t) and U (t) are nominal state and input trajectories which are necessary for tracking. We shall not require x*(t) and u*(t) to satisfy a linear differential equation such as the one given in assumption (A.2). Instead, it shall be assumed that both i*(t) and Ti*(t) can be generated by external means so that they are readily available. In

practice, an exact generation of these signals may not always be
possible, however, discussion of this circumstance is deferred until a later section on robustness.

In certain applications it may be desirable to change the reference
signal to a value other than F*(t). (One such application for the









robotic manipulator is discussed in Chapter Six.) The true reference

signal shall be denoted r*(t). Also, it is likely that the actual

disturbance, denoted w*(t), will not be the same as the nominal disturbance w (t). When r*(t) and w*(t) are applied to the system then,
if assumption (A.1) holds, tracking will occur provided that a certain state and input trajectory are present. Denoting this state and input as x *(t) and u*(t) respectively, the following definitions can be made


r (t) r*(t) -*(t)


x (t) x *(t) - *(t) (5-3)
U t u* (t)



where r*(t), *(t), k*(t), and u* (t) denote the deviations of the true

signals from the nominal signals. Note that because the system under consideration is nonlinear, Mx(t) and U (t) are not necessarily the state and input trajectories which give tracking of r (t) with the disturbance W (t).
Now as before, the objective of the control will be to cause the actual state trajectory x(t) to asymptotically approach the state trajectory x*(t). In addition, this will be accomplished with an input u(t) which asymptotically approaches u*(t).
Consider the control scheme given in Figure 5-1. In relation to this scheme we have used the following definitions


(t) x(t) - *(t)
(5-4)
^U(t) UM ut -u u(t)











w (t)


Closed-loop control system


Figure 5-1.









It is apparent that '(t) and %(t) are simply the deviations of the true state and input from the nominal state and input.

As indicated, the tracking error will be zero whenever the state and input of the plant are x*(t) and u*(t) respectively. Assuming there

exists a state trajectory n (t) for the internal model system which allows this to happen, we must then have


K2(t)n*(t) = -K1(t)R*(t) - %*(t) (5-5)


The above equation is a mere consequence of definition (5-3) and the structure of the controller.

Although K1(t) and K2(t) are shown to be functions of time, for the

present, assume that they are constant. Also, assume that x (t) and

S*(t) satisfy a linear differential equation of the form given in assumption (A.2). It then readily follows (see Chapter Two) that if the internal model system is chosen to contain the modes of x*(t) and * (t) and if the pair (A, K2(t)) is observable, tracking will occur for some

initial state [x*(O), n (0)]. Consequently, we shall require that x Wt) and u*(t) satisfy the differential equation given in (A.2). The internal model system is designed accordingly.
One advantage of the new requirement is that, effectively, the class of signals for which x*(t) and u*(t) are allowed to belong is increased. For example, Figure 5-2 shows a state trajectory x*(t) which

consists of a sinusoidal trajectory Z *t) superimposed on some nominal trajectory R*Ct). The trajectory - (t) is not restricted by assumption (A.2).






79


























/-' /
*(t)



t
















Figure 5-2. Nominal and desired state trajectories









We have indicated that tracking occurs provided the initial state is correct. It is also necessary to show asymptotic stability of the system which models the transient dynamics. Since the formulation used

here is roughly the same as in previous derivations, the system NCT given in Chapter Two is still a valid model.


Feedback Gain Selection

In this section feedback gain calculation for the control scheme of

Figure 5-1 shall be discussed. Again, as in the previous chapter,

linearization procedures shall be employed so that Liapunov's indirect method can be used to determine stability.

Before proceeding, it is necessary to discuss the time-varying feedback law which has been chosen for the control scheme of Figure 5-1. Time-varying feedback is in contradiction with the requirements imposed to solve the servomechanism problem; however, it may be true that the time-varying gains vary slowly enough to treat them as constant for all practical purposes (the quasi-static approach). Here we assume this to be the case. Later is will become apparent that selecting timevarying feedback provides better compensation over the nominal trajectory. If the signals comprising the nominal trajectory vary

slowly enough then it is likely that the feedback gains K1(t) and K2(t) can be chosen to vary slowly.

Now consider the stabilization problem. Let us assume that all conditions for tracking and disturbance rejection have been met and the internal model system for the scheme of Figure 5-1 has been chosen in accordance with the theory of Chapter Two (using x*(t) and u*(t) in (A.2)). The linearization of the transient system NCT has been given previously but is repeated here for convenience.










XA(t) FA(t)xA(t) (5-6)
where
[x(t)- x*(t
xA(t) [n(t) - *(t)] (5-7)

and
F F * (t) - G*(t)KI(t) -G*(t)2(t
FA(t) -BH (5-8)


The Jacobian matrices F*(t) and G*(t) are evaluated at the signals x*(t), u*(t), and w*(t). More precisely, we may write




Ft* ( - f(x,u,w) ( G (t) = af(xuw)*
ax x x (t) au x x (t)
u = (t) u u* (t) (5-9)
w (t) w w (t)


Usually these Jacobian matrices cannot be evaluated apriori since x*(t), u*(t), and w*(t) are not known. Consequently, to show stability of the linearized system we shall use a technique already presented in Chapter Four; namely, the Poincare-Liapunov theorem. Let us write the
linearized equation given by (5-6) as


A(t) s FA(t)xA(t) + [FA(t) - FA(t)]xA(t) (5-10)









where

_. F*(t) - G*(t)Kl(t) -G*(t)K2(t)
A -BH A

and




T*(t) := f(x,u,w) G*t = I f(x,u,w)
ax x au() x M
U () uu U*(t) (5-12)
w *t w

Now (see Theorem 4.1) if the system


xA(t) = FA(t)xA(t) (5-13)


is the exponentially asymptotically stable and sup IIF*(t) - FA(t)li. is


suitably small then (5-10) is exponentially asymptotically stable as desired.
First consider conditions under which the quantity sup 1iFA(t) - FA(t)ii is sufficiently small. When the true reference t>O

and disturbance signals are precisely equal to the nominal reference and

disturbance signals then obviously this quantity is zero. If the true signals deviate from the nominal signals then sup IIF*(t) -FA(t)i is t)O
not zero, however, it is generally small (assuming FA(t) depends

continuously on the reference and disturbance) whenever the deviations from the nominal signals are small. Hence, the control law developed here will be effective when the true reference and disturbance signals are close to the nominal reference and disturbance signals.









Now consider the exponential stability of the system (5-13). In

order to achieve this condition with time-varying feedback, it is necesssary that the pair




-BH A 0(5-14)



be stabilizable. The feedback gains could then be selected, for example, by optimal control techniques [20]. Recall, however, that the rate of variation of the gains Kl(t) and K2(t) must be slow if the quasi-static approach is to remain justifiable. Such a condition is

likely when the nominal signals r*(t) and ; (t) are themselves, slowly varyi ng.

Another method for obtaining the needed stability is to choose the feedback gains K1(t) and K2(t) in such a way that the eigenvalues of FA(t) lie in the left half-plane for all t. This approach is valid [17] under the assumption of a slowly time-varying system (i.e.,
d -*
11-FA(t)Ili should be suitably small). Obviously, the slowly timevarying condition is only likely to occur when the nominal signals are slowly time-varying. Assuming this to be true, the resulting feedback gains will also be slowly time-varying as required previously.

As a final point, note that if the nominal signals (t) and i (t) are constant and if the original system is autonomous, then the matrices given in (5-14) will be constant. In this circumstance, constant

feedback gains can be employed, thus eliminating any concern that the slowly-varying feedback gain assumption may not be justified.









Robustness with Respect to Generation of the Nominal Signals

It was previously indicated that, in practice, the nominal signals i*(t) and ii*(t) used as open-loop commands may not be generated

correctly. This could be due to modeling errors in the nonlinear system or even to imperfections in the actual generating mechanism.

First consider the case when only the input is not generated correctly. This is the more important case since often the nominal state trajectory is known exactly while the corresponding nominal input is only approximate due to modeling errors. Suppose that the nominal

input actually supplied to the system is -a(t) while, as before, the input needed to obtain the nominal trajectory is U (t). Now make the

definition


ud(t) = 6*(t) -,a(t) (5-15)



where Ud(t) will be referred to as an input disturbance.

In terms of the definition given by equation (5-15) we may look at the problem from a different point of view. That is, assume the input u*(t) is being generated correctly, however, also assume that there is a disturbance -6d(t) acting in the input channel so that, effectively, iG*(t) - Gd(t) = a(t) is the true signal supplied to the system. This is shown in Figure 5-3. By looking at the problem from the new perspective it is evident that the method described in the previous section can still be applied by simply modeling id (t) as part of the disturbance. It can readily be deduced from Figure 5-3 that this translates into Td(t) satisfying a linear differential equation of the form given in assumption (A.2). Consequently, the dynamics associated

















Za(t)


- d(t)


- T*(t)_--d(t) =a(t)


Modeling incorrect nominal inputs


Figure 5-3.









with T5d (t) must be included in the internal model system (assuming they have not already been included). For example, if the nominal input

supplied to the system differs from the required nominal input by a constant, the internal model system must contain integrators.

Now consider the case when the nominal state is not generated correctly. Let us write


_d * . a
a (t) = R*(t) (t) (5-16)


where a(t) is the nominal state which is actually supplied to the system, x*(t) is the correct nominal state which should have been supplied, and . d(t) is the disturbance representing the difference between the correct and actual signals. Since the nominal state is fed through to the input via a linear feedback gain matrix (see Figure 5.1) it is apparent that d (t) can be modeled as an input disturbance. Hence, we may conclude that the dynamics associated with d(t) must

also be included in the internal model system.

To summarize, we have shown that robustness with respect to the open-loop signals R (t) and U (t) is obtained provided that any deviations from these signals are sucessfully modeled in the dynamics of the internal model system.


Digital Implementation

In the previous treatment of the servomechanism problem there has been an underlying assumption that the control will be implemented via continuous-time methods. Often it is desirable to implement the control using a digital computer and hence a discrete-time control law is









necessary. In this section we give a brief discussion as to how to devise a satisfactory discrete-time algorithm. It is assumed that the discrete-time control algorithm will closely approximate the performance of the already developed continuous-time algorithm. Consequently,

essentially no new theory will be needed. In addition, since discretetime control is a well known subject area [22], [23] much unnecessary detail shall be omitted from this discussion.

Discrete-time control requires sampling of the various outputs (or states) of the plant and if T denotes the spacing between samples, then

sampling occurs at the times t = kT, k = 0,1,2. It is necessary

that the sample rate (l/T) be chosen high enough so that, for all practical purposes, the resulting control algorithm will behave as a continuous-time control law. For example, if the state x*(t) and the input u*(t) consist of sinusoidal signals then obviously the sample rate should be higher than the highest frequency in the sinusoidal signals.

With the above comments in mind we give a typical digital

implementation of the control scheme shown in Figure 5-1. This is shown in Figure 5-4. Notice that the needed continuous-time signals from the plant are converted into discrete-time signals by sampling so that they may be processed digitally. Additionally, the control u(t) to the plant

is produced by converting the discrete-time signal u(k) into an analog signal. This is accomplished by means of a zero-order hold (z.o.h.) which can be thought of simply as a digital-to-analog convertor

providing a piecewise constant version of u(k).

The remainder of Figure 5-4 is basically self-explanatory, however,

we shall discuss two issues in more detail; namely, construction of the internal model system and selection of the feedback gains.







































Figure 5-4. Discrete-time control algorithm


u (k)


w (t)








First consider the internal model system. Since it is to be

implemented digitally a discrete-time model is required. Rather than

discretizing the continuous-time internal model system it is convenient to reformulate the problem in terms of discrete-time signals. If

assumption (A.2) holds then the elements of both the sampled state x*(k) and the input u*(k) will satisfy the difference equation


s(k+r) + drIs(k+r-1) + . + dls(k+l) + d0s(k) = 0 (5-17)


where s(j), j = k, k+1, ., k+r denotes either an element of x*(j) or an element of u*(j). This result is readily obtained using z-transform theory and later an example will be given demonstrating how to obtain the scalars dj, j = 0,1,., r-1 using z-transforms.

The internal model system then takes the form


n(k+l) = adn(k) + Bde(k)

(5-18)

e(k) = r(k) - y(k)


The matrices Ad and Bd are defined as


Ad = T-1 block diag. [Cd, Cd, ., Cd] T (5-19)


Bd = T-1 block diag. [T, T, .,T (5-20)


where






90




00 1 0 . 0 0

Cd T[ ] (521)
0 0 o . 6
-do -dI -d2 .-dr_11 1



and T is arbitrary but nonsingular.

It is not difficult to show that the eigenvalues of the matrix Ad will be identical to the eigenvalues obtained by discretizing the continuous-time internal model system given by equations (2-25) through (2-28). Consequently, the performance obtained using either the system of (5-18) or the direct discretization will be roughly the same. The

system of (5-18) is, however, often easier to obtain and implement.

The following example shows how to calculate the coefficients of the difference equation used to define the discrete-time internal model system.


Example 5.1


Suppose x*(t) = a1

u (t) = a2sinwt (5-22)



where a, and a2 are constants. The sampled signals are


x (k) = a1

u*(k) = a2sinwkT (5-23)


where sampling occurs at every t = kT seconds. Letting X*(z) denote the








z-transform of x*(k) and U*(z) denote the z-transform of u*(k) we have


X*(z) - 1 z- 1


(5-24)


U*(z) a _2z
z- 2zcoswT + 1


The minimum polynomial having roots corresponding to the poles of both X*(z) and U*(z) determines the difference equation (5-17). In our

example, this polynomial is obtained by multiplying together the denominator polynomials of X*(z) and U*(z). The result is the following


(z-1)(z2- 2zcoswT + 1) =


(5-25)


z3 - [l+2coswT]z2 + [l+2coswT]z - 1


Hence, the difference equation is


s(k+3) - [1+2coswT]s(k+2) + [1+2coswT]s(k+1) - s(k) = 0 (5-26) s(j) = x*(j) or u*(j)


and the matrix Cd is


1
0
-(1+2coswT)


0 1
1
(1+2coswT)j


We now consider selection of the feedback gains. Notice in Figure 5-4 that the feedback control law is


Cd 0


(5-27)











'(k) = -Kl(k)'(k) - K2(k)n(k) (5-28)


To be consistant with the time-varying feedback law required for the

quasi-state approach, the gains are shown to be functions of the sample integer k. From (5-28) and the assumption that a zero-order hold will be employed, the actual control input to the plant can be expressed as


"(t) = -Kl(k)^(k) - K2(k)n(k) , kT < t < (k+l)T (5-29)


A necessary requirement for the discrete-time control scheme to be satisfactory is that the control law of (5-29) must result in asymptotic stability of the closed-loop transient system.

One possible method [24], [25] of selecting the feedback is based on a discretized model for the nominal linearized system. This nominal linearized system is given in continuous-time form by equation (5-13). Since the internal model system has already been given in discrete-time form, only the part of the linearized system corresponding to the plant must be discretized. Assuming that the continuous-time system (5-13) is slowly time-varying, we may write the discrete-time equation approximating the dynamics of (5-13) as


xA(k+l) Fd(k)XA(k) (5-30)


where *F(k) - Gd(k)Kl(k) -Gd(k)K2(k)

TA*d(k) dI (5-31)
Ad-BdH Ad I









The discrete-time matrices Fd(k) and Gd(k) corresponding to the plant are obtained [26] by the relationships



F(k) = F(kT)T (5-32)



Gd(k) [ e F (kT)tdt]G*(kT) (5-33)
d 0


Where T is again the sample period. By defining Fd(k) and Gd(k) in this manner it is implicitly assumed that the dynamics of the linearized time-varying system (5-13) do not change over any given sample period. Thus, excluding the case when the reference and disturbance signals are constant, equation (5-30) is indeed only an approximation.

Assuming that our discrete-time model is reasonably accurate, the feedback gains Kl(k) and K2(k) are selected to give stability of the discrete-time system (5-30). The actual mechanism for selecting the feedback gains shall not be discussed, however, solving an algebraic Riccati equation would be one approach [24]. In any event there is a stabilizability (controllability) requirement for the discretized system which must be met. In general, the controllability requirement will be met whenever the continuous-time system is controllable [27] so that controllability of the pair given in equation (5-14) is often sufficient.

















CHAPTER SIX
APPLICATION TO THE ROBOTIC MANIPULATOR

Several key ideas concerning the solution of the servomechanism problem for the robotic manipulator system are presented in this chapter.

The first of the chapter contains a brief summary of the dynamic equations modeling the robotic manipulator. Next the overall structure for the control system is given. In order to effectively treat the

manipulator problem, discussion is given in regards to determining the dynamics which must be included in the internal model system as well as finding a stabilizing feedback law. The controller design is based

primarily on the results of Chapter Five

The final portion of this chapter deals with compensating for structural flexibilities in the manipulator system. It is shown that the proposed control algorithm can correct for end-effector deviations caused by slowly-varying external forces provided that the forces can be measured.



Manipulator Dynamics

In this section we discuss the dynamic equations modeling a rigidlink serial manipulator having revolute joints. A more thorough

treatment of this topic can be found in Thomas and Tesar [281.

Using Lagrange's equation of motion, it is possible to obtain the following dynamic representation for the manipulator.










J(6e + Tv(o,6) = TA(t) + Tg(e) + Td (w(t),e) (6-1)


For a manipulator having N links, e(t), 6(t) and '6(t) are the vectors of length N defining angular positions, velocities, and accelerations of the actuator joints. The matrix J(e) . RNx is the inertia matrix which depends on the manipulator's configuration (i.e., the joint angles e(t)). It can be shown that J(o) is positive definite for all e (see [24]) and is thus always invertible. The inertia torque vector

TV(6,6) : RN corresponds to dynamic torques caused by the velocities of the manipulator's links. Denoting the j-th component of the intertia torque vector as Tv(e,6). we have the following


TV(0,6) = 6'PJ(e)b (6-2)


where PJ(e) . RNxN is a purely configuration dependent matrix referred to in [28] as the intertia power modeling matrix. The term TA(t) 6 RN is the control torque vector which is typically supplied to the actuator joints by electric motors. The torques resulting from gravitational loading are designated by Tg (e) e RN which is a configuration

dependent vector. Finally, Td(w(t),O) E RN is a torque vector resulting from external uncontrollable forces. It is possible to write Td(w(t), e) in the following form
Td(w(t),e) = D(e)w(t) (6-3)


where D(e) : RNxd depends only on the manipulator's configuration and w(t) e Rd denotes the disturbance force vector.




Full Text
147
Correcting for Flexibilities
In this section we present simulations obtained for a manipulator
with flexible links. Again, as in the last section, the manipulator is
to move over a nominal trajectory. Here we wish the hand to actually
follow the nominal trajectory and this is to occur even when a distur
bance force is introduced which causes bending in the links. The scheme
to accomplish this task was discussed in Chapter Six and requires that
the joint angles track reference signals which are computed to offset
link flexibilities.
Figure 7-22 shows the way in which the bending of a link is
modeled. The constants Kv>p, KVjy, K^p, and K^y are taken to be the
same for each link and are given as follows
Kv>p = 2000 m/N
Kv y = 800 m/N-m
K^p = 800 m/N
K^y = 240 rad/N-m
It is assumed that the bending which occurs in the links is a result of
the vertical force acting downward on the hand.
The results of the simulations are given in figures 7-23 through
7-25. Here we consider only a constant disturbance and hence the
internal model system is constructed using integrators. Figure 7-23
shows the actual and desired hand trajectories which occur when a 50 N
disturbance is applied and no compensation is used (i.e., corrective
reference signals are not supplied). It is seen that a significant
amount of error is present. In Figure 7-24 we then supply


112
between each sampling interval of the controller. Since the controller
samples every 40 milliseconds, the Runge-Kutta algorithm updates every 4
milliseconds. The total simulation time is always 4 seconds.
Now consider the disturbance and reference signals which are
applied in the simulations. These shall usually be sinusoidal in
nature. As indicated, the disturbance force acts vertically downward on
the hand. In addition, we shall always choose the reference joint
angles 0^(t) and e,,(t) to be identical. For example, if e^(t) =
10sint then 1)2(t) = 10sint (note that the amplitudes of the
sinusoidal signals are given in degrees). Finally, we point out that in
all simulations both the reference and disturbance signals are applied
at 0.25 seconds.
Control about a Stationary Configuration
In this section we present simulations which have been obtained
when the manipulator operates relative to a fixed nominal position. In
"k k
all cases, the nominal position is = -18, 02 = 36. Since the
control is designed relative to a fixed position, the stabilizing
feedback gains are constant. The various responses are shown in Figure
7-2 through Figure 7-10.
We have adopted a certain notation for describing the figures. For
example, if the figure caption reads "ref. = 6 Hz sinusoid with
amplitude of 10" then it will mean that the reference joint angles
^(t), i = 1,2 are both equal to the function 10sin (2ir 6(t-0.25))
(recall that the reference is always applied at 0.25 seconds). In
addition, the notation "I.M. frequency = F Hz" will indicate that the
internal model system has eigenvalues at e*j2TrFT where T is the sample


n(t)
Figure 2-1. Closed-loop control system NC
<£>


135
O
(a) Tracking error
Figure 7-16. Manipulator's response: ref. = 2.0 Hz sinusoid with
amplitude of 10; dist. =0; I.M. frequencies: 0 Hz
and 2.0 Hz


132
O
O
07
(b) Actual and desired trajectories of the hand
Figure 7-14 continued


49
Adhering to this more compact notation, we may write the linearized
approximation to NCT as
where
and
xA(t) FA(t)xA(t)
F*(t) G*(t)K1 -G*(t)K2
-BH A
(4-3)
(4-4)
F*(t)
3f(x,U,w)
3X
*
X (t)
u*(t)
w*(t)
G*(t)
_ 3f(x,U,w)
3U
*
X (t)
u*(t)
w*(t)
(4-5)
Note that (4-3) is the linearized system needed in conjunction with
Liapunov's indirect method (Theorem 2.3) and was originally given as
equation (2-40).
Although by applying Liapunov's indirect method we reduce the
problem from stabilizing a nonlinear system to stabilizing a linear
system, the time-dependency of this linear system can create compli
cations. Equations (4-4) and (4-5) show how this time-dependency enters
ic
into the linearized system due to the time-varying signals x (t), u (t),
and w*(t). To further complicate the stability problem, it is very
likely that the matrices F*(t) and G*(t) will not even be known. This
is because both F*(t) and G*(t) are implicitly dependent on the


11
problem is avoided here, however, since we consider systems which have
the same number of inputs as outputs.
The second assumption is perhaps the most restrictive. It is
different from the assumption commonly made in the linear servomechanism
problem; namely, that the disturbance w*(t) and the reference r*(t) both
satisfy a linear differential equation of the form given by (2-4). Here
we are concerned with this class of disturbance and reference signals;
however, in the nonlinear case it is important to work also with the
acuta! state and input trajectories which arise during tracking.
If we assume that r*(t) satisfies a differential equation of the
form given by (2-4), it is actually not unreasonable to assume that
x (t) will satisfy the same equation. This is because the output y (t),
jlf
which must be identically equal to r (t) during tracking, is taken to be
a linear combination of the state x (t). Consequently, if all elements
of the state are reflected in the output, these elements must satisfy
(2-4). The assumption on the input signal u (t) is then the assumption
which needs further discussion. In the nonlinear servomechanism
problem, it is often the case that u (t) will contain terms (e.g.,
sinusoids) not present in either w*(t) or r*(t). To help clarify this
point, consider the following proposition.
Proposition 2.1: Given the autonomous system
x(t) = f(x(t) u(t), w(t)) (2-5)
assume there exists an input u(t) such that x(t) = Xp(t) is the solution
to (2-5) with initial state xp(0) and with disturbance w(t) = wp(t).


ERROR IN JOINT 2 (DEGREES) ERROR IN JOINT 1 (DEGREES)
-1.00 -2.00 0.00 2.00 1.0 -2.00 -1.00 0.00 1.00 2.00
117
Figure 7-4
Tracking error: ref. = 0; dist. =
amplitude of 50 N; I.M. frequency
1.5 Hz sinusoid with
= 0 Hz


6
conditions will allow the true state trajectory to asymptotically
converge to the trajectory which gives zero tracking error. These
stability conditions are easily checked using Liapunov's indirect
method. It is noted, however, that with Liapunov's approach, the
tracking error may only asymptotically converge to zero for a limited
range of initial states. Roughly speaking, this can be considered
equivalent to requiring that the disturbance and reference signals
remain small.
In Chapter Three, using the approach developed for the nonlinear
problem, we rederive the well known conditions imposed for a solution to
the linear servomechanism problem.
In Chapter Four, selection of locally stabilizing feedback based on
linearization techniques is discussed in detail. Due to the complexity
of the stability problem, the control law derived here is for time-
invariant systems which are acted upon by small reference and distur
bance signals. Simulations of a nonlinear system are provided which
verify the design technique. Also discussed in Chapter Four is the
interpretation of using optimal control techniques to arrive at the
feedback law required for the linear servomechanism problem. In a
nonlinear system, however, a certain degree of stability is often
desired. Consequently, in order to achieve this stability using optimal
control theory, a well known technique due to Anderson and Moore [16] is
presented.
In Chapter Five, we develop a controller designed to solve the
nonlinear servomechanism problem when a nominal input and state
trajectory are supplied as open-loop commands. Here essentially no new
theory is needed since the control problem can actually be treated using


SIX APPLICATION TO THE ROBOTIC MANIPULATOR 94
Manipulator Dynamics.............. 94
Actuator Driving Torques 96
Feedback Control System for Tracking and Disturbance
Rejection.............. 97
Determining the Dynamics of the Internal Model System... 98
Feedback Gain Calculation 101
Compensation for Flexibilities in the Manipulator's
Links 106
SEVEN SIMULATION RESULTS 109
The Simulated System 109
Control about a Stationary Configuration 112
Control over a Time-varying Nominal Trajectory 124
Correcting for Flexibilities 147
EIGHT CONCLUSIONS AND OPEN PROBLEMS 153
REFERENCES 155
BIOGRAPHICAL SKETCH..... 158
IV


7
previously developed ideas. Designing the controller about a nominal
trajectory is a standard technique often used in dealing with nonlinear
systems. This technique usually reduces the stability requirements of
the feedback law since it is assumed that the state trajectory will
never deviate far from a prescribed nominal trajectory.
Another topic discussed in Chapter Five is that of using a
discrete-time control law to approximate the already developed
continuous-time control law.
In Chapter Six we consider the robotic manipulator as a system for
which to apply the results derived in this paper. The dynamic equations
modeling the manipulator are analyzed in view of the servomechanism
problem. It is shown that the requirements needed for a solution to the
servomechanism problem are satisfied for the robotic system. Also
considered for the manipulator is an application of tracking where the
signals being tracked are used to compensate for errors in end-effector
location due to link flexibilities.
Chapter Seven shows results, obtained using simulations, of
applying the control scheme to a 2-link manipulator. These results
verify that the method will improve, or sometimes eliminate, steady-
state tracking error.


REFERENCE FOR JOINT 2 (DEGREES) REFERENCE FOR JOINT 1 (DEGREES)
,-10. cm -5.00 0.00 5.00 10.00 -10. GO -5.00 0.00 5.00 10.00
152
Figure 7-25. Corrective reference signals


42
Example 3.1:
Let us take
R*(s)
is-V
a +
(S-X1)
b +
(s-L)
(3-8)
where a, b, c e RP are constant vectors. Also, for simplicity of this
example let
W*(s) = 0
(3-9)
We now show that when conditions (1), (2), and (3) used in Theorem 3.2
ie
hold true, tracking will occur for some initial state xQ and input
U(s). The input U(s) can be chosen as
u(s) = w + ,
(s V (s-xj)
Vv +XCTr
where w, v, r e Rm are vectors given by
v = {H[x1I-F]1G}"1b
w = {H[X1I-F]_1G}"1 {a + H[x1I-F]"2Gv}
r = {H[x2I-F]'1G}'1c
The initial state
should be taken as
(3-10)
(3-11)
x* = -{ [F-x1I]"1Gw + [F-x1I]"2Gv + [F-X2I]1Gr }
(3-12)


ERROR IN JOINT 2 (DEGREES) ERROR IN JOINT 1 (DEGREES)
-2.00 -1.00 0.00 1.00 2.00 -0.6O -0.30 0.00 0.30 0.60
116
Figure 7-3. Tracking error: ref. = 0; dist. = 1.5 Hz sinusoid with
amplitude of 50 N plus constant with amplitude of 25 N;
I.M. frequencies = 0 Hz and 1.5 Hz


105
fact, obvious since complete controllability of the linearized system
has already been asserted.
One method of obtaining Mj, M2 and M3 is simply to select
stabilizing feedback gains for some arbitrary time, say tQ. This yields
MX = F1(t0) -
M2 F2(t0) (6-19)
M3 -J_1(e*)K2(t0)
Then, the feedback proposed to make the system behave as though it were
time-invariant is the following
Kl,l(t) = J(e*(t)) [Fx(t) Mx]
K1>2(t) = J(0*(t)) [F2(t) M2] (6-20)
K2(t) = -J(0*(t)) M3
Substituting the above gains into (6-17) will immediately verify that
they are indeed correct.
We note that the feedback gains derived here can be evaluated at
any instant of time using the nominal position and velocity of the
manipulator. One method of computing the gains in real-time, however,
might be based on actual rather than nominal trajectories. This is
especially true if the nominal trajectory is not known in advance.


65
(a) Design based on nonlinear servomechanism theory
(b) Design based on linear servomechanism theory
Figure 4-4
Tracking error: reference = 2sin(t), disturbance = 1.0


44
"k *k
Consequently, x (t), u(t) and u (t) all satisfy the differential
equation
(.)(3) + C-:A2-2X13 (-)(2) + [2x1x2 + Xj2] (*)(1) + C-A2 X2] () = 0
(3-16)
This completes the example. Although a disturbance was not considered,
the inclusion of a disturbance would have led to similar results.
Next we consider another requirement which was needed to solve the
nonlinear servomechanism problem and relate it to the linear problem.
This requirement is that the closed-loop transient system NCT given by
(2-38) must be asymptotically stable. When the linear system is
considered, (2-38) takes the following form
LCT:
\t)
F 0
"x(t)
G
s
+
_n(t)
-BH A
_5(t)_
0
u(t) = -K^(t) K2(t) (3-17)
Now if conditions (B.l) and (B.2) both hold then it is possible to
select ! consequence of the following well known theorem (see [2] or [3]).
Theorem 3.3: If (B.l) and (B.2) both hold and the eigenvalues of A
correspond exactly to A., i =1, 2, ..., r given by condition (B.2)


114
introduce additional poles into the I.M. system at 3 Hz. The resulting
response is shown in Figure 7-6. Although the sinusoidal ripple has
been virtually eliminated, there is a very small dc error. This dc
error would not be expected in linear servomechanism problems, however,
it occurs here because the manipulator system is nonlinear. In order to
eliminate the dc error we then introduce integration capabilities into
the I.M. system. The resulting error curves are shown in Figure 7-7 and
the steady-state error is essentially zero.
Figures 7-8 through 7-10, show the responses obtained when both
tracking and disturbance rejection is to occur. Figure 7-8 shows the
result of using linear servomechanism theory. That is, only the poles
corresponding to the frequencies of the reference and disturbance
signals (3 Hz and 1 Hz respectively) have been included in the I.M.
system. A steady-state sinusoidal error at a frequency of 6 Hz is seen
in the plots. Figure 7-9 shows the response obtained when the I.M.
system is adjusted to include poles corresponding to a 6 Hz sinusoidal
signal. A small ripple still occurs at an even higher frequency;
however, for the most part, the error consists only of a dc component.
Consequently, in the next phase of the design, integrators are
incorporated into the I.M. system. The response is given in Fiugre 7-10
and we note that the steady-state error in both joints is now very
small.
The simulations presented in this section show that if small
steady-state tracking error is required, it may be necessary to
introduce dynamics into the internal model system which may not have
been included in a design based only on linear servomechanism theory.


53
We see that F is simply the matrix resulting from a linearization of
the system NCT about the origin. In addition, because the original
system is time-invariant F, G, and F are constant matrices.
Let us investigate the stability of NCT using the Poincare-Liapunov
theorem. To do this we give the following result.
Theorem 4.2: Suppose that for a particular time-invariant system
assumptions (A.l) and (A.2) are satisfied and that the conditions needed
to apply Liaponov's indirect method to the system NCT (see eq. (2-47)
and (2-48)) hold true. In addition, assume that the following
conditions are satisfied.
(i)sup iiF*(t) Fii. = e
t>0 1
(ii)sup llG*(t) GII. = e,
t>0 1
where and e^ are positive constants.
(iii)The pair [F, G] is stabilizable
(iv)
rank
xiI F
-H
n + p
for all which are characteristic roots of the differential
equation given in assumption (A.2).


63
Y
i
?.
=r
(a) Design based on nonlinear servomechanism theory
(b) Design based on linear servomechanism theory
Figure 4-2
Tracking error: reference = 2sin(t), disturbance =1.0


59
Using (4-33) and (4-34) it readily follows that
and
* o
sup IlF (t) F II = (2ou + 40,00 = s
t>0 lie.
sup IIG (t) G^ II = 0 = e,
t>0
(4-38)
(4-39)
In order to show stability by the Poincare-Liapunov theorem we must then
have
e^m a < 0 (4-40)
where -a corresponds to the real part of the right-most eigenvalue of
the matrix given in equation (4-35) when F*(t) is replaced by F. The
constant m depends on the eigenvectors of this matrix.
In order to meet the stability requirement, the feedback gains
and K2 are selected using a standard technique for eigenvalue assign
ment. (Note: it can be shown that the linearized system is control
lable and hence arbitrary eigenvalue assignment can be made). The
closed-loop eigenvalues are chosen to be:
-4.0
-5.0
-5.0 j3.0
-4.0 j2.0
This gives a = 4, however, we shall not concern ourselves with the
calculation of m. Just as an example, let us assume m = 1, = 1/2


102
% 3V*taw)
<()=()) (t)
*
w=w (t)
0
Fx(t)
!NxN
F2(t)
(6-12)
-*
3f(j)(<()JA,w)
3Tn
= (t)
TA=T*(t)
*
w=w (t)
-1 _*
J 1(e )
(6-13)
It is not difficult [24] to derive equations which explicitly relate the
matrices Fj(t) and ^(t) to the nominal trajectory, however, in the
interest of brevity, we omit these derivations.
Now consider the feasibility of selecting the feedback gains to
stabilize the linearized closed-loop system. Assuming that the slowly
time-varying approach will apply, the eigenvalues of the linearized
matrix must lie in the open left half-plane for all t. Theorem 3.3 can
be used to determine when it is possible to meet this goal using an
appropriate feedback law. Interpreting Theorem 3.3 for the slowly time-
varying system, the following conditions must hold:
(1) [Fx(t), G,(t)] must be controllable at each t.

(2)
Xi ¡2Nx2N
-H.
%{t)
K(t>
0
must have full rank (rank 3N) for
each t and for each which is an eigenvalue of the internal model
matrix A.


27
half-plane, all modes associated with A must be forced to zero or NCT
will not be asymptotically stable. From (2-38) we see that 'x(t) is the
only signal which can accomplish this task. Consequently, x(t) should
be a function of the modes in n(t) induced by the eigenvalues of A.
Since x(t) depends on 'ri(t) only through the feedback coupling from the
gain K2 and (A, K2) is not observable, it is impossible for x(t) to
depend on the unobservable modes. Thus, we can conclude that
observability of the pair (A, K2) is necessary for asymptotic stability
of the system NCT. We therefore have the following result.
Proposition 2.3: Controllability of (A, B) and observability of (A, K2)
are necessary conditions for asymptotic stability (either local or
global) of NCT.
Although the above result is important, it is even more important
that a practical method is available which allows one to ascertain
directly whether or not NCT is asymptotically stable. It has already
been indicated that the local stability results of Theorem 2.2 will most
often apply. One convenient method for showing local stability is
Liapunov's indirect method which can be found in standard texts on
nonlinear systems (e.g., see [17]). The required linearization of the
closed-loop transient system NCT about the equilibrium point x(t) = 0,
ri(t) = 0 is
£(t)
F*(t) G*(t)K1
-G*(t)K2
x(t)
S(t)
-BH
A
(t)


34
Summary
In this chapter, a nonlinear system was considered with the number
of inputs equal to the number of outputs and with the output taken as a
linear combination of the system's state. In the last part of the
chapter, conditions were given so that a nonlinear system with more
inputs than outputs could be treated.
Prior to developing the theory for the nonlinear servomechanism
problem, two major assumptions were made. The first of these, assump
tion (A.l), was absolutely necessary since without it, the
servomechanism problem could not be solved under any circumstance. This
being the case, the primary attention was focused on the second assump
tion, assumption (A.2). Here, the requirement was made that the input
and state trajectories which occured during tracking were to satisfy a
linear differential equation. It was noted that in practice, such an
assumption may only be approximate, however, a design based on the
approximation could be perfectly adequate. Typically, truncated Fourier
series expansions approximating the true signals would be used for
design purposes.
In the first part of the controller design we dealt with the
development of an internal model system. It was indicated that this
internal model system would have to contain dynamics which matched the
dynamics of the state and input which are necesary for tracking. The
importance of such an internal model system becomes evident when it is
compared to a standard alternative. A typical approach to solving the
nonlinear servomechanism problem is to first linearize the nonlinear
system and then design a controller using linear servomechanism
theory. This leads to an internal model system containing dynamics


CHAPTER SEVEN
SIMULATION RESULTS
The purpose of this chapter is to demonstrate the performance of
controllers designed to solve the nonlinear servomechanism problem for
the robotic manipulator system. The particular manipulator chosen for
the demonstrations is a 2-link planar manipulator. Certain tracking and
disturbance rejection applications are considered and the resulting
control schemes are tested using computer simulations.
The Simulated System
As indicated, a 2-link planar manipulator is used to evaluate the
performance of the control schemes. A schematic of this manipulator is
given in Figure 7-1. It is important to note that the manipulator has
been selected to operate in the vertical plane with gravity acting down
ward. In addition, a disturbance force which acts vertically downward
on the hand has been included. This will be the nature of all disturb
ances used in the simulations.
We have chosen a discrete-time controller as a means of demon
strating the control algorithms. Basically, the results from Chapter
Five on digital implementation have been applied. In all cases, a 40
millisecond sample period is used. The combined plant and controller
can be modeled by the discrete-time version of the scheme given in
Figure 6-1.
109


ACKNOWLEDGEMENTS
I would like to express my deep appreciation to Dr. Edward W. Kamen
for the suggestions and encouragement he provided over the duration of
my graduate studies. It is safe to say that without his continuous
support, this research would not have been completed. I would also like
to thank Dr. Thomas E. Bullock for his valuable comments concerning this
work. These comments were needed, and the time he spent reviewing the
paper is greatly appreciated. Special thanks also go to Dr. Irena
Lasiecka, Dr. Peyton Z. Peebles, and Dr. George N. Sandor for serving on
my Supervisory Committee; all three are professors under whom I have had
the pleasure of taking courses.


51
(2) f^(t, x) satisfies the criterion
sup Ilf. (t,x)n < LlixII L > 0 (4-10)
t>t
o
Then if we have L suitably small so that
(mL a) < 0, (4-11)
the system (4-6) is exponentially asymptotically stable for the
equilibrium point x = 0.
Proof: Let $(t,tQ) be the state transition matrix for the system
(4-8). Consequently, we can write the solution to (4-6) as
t
x(t) = *(t,t )x + / *(t, t)f,(t, x(t))dx (4-12)
*o
By taking the norm of both sides of (4-12) and using (4-9) it easily
follows that
IIx(t) II < me"a^"^o^ llxQII + / mea^"x^L IIx(x) IIdr (4-13)
Multiplying through by e gives
. at t
eaT* iix(t) il < me 0 llxQ ll + mL / eaT IIx(x) lldx
t_
(4-14)
We may now apply the Bellman-Gronwall inequality (see [17]) to obtain


38
Consider the linear time-invariant system
L: x(t) = Fx(t) + Gu(t) + Ew(t)
y(t) = Hx(t) (3-1)
e(t) = r(t) y(t)
where x(t) e Rn is the state, u(t) e Rm is the input, w(t) e R* is a
disturbance, y(t) e Rp is the output, and e(t) e Rp is the error which
arises in tracking the reference signal r(t) e Rp. Conditions shall be
given as to when it is possible to design a controller such that
e(t) + 0 as t + . It is assumed that the elements of the reference
r(t) as well as the disturbance w(t) satisfy the linear differential
equation
() + vi(')(r'1) + + n(-)<1) + = 0 (3-2)
where the characteristic roots of (3-2) are assumed to be in the closed
right half-plane. We shall let i = 1, 2 r denote the
distinct characteristic roots of (3-2) where r < r due to multi
plicities. The following well known result gives conditions under which
the linear servomechanism problem can be solved.
Theorem 3.1: Assume the state x(t) is available for feedback. A
necessary and sufficient condition that there exists a linear time-
invariant controller for (3-1) such that e(t) + 0 as t + for all
r(t) and w(t) with elements satisfying (3-2) is that the following two
conditions both hold.


83
Now consider the exponential stability of the system (5-13). In
order to achieve this condition with time-varying feedback, it is
necesssary that the pair
F*(t)
0
~G*(t)
-BH
A
*
0
(5-14)
be stabilizable. The feedback gains could then be selected, for
example, by optimal control techniques [20]. Recall, however, that the
rate of variation of the gains K^(t) and ^(t) must be slow if the
quasi-static approach is to remain justifiable. Such a condition is
likely when the nominal signals r (t) and w (t) are themselves, slowly
varying.
Another method for obtaining the needed stability is to choose the
feedback gains K^(t) and ^(t) in such a way that the eigenvalues of
k
F^(t) lie in the left half-plane for all t. This approach is valid
[17] under the assumption of a slowly time-varying system (i.e.,
ll^jrFA(t) ll.. should be suitably small). Obviously, the slowly time-
varying condition is only likely to occur when the nominal signals are
slowly time-varying. Assuming this to be true, the resulting feedback
gains will also be slowly time-varying as required previously.
k k
As a final point, note that if the nominal signals r (t) and w (t)
are constant and if the original system is autonomous, then the matrices
given in (5-14) will be constant. In this circumstance, constant
feedback gains can be employed, thus eliminating any concern that the
slowly-varying feedback gain assumption may not be justified.


TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT v
CHAPTERS
ONE INTRODUCTION 1
TWO TRACKING AND DISTURBANCE REJECTION FOR
NONLINEAR SYSTEMS 8
Notation 8
Main Results for the Nonlinear Servomechanism Problem... 9
Stability of the Closed-Loop Transient System 26
The Relation Between the Dimension of the Internal
Model System and the Input/Output Dimensions 29
Summary 34
THREE APPLICATION TO LINEAR SYSTEMS 37
Review of Linear Servomechanism Results 37
Solution to the Linear Problem via the
Nonlinear Formulation 39
FOUR FEEDBACK CONTROL 47
Stabilization Using the Linearized Equation 47
Optimal Feedback for the Linear Servomechanism Problem.. 66
Increased Degree of Stability Using the Optimal
Control Approach 71
FIVE PRACTICAL CONSIDERATIONS 74
Controller Based on the Nominal Trajectories 74
Feedback Gain Selection 80
Robustness with Respect to Generation of the
Nominal Signals 84
Digital Implementation 86
i i i


32
D = QB
(2-54)
By the assumption p < q there exists a row vector z' e which is
orthogonal to the range D. Hence
z'D = 0 (2-55)
Consequently, by letting v = z'Q it is evident that (2-50) holds.
In the previous sections, it was shown that stability of the
closed-loop transient system NCT is a key requirement in the solution of
the nonlinear servomechansim problem. To achieve such stability, the
conditions that the pair (A, B) be controllable and the pair (A, K2) be
observable were shown to be crucial. Hence, from Proposition 2.4 we can
conclude that the number of blocks q in the internal model system must
be such that q < min(m,p).
Now recall a major earlier assumption; namely, the open-loop input
u (t) which forces the nonlinear system N to track r (t) satisfies the
differential equation (2-4). Ultimately, such an input is generated by
the internal model system as can be seen from Theorem 2.1 or equation
(2-35). If u*(t) e Rm is arbitrary (aside from satisfying (2-4)) then
it is not difficult to see that the internal model system must have at
least m blocks. In otherwords, we must have the condition q > m.
Because of earlier condition that q < min(m,p) and because of the
assumption that m > p we are forced to consider systems with m = p.
The modes of the various elements of u (t) have a one to one
corresondence with the eigenvalues of the C matrices which makg up the
block diagonal A matrix. Since we assume all m elements of u (t) are
independent of one another then m separate blocks will be needed in A to
insure this independence. For further insight, see the proof of
Proposition 2.2.


87
necessary. In this section we give a brief discussion as to how to
devise a satisfactory discrete-time algorithm. It is assumed that the
discrete-time control algorithm will closely approximate the performance
of the already developed continuous-time algorithm. Consequently,
essentially no new theory will be needed. In addition, since discrete
time control is a well known subject area [22], [23] much unnecessary
detail shall be omitted from this discussion.
Discrete-time control requires sampling of the various outputs (or
states) of the plant and if T denotes the spacing between samples, then
sampling occurs at the times t = kT, k = 0,1,2... It is necessary
that the sample rate (1/T) be chosen high enough so that, for all
practical purposes, the resulting control algorithm will behave as a
if
continuous-time control law. For example, if the state x (t) and the
input u (t) consist of sinusoidal signals then obviously the sample rate
should be higher than the highest frequency in the sinusoidal signals.
With the above comments in mind we give a typical digital
implementation of the control scheme shown in Figure 5-1. This is shown
in Figure 5-4. Notice that the needed continuous-time signals from the
plant are converted into discrete-time signals by sampling so that they
may be processed digitally. Additionally, the control u(t) to the plant
is produced by converting the discrete-time signal u(k) into an analog
signal. This is accomplished by means of a zero-order hold (z.o.h.)
which can be thought of simply as a digital-to-analog convertor
providing a piecewise constant version of u(k).
The remainder of Figure 5-4 is basically self-explanatory, however,
we shall discuss two issues in more detail; namely, construction of the
internal model system and selection of the feedback gains.


84
Robustness with Respect to Generation of the Nominal Signals
It was previously indicated that, in practice, the nominal signals
x (t) and (t) used as open-loop commands may not be generated
correctly. This could be due to modeling errors in the nonlinear system
or even to imperfections in the actual generating mechanism.
First consider the case when only the input is not generated
correctly. This is the more important case since often the nominal
state trajectory is known exactly while the corresponding nominal input
is only approximate due to modeling errors. Suppose that the nominal
input actually supplied to the system is a(t) while, as before, the
*
input needed to obtain the nominal trajectory is u (t). Now make the
definition
d(t) = *(t) a(t) (5-15)
where lid(t) will be referred to as an input disturbance.
In terms of the definition given by equation (5-15) we may look at
the problem from a different point of view. That is, assume the input
u (t) is being generated correctly, however, also assume that there is a
disturbance d(t) acting in the input channel so that, effectively,
(t) ua(t) = Iia(t) is the true signal supplied to the system. This
is shown in Figure 5-3. By looking at the problem from the new perspec
tive it is evident that the method described in the previous section can
still be applied by simply modeling d(t) as part of the
disturbance. It can readily be deduced from Figure 5-3 that this
j
translates into Tr(t) satisfying a linear differential equation of the
form given in assumption (A.2). Consequently, the dynamics associated


55
II 4>(t, tQ) II.
< me
-a(t-tQ)
(4-23)
for some positive constants m and a. Hence, to show condition (2) of the
Poincare-Liapunov theorem and also to conclude stability we must have
sup HF*(t)
t>0 A
< L
where L is small enough to insure that
(4-24)
(mL a) < 0
(4-25)
Equation (4-24) can be verified by using the definitions for
k o
F^(t) and F^ to obtain the relationship
llF*(t) Fa"i < llF*(t) F"i + gVhiCKj, K2]ni (4-26)
Using (i) and (ii) then gives
sup llF*(t) Fll < ej + e2H[K1, K2]IIi (4-27)
If and e2 are small enough, then condition (2) of the Poincare-
Liapunov theorem is satisfied and the system NCT is asymptotically
stable.
The final condition which needs to be verified for a solution to
the servomechanism problem is the observability of the pair (A, K2).
This condition presents no problem, however, since it immediately
follows from Theorem 3.4 that (A, K2) is observable whenever the system


137
i
|
O
o
o
o
Figure 7-17. Manipulator's response: ref. = 2.0 Hz sinusoid with
amplitude of 10 ; dist. = 0; I.M. frequencies = 0 Hz,
2.0 Hz, and 4.0 Hz


40
Proof: We shall show the existence of u*(t) and x*(t).
Let
u*(t) = -Kx*(t) + u(t) (3-3)
where K e Rmxn and Tj(t) e Rm are still to be defined. Also let the
Laplace transforms of r*(t), w*(t), x*(t), and U(t) be denoted as
R (s), W (s), X (s), and U(s) respectively. Then, if tracking is to
occur, it can be verified using (3-1) that the following relationships
must hold.
X*(s) = [sI-F]_1x* + [sI-F]-1Glj(s) + [sI-F]_1EW* (s) (3-4)
and
R*(s) = H[sI-F]-1x* + H[sI-F]-1G(s) + H[sI-F]_1EW*(s) (3-5)
where
F:= F GK (3-6)
Now consider the following conditions
(1) m > p
(2) rank [ x.I-F] = n for all x^ i = 1,2, ..., r
(3) rank H[X^I-F]G = p for all X^ i = 1,2 r
where x.., i = 1,2 r are the characteristic roots of the linear
differential equation (3-2).
When (1), (2), and (3) are satisfied then both (3-4) and (3-5) will
hold true (i.e., tracking will occur). Furthermore, when these three


106
Real-time computations would have to be performed very quickly and would
most likely require techniques such as those described by Wander [30]
where array processors have been employed.
Compensation for Flexibilities in the Manipulator's Links
The desire for greater accuracy and reduced weight in robotic
manipulators has spurred interest in control algorithms capable of
compensating for structural flexibilities. Here we develop a control
strategy for the manipulator which provides compensation for end-
effector deviations resulting from link flexibilities. This control
strategy is based on the assumption that the external disturbance forces
which cause link flexing can be measured by force sensing equipment. In
addition, it is assumed that the external disturbances are of a very low
frequency content and that the manipulator is not undergoing rapid
motion. This means that the manipulator dynamics can still be modeled
as though it were rigid as opposed to more complex models [31-32],
devised for flexible manipulators.
The proposed control strategy involves little more than providing a
suitable corrective reference signal ^(t) for the scheme shown in
Figure 6-1. Previously it was indicated that if tracking of a nominal
trajectory is desired, then e^t) should be taken as zero since a
nonzero e (t) gives a deviation from the nominal trajectory. Here,
however, this deviation is used to adjust the joint angles in such a way
that the end-effector remains on a specified path, even when bending
occurs in the links.
We now give a brief description of the method used to calculate the
corrective reference signal ^(t). By considering the stress-strain


r (k) +/~\ e(k)
\ly
A -
n(k+l)=Adn(k)+B(1e( k)
n(k)
K>(k)
u(k)
y(k)
IS
sampler
Figure 5-4. Discrete-time control algorithm
00
00


r (t) = r (t)+r (t)^ e(t)
a.
= An + Be
n( t)
K^t)
Figure 5-1. Closed-loop control system
I


45
then the pair
F
0
6
-BH
A
>
0
L
(3-18)
is stabilizable. If in addition, the word "stabilizable" in (B.l) is
replaced by "controllable" then the pair of (3-18) is controllable.
The next theorem shows that when and Kg are selected to
stabilize (3-17) the pair (A, Kg) is observable. This is the precise
condition needed for Theorem 2.1 and the final condition required for
our discussion.
Theorem 3.4: If all eigenvalues of A are in the closed right half-plane
and the system LCT described by (3-17) is asymptotically stable then
the pair (A, Kg) is observable.
Proof: We use contradiction. Suppose that the system LCT is
asymptotically stable but (A, Kg) is not observable. This implies that
there exists a vector v such that
XI-A
k2
v = 0
(3-19)
for some x which is an eigenvalue of A. Consequently, we can write
"xi F + GKX GKg
~ o'
BH XI-A
V
(3-20)


26
Stability of the Closed-Loop Transient System
In this section we investigate the stability of the system NCT.
First, the previously defined condition of controllability for the pair
(A,B) and the required observability of the pair (A, K2) are related to
the stability of NCT. Next, a method for checking local stability of
NCT using Liapunov's indirect method is presented.
Now consider controllability of the pair (A,B) which has already
been insured by the chosen structure for the internal model system.
Suppose, for the sake of example, that the pair (A,B) is not
controllable. This implies that the pair (A, BH) is not controllable.
Consequently, there exists a linear transformation matrix P such that
~A1
a2~
i ,
a4~
9
P ^BH =
_0
a3_
_0 _
(2-39)
Since the eigenvalues of A are in the closed right half-plane, the
eigenvalues of A3 are in the closed right half-plane. It is apparent
|f
that the modes associated with A3 are not affected by any control
law. Thus, we can conclude that when (A,B) is not controllable, the
system NCT can not be made asymptotically stable. This points out one
of the reasons behind the structure chosen for the internal model
system.
Now consider the situation which arises when the pair (A, K2) is
not observable. Since the eigenvalues of A are in the closed right
* Modes are components of the form t^ext which appear in the
solutions to linear differential equations. For example, given the
system x(t) = Fx(t), x(0) = xQ with solution x(t) = $(t,0)xo; the
elements of the state transition matrix $(t,0) are made up of modes of
the form tke Here X, which is generally complex valued, represents
an eigenvalue of the matrix F.


98
In normal circumstances, the function of the controller will be to
insure that the manipulator actually maintains the nominal trajectory

(t) even in the presence of disturbances.
The input torque TA(t) which gives the nominal trajectory can be
calculated using (6-7). Consequently, with the nominal state, input,
and disturbance known, it is possible to use a controller of the form
shown in Figure 5-1. This control scheme is given again in Figure 6-1
with the notation switched to correspond to the manipulator system. It
is apparent from the figure that the joint angle vector e(t) is chosen
as the output. This is because when tracking occurs for e(t) it must
also occur for 0(t). Typically, the goal is to have e(t) equal to i*(t)
so that (t) will usually be zero. In certain instances, however, it
may be necessary to adjust e (t) on-line to accommodate for
flexibilities in the manipulator's links. More will be said on the
flexibility problem in a later section.
Since most of the background for the controller design has been
given in previous chapters, it shall not be repeated here. In the next
two sections, however, we investigate in more detail both the design of
the internal model system and calculation of the feedback gains.
Determining the Dynamics of the Internal Model System
The structure of the internal model system has been given in
previous chapters. It is still necessary to evaluate the input and
state trajectories which occur during tracking so that the internal
model system can be designed accordingly. Since the state trajectory is
Actually, it is the deviations of the state and input
trajectories from the nominal trajectories which are important.


119
O
o
o
o
Figure 7-6
Tracking error: ref. = 1.5 Hz sinusoid with amplitude
of 15 ; dist. = 0; I.M. frequencies = 1.5 Hz and 3.0 Hz


97
give a desired end-effector location. In fact, e is not unique [29].
Once 0 is known, obtaining the joint velocities and joint accelerations
given the end-effector velocities and accelerations is rather easy.
Suppose that the desired end-effector trajectory has been specified
and converted into the corresponding trajectory of joint positions,
velocities, and accelerations. The symbols 6*(t), 0*(t), and e*(t) will
be used to denote this trajectory where we are assuming that 0*(t) is
twice differentiable. The torque T^(t) which must be supplied to the
actuators to produce the desired trajectory can be calculated from the
relationship
k 99k \t k *k n k A k
TA(t) = J(9 )0 + TV(0 ,0 ) T9(0 ) T(w(t),0 ) (6-7)
Here it is assumed that the disturbance w(t) is known. If this is not
the case, the expected (nominal) disturbance w*(t) would generally be
substituted into (6-7).
Feedback Control System for Tracking and Disturbance Rejection
The control strategy discussed in Chapter Five shall be adopted for
the control of the robotic manipulator. Thus, it will be necessary to
know in advance the nominal trajectory as well as the input which gives
this trajectory. The nominal trajectory needed for the control algo
rithm consists of only the joint angles and the joint velocities. If
m~k
the nominal joint angles and joint velocities are denoted 0 (t) and
Xk
0 (t) respectively, it is appropriate to define the nominal state
trajectory as
$*(t) =
0*(t)
0*(t)
(6-8)


124
Control Over a Time-varying Nominal Trajectory
In this section we present simulations to show the performance of
the control schemes when the manipulator is traveling over a prescribed
nominal trajectory. Tracking is relative to the nominal trajectory.
Since the nominal linearized system is time-varying it is necessary to
use time-varying feedback gains. Consequently, we assume the quasi
static model for the feedback gains is adequate, however, a slight
reduction in performance is expected relative to the case of constant-
gain feedback.
The nominal trajectory is a straight-line path of the end-effector
(hand) starting at the coordinates (x = 1.0 m, y = 0.1 m) and ending at
the coordinates (x = -0.35 m, y = 0.95 m). This is shown in Figure
7-11. The trajectory is chosen so that the manipulator both starts and
ends at rest. In order to achieve the desired motion, a certain amount
of acceleration and deceleration is needed. Figure 7-12 shows the
position, velocity, and acceleration profiles which have been selected
for the nominal trajectory of the end-effector.
It is important to note that when a reference signal is applied,
the nominal trajectory of the hand will no longer be the desired
trajectory of the hand. The reference signal determines the deviation
from the nominal trajectory (i.e., the reference is relative to the
nominal trajectory). In addition, since the reference signal is
actually for the joint-angles rather than the hand position, the
resulting hand trajectory will not exactly match the reference signal.
For example, a sinusoidal reference will cause a somewhat distorted
sinusoid to appear in the trajectory of the hand. Such reference
signals are only for demonstration purposes.


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
George N. Sandor
Professor of Mechanical Engineering
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
August 1985
Dean, College of Engineering
Dean, Graduate School


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REFERENCES
[1]. C.-T. Chen, Linear System Theory and Design. New York, NY: Holt,
Rinehart and Winston, 1984.
[2], C.A. Oesoer and Y.T. Wang, "Linear Time-Invariant Robust
Servomechanism Problem: A Self-Contained Exposition," Control and
Dynamic Systems, Vol. 16 (C.T. Leondes, Ed.), New York: Academic
Press, pp. 81-129, 1980.
[3]. E.J. Davison, "A Genralization of the Output Control of Linear
Multivariable Systems with Unmeasurable Arbitrary Disturbances,"
IEEE Trans. Automat. Contr., Vol. AC-20, pp. 788-792, Dec. 1975.
[4]. E.J. Davison, "The Robust Control of a Servomechanism Problem for
Linear Time-invariant Multivariable Systems," IEEE Trans.
Automat. Contr., Vol. AC-21, pp. 25-34, Feb. 1976.
[5]. E.J. Davison and A. Goldenberg, "Robust Control of a General
Servomechanism Problem: The Servo Compensator," Automtica, Vol.
11, pp. 461-471, 1975.
[6]. B.A. Francis, "The Linear Multivariable Regulator Problem," SIAM
J. Contr., Vol. 15, No. 3, pp. 486-505, May 1977.
[7], B.A. Francis, "The Multivariable Servomechanism Problem from the
Input-Output Viewpoint," IEEE Trans. Automat. Contr., Vol. AC-22,
pp. 322-328, June 1977.
[8]. C.D. Johnson, "Accommodation of External Disturbances in Linear
Regulator and Servomechanism Problems," IEEE Trans. Automat.
Contr., Vol. AC-16, pp. 635-644, Dec. 1971.
[9]. C.D. Johnson, "Theory of Disturbance-accomodating Controllers,"
Control and Dynamic Systems, Vol. 12 (C.T. Leondes, Ed.), New
York: Academic Press, pp. 287-489, 1976.
[10]. W.A. Wolovich and P. Ferreira, "Output Regulation and Tracking in
Linear Multivariable Systems," IEEE Trans. Automat. Contr., Vol.
AC-24, pp. 460-465, June 1979.
[11]. W.M. Wonham, "Towards an Abstract Internal Model Principle," IEEE
Trans. Syst. Man Cybernetics, Vol. SMC-6, pp. 735-740, Nov. 1976.
[12]. C.A. Desoer and Y.T Wang, "The Robust Non-linear Servomechanism
Problem," Int. J. Control, Vol. 29, No. 5, pp. 803-828, 1979.
155


Y-POSITION OF HAND (METERS) X-POSITION OF HAND (METERS)
,-0.05 0.19 0.95 0.70 0.95 -0.35 -0.01 0.39 0.G9 1.03
151
Figure 7-24. Actual and desired trajectories of the hand with
compensation for flexible links


3
single-input single-output systems. As expected, the resulting
controller requires integrators in the feedback loop.
Solomon and Davison [13] have used state-space techniques to treat
the servomechanism problem for a certain class on nonlinear systems.
They too have considered only constant reference and disturbance
signals. In addition, the nature of the disturbance is such that it
affects the output directly without affecting the dynamics of the
nonlinear system. It can be shown that such a disturbance can be
regarded as simply a change in the level of the reference signal.
Various assumptions are made and conditions are given stating when it is
possible to solve this servomechanism problem. The resulting control
law employs integrators in the feedback loop and nonlinear feedback is
used to give global stability. Although global stability is obtained,
the range in amplitude of the reference and disturbance signals which
can be applied is limited.
Some appealing results, again only for the case of constant
reference and disturbance signals, are derived in Desoer ad Lin [14] and
in Anantharam and Desoer [15]. Desoer and Lin have shown that if the
nonlinear plant has been prestabilized so that it is exponentially
stable and if the stabilized plant has a strictly increasing, dc,
steady-state, input-output map then the servomechanism can be solved
with a simple proportional plus integral controller. Using such a
control scheme, it is necessary that the gains of the integrators be
sufficiently small and that the proportional gain be chosen
appropriately. Anantharam and Desoer have derived results virtually
identical to those found in Desoer and Lin. In their paper, however,
the proof is somewhat different and a nonlinear discrete-time system is


127
The various responses obtained when the manipulator is tracking
relative to the nominal trajectory are shown in figures 7-13 through
7-21. Each figure is presented so that part (a) depicts the error in
the joint angles while part (b) shows the actual and desired trajec
tories of the hand. The trajectories of the hand are given for both the
x- and y- displacements. The notation used here for the figure captions
is identical to the notation used in the previous section.
Figures 7-13 and 7-14 show the error which results when a
sinusoidal disturbance is applied. In Figure 7-13 the I.M. system
contains poles at the frequency of the disturbance. The tracking error
is very small; however, a slight ripple, most likely due to the time-
varying feedback gains, is evident in the joint-angle error curves.
Nevertheless, it is difficult to distinguish the actual hand trajectory
from the desired hand trajectory. Figure 7-14 is included mainly to
show that the I.M. system should contain poles corresponding to the true
frequencies of the disturbance. Here only integrators are used. As can
be seen from either the joint-angle error curves or the curves showing
the trajectory of the hand, a sinusoidal tracking error at the frequency
of the disturbance signal is present.
Figures 7-15 through 7-18 show the responses obtained when a
reference signal (no disturbance) is applied. Figure 7-15 illustrates
the response obtained when the poles of the I.M. system are at the
frequency of the disturbance signal (i.e., 2 Hz). A sinusoidal error at
a frequency of 4 Hz and with a slight dc offset is seen to occur. To
eliminate the dc offset, integrators are included in the I.M. system and
the resulting response is shown in Figure 7-16. As expected, the 4 Hz
sinusoid is still present in the steady-state tracking error. In order


146
O
O
(b) Actual and desired trajectories of the hand
Figure 7-21 continued


92
(k) = -K1(k)x(k) K2(k)n(k)
(5-28)
To be consistant with the time-varying feedback law required for the
quasi-state approach, the gains are shown to be functions of the sample
integer k. From (5-28) and the assumption that a zero-order hold will
be employed, the actual control input to the plant can be expressed as
u(t) = -K1(k)x(k) K2(k)n(k) kT < t < (k+l)T (5-29)
A necessary requirement for the discrete-time control scheme to be
satisfactory is that the control law of (5-29) must result in asymptotic
stability of the closed-loop transient system.
One possible method [24], [25] of selecting the feedback is based
on a discretized model for the nominal linearized system. This nominal
linearized system is given in continuous-time form by equation (5-13).
Since the internal model system has already been given in discrete-time
form, only the part of the linearized system corresponding to the plant
must be discretized. Assuming that the continuous-time system (5-13) is
slowly time-varying, we may write the discrete-time equation
approximating the dynamics of (5-13) as
xA(k+l) F*>d(k)xA(k)
(5-30)
where
F*(k) GjtkiKjik)
-BdH
Gd(k)K2(k)
(5-31)


122
I.
O
o
Figure 7-9. Tracking error: ref. = 3.0 Hz sinusoid with amplitude
of 15; dist. = 1.0 Hz sinusoid with amplitude of 50 N;
I.M. frequencies = 1.0 Hz, 3.0 Hz, and 6.0 Hz


CHAPTER FOUR
FEEDBACK CONTROL
In this chapter various concepts related to selecting the feedback
gains for the servomechanism problem are discussed. The beginning of
the chapter deals primarily with the nonlinear servomechanism problem
while some results for the linear servomechanism problem are given in
later sections.
Since it is often difficult to show global stability in nonlinear
systems, here the emphasis is placed on achieving only local
stability. Conditions are given showing when it is possible to obtain a
time-invariant feedback control law which stabilizes the system NCT.
These conditions apply mainly to the case when the reference and
disturbance signals are small. An example is provided to demonstrate
the method.
Also discussed in this chapter is the interpretation of using
optimal control techniques in the selection of the feedback gains for
the linear servomechanism problem. The optimal control approach can be
applied to the nonlinear servomechanism problem when the linearized
equations are used; however, the interpretation is less precise.
Stabilization Using the Linearized Equation
To solve the tracking and disturbance rejection problem it is
necessary to have asymptotic stability of the system NCT. Furthermore,
47


39
(B.l) (F, G) is stabilizable
(B.2) rank
V F
-H
G
0
n + p i=l,2,...,r
Conditions (B.l) and (B.2) are essential for a solution to the
linear servomechanism problem. Therefore, when the linear problem is
solved using the framework developed for the nonlinear problem,
conditions (B.l) and (B.2) should play important roles.
Solution to the Linear Problem via
the Nonlinear Formulation
We now proceed to show that when conditions (B.l) and (B.2) are
satisfied, the conditions given in Chapter Two for the nonlinear
formulation are also satisfied.
First consider assumptions (A.l) and (A.2) when applied to a linear
system. The following theorem will relate these assumptions to
conditions (B.l) and (B.2).
Theorem 3.2: Consider the linear system L and assume both the reference
r*(t) and the disturbance w*(t) satisfy the linear differential equation
(3-2). Then, if conditions (B.l) and (B.2) are both satisfied, there
exists an input u*(t) and an initial state x*(0) = x* such that
tracking occurs. Furthermore, u (t) and xQ can be chosen so that the
resulting state trajectory x*(t) and the input u*(t) satisfy the linear
differential equation (3-2) (i.e., assumptions (A.l) and (A.2) are
satisfied).


89
First consider the internal model system. Since it is to be
implemented digitally a discrete-time model is required. Rather than
discretizing the continuous-time internal model system it is convenient
to reformulate the problem in terms of discrete-time signals. If
assumption (A.2) holds then the elements of both the sampled state x (k)
fa
and the input u (k) will satisfy the difference equation
s(k+r) + d^sik+r-l) + ... + d^s (k+1) + dQs(k) = 0 (5-17)
where s(j), j = k, k+1, ..., k+r denotes either an element of x (j) or
an element of u*(j). This result is readily obtained using z-transform
theory and later an example will be given demonstrating how to obtain
the scalars dj, j = 0,1,..., r-1 using z-transforms.
The internal model system then takes the form
n(k+l) = A^nCk) + B^k)
(5-18)
e(k) = r(k) y(k)
The matrices Ad and Bd are defined as
Ad = T1 block diag. [Cd, Cd, ..., Cd] T (5-19)
Bd = T"1 block diag. [t, t, ..., x] (5-20)
where


68
been constructed in agreement with equation (4-44). Since asymptotic
tracking and disturbance rejection will occur, there must be a steady-
state solution to (4-44). Consequently, if both w(t) and r(t) are
applied to the system at the time t = 0 then there exist initial states
x(0) = xQ and n(0) = n0 such that no transients appear in the state
trajectory and the tracking error is zero. We denote this trajectory by
the pair [xs$(t), n$s(t)]. In terms of this notation, (4-44) becomes
s
F
o'
\s(ti
+
G
uss(t> +
E '
w(t) +
0
."ssW
-BH
A
."ss(t)_
0
* .
0
B
uss(t) = -KjXss(t) -K2nss(t) (4-45)
yss(t) = r(t) = Hxss(t)
If (4-45) is subtracted from (4-44) the following equation results:
(t)
F 0
x(t)"
G

as
+
n(t)
-BH A
(t)
0
where
x(t) = x(t) xss(t)
n(t) n(t) nss(t)
u(t) = u(t) us$(t) = -Kjxt) K2(t)
(4-47)


BIOGRAPHICAL SKETCH
The author was born on October 17, 1958 in Rockledge, Florida. He
received a B.S. in mechanical engineering and an M.S. in electrical
engineering from the University of Florida in 1982 and 1984
respectively. He is currently working on a Ph.D. in electrical
engineering. The author hopes to one day become more like the cat in
reference [35],
158


156
[13], A. Solomon and E.J. Davison, "Design of Controllers to Solve the
Robust Servomechanism Problem for a class of Nonlinear Systems,"
in Proc. 22nd IEEE Conf. Decision Contr., San Antonio, TX, pp.
335-341, 1983.
[14], C.A. Desoer and C.-A. Lin, "Tracking and Disturbance Rejection of
MIMO Nonlinear Systems with PI Controller," IEEE Trans. Automat.
Contr., Vol. AC-30, pp. 861-867, Sept. 1985.
[15]. V. Anantharam and C.A. Desoer, "Tracking and Disturbance
Rejection of MIMO Nonlinear Systems with a PI or PS Controller,"
in Proc. 24th IEEE Conf. Decision Contr., Ft. Lauderdale, FL, pp.
1367-1368, 1985.
[16]. B.D.O. Anderson and J.B. Moore, Linear Optimal Control, Englewood
Cliffs, NJ: Prentice-Hall, 1971.
[17], M. Vidyasagar, Nonlinear System Analysis, Englewood Cliffs, NJ:
Prentice-Hall, 1978.
[18], C. Corduneanu, Principles of Differential and Integral Equations,
Boston, MA: Allyn and Bacon, 1971.
[19], M.R. Spiegel, Applied Differential Equations. Englewood Cliffs,
NJ: Prentice-Hall, 1981.
[20], H. Kwakernaak and R. Si van, Linear Optimal Control Systems, New
York, NY: John Wiley & Sons, Inc., 1972.
[21]. B.D.O. Anderson and J.B. Moore, Optimal Filtering, Englewood
Cliffs, NJ: Prentice-Hall, 1979.
[22]. G.F. Franklin and J.D. Powell, Digital Control of Dynamic
Systems, Reading, MA: Addison-Wesley, 1980.
[23], B.C. Kuo, Digital Control System. New York, NY: Holt, Rhinhart
and Winston, Inc. 1980.
[24]. M.L. Whitehead, "Control of Serial Manipulators with Emphasis on
Disturbance Rejection," Master's Thesis, University of Florida,
1984.
[25]. M.L. Whitehead and E.W. Kamen, "Control of Serial Manipulators
with Unknown Variable Loading," in Proc. 24th IEEE Conf. Decision
Contr., Ft. Lauderdale, FL, pp. 362-363, 1985.
[26]. K. Ogata, State Space Analysis of Control Systems, Englewood
Cliffs, NJ: Prentice-Hall, 1967.
[27]. R.E. Kalman, Y.C. Ho, and K.S. Nerendra, "Controllability of
Linear Dynamical Systems," Contributions to Differential
Equations, Vol. 1, New York: John Wiley & Sons, Inc., pp. 189-
213, 1963.


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Edward W. Kamen, Chairman
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Thomas E. Bullock
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Peyton Z. Peebles, Jr.
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Irena Lasiecka
Professor of Mathematics


113
period. This would correspond to eigenvalues at j2irF for a
continuous-time implementation. If F is zero then we wish to indicate
that the internal model system contains integrators (i.e., eigenvalues
at 1.0 for the discrete-time implementation).
The responses shown in Figure 7-2 demonstrate the ability of the
controller to reject a sinusoidal disturbance force. No reference is
used. Recall from Chapter Six that the controller only needs to contain
poles (eigenvalues) at the frequencies of the disturbance in order to
achieve perfect tracking. The poles in the internal model (I.M.) system
correspond to the 1.5 Hz disturbance and it is clearly seen that the
error does indeed go to zero. Figure 7-3 shows a similar test, however,
in addition to a sinusoidal disturbance a constant disturbance is also
introduced. The poles of the I.M. system are adjusted accordingly
(i.e., integrators are added) and the error once again goes to zero in
steady-state. Figure 7-4 is included to demonstrate that a simple
scheme employing just integrators in the I.M. system will not adequately
compensate for sinusoidal disturbances. Here we see that the steady-
state error to a sinusoidal disturbance is a sinusoid at the frequency
of the disturbance.
Figures 7-5 through 7-7 show the error which occurs when a
sinusoidal reference signal is introduced. A disturbance force in not
considered in these tests. First, Figure 7-5 shows the performance of
the design which results from using only linear servomechanism theory.
The I.M. system contains poles at the frequency of the reference signal
(1.5 Hz in this case). It is evident that the steady-state error is
sinusoidal in nature with a frequency of 3 Hz; twice that of the
reference signal. In order to improve the steady-state performance we


Y-POSITION OF HAND (METERS) X-POSITION OF HAND (METERS)
-0.21 0.1L 0.02 0.73 1.04 a-0.55 -0. 16 0.24 0.03 1.03
144
(b) Actual and desired trajectories of the hand
Figure 7-20 continued


104
Now consider one method of selecting feedback gains which will make
the linearized system behave as though it were time-invariant. First
make the definition
Kl(t) := [K1#1(t), Klf2(t)] (6-16)
where Kj^it) corresponds to feedback of the joint positions and K-Lj2(t)
corresponds to feedback of the joint velocities. Using (6-11) through
(6-13) it is possible to write the linearized system matrix as
F*(t) G*(t)Kx(t) -G*(t)K2(t)
-BH,
A
(6-17)
I
Fj(t) J1(*)K1^1(t)
F2(t)
NxN
1,-X*,
J "(0 )K1>2(t)
J(9 )K2(t)
We desire a feedback law which will make the right-hand side of
(6-17) take the following form
0 *NxN 0
m2 m3
-BOA
(6-18)
where e R^xN, M2 e RNxN, and M3 e R^xr^ are constant matrices (note
that A e R^NxrN). can ke that the eigenvalues of (6-18) can be
assigned arbitrarily by proper choice of Mj, M2 and M3. This is, in


ERROR IN JOINT 2 (DEGREES) ERROR IN JOINT 1 (DEGREES)
3-21.00 -16.00 -8.00 0.00 8.00 -27.00 -18.00 -9.00 0.00 9.00
118
Figure 7-5. Tracking error: ref. = 1.5 Hz sinusoid with amplitude
of 15 ; dist. = 0; I.M. frequency = 1.5 Hz


138
O
(b) Actual and desired trajectories of the hand
Figure 7-17 continued


100
often specified in advance, there is little difficulty associated with
introducing the dynamics of the state trajectory into the internal model
system. For example, if the desired trajectory consists of sinusoidal
signals then the internal model system must have poles (eigenvalues of
the A matrix) at the frequencies of the sinusoidal signals.
The dynamics which must be included in the internal model system in
order to generate the appropriate input signal are investigated here.
The input needed to produce a certain state trajectory can be calculated
from (6-7). This input is seen to be a function of the state trajectory
(i.e., 0 (t), e (t) and e (t)) as well as the disturbance w(t).
Unfortunately, the disturbance is not generally known in advance so that
determining an exact input may not be possible. In many situations,
however, one has an idea of the frequencies and also the ranges of
amplitudes which might occur in the disturbance signal. Consequently,
*
the input T^(t) can be evaluated for various combinations of the
anticipated disturbance signals. Using Fourier analysis, the dominant
frequencies in the resulting inputs can then be identified and the
internal model system can be designed by placing eigenvalues in loca
tions corresponding to these anticipated frequencies.
Now consider a very important situation which does not require
Fourier analysis; namely tracking a constant (or step) reference signal
while sinusoidal disturbances are being applied. When e (t) is constant
then 9*(t) and 0*(t) are both zero. Consequently, equation (6-7) can be
written in the following form
T¡J(t) = -Tg(0*) D(0*)w(t)
(6-9)


93
_*
.*
The discrete-time matrices F^k) and G^k) corresponding to the plant
are obtained [26] by the relationships
F*(k) e^kT)T
(5-32)
0
(5-33)
Where T is again the sample period. By defining F^k) and G^k) in
this manner it is implicitly assumed that the dynamics of the linearized
time-varying system (5-13) do not change over any given sample period.
Thus, excluding the case when the reference and disturbance signals are
constant, equation (5-30) is indeed only an approximation.
Assuming that our discrete-time model is reasonably accurate, the
feedback gains K^k) and K2(k) are selected to give stability of the
discrete-time system (5-30). The actual mechanism for selecting the
feedback gains shall not be discussed, however, solving an algebraic
Riccati equation would be one approach [24]. In any event there is a
stabilizability (controllability) requirement for the discretized system
which must be met. In general, the controllability requirement will be
met whenever the continuous-time system is controllable [27] so that
controllability of the pair given in equation (5-14) is often
sufficient.


139
O
O
Figure 7-18
Manipulator's response: ref.
amplitude of 10; dist. = 0;
2.0 Hz sinusoid with
M. frequency = 0 Hz


16
Solving for u (t) gives
fr OI p 1
U (t) = a^toCOSiot 3a^Sl'no)t + (-g + agJcosZwt (tj- + 2) 4ct2
(2-22)
The characteristic polynomial for U (s) is
(s2 + 4(o2) (s2+oj2) (s) = s^ + 5u2s^ + 4 Hence, both u (t) and x (t) satisfy
^ () + 52-4 () + () = 0
dt3 dt3 az
(2-24)
Now that assumptions (A.l) and (A.2) have been justified, we
proceed by introducing an internal model system. In the literature, an
internal model system is usually taken as a system which replicates the
dynamics of the reference and disturbance signals. Here it will take on
a slightly different meaning which is made more precise by the following
definitions. Let C e Rrxr and t e Rr be defined as follows
II
0
"0 1 0 ... 0
0 0 1 ... 0


, t : =
1
0 0
1

0 0 0 ... 1
_-a0 ~al -a2 "ar-l
r
1* 0
1
with the coefficients aj, j = 0,1, ...,
r-1 defined by (2-4).


TRACKING AND DISTURBANCE REJECTION FOR NONLINEAR SYSTEMS
WITH APPLICATIONS TO ROBOTIC MANIPULATORS
By
MICHAEL L. WHITEHEAD
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1986


64
(a) Design based on nonlinear servomechanism theory
(b) Design based on linear servomechanism theory
Figure 4-3. Tracking error: reference = 2sin(t), disturbance = 2.0


121
r
o
o
o
o
Figure 7-8. Tracking error: ref. = 3.0 Hz sinusoid with amplitude
of 15; dist. = 1.0 Hz sinusoid with amplitude of 50 N;
I.M. frequencies = 1.0 Hz and 3.0 Hz


52
at atn mL(t-t )
eai'iix(t) ii < me 0 nxQlie
or
nx(t)n < m iixQlie
(mL-a)(t-t )
(4-15)
(4-16)
If L is sufficiently small so that
(mL-a) < 0
(4-17)
then we have the desired result.
Consider again the linearization of NCT given by (4-3). This can
be written as
xA(t) FxA(t) + FA^xA^^
(4-18)
The matrix F is defined as follows
F =
hA
F GK1 -GK2
-BH A
where
(4-19)
Fo = lf(x,.u>w)
3X
G0 = i.f(x,_u,wl
9U
x = 0
u = 0
w = 0
9
x = 0
u = 0
w = 0
(4-20)


79
Figure 5-2. Nominal and desired state trajectories


ERROR IN JOINT 2 (DEGREES) ERROR IN JOINT 1 (DEGREES)
-27.00 -18.00 -9.00 0.00 9.00 -27.00 -18.00 -9.00 0.00 9.00
141
(a) Tracking error
Figure 7-19. Manipulator's response: ref. = 3.0 Hz sinusoid with
amplitude of 10; dist. = 4.5 Hz sinusoid with
amplitude of 50 N; I.M. frequencies = 3.0 Hz and
4.5 Hz


107
relationship of the manipulator's links [33] we can write the following
equation
r(t) = h(e(t), f(t)) (6-21)
where r(t) is the vector locating the end-effector of the manipulator
and h(e,f) is a function of the manipulator's joint angle vector e(t)
and of the force vector f(t), both of which are assumed measurable.
This equation is only valid in the quasi-static case since otherwise,
the end-effector location would also depend on e(t) and e(t).
When (6-21) is linearized about a nominal trajectory, the following
result is obtained
r(t) HjUJeU) + H2(t)f(t) (6-22)
where
Hx(t) =
9h(e,f)
ae
e,f=nominals
H2(t) =
ah(e,f)
af
e,f=nominaIs
(6-23)
Here r(t), Q{t), and f(t) denote the deviations of r(t), e(t) and f(t)
from their nominal values. We note that H^(t) is actually the standard
Jacobian matrix [28] often used to relate the end-effector velocity to
the joint-angle velocities.
If the end-effector is to maintain a specified trajectory, r(t)
must be zero. Consequently, e (t) can be obtained by setting r(t) equal
to zero, solving for e(t), and letting e*(t) equal e(t). Provided that
Hj(t) is nonsingular this gives


145
O
O
O
o
(a) Tracking error
Figure 7-21
Manipulator's response: ref. = 3.0 Hz sinusoid with
amplitude of 10; dist. = 4.5 Hz sinusoid with
amplitude of 50 N-, I.M. frequencies = 0 Hz, 3.0 Hz,
4.5 Hz, and 6.0 Hz


29
(2-46)
We now present a theorem based on Liapunov's indirect method which can
be used to show local tracking. In order to apply Liapunov's indirect
method, the following two technical conditions are required
lim ,sup
ixAi+0 lt>0
(2-47)
Fa(*) is bounded
(2-48)
It is mentioned that the above conditions are almost always satisfied in
practical systems.
Theorem 2.3: Suppose that the hypotheses of Theorem 2.1 are satisfied
and also assume that conditions (2-47) and (2-48) hold true. If in
addition, the system (2-40) is asymptotically stable then, with the
control scheme defined by system NC, local tracking of r*(t) with
disturbance w*(t) will occur.
Extensive use will be made of Theorem 2.3 in later chapters.
The Relation Between the Dimension of the
Internal Model System and the Input/Output Dimensions
In the previous sections, the servomechanism problem was treated
where it was assumed that the number of inputs to the plant was the same


82
where
and
-*
pA(t)
F*(t) G^tiK^t)
-BH
-G*(t)K2(t)
A
(5-11)
F*(t)
x = x*(t)
u = u*(t)
"k
w = w (t)
G*(t)
9f(x,U,w)
9U
x*(t)
u*(t)
_*
W (t)
(5-12)
Now (see Theorem 4.1) if the system
xA(t) = F*(t)xA(t) (5-13)
k w,^k
is the exponentially asymptotically stable and sup nF.(t) F.(t)n. is
t>0 at
suitably small then (5-10) is exponentially asymptotically stable as
desired.
First consider conditions under which the quantity
k ^~k
sup iiF.(t) F. (t) II. is sufficiently small. When the true reference
t>0 at
and disturbance signals are precisely equal to the nominal reference and
disturbance signals then obviously this quantity is zero. If the true
k i
signals deviate from the nominal signals then sup nF.(t) F.(t)n. is
t>0 at
k
not zero, however, it is generally small (assuming ^(t) depends
continuously on the reference and disturbance) whenever the deviations
from the nominal signals are small. Hence, the control law developed
here will be effective when the true reference and disturbance signals
are close to the nominal reference and disturbance signals.


103
Controllability is used in condition (1) since we wish to have complete
freedom in selecting the eigenvalue locations.
It is not difficult to show that for the robotic manipulator,
conditions (1) and (2) always hold true. To verify condition (1) we
evaluate the first two blocks of the standard controllability matrix
[20] for the pair [F,(t), G*(t)]. This gives
9 [G*(t), F*(t)G*(t)]
9 9 9
0 J"1(0*)
J-1(0*) F2(t)J'1(0*)_
(6-14)
Since J(0 ) is nonsingular, the controllability matrix has full rank
(i.e., rank 2N for the N-link manipulator) and hence the pair
[F (t), G,(t)] is controllable.
9 9
Now consider condition (2). Using (6-11) through (6-13) it is
possible to write
*
X1!2Nx2N V1'
*<*>"
_
Xi!NxN
-F^t)
-INxN
Vnxn
0
_1 *
J )
- -H*
0 .
_*NxN
0
0
(6-15)
The right-hand side of (6-15) is a 3Nx3N matrix which is readily seen to
have full rank for all Aj. The fact that this matrix is full rank for
any A^ means that the stability condition can be achieved regardless of
the eigenvalues of the internal model system. In otherwords, the
frequencies of the reference and disturbance signals will not be a
factor in deciding whether or not the stability condition can be met.
By showing that both conditions (1) and (2) hold, we have proven
that it is possible to stabilize the linearized manipulator system
(assuming the slowly time-varying approach will apply).


54
i
Then, provided and e£ are suitably small, there exist feedback
gains K]^ and K2 so that with the controller given as system NC,
(eq. (2-29)) local tracking of r (t) with disturbance w (t) will occur.
Proof: Our main concern here will be to show asymptotic stability of
the system NCT and observability of the pair (A,!^). With these
conditions verified, the remainder of the proof is immediate from the
results obtained in Chapter Two.
First let us compare (4-18) to (4-6) letting F take the role of
F^t) and [F^(t) ~ F^]XA(t) take the role of f-^t.x). Condition (1)
of the Poincare-Liapunov theorem then requires exponential stability of
the system
x
A
(4-21)
In order to meet this stability requirement, the pair
(4-22)
must be at least stabilizable. Since the matrices given in (4-22) are
constant, Theorem 3.3 can be employed. More specifically, conditions
(iii) and (iv) imply that the pair given in equation (4-22) is
stabilizable and hence, proper selection of the feedback and l<2 will
give exponential stability to the system (4-21).
Assuming that suitable feedback gains have been selected, the state
transition matrix $(t,to) associated with (4-21) will satisfy the
inequality


115
TIME (SECONDS)
Figure 7-2
Tracking error: ref. = 0; dist. =
amplitude of 50 N; I.M. frequency
1.5 Hz sinusoid with
= 1.5 Hz


24
It is seen that the system NCT has an equilibrium point at
x(t) = n(t) = 0
Theorem 2.2: Suppose that the hypotheses of Theorem 2.1 are satisfied
and that for some choice of Kj and the system NCT is locally
asymptotically stable. Then, with the control scheme defined by system
NC, local tracking of r (t) with disturbance w (t) will occur. If in
addition, system NCT is globally asymptotically stable then global
tracking of r*(t) with disturbance w*(t) will occur.
Proof: Obvious since x(t) + 0 as t + and e(t) = r (t) Hx(t) =
H[x*(t) x(t)] = -Hx(t).
Since global stability is often difficult to obtain in many
practical systems using constant-gain feedback, the local stability
result of Theorem 2.2 will most often apply. Consequently, success of
the control scheme will depend on the initial state of the original
system and of the internal model system. This will usually mean that
tracking and disturbance rejection can be achieved only if the reference
signal and the disturbance signal are not excessively large.
Stability of the system NCT will be a major topic of the next
section as well as subsequent chapters. At this point, however, it is
appropriate to generalize the results obtained so far. This is
important since previous results have been developed with the assumption
that only one particular reference signal r*(t) and one particular
"if
disturbance signal w (t) will be applied to the system.
The


Y-POSITION OF HAND (METERS) X-POSITION OF HAND (METERS)
0.30 0.52 0.73 0.35 -0.35 -0.01 0.33 0.06 1.00
130
i
1
(b) Actual and desired trajectories of the hand
Figure 7-13 continued


23
Definition: If the state trajectory for the closed-loop system NC
k
converges to [x (t), n (t)] for a set of initial states in the
neighborhood of [xQ, nQ] then we say there is local tracking of r (t)
fc
with disturbance w (t). If this convergence occurs for all initial
states then we say there is global tracking of r*(t) with disturbance
w*(t).
To give conditions under which global or local tracking will occur,
we first define a new set of state vectors as follows
x(t) = x(t) x*(t)
*n(t) = n(t) n*(t)
(2-37)
where [x(t), n(t)] is the state trajectory of NC resulting from an
"k "k
arbitrary initial state and [x (t), n (t)] is the trajectory which
"k 'k
gives e(t) = 0, t > 0 and results from the initial state [x n ].
o o
Since it is our goal to have the trajectory [x(t), n(t)] converge,
* k
eventually, to the trajectory [x (t), n (t)] we may think of
Cx(t), (t)] as the transient trajectory. Using (2-29) and (2-35), it
is then possible to write a dynamic equation modeling the transient
response of the closed-loop system. This will be referred to as the
closed-loop transient system NCT. The system NCT is given by
NCT:
x(t) = f(x*(t)+x(t), u*(t)+u(t), w*(t)) f(x*(t), u*(t), w*(t))
^{t) = Ari(t) BHx(t)
u(t) = -Kxx(t) K2n(t)
(2-38)


xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008243000001datestamp 2009-02-16setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Tracking and disturbance rejection for nonlinear systems with applications to robotic manipulatorsdc:creator Whitehead, Michael L.dc:publisher Michael L. Whiteheaddc:date 1986dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082430&v=0000116568635 (oclc)000938559 (alephbibnum)dc:source University of Floridadc:language English


101
Here we have used (6-2), (6-3) and the fact that = e* = 0. Since e
is a constant, both T9(e*) and D(e*) are also constant. This means that
k
the frequencies of TA(t) are identical to the frequencies of w(t) and
hence only the dynamics of the disturbance signal need to be included in
the internal model system. In this case, assumption (A.2) is satisfied
exactly. Note that the internal model system should also include
dynamics (i.e., integrators) to accomodate for the constant
gravitational torque T9(e*) if this torque is not supplied separately.
Feedback Gain Calculation
In this section we discuss the feedback gain calculation for the
manipulator system when the linearized model is used. In conjunction
with the chosen control scheme, this linearized model is evaluated over
the nominal trajectory. The linearized closed-loop system takes the
following form
(t) $*(t)
m n*(t)
F*(t) G*(t)K1(t)
-G*(t)K2(t)
A
<|>(t) n(t) n*(t)
(6-10)
Since only the first N components of (i.e., e) are taken as the
output, is defined as
% cw (6-H)
where 1^ is the NxN identity matrix. Evaluation of the nominal
Jacobian matrices using a linearization of (6-6) leads to


154
that the reference will be known in advance so that the controller can
be designed accordingly. For instance, the feedback law can be chosen
to give stability relative to a nominal trajectory as was discussed in
Chapter Five.
There are several aspects of the approach taken here which deserve
investigation. Some of these are:
(1) Analyze the situation which occurs when the internal model
system's poles are not placed at the correct locations. For
example, what is the consequence of placing poles at j2 when they
are actually needed at j2.1.
(2) Find a way to bound the tracking error when (A.2) is not satis
fied. Also show the relationship between the error bound and the
number of terms taken from a Fourier series expansion of the true
state and input.
(3) Develop a feedback law which either gives global stability or
increases the region of stability for the system NCT. Since the
resulting feedback may be nonlinear, further conditions are likely
to be imposed for a solution to the servomechanism problem. One
scheme which warrants investigation is the use of prestabilizing
nonlinear feedback on the plant.
(4) Investigate the effect of the proposed scheme on the transient
behavior of the system. Although it is true that the steady-state
error can be improved with this scheme, the use of an internal
model system of large dimension may degrade transient performance.


CHAPTER THREE
APPLICATION TO LINEAR SYSTEMS
In this chapter the linear servomechanism problem is considered.
Since the linear problem can be regarded as a special case of the
nonlinear problem, the methods developed in the previous chapter apply
here as well.
When using the methods of the previous chapter, specific conditions
must be met in order to guarantee a solution to the servomechanism
problem. Here, it will be shown that these conditions are satisfied
whenever the well established conditions [1-7] imposed for the linear
problem are satisfied. Of course this is obvious; however, the insight
obtained by approaching the problem from a different point of view will
prove beneficial. In fact, many of the results obtained in this chapter
will be used in subsequent chapters where feedback gains are selected
for the nonlinear problem via a linearized model.
Review of Linear Servomechanism Results
In this section we give a brief review of the well known results
for the linear servomechanism problem. A more general discussion of
this topic can be found in [2] or [3].
37


36
closed-loop transient system. Again, the approach taken here is seen to
be drastically different from the approach taken in the well known
linear servomechanism problem. Asymptotic stability of the closed-loop
transient system was allowed to be either global or local; however, with
local stability it was indicated that tracking and disturbance rejection
would occur only for certain initial states. These initial states were
restricted to the neighborhood of the particular initial state which
defined the equilibrium point of the closed-loop transient system.
The dynamic equations modeling the closed-loop transient system
were seen to be nonlinear and somewhat complicated. In order to apply
Liaponov's indirect method, a much simpler dynamic system was derived
through linearizations. Although the linearized model is much more
suitable for feedback gain selection, stability of the linearized model
only insures local stability of the true system.
In this chapter, no discussion was given as to possible means of
determining the stabilizing feedback gains. This topic is the subject
of Chapters Four and Five.


123
\
O
o
Figure 7-10. Tracking error: ref. = 3.0 Hz sinusoid with amplitude
of 15; dist. = 1.0 Hz sinusoid with amplitude of 50 N;
I.M. frequencies = 0 Hz, 1.0 Hz, 3.0 Hz, and 6.0 Hz


95
J(e)e + Tv(e,) TA(t) + Tg(e) + Td(w(t),e) (6-1)
For a manipulator having N links, e(t), e(t) and *e(t) are the vectors of
length N defining angular positions, velocities, and accelerations of
the actuator joints. The matrix J(e) e R^ is the inertia matrix which
depends on the manipulator's configuration (i.e., the joint angles
e(t)). It can be shown that J(e) is positive definite for all 9 (see
[24]) and is thus always invertible. The inertia torque vector
Tv(0,0) e corresponds to dynamic torques caused by the velocities
of the manipulator's links. Denoting the j-th component of the intertia
torque vector as Tv(e,e). we have the following
J
TV(e,9), = 0'PJ(0)0 (6-2)
J
where P^(0) e R^ is a purely configuration dependent matrix referred
to in [28] as the intertia power modeling matrix. The term T^(t) e R^
is the control torque vector which is typically supplied to the actuator
joints by electric motors. The torques resulting from gravitational
loading are designated by T^ (0) e R^ which is a configuration
dependent vector. Finally, T^(w(t),e) e R^ is a torque vector resulting
from external uncontrollable forces. It is possible to write Td(w(t),
0) in the following form
Td(w(t),0) = D(e)w(t) (6-3)
where D(0) e R^xd depends only on the manipulator's configuration and
w(t) e R^ denotes the disturbance force vector.


149
the corrective reference signals to the joint angles. The steady-state
tracking error is reduced considerably so that the actual and desired
trajectories almost coincide. Figure 7-25 depicts the joint angle
corrections which have been supplied as the reference signals. These
signals, although not constant, vary slowly enough to justify the use of
integrators in the internal model system.


73
A new performance index is taken as
0 = 1 Cfr'(t)QS(t) + tr* (t)R(t)]dt (4-58)
o
The standard algebraic Riccati equation may be solved yielding an
optimal feedback gain K for the system (4-56). It just so happens that
if this same feedback gain is used for the original system then the
integrand of (4-55) is minimized with respect to the original system.
In addition, the eigenvalues of the closed-loop system will all have
real parts less than -a. This is the desired result. A similar
technique can be used in discrete-time control [21].


4
treated where proportional plus sum (discrete-time integral) control is
employed.
Other than the case of constant reference and disturbance signals,
it appears that there were no satisfactory results for the nonlinear
servomechanism problem. One could attempt to linearize the nonlinear
system and design a controller based on linear servomechanism theory.
However, such an approach will usually lead to steady-state tracking
error.-
In this dissertation, the servomechanism problem is solved for a
class of multi-input, multi-output, nonlinear systems. Here the results
are valid for reference and disturbance signals which belong to a much
wider class of signals than simply those which tend to constants.
The major contributions of this research are:
1. Conditions are given for a solution to the nonlinear servomechanism
problem. When these conditions are not satisfied exactly,
employing the type of controller developed here still makes
intuitive sense.
2. The problem is solved in the time-domain using a completely new
approach. A time-domain approach is necessary because standard
techniques (i.e., frequency domain analysis) used for solving the
linear problem are not applicable to nonlinear systems.
3. It becomes apparent that the idea of an internal model system which
contains the dynamics of only the reference and disturbance signals
is not complete. It is shown that actually, the internal model
system should include the dynamics found in both the input and in
the state which must be present during successful tracking.


78
It is apparent that 'X'(t) and ui(t) are simply the deviations of the true
state and input from the nominal state and input.
As indicated, the tracking error will be zero whenever the state
and input of the plant are x*(t) and u*(t) respectively. Assuming there
exists a state trajectory n (t) for the internal model system which
allows this to happen, we must then have
K2(t)n*(t) = -K^tj/ft) lT*(t) (5-5)
The above equation is a mere consequence of definition (5-3) and the
structure of the controller.
Although K-^(t) and ^(t) are shown to be functions of time, for the
present, assume that they are constant. Also, assume that x (t) and
ii*(t) satisfy a linear differential equation of the form given in
assumption (A.2). It then readily follows (see Chapter Two) that if the
internal model system is chosen to contain the modes of x (t) and u (t)
and if the pair (A, ^(t)) is observable, tracking will occur for some
initial state [x (0), n (0)]. Consequently, we shall require that
x*(t) and D*(t) satisfy the differential equation given in (A.2). The
internal model system is designed accordingly.
One advantage of the new requirement is that, effectively, the
class of signals for which x*(t) and u*(t) are allowed to belong is
increased. For example, Figure 5-2 shows a state trajectory x (t) which
consists of a sinusoidal trajectory "x*(t) superimposed on some nominal
trajectory x*(t). The trajectory x*(t) is not restricted by assumption
(A.2).


CHAPTER TWO
TRACKING AND DISTURBANCE REJECTION FOR NONLINEAR SYSTEMS
In this chapter we derive a method to achieve tracking and
disturbance rejection for certain nonlinear multi-input, multi-output
systems. Conditions are given which reveal when the problem can be
solved. An internal model system is used as a basis for the design,
however, unlike the case of the linear system, the internal model
contains dynamics which may not appear in either the reference or
disturbance signals.
Notation
Given a positive integer n, let Rn denote the set of n-dimensional
vectors with elements in the reals and let Rmxn be the set of matrices
of dimension mxn with elements in the reals. The symbol II II shall
denote the Euclidian norm of a given vector. For a matrix A, the symbol
llAll^ will be the induced norm defined as
Vz
HAII- := sup llAxll = [X (A'A)] (2-1)
1 lixii=l max
The symbol := will mean equality by definition and the notation A1
signifies the transpose of the matrix A.
When referring to square matrices, the notation A>0, A > 0 A<
0 will mean that A is positive definite, positive semidefinite, and
negative definite respectively.
8


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
TRACKING AND DISTURBANCE REJECTION FOR NONLINEAR SYSTEMS
WITH APPLICATIONS TO ROBOTIC MANIPULATORS
By
MICHAEL L. WHITEHEAD
December 1986
Chairman: Dr. Edward W. Kamen
Major Department: Electrical Engineering
The servomechanism problem (i.e., output tracking in the presence
of disturbances) is considered for a class of nonlinear systems.
Conditions are given which guarantee the existence of a solution to the
problem. The resulting controller requires an internal model system in
the feedback loop; however, due to the nonlinearity of the system, the
internal model must contain dynamics other than those found in the
reference and disturbance signals.
A robotic manipulator system has been considered as one possible
application for the proposed control scheme and various hypotheses are
tested with respect to this system. Simulations are provided which
demonstrate the performance of the control scheme when applied to a 2-
link manipulator.
v


157
[28], M. Thomas and D. Tesar, "Dynamic Modeling of Serial Manipulator
Arms," Trans. ASME, Vol. 104, pp. 218-228, Sept. 1982.
[29]. J. Duffy, Anaylsis of Mechanisms and Robot Manipulators. New
York, NY: John Wiley & Sons, Inc., 1981.
[30]. J.P. Wander, "Real-time Computation of Influence Coefficient
Based Dynamic Modeling Matrices for Improved Manipulator
Control," Master's Thesis, University of Florida, 1985.
[31]. R.P. Judd and D.R. Falkenburg, "Dynamics of Nonrigid Articulated
Robot Linkages," IEEE Trans. Automat. Contr., Vol. AC-30, pp.
499-502, May, 1985.
[32]. F.A. Kelly and R.L. Huston, "Modeling Flexibility Effects in
Robot Arms," in Proc. 1981 Joint Automat. Contr. Conf., Vol.l.
[33]. J.E. Shigley, Mechanical Engineering Design. New York, NY:
McGraw-Hill, 1977.
[34]. R.L. Burden, J.D. Faires, and A.C. Reynolds, Numerical Analysis,
Boston, MA: Prindle, Weber & Schmidt, 1981.
[35]. Dr. Seuss, The Cat and the Hat, Boston, MA: Random House, 1957.


15
impulse type terms we have
J dj
I a. T- u(t) =0 t > 0 (2-18)
j=0 J dtJ K
This is exactly of the form given by (2-4).
We now give further motivation for the assumption that u (t)
satisfies equation (2-4) by showing an example of a nonlinear system
where this is indeed the case.
Example 2.2
Consider the system
x(t) = 3x(t) + x2(t) + (2x2(t) + 4)w(t) + u(t)
y(t) = x (t)
(2-19)
where we desire y*(t) = r*(t)
constant. Thus
djSincot and the disturbance is
x (t) = o^sintot
k, .
w (t) = ou
(2-20)
where and are constants. Substituting (2-20) into (2-19) yields
the following
12 2 *
ctjWCOSwt = 3a^sin)t +(l-COS2ait) + [a^ (l-cos2a>t) + 4]a^ + U (t)
(2-21)


12
Furthermore, assume that xp(t) and wp(t) are periodic with a common
period of T. Under these conditions, for the same inital state xp(0)
and disturbance wp(t), there exists a periodic input up(t) having a
period of T which results in the state trajectory xp(t).
Proof: Make the definition
uT( t)
u(t)
0
0 < t < T
otherwise
Then let
co
u (t) = £ uT(t-nT)
p n=0 1
(2-6)
(2-7)
Since xp(0) = xp(T) = xp(2T) = ... and wp(t) repeats itself over every
interval T, the result is obvious.
Let xp(t), up(t), and wp(t) be periodic with a common period T and
assume that the following differential equation is satisfied.
xp(t) = f(xp(t), up(t), wp(t)) (2-8)
In Proposition 2.1 we have already asserted that a periodic up(t) will
exist whenever xp(t) and wp(t) are periodic with a common period.
Assuming up(t) is integrable over any period, let the Fourier series
expansion of up(t) be
00
Un(t) = a + E a. cos(kwt + <(>. )
P 0 K K
(2-9)


43
Note that for the matrix inverses given in (3-11) to exist, conditions
(1), (2), and (3) are needed. In addition, if H[xl F]~*G is not
square, any right inverse can be used.
We shall make use of the following formula which can be obtained
using a partial fraction expansion
Csl-F]"1
(s-X)'
N i
- { I (-1)J
j=l
L[AI-F]"j-
(S-A)
N+l-j
+ ( [sI-F]-1 [F-AirN }
(3-13)
where it is assumed that A is not an eigenvalue of F. Using (3-13) and
(3-10) it is then possible to write (3-5) as
R*(s) = HCSI-F]-1 { x* + [F-A^rtw + [F-A1I]"2Gv + [F-A^l'V }
t + H[X1I-F]_1Sv^-i- HCAjI-F^Gv^
T
(3-14)
When (3-11) and (3-12) are substituted into (3-14) we obtain the desired
k .
R (s).
It also follows from (3-4), (3-10), and (3-12) that the X*(s) which
occurs during tracking can be expressed as
X*(s) = [A1I-F]"1Gwtilrr + [A1I-F]1Gvy-^- ^
(s-Aj)
-,-2... 1
1
- [A.I-F]" Gv-^y-y + [A2I-F]-GrT^x-T
(3-15)


85
3(t)
Figure 5-3. Modeling incorrect nominal inputs


58
f(x,u,w)
3x +
x2
(2x2 +
4)w + u
(4-32)
so that
F*(t)
G*(t)
3f(x,u,w)
3X
3f(x,U,w)
3U
x=ajSint
w=cu
x=c^sint
w=a0
= (3+2x+4wx)
x=a^sint
W=a0
= 1
3+ (2a^+4a j otg) s i n t
(4-33)
(4-34)
The linearization of NCT may be written as
xA(t)
[F (t)Kj3 [
0
0
0
0
-1
0
0
0
0
0
1
0
0
0
-4
-K2
0
1
0
0
0
0
0
1
0
-5
0
0
0
1
0
Xfl(i)
(4-35)
where
xA(t)
x(t)
(t)
(4-36)
It is not difficult to show that all conditions needed to apply Theorem
4.2 are satisfied. Evaluation of the linearized system about the origin
gives
F = 3
9
G = 1
(4-37)



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5
4. The controller is rather simple to implement. The internal model
system is linear and stabilization is accomplished by the use of
well known linearization techniques.
The main results of this paper are contained in Chapter Two. Here
the servomechanism problem is solved for a nonlinear system having the
same number of inputs as outputs. Later, a method is introduced so that
the results can be extended to a nonlinear system having more inputs
than outputs.
The assumptions needed in the derivations are that a solution to
the problem does indeed exist and that when tracking does occur, both
the state and input will satisfy a linear differential equation.
Although the latter assumption is restrictive, when it does not hold, a
design based on such an approximation may still result in very small
tracking error.
After the assumptions are stated, an internal model system is
introduced. This internal model system replicates the dynamics found in
the state and input signals which are necessary to achieve tracking.
The concept of including the dynamics of the state and input rather than
the common practice of including the dynamics of the reference and
disturbance is believed to be new.
The next step in the design involves the use of constant gain
feedback with the internal model systm incorporated into the feedback
loop. It is shown that observability of the internal model system,
through its associated feedback gain, insures that zero tracking error
will occur for all time provided the initial state of the combined plant
and controller has the correct value. Since such an initial state is
unlikely to occur in practice, it is next shown that certain stability


120
Figure 7-7. Tracking error: ref. = 1.5 Hz sinusoid with amplitude
of 15 ; dist. =0; I.M. frequencies = 0 Hz, 1.5 Hz,
and 3.0 Hz


21
This gives
1 k A-t* 1 k
K2Cn (t) n (t)] = K2eax[n0 nQ] = 0 (2-34)
1
and the vector [nQ nQ] is not observable which is a contradiction.
Theorem 2.1 Given the p-input, p-output system N suppose that for a
particular reference r*(t) and a particular disturbance w*(t)
assumptions (A.l) and (A.2) both hold. In addition, suppose that the
pair (A,B) defines an internal model system of r*(t) and w*(t) with
respect to N. Furthermore, assume K2 of the system NC is such that the
pair (A,K2) is observable and let Kj be arbitrary. Under these
if
conditions, there exist initial states x(0) = xQ and n(0) = nQ such
that in the closed-loop system NC, e(t) = [r*(t) y(t)] = 0 for all
if if
t > 0 when the exogenous signals r (t) and w (t) are present.
Proof: To prove Theorem 2.1 it is necessary to show that there exists
an initial state* for the system NC such that perfect tracking occurs.
Let [x0, n*] denote this initial state and let [x*(t), n*(t)] be the
corresponding state trajectory. The following relationship must then
hold for the system NC
x*(t) = f(x*(t), u*(t), w*(t))
n*(t) = An*(t) (2-35)
u*(t) = -K]X*(t) K2n*(t)
e(t) = r*(t) Hx*(t) = 0
Henceforth, the initial state
controller will be grouped in a pair as [x
which results from this initial state will
of the combined plant and
nQ]. The state trajectory
grouped as [x(t), n(t)].
&


2
The servomechanism problem has been successfully dealt with for
linear, time-invariant systems. Many results are available [1-11] and
an excellent summary is provided by Desoer and Wang [2]. A more
abstract discussion is give by Wonham [11]. It has been shown that an
essential ingredient in a controller designed to solve the servo
mechanism problem is an internal model system. This internal model
system is a system which replicates the dynamics of the exogenous
signals (i.e., reference and disturbance) in the feedback loop.
Because any real system is seldom linear, it is important to
consider the servomechanism problem for nonlinear systems. Some results
exist for the nonlinear problem [12-15]; but for the most part, the
results apply only when the reference and disturbance signals are
constant.
Desoer and Wang [12] have approached the problem using input-output
techniques. They have considered a linear system with nonlinearities
both preceeding it (input channel nonlinearities) and following it
(output channel nonlinearities). They first treat the case of input
channel nonlinearities (such as a sensor nonlinearity). Conditions are
given as to when tracking (disturbance rejection is not considered) will
occur. Although conditions are given, no method is provided which will
enable one to construct a suitable controller nor is discussion given as
to ways of testing the conditions. The main results derived by Desoer
and Wang, however, are for memoryless nonlinearities (both input and
output channel). These results are valid only for reference and
disturbance signals which tend to constants. The conditions given for a
solution to the problem are precise, however, it appears that the
algorithm recommeded for selecting the control law is useful only for


91
z-transform of x (k) and U (z) denote the z-transform of u (k) we have
JL OtrtZ
u (z) = 2 (5-24)
z 2zcoswT + 1
The minimum polynomial having roots corresponding to the poles of both
jlf
X (z) and U (z) determines the difference equation (5-17). In our
example, this polynomial is obtained by multiplying together the
denominator polynomials of X*(z) and U*(z). The result is the following
(z-l)(z2- 2zcoscdT + 1) =
z3 [l+2coso)T]z2 + [l+2coswT]z 1
(5-25)
Hence, the difference equation is
s(k+3) [l+2coswT]s(k+2) + [l+2cosuT]s(k+1) s(k) = 0 (5-26)
s(j) = x*(j) or u*(j)
and the matrix is
Ch =
0
0
1
1
0
0
1
-(l+2coswT) (1+2cosiT)
(5-27)
We now consider selection of the feedback gains. Notice in Figure
5-4 that the feedback control law is


31
(1) If p < q the pair (A, B) is not controllable
(2) If m < q the pair (A, K2) is not observable
Proof: The proofs to properties (1) and (2) are similar so we only
prove (1). This will be accomplished by showing the existence of a row
vector v' such that
v'[XI A B] = 0
(2-50)
for some x which is an eigenvalue of A.
From the structure of A, it is apparent that x is an eigenvalue of
C and hence there is a row vector w' such that
w'CxI C] = 0
(2-51)
Now define a matrix Q e Rclx Q =
w1 0
0 w'
0
0
(2-52)
0 0
w
It can be readily seen that
Q Cxi A] = 0
(2-53)
Now let D e RC'XP be the matrix product of Q and B. That is


90
0 1 0
0 0 1

0
o
o
o
(5-21)
0 0 0
dQ -ct1 -d2
1
0
1
and T is arbitrary but nonsingular.
It is not difficult to show that the eigenvalues of the matrix
will be identical to the eigenvalues obtained by discretizing the
continuous-time internal model system given by equations (2-25) through
(2-28). Consequently, the performance obtained using either the system
of (5-18) or the direct discretization will be roughly the same. The
system of (5-18) is, however, often easier to obtain and implement.
The following example shows how to calculate the coefficients of
the difference equation used to define the discrete-time internal model
system.
Example 5.1
Suppose x* (t) = aj_
(5-22)
where and a2 ace constants. The sampled signals are
(5-23)
where sampling occurs at every t = kT seconds. Letting X (z) denote the


YPOSITION OF HAND (METERS) X-POSITION OF HAND (METERS)
,-0.07 0.19 0.99 0.70 0.95 =-0.9l -0.05 0.3L 0.S7
150
ro
o
Figure 7-23. Actual and desired trajectories of the hand without
compensation for flexible links


86
with d(t) must be included in the internal model system (assuming
they have not already been included). For example, if the nominal input
supplied to the system differs from the required nominal input by a
constant, the internal model system must contain integrators.
Now consider the case when the nominal state is not generated
correctly. Let us write
xd(t) = x (t) xa(t) (5-16)
where xa(t) is the nominal state which is actually supplied to the
*
system, x (t) is the correct nominal state which should have been
supplied, and xd(t) is the disturbance representing the difference
between the correct and actual signals. Since the nominal state is fed
through to the input via a linear feedback gain matrix (see Figure 5.1)
it is apparent that xd(t) can be modeled as an input disturbance.
Hence, we may conclude that the dynamics associated with xd(t) must
also be included in the internal model system.
To summarize, we have shown that robustness with respect to the
ic *
open-loop signals x (t) and u (t) is obtained provided that any
deviations from these signals are sucessfully modeled in the dynamics of
the internal model system.
Digital Implementation
In the previous treatment of the servomechanism problem there has
been an underlying assumption that the control will be implemented via
continuous-time methods. Often it is desirable to implement the control
using a digital computer and hence a discrete-time control law is


30
as the number of outputs. In this section, further insight into this
assumption is presented by showing its relation to the chosen controller
structure. In addition, sufficient conditions will be given to allow
one to consider a system with more inputs than outputs. The case where
the input dimension is less than that of the output shall not be
considered since, in this circumstance, a solution to the servo-
mechansism problem does not generally exist. The intuitive reason for
this is that it requires at least p independent inputs to control p
degrees of freedom independently.
Let us now consider a nonlinear system with input u(t) e Rm and
output y(t) e RP, where m > p. Assume that the controller is im
plemented in essentially the same way as the previously discussed
controller except now consider changing the dimension of the internal
model system. It is assumed that the matrix A of the internal model
system has q blocks on the diagonal rather than p blocks as before.
Consequently, we now have A e Rclrxclr, B e R^P and l<2 e RmX(lr. The
exact change in the A matrix is shown by the following equation
A = T-1 block diag. [C, C, ..., C] T (2-49)
y -j
q blocks
where C is again defined by (2-25). The corresponding change in the B
matrix does not need to be considered in this analysis.
Proposition 2.4: Given the triple (A, B, 1^) with A e R (2-49), B e R(lrxP arbitrary, and K2 e Rmxclr arbitrary. The following
properties are true.


67
derived assuming u(t) is the only external input to the system and no
consideration is made for uncontrollable inputs such as a disturbance or
reference signal.
In the linear servomechanism problem, the dynamics of the plant and
the controller can be modeled by the following equation
LC:
x(t)
n(t)
F 0
-BH A
x(t)
n(t)
u(t) +
E
w(t) +
0
0
r(t)
B
u(t) = -Kjxit) K2n(t)
y(t) = Hx(t)
(4-44)
where x(t) e Rn is the state of the plant, n(t) e RPr is the state of
the internal model system, u(t) e Rm is the input, y(t) e RP is the
output, w(t) e R^ is a disturbance, and r(t) e RP is the reference. The
system LC is essentially the linear version of the system NC given in
equation (2-29) for the nonlinear servomechanism problem. Note that for
the linear system, it is not necessary for the dimension of the input
and the dimension of the output to be the same.
As already discussed, if the internal model system is chosen
appropriately and the feedback gives asymptotic stability to the system
LC without the exogenous inputs w(t) and r(t); then tracking and
disturbance rejection will occur when w(t) and r(t) are applied. The
precise conditions as to when it is possible to obtain a stabilizing
feedback are conditions (B.l) and (B.2).
Suppose that the conditions for a solution to the linear servo
mechanism problem have been satisfied and the closed-loop system LC has


to
components which are either constant or sinusoidal in nature. To
simplify the development, we shall consider a particular disturbance
signal, say w (t), and a particular reference signal, say r (t), to be
represenative signals from a given class of functions. Once the
controller is derived with respect to these signals, the results can be
generalized to cover a class of functions for which r*(t) and w*(t) are
assumed to belong.
We now make the following assumptions:
(A.l) For some chosen reference signal r (t) and a particular
"Jc "k
disturbance w (t) there exists an open-loop control u (t) and an
initial state x (0) = x such that
o
x*(t) = f(x*(t), u*(t), w*(t))
y*(t) = Hx*(t) = r*(t) (2-3)
e(t) = r*(t) y*(t) = 0 for all t > 0
-k k
(A.2) The elements of both x (t) and u (t) satisfy the scalar, linear
differential equation
()(r) + ar-l(*)(r_1) + + ai(*)U) + a0(,) = 0 (2"4)
where the characteristic roots of (2-4) are all in the closed right
half-plane.
The first assumption is merely a way of stating that it is possible
to provide output tracking. A typical example where (A.l) would not
hold is for a system having more outputs than inputs. This particular


13
where m is the fundamental frequency.
Given a positive integer K, let u^(t) denote the truncation
K
UK(t) = aQ + E a^cos(kcot + <¡>k) (2-10)
k-1
The truncation u^(t) satisfies a differential equation of the form given
by (2-4). Example 2.1 will show how to obtain the specific differential
equation using Laplace transform theory.
Now let X|<(t) denote the solution to
xK(t) = f(xK(t), uK(t), wp(t)) (2-11)
(assuming the solution x^(t) exists)
If HXp(t) x^(t)ii is suitably small for t > 0, the assumption
that the input satisfies (2-4) is reasonable. Often, either by using
simulations or actual tests, it is possible to determine apriori how
small iiXp(t) x|^(t)ii is for a given value of K. Also note that in
practice there is always some error, so that demanding
nx (t) x(t)n = 0 is not reasonable.
P k
We now mention an important practical point which was overlooked in
the preceeding discussion. For iixp(t) x^(t) ii to be suitably small,
the nonlinear system N must be stable in the sense that bounded inputs
give bounded outputs. If this is not the case, it would be necessary to
use a pre-stabilizing feedback so that the unstable portion of xK(t)
could be eliminated. This allows one to make the most meaningful
assessment of how "good" the input uK(t) acutally is. The use of such a
stabilizing feedback would be needed only in simulations and testing


81
where
and
xA(t) F*(t)xA(t)
xA(t)
x(t) x*(t)
n(t) n*(t)
F*A(t)
F*(t) G*(t)K1(t)
-BH
-G*(t)K2(t)
A
(5-6)
(5-7)
(5-8)
The Jacobian matrices F*(t) and G*(t) are evaluated at the signals
x (t), u (t), and w (t). More precisely, we may write
F*(t)
x = x (t)
u = u*(t)
*
w = w (t)
G*(t)
3f(X,U,W)
3U
X = X (t)
U = U*(t)
*
w = w (t)
(5-9)
if
Usually these Jacobian matrices cannot be evaluated apriori since x (t),
fa ^
u (t), and w (t) are not known. Consequently, to show stability of the
linearized system we shall use a technique already presented in Chapter
Four; namely, the Poincare-Liapunov theorem. Let us write the
linearized equation given by (5-6) as
xA(t) FA(t)xA(t) + CFA(t) FA(t)]xA(t)
(5-10)


60
and 2=1. Then, from (4-38), we get ej = 3 so that e^m-a = -1 < 0
and the system NCT is locally asymptotically stable.
Various simulations have been obtained using a Runge-Kutta
algorithm [19] to numerically integrate the closed-loop nonlinear
system. In some of these simulations, an internal model system has been
used which does not contain dynamics corresponding to the second
harmonic of the reference signal. Such a design results when the
internal model system is chosen in accordance with linear servomechanism
theory. To give a fair comparison between the two control schemes, the
closed-loop eigenvalues of the design based on linear theory are identi
cal to those given above, except that the eigenvalues at -4 j2 are no
longer needed due to a reduction in system order.
Figure 4-1 shows the responses obtained for both design approaches
when only a reference signal is applied. The proposed control design
works well (see Fig. 4-l(a)) and the tracking error is completely
eliminated in steady-state. The design based on linear servomechanism
theory, however, results in a steady-state tracking error which is
sinusoidal with a frequency twice that of the reference signal.
Figure 4-2 again shows the responses obtained using both design
approaches. Here, however, a constant disturbance has been introduced
in addition to the sinusoidal reference signal.
In Figure 4.3 the amplitude of the disturbance has been increased
and as a result, the transient response in the system designed by linear
theory is very poor. Note that in order to show the complete response,
the scaling of the plot in Figure 4-3(b) is different from the scaling
used in previous figures.


69
Equation (4-46) is a linear dynamic equation modeling the transient part
of the state trajectory of the system LC. Figure 4-5, shows a typical
illustration of the actual, steady-state, and transient trajectories
which might result when a sinusoidal signal is being tracked. It is
important to note that neither w(t) nor r(t) appear in (4-46) so that
these exogenous signals play no role in the transient trajectory or, in
otherwords, how fast tracking occurs. In addition, without the exo
genous signals, (4-46) is of a form which makes possible the interpre
tation of using optimal control techniques for the selection of the
feedback gains.
Now let us assume that the stabilizing gain K := [K^, 1^] is found
by solving the algebraic Riccati equation for optimal control. That is,
the positive semidefinite P satisfying
Q + PFa + F^P PGaR_16P = 0 (4-48)
is obtained and K is selected as
K = R_1G^P
(4-49)
Here, Q > 0 is a symmetric matrix of dimension n+pr x n+pr, R > 0 is a
symmetric matrix of dimension m x m, and
FA =
F
-BH
0
A
9
G
A
G
0
(4-50)
If the reference signal r(t) and the disturbance w(t) are applied at
time t=0 then the following quadratic performance index is minimized


Y-POSITION OF HAND (METERS) X-POSITION OF HAND (METERS)
,-0.18 -0.09 0.29 0.67 1.05 -0.55 -0.16 0.21 0.81 1.03
142
(b) Actual and desired trajectories of the hand
Figure 7-19 continued


61
As a final test of the control scheme, a sinusoidal reference at a
frequency slightly different from the intended frequency is applied to
the system. Figure 4-4 depicts the resulting responses. Although we
have not considered this particular situation from a theoretical
standpoint, it is seen that the controller which contains a second
harmonic works considerably better than the controller designed
according to linear servomechanism results.


Y-POSITION OF HAND (METERS) X-POSITION OF HAND (METERS)
,-0.13 0.16 0.05 0.7U l.OU -0.55 -0. L$ 0.23 0.62 1.01
140
I
(b) Actual and desired trajectories of the hand
Figure 7-18 continued


128
to eliminate the sinusoidal error additional poles are introduced into
the I.M. system corresponding to the 4 Hz signal. The response is given
in Figure 7-17. Notice that the steady-state error is now virtually
zero. Figure 7-18 is presented simply to show the poor performance which
results from using an I.M. system consisting of just integrators.
Figures 7-19 through 7-21 illustrate the effects of applying both a
reference and a disturbance signal. First, poles at only the
frequencies of the reference and disturbance signals are included in the
I.M. system. The resulting response is shown in Figure 7-19. Since the
steady-state error in joint 2 has a large dc offset, integrators are
then incorporated into the I.M. system. The response is shown in Figure
7-20. Notice that the dc error has been eliminated, however a ripple at
a frequency of 6 Hz is present. The I.M. system is then modified to
accommodate for the 6 Hz signal and the response is given in Figure
7-21.
This concludes the tests of the rigid-link manipulator. In the
next section we consider a manipulator with flexible links.


41
conditions are met, x*(t), U(t) and hence u*(t) can be chosen to
satisfy the differential equation (3-2).
The proof to the above statement is quite tedious and will be
omitted; however, an example shall be given later which will make the
statement obvious.
Now it is only necessary to show that when (B.l) and (B.2) hold,
conditions (1), (2), and (3) are satisfied. It immediately follows from
condition (B.2) that (1) must be true. Also, by condition (B.l) we can
select K so that A.., i = 1,2, ..., r is not an eigenvalue of [F GK]
and hence (2) holds. Assuming that such a K has been chosen, it is easy
to show that
rank
X1I F
-H
G
0
rank
A-jI F + GK
H[XiI F + GK]-1G
(3-7)
This is accomplished by premultiplying and postmultiplying the left-hand
side of (3-7) by
HCA-jl F + GK]'1 Ipxp
respectively. Here Inxn denotes the identity matrix of dimension nxn.
From condition (B.2) and equation (3-7) the matrix H[A.jI F + GK]G must
have rank p for all A.¡, i = 1,2, ..., r. This gives (3) and the proof
is complete.
The following example helps to verify the statement given in
nxn
K
lmxm
Theorem 3.2.


CHAPTER SIX
APPLICATION TO THE ROBOTIC MANIPULATOR
Several key ideas concerning the solution of the servomechanism
problem for the robotic manipulator system are presented in this
chapter.
The first of the chapter contains a brief summary of the dynamic
equations modeling the robotic manipulator. Next the overall structure
for the control system is given. In order to effectively treat the
manipulator problem, discussion is given in regards to determining the
dynamics which must be included in the internal model system as well as
finding a stabilizing feedback law. The controller design is based
primarily on the results of Chapter Five
The final portion of this chapter deals with compensating for
structural flexibilities in the manipulator system. It is shown that
the proposed control algorithm can correct for end-effector deviations
caused by slowly-varying external forces provided that the forces can be
measured.
Manipulator Dynamics
In this section we discuss the dynamic equations modeling a rigid-
link serial manipulator having revolute joints. A more thorough
treatment of this topic can be found in Thomas and Tesar [28].
Using Lagrange's equation of motion, it is possible to obtain the
following dynamic representation for the manipulator.
94


CHAPTER FIVE
PRACTICAL CONSIDERATIONS
In this chapter practical aspects dealing with implementing the
control algorithm for the nonlinear servomechanism problem are treated.
First discussed is a modification of the already proposed control
algorithm which is specifically tailored for tracking and disturbance
rejection with respect to a nominal trajectory. The modified design is
based on deviations from the nominal trajectory and since these devia
tions are generally small, the demand on the stabilizing capabilities of
the controller is minimized. Another benefit of the design is that
tracking of a wide class of reference signals is made possible by sup
plementing the closed-loop feedback with open-loop control inputs.
The final section of this chapter deals with discrete-time
techniques so that the control algorithm can be implemented using a
digital computer.
Controller Based on the Nominal Trajectories
In this section we develop a controller for the servomechanism
problem which operates relative to nominal reference and disturbance
signals. The design presented here employs not only feedback control
but also feedforward control to achieve the desired trajectory.
Although feedforward control is introduced, the control scheme is
formulated in essentially the same way as was done in previous treatment
of the nonlinear servomechanism problem.
74


17
Definition: Given the system N described by (2-2), suppose that for a
particular r (t) and a particular w (t) assumptions (A.l) and (A.2) both
hold. Then an internal model system of r (t) and w (t) with respect to
N is a system of the following form:
n(t) = An(t) + Be(t)
e(t) = r*(t) y(t) = H[x*(t) x(t)] (2-26)
where
A = T"* block diag. [C, C, ..., C] T
v_
p blocks
(2-27)
r-1
block diag. [t, t]
p blocks
(2-28)
where the state n(t) e Rpr, T is an arbitrary nonsingular matrix,
B e Rprxp, and the pair (A,B) is completely controllable. In
practice, T is usually taken as the identity matrix.
Roughly speaking, the above internal model system is seen to
contain p copies of the dynamics of the state and input signals which
must occur during tracking. This is different from the internal model
system used in the linear servomechanism problem where only the dynamics
of the reference and disturbance signals are included. The dynamics of
the reference and disturbance will inevitably be included in (2-26);
however, the nonlinear structure of N may necessitate the introduction
of additional dynamics.


25
generalization is rather obvious. If tracking and disturbance rejection
is to hold for a certain class of signals r(t) and w(t), two conditions
are required: 1) the internal model system must contain the necessary
dynamics to cover the entire class of signals, and 2) the closed-loop
tranasient system NCT must remain locally (or globally) asymptotically
stable over this class of signals.
Often in practice, the precise reference and disturbance signals
acting on the system are not known in advance and hence neither are
Vc Vr
x (t) and u (t). In order to determine the dynamics which must be
included in the internal model system it is necessary to have some
apriori knowledge of the state and input signals which will occur during
tracking. Usually, knowledge of the frequencies of the anticipated
disturbance and reference signals is available. Generally, the
"fc
frequencies of the reference r (t) and the disturbance w (t) will
directly affect the frequencies of the corresponding state x (t) and
jlp
input u (t). Assuming this to be true, the mathematical model
describing the nonlinear system can be used to determine x (t) and u (t)
for various combinations of r (t) and w (t). Fourier analysis can then
be used to determine the dominant frequencies in the signals comprising
the various x (t) and u (t) and the internal model system can be
designed accordingly. Even when the mathematical is not used, an
educated guess or perhaps trial and error can enable one to design an
internal model system with the appropriate dynamics. For example, if
sinusoidal disturbance and reference signals are expected, it might be
advisable to design the internal model system to accomodate for various
harmonics and subharmonics of the anticipated sinusoidal signals.


9
Usually, capital letters (e.g., A, B, F) will denote matrices,
while lower case letters (e.g., x, y, z) shall denote vectors. Both
standard lower case letters and Greek lower case letters shall indicate
a scalar. Any deviations made from the notation for matrices, vectors,
and scalars will be clear from context.
Main Results for the Nonlinear Servomechanism Problem
Consider the nonlinear system
N: x(t) = f(x(t), u(t), w(t))
y(t) = Hx(t)
e(t) = r(t) y(t) (2-2)
where x(t) e Rn is the state of the system, u(t) e RP is the control
input to the system, w(t) e Rd is a disturbance signal, y(t) e RP is the
output of the system, and e(t) e RP is the error which occurs when
tracking the reference signal r(t) e RP. Note that the output is
assumed to be a linear function of the state and the dimension of the
input is the same as that of the output.
It is our objective to design a closed-loop controller which will
asymptotically regulate against disturbances and also asymptotically
track a reference signal. In particular, we desire e(t) = r(t) y(t)
+ 0 as t + where r(t) is a specified output chosen from a given
class of functions.
The primary concern here will be tracking and disturbance rejection
when the disturbance w(t) and the reference r(t) are comprised of


CHAPTER EIGHT
CONCLUSIONS AND OPEN PROBLEMS
A control scheme has been developed which solves the servomechanism
problem for nonlinear systems satisfying assumptions (A.l) and (A.2).
Existence of a solution is guaranteed by (A.l) and (A.2) requires the
state and input (which occur during tracking) to satisfy a linear
differential equation.
It has been argued that when (A.2) does not hold it is still
possible to design a satisfactory controller. For example, when the
reference and disturbance signals are periodic then the state and input
signals must be periodic if tracking is to occur (assuming an autonomous
system). Therefore, since many systems attenuate high frequency
signals, a truncated Fourier series expansion of the true input and
state is likely to be a reasonable approximation from which to base the
design. The state and input resulting from such an approximation do
satisfy a linear differential equation. The above argument has also
been verified using simulations of a 2-link robotic manipulator.
Tracking will be local if only local stability of the system NCT
can be shown (see the definitions given in Chapter Two). Furthermore,
when the feedback giving local stability is obtained using a lineariza
tion of the closed-loop system about the origin; tracking will occur for
only small reference and disturbance signals. In many cases it is not
necessary to have tracking for large disturbance signals, since by
nature, disturbance signals are frequently small. Also, it is likely
153


96
In order to write the dynamic equation modeling the manipulator in
standard state-space form, we make the definition
(t)
0(t)
0(t)
(6-4)
From equation (6-1) it then follows that
(t) = f+U(t), TA(t), w(t)) (6-5)
where
f+U(t), TA(t), w(t))
e
J_1(0)CTA(t) + Tg(0) + Td(w(t),0) Tv(0,0)]
(6-6)
Equation (6-5) is a nonlinear dynamic equation which can be used either
in performing simulations or in controller design.
Acuator Driving Torques
Often one wishes to compute the torques required to drive the
manipulator over some trajectory. This trajectory is usually specified
in terms of the position, velocity, and acceleration of the manipu
lator's end-effector (i.e., hand). It is usually necessary to convert
the end-effector trajectory into the corresponding trajectory for the
joint angles, joint velocities, and joint accelerations. This task
shall not be discussed in detail here since it will not be of major
importance in demonstrating the control algorithm. We mention, however,
that it can be difficult to obtain the joint angle vector 0 which will


33
The above restriction is actually somewhat misleading.
Specifically, consider a system having more inputs than outputs (i.e.,
m > p). It is quite possible that when only p out of the m inputs are
used, all conditions for solving the servomechanism problem will be
satisfied. In this case, we can define a new system NP which is nothing
more than the original system N operating with only p inputs. This is
shown by the following equations:
NP:
x(t) = f(x(t), u(t), w(t)) = fp(x(t), up(t), w(t))
y(t) = Hx(t) (2-56)
u(t) = Mup(t)
where M e RmxP is of full rank. If there exists an M such that the
system Np with input Up(t) meets all conditions given in the previous
sections, then the problem can be solved. Working with the input Up(t),
let the feedback law which stabilizes the closed-loop transient system
be
uP(t)
x(t) K ?n(t)
K 9C
(2-57)
In terms of the original system N, the feedback law will then be
where
u(t) = -K-^xit) K2n(t)
K1 "
MK
P1
K9 = MK 9
2 p,2
(2-58)
(2-59)


76
robotic manipulator is discussed in Chapter Six.) The true reference
signal shall be denoted r*(t). Also, it is likely that the actual
ft
disturbance, denoted w (t), will not be the same as the nominal
disturbance w (t). When r (t) and w (t) are applied to the system then,
if assumption (A.l) holds, tracking will occur provided that a certain
state and input trajectory are present. Denoting this state and input
ic 'if
as x (t) and u (t) respectively, the following definitions can be made
?*(t)
:= r*(t)
- r (t)
tf(t)
w*(t)
- w*(t)
X (t)
x*(t)
- X (t)
(5-3)
/N*, x
U (t)
:= u*(t)
- U (t)
where r*(t), w^t), x^t), and 1j*(t) denote the deviations of the true
signals from the nominal signals. Note that because the system under
consideration is nonlinear, x^t) and "ij*(t) are not necessarily the
state and input trajectories which give tracking of >r*(t) with the
disturbance w*(t).
Now as before, the objective of the control will be to cause the
actual state trajectory x(t) to asymptotically approach the state
'if
trajectory x (t). In addition, this will be accomplished with an input
if
u(t) which asymptotically approaches u (t).
Consider the control scheme given in Figure 5-1. In relation to
this scheme we have used the following definitions
x{t) := x(t) x*(t)
tiit) := u(t) u*(t)
(5-4)


14
since later, a stabilizing feedback law will be developed for the actual
implementation.
The following example shows how a differential equation of the form
given by (2-4) is derived from a truncated Fourier series
Example 2.1
Suppose
K
UK(t) = aQ + £ a^cos( knit + <(>k) (2-12)
Taking Laplace transforms of both sides we get
Ms) = K^ (2-13)
l\ l\ rt 0 0
s[ n (s + K ai )]
k=l
where N(s) is a polynomial in s. Equation (2-13) can be expressed as
s[ n (s2 + k2a)2)]UK(s) = N(s) (2-14)
Next, by writing
we have
s[ n (s2+ k2o>2)]
k=l
J
£ a.s3 J = 2K + 1
j=0 3
[ £ a.sJ] U,,(s) = N(s)
j=0 3 K
(2-15)
(2-17)
Now taking inverse Laplace transforms and noting that since N(s) is a
polynomial in s and hence has an inverse Laplace transform consisting of


136
!
C7
O
(b) Actual and desired trajectories of the hand
Figure 7-16 continued


80
We have indicated that tracking occurs provided the initial state
is correct. It is also necessary to show asymptotic stability of the
system which models the transient dynamics. Since the formulation used
here is roughly the same as in previous derivations, the system NCT
given in Chapter Two is still a valid model.
Feedback Gain Selection
In this section feedback gain calculation for the control scheme of
Figure 5-1 shall be discussed. Again, as in the previous chapter,
linearization procedures shall be employed so that Liapunov's indirect
method can be used to determine stability.
Before proceeding, it is necessary to discuss the time-varying
feedback law which has been chosen for the control scheme of Figure
5-1. Time-varying feedback is in contradiction with the requirements
imposed to solve the servomechanism problem; however, it may be true
that the time-varying gains vary slowly enough to treat them as constant
for all practical purposes (the quasi-static approach). Here we assume
this to be the case. Later is will become apparent that selecting time-
varying feedback provides better compensation over the nominal
trajectory. If the signals comprising the nominal trajectory vary
slowly enough then it is likely that the feedback gains K^t) and K2(t)
can be chosen to vary slowly.
Now consider the stabilization problem. Let us assume that all
conditions for tracking and disturbance rejection have been met and the
internal model system for the scheme of Figure 5-1 has been chosen in
accordance with the theory of Chapter Two (using x*(t) and "u*(t) in
(A.2)). The linearization of the transient system NCT has been given
previously but is repeated here for convenience.


no
Force
(a) Schematic
X
link
link
lengths
Centroid
locations
link
masses
link
inertias
1
a12 = m
dj = 0.25 m
m^ = 4.5 Kg
Ij = 1.0 Kg-m2
2
a23 = 0.5 m
d2 = 0.23 m
m2 = 3.8 Kg
I2 = 0.6 Kg-m2
(b) Physical characteristics
Figure 7-1. The 2-link planar manipulator


57
Example 4.1
Consider again the system of Example 2.2
x(t) = 3x(t) + x^(t) + (2x^(t) + 4)w(t) + u(t)
y(t) = x(t) (4-28)
Assume that the disturbance is a constant with a value of a2 and
that the reference is a sinusoid with amplitude and frequency u> =
1. Thus
r*(t) = ajsint
w*(t) = a2 (4-29)
Vf 'if
It was found in Example 2.2 that the input u (t) and state x (t) satisfy
the differential equation
5 3
d_ (.) + 51_ (.) + 4d_ (.} = o (4-30)
dt dt
The internal model system is thus
n(t)
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
n(t) +
0
0
0
0
0
1
0
0
-4
0
-5
0
1
e(t)
(4-31)
For the nonlinear system (4-28) we have


134
O
(b) Actual and desired trajectories of the hand
Figure 7-15 continued


129
(a) Tracking error
Figure 7-13. Manipulator's response: ref. = 0; dist. = 2.0 Hz
sinusoid with amplitude of 50 N; I.M. frequency =
2.0 Hz


71
oo
J = / [x^(t)QxA(t) + u' (t)Rtf(t)]dt
(4-51)
where
and
x(t) xss(t)
n(-t) riss(t)
(4-52)
u(t) = Cu(t) uss(t)] = Cu(t) + KlXss(t) + K2nss.(t)] (4-53)
The above observation simply points out that the transients behave in
some optimal fashion.
Now consider the state trajectory [x$s(t), nss(t)] which is unique
for a specific feedback gain K = [K-^, K2]. By assumption, this gain has
been selected by optimal control techniques. If this is not the case,
it is quite possible that the steady-state trajectory [xss(t), nss(t)]
as well as the input uss(t) might be improved in some respect. For
example, a different choice of gains might actually lessen the average
power required to maintain tracking of a certain reference signal.
Increased Degree of Stability Using the
Optimal Control Approach
In the previous section we discussed using the optimal control
approach to obtain stabilizing feedback gains for the linear servo
mechanism problem. The method can also be employed in the nonlinear
servomechanism problem when the linearized equations are used; however,
the control can no longer be considered optimal. It is sometimes
advantageous to make a slight modification on the performance index to
give a higher degree of stability to the linearized model. It was
previously indicated that when the linearization of NCT about the origin


46
This means that x is also an eigenvalue of the matrix
F GKX -GK2
-BH A
Since the system LCT can be written as
*(t)~
F GKj
-gk2
~x(t)~
Ti(t)_
-BH
A
jn(t)_
and x is, by assumption, in the closed right half-plane then LCT is not
asymptotically stable. This is a contradiction to the assertion that
LCT is asymptotically stable and the proof is complete.
Theorems 3.2, 3.3, and 3.4 show that when conditions (B.l) and
(B.2) are true, then all requirements for a solution to the linear
servomechanism problem are satisfied. Furthermore, this has been
accomplished in the framework developed for the nonlinear servomechanism
problem. In the next chapter, we shall apply some of these results to
obtaining the stability required for the nonlinear servomechansim
problem using the linearization approach.


18
The closed-loop control scheme proposed to solve the nonlinear
servomechanism problem has, incorporated into the feedback, the internal
model system of the disturbance and reference signals with respect to
the nonlinear system N. The implementation of this closed-loop
controller is shown in Figure 2-1. The equations modeling the closed-
loop system are the following:
NC: x(t) = f(x(t), u(t), w*(t))
n(t) = An(t) + BH[x*(t) -x(t)]
(2-29)
u(t) = -K^t) K2n(t)
e(t) = H[x*(t) -x(t)]
where e Rpxn and K^e Rpxpr are constant feedback matrices. It is
assumed that the state x(t) is available for feedback.
We will show in Theorem 2.1 that there exists an initial state for
the closed-loop system NC such that tracking occurs with e(t) = 0 for
t > 0. The following proposition shall be required in the proof of
this theorem.
Proposition 2.2 Let z(t) e Rp be any vector with elements satisfying
the linear differential equation
()^ + ar_i(*)(r ^ + + ai(*)^ + 3q(*) = 0 (2-30)
and let C e Rrxr be a matrix whose eigenvalues, including
multiplicities, exactly match the characteristic roots of (2-30). If in
addition, the pair (A, K2) is observable with the constant matrix K2
e Rpxpr and A e Rprxpr defined by


CHAPTER ONE
INTRODUCTION
One of the most important problems in applications of feedback
control is to provide output tracking in the presence of external
disturbances. This is commonly referred to as the servomechanism
problem. More precisely, given a certain system, the servomechanism
problem involves the design of a controller which enables the output to
asymptotically track a reference signal r(t), in the presence of a
disturbance w(t), where r(t) and w(t) belong to a certain class of
functions. The class of functions might be, for example, combinations
of step, ramp and sinusoidal signals. The frequency of the signals is
usually assumed to be known. Typically, enough freedom is allowed,
however, so that it is not necessary to have apriori knowledge of the
amplitude or phase of either the disturbance or the reference.
The assumption of known frequency but unknown amplitude and phase
provides a realistic model for many reference and disturbance signals
encountered in practice. For example, an imbalance in a piece of
rotating machinery might cause a sinusoidal disturbance force to act on
a certain system. Although the frequency of this force might be easy to
predict, it is doubtful that the exact amplitude could be determined.
Even if the amplitude was known exactly, modeling errors in the plant
would make such schemes as open-loop compensation unreliable. This
leads to an important feature of a controller design to solve the
servomechanism problem. Namely, there should be a certain amount of
robustness with respect to plant variations and with respect to
variations in signal level.
1


70
|
Figure 4-5. Actual and steady-state trajectories


108
3*(t) = -Hj1(t)H2(t)f(t) (6-24)
It is perhaps better to express (6-24) in terms of the measured force
f(t) and the nominal force f(t). More precisely, we may write
e*(t) = -H1(t)H2(t)[f(t)-f(t)] (6-25)
By supplying e (t) as the corrective reference signal for the
control scheme of Figure 6-1, end-effector tracking can be improved
considerably. In order to achieve this objective, it is necessary that
the internal modal system contain the appropriate dynamics. If the
external forces are not varying rapidly, it is likely that an internal
model system consisting of just integrators will be sufficient. As a
final point note that perfect end-effector tracking will not occur since
equation (6-25) is only approximate.
In the next chapter, simulations are provided to demonstrate the
performance of the control schemes discussed in this chapter when
applied to a 2-link manipulator.


ACCELERATION VELOCITY POSITION
a. 00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 0.00
TIME (SECONDS)
Figure 7-12. Position, velocity, and acceleration profiles for
the nominal trajectory


\\\\\\\
148
Force
Figure 7-22. Modeling the flexible links.


22
In order to verify (2-35) we first note that by assumption (A.l)
there is an initial state x*(0) = x* and an input u*(t) such that e(t)
= 0 for all t > 0. Hence, it must be shown that for some initial
^ it ^
state n (0) = nQ of the internal model system, the input u (t) can be
produced by feedback of the form
u*(t) = -KlX*(t) K2n*(t) (2-36)
From assumption (A.2) we know that the elements of u*(t) and x*(t) will
satisfy the differential equation (2-4) (or equivalently, equation
(2-30)). Also observe that because e(t) = 0 in (2-35), the internal
model system is completely decoupled from the original system. This
decoupling allows us to apply Proposition 2.2. Specifically, we may
verify (2-36) by letting z(t) = -u*(t) K^x*(t) in Proposition 2.2.
This completes the proof.
We have shown that if certain conditions have been met, then when
ic it
the exogenous signals r (t) and w (t) are acting on the closed-loop
it it
system NC, there exists an initial state [xQ, n ] such that perfect
tracking occurs. However, if the initial state [x(0), n(0)] differs
it it
from [x f nQ], the resulting state trajectory [x(t), n(t)] may not
it ^
converge to [x (t), n (t)] as t > . To achieve (asymptotic)
it ft
tracking, we want [x(t), n(t)] to converge to [x (t), n (t)] for some
range of initial states [x(0), n(0)]. This leads to the following
notation.


ERROR IN JOINT 2 (DEGREES) ERROR IN JOINT 1 (DEGREES)
-eo.oo -la.oo o.ao 10.00 2a.oo a-2o.oo -la.oo o.ao 10.00 aa.oo
143
00 0.50 1.00 1.50 2.00 2.5D 3.00 3.50 li.OO
TIME (SECONDS)
(a) Tracking error
Figure 7-20.
Manipulator's response: ref. = 3.0 Hz sinusoid with
amplitude of 10; dist. = 4.5 Hz sinusoid with
amplitude of 50 N; I.M. frequencies = 0 Hz, 3.0 Hz,
and 4.5 Hz



Figure 6-1. Control system for robotic manipulator


133
O
O
o
(a) Tracking error
Figure 7-15
Manipulator's response
amplitude of 10; dist
ref. = 2.0 Hz sinusoid with
= 0; I.M. frequency = 2.0 Hz


ERROR IN JOINT 2 (DEGREES) ERROR IN JOINT 1 (DEGREES)
,-6.00 -3.00 0.00 3.00 6.00 -3.00 -1.50 0.00 1.50 3.00
131
(a) Tracking error
Figure 7-14
Manipulator's response: ref. = 0; dist. = 2.0 Hz
sinusoid with amplitude of 50 N; I.M. frequency = 0 Hz


48
only state feedback with constant gain matrices can be used to achieve
this stability. The system NCT is given again here for convenience.
NCT:
x(t) = f(x*(t)+x(t), u*(t)+u(t), w*(t)) f(x*(t), u*(t), w*(t))
'n(t) = Aff(t) BHx(t)
u(t) = -K^Sc(t) Kg'nit) (4-1)
Two forms of asymptotic stability are actually considered in the
solution to the servomechanism problem: global stability and local
stability. It was indicated that local stability allows tracking and
disturbance rejection to occur only for certain initial states whereas
global stability allows tracking and disturbance rejection for all
initial states. Global stability is thus the most desirable form of
stability, however, due to the diversity which can occur in the system
NCT, we shall limit our concern to the local stability problem. In
addition, the original nonlinear plant will be taken as time-invariant
(i.e., the function f(x,u,w) is independent of time). These
restrictions will allow us to obtain a time-invariant feedback law which
gives local tracking and disturbance rejection for small reference and
disturbance signals.
For convenience, when discussing the system NCT, the following
definition shall be used
~x(t)
n(t)
xA(t) :
(4-2)


75
Consider again the nonlinear system N
N: x(t) = f(x(t), u(t), w*(t))
y(t) = Hx(t)
e(t) = r*(t) y(t) (5-1)
Here r (t) and w (t) indicate a particular reference and disturbance out
of the class of signals r(t) and w(t).
Now assume that for a certain nominal reference r*(t) and a nominal
(anticipated) disturbance w*(t) tracking can be achieved. In
otherwords, assumption (A.l) holds for the nominal signals so that the
following solution for (5-1) exists
x (t) = f(x (t), U (t), w (t))
y*(t) Hx*(t) (5-2)
0 = r*(t) y*(t)
^ dp
where x (t) and (t) are nominal state and input trajectories which
are necessary for tracking. We shall not require x*(t) and *(t) to
satisfy a linear differential equation such as the one given in assump-
up dp
tion (A.2). Instead, it shall be assumed that both x (t) and u (t) can
be generated by external means so that they are readily available. In
practice, an exact generation of these signals may not always be
possible, however, discussion of this circumstance is deferred until a
later section on robustness.
In certain applications it may be desirable to change the reference
signal to a value other than r*(t). (One such application for the


125
Figure 7-11. Nominal path of the manipulator


Ill
In order to obtain the stabilizing feedback gains, an algebraic
Riccati equation is solved relative to the discretized linear system.
Furthermore, the quadratic performance index is exponentially weighted
(see Chapter Four) so that a high degree of stability is achieved. For
our design purposes, the degree of stability is chosen so that all poles
of the closed-loop linearized system are guaranteed to lie within a
circle of radius 0.7.
Solving an algebraic Riccati equation is a standard procedure used
in selecting feedback gains for time-invariant linear systems. However,
when the nominal trajectory is chosen to give translation of the hand,
the linearized system becomes time-varying. In this case we assume that
the slowly time-varying approach can be employed and hence, the feedback
gains are selected to give stability to the family of frozen-time
systems. Here we employ a technique given in [24] which requires that
an algebraic Riccati equation be solved for only several (6 in this
case) frozen-time systems. Lagrange interpolation [34] is then used for
calculating the gains at intermediate points.
Let us now consider the simulation procedure. Because the con
troller is a discrete-time system, updating the dynamics of the con
troller is easily accomplished using computer simulations. In order to
update the dynamics of the plant (i.e., the manipulator) it is necessary
to employ numerical integration techniques. Here we use a Runge-Kutta
numerical integration algorithm [19] which numerically integrates the
nonlinear dynamic equation modeling the manipulator. To make the simu
lations realistic, the Runge-Kutta algorithm takes 10 steps to iterate
The term "frozen-time system" is used to denote the time-
invariant system obtained by "freezing" all time-varying parameters of a
time-varying system.


66
Optimal Feedback for the Linear Servomechanism Problem
In this section we discuss the consequences of using optimal
control theory as a means of determining the stabilizing feedback gains
for the linear servomechanism problem. Only the linear problem shall be
treated since the interpretation of the results for the nonlinear
problem is not clear. It is true, however, that the actual method of
feedback gain selection discussed in both this and the succeeding
section can be applied to the nonlinear problem when linearization
techniques are used.
Now consider the well known linear optimal control problem [16],
[20]. That is, given the linear time-invariant system
x(t) = Fx(t) + Gu(t) (4-41)
select the control u(t) to minimize the quadratic performance index
oo
J = / [x'(t)Qx(t) + u'(t)Ru(t)]dt (4-42)
o
Where Q 5 0 and R > 0 are symmetric matrices of appropriate
dimension. The optimal control law is found to be of the form
u(t) = -Kx(t) (4-43)
where K is a time-invariant feedback gain determined by solving an
algebraic Riccati equation.
The question answered here concerns the interpretation of applying
such a control to the linear servomechanism system. The reason that an
interpretation is considered necessary is that the optimal control is


56
given by (4-21) is exponentially stable. This completes the proof.
In order to interpret Theorem 4.2 we need certain continuity
"fe *Hp
conditions to hold. That is, F (t) and G (t) should be continuous
functions of the reference r (t) and the disturbance w (t). Then
assumptions (i) and (ii) are realistic since both iiF (t) Fn^ and
iiG (t) G II.. will be small whenever nr (t)ii and iiw (t) it are
small. Furthermore, because F (t) and G (t) are often periodic due to
'ic "fc
the periodicity of x (t), u (t), and w (t), and will be,
* o
respectively, the maximum values that nF (t) F n.. and
* o
nG (t) G n.. assume over one period.
As a final point, note that if in condition (iii) of Theorem 4.2
the word "stabi.1 izable" is replaced by "controllable" then the feedback
gains Kj and l<2 can be selected to arbitrarily assign the eigenvalues of
the system (4-21). This may, in turn, make it possible to obtain a
large ratio of a/m with suitably chosen feedback gains. Then, provided
that nK^, Kg]ii- does not become too large, L will increase and hence
larger reference and disturbance signals will be allowed. Note also
that if the input enters into the nonlinear system by a linear time-
invariant mapping, will be zero so that increasing the ratio a/m will
always increase L.
We now give a rather lengthy example which makes use of many of the
results obtained so far for the nonlinear servomechanism problem. In
this example, simulated test results are provided to show the per
formance of the control algorithm. Also, simulations are provided which
show the consequence of using a controller based on linear servo
mechanism theory.


72
is exponentially stable, tracking and disturbance rejection will occur
for small reference and disturbance signals. It is also true that in
certain cases, increasing the degree of stability of this system will
allow for a larger range of reference and disturbance signals. To meet
this condition using optimal control theory, a well known technique due
to Anderson and Moore [16] can be used.
Consider the linear system
x(t) = Fx(t) + Gu(t) (4-54)
where the pair (F,G) is completely controllable. Equation (4-54) could
model the linearized servomechanism equations.
Consider also the exponentially weighted performance index
j = J e2oit[x'(t)Qx(t) + u* (t)Ru(t)]dt (4-55)
01 o
where Q > 0 and R > 0 are symmetric matrices with (F1,Q)
stabilizable. It can be shown that minimizing (4-55) with respect to
the system (4-54) results in a feedback law u(t) = -Kx(t). Furthermore,
the degree of stability of the closed-loop system is increased relative
to using a performance index without exponential weighting.
The problem is actually easier to solve by defining a new system
given as
x(t) = FS(t) + GQ(t)
(4-56)
where
? = F + al ,G = G
(4-57)


35
corresponding only to the modes present in the reference and disturbance
signals. From the results derived here, it is obvious that this type of
approach may not be adequate. In fact, tracking error will always occur
when modes which are required to be present in the input for tracking
are not incorporated into the dynamics of the internal model system.
Hence, our idea is to incorporate enough modes into the internal model
system's dynamics to insure that the tracking error is indeed small.
These modes, if sinusoidal, could actually be sinusoids at frequencies
which are harmonics or subharmonics of the frequencies found in the
reference and disturbance signals.
Once the formulation for the internal model system was complete,
the controller design was given. In this design, state feedback was
used and the internal model system was incorporated into the feedback
loop. It was shown that a necessary condition for a solution to the
servomechanism problem (for arbitrary K^) was observability of the
internal model system's state through its feedback gain matrix. This
was a key requirement which has not been postulated for the linear
servomechamism problem, but was needed here due to the different
approach used in solving the nonlinear servomechansim problem. Later it
was indicated that the observability condition would also be required
for stability of the closed-loop system. Consequently, it is enough to
consider only the stability problem since the observability condition is
satisfied automatically whenever stability is achieved.
The stability requirement for the nonlinear servomechanism problem
was imposed upon a dynamic system which modeled the difference between
the actual state trajectory and the desired state trajectory of the
closed-loop system. Thus, the dynamical model was referred to as the


28
where
F*(t)
3f(x,u,w)
ax
*
x (t)
k
U (t)
*
W (t)
(2-41)
and
G*(t)
af(x,u,w)
au
x*(t)
u*(t)
w*(t)
(2-42)
'fc ^
Notice that the Jacobian matrices F (t) and G (t) are evaluated along
the trajectory which gives tracking of r*(t) with disturbance w*(t).
Often this trajectory is not known in advance, however, we shall defer a
more detailed discussion of this problem until a later chapter.
Let us make the following definitions:
xA(t)
x(t)'
'n(t)
(2-43)
FJ(t)
f^U, xA)
F*(t) -G*(t)Kx -G*(t)K2
-BH A
ff(x*(t)+x, u*(t)+u, w*(t)) f(x*(t), u*(t), w*(t))
(2-44)
(2-45)


20
A = T^ block diag. [C, C, ..., C] T
p btocks
then for some n* e Rpr, z(t) can be generated as follows
(2-31)
z(t) = K2n (t)
n*(t) = An*(t) n*(0) = n*
(2-32)
Furthermore, the initial state nQ is unique for any given z(t).
Proof: By the conditions given in Proposition 2.2 each of the p
components of z(t) must satisfy (2-30) and hence the entire vector z(t)
can be expressed uniquely in terms of a pr dimensional initial condition
vector. By assumption, this vector can lie anywhere in pr dimensional
space. From the definition of the A matrix, each element of the vector
*
K2n (t) must also satisfy (2-30). Hence, if we show the initial
k
condition vector representing (t) can be made to lie anywhere in
k
pr dimensional space by appropriate choice of the initial state nQ,
the proof will be complete. Such an initial state can be shown to exist
k
by noting that any state nQ can be observed through the output
k
K2n (t). Consequently, a linearly independent set of initial states
k
must result in a linearly independent set of outputs K2n (t). Thus,
*
since nQ spans pr dimensional space, the initial condition vector
k
defining K2n (t) will span pr dimensional space.
To prove uniqueness, let n* be another initial state such that
z(t) = K2n (t) (t) = An (t) n (0) = nQ
(2-33)


50
reference and disturbance signals and these signals (especially the
disturbance) are usually not known in advance.
At this point we consider one method of dealing with the stability
problem which will be applicable when the reference and disturbance
signals are small. This assumption, although restrictive, is necessary
to show stability of the system NCT when the feedback gains are kept
constant. We first give what is known as the Poincare-Liapunov theorem
[18].
Theorem 4.1: Consider the system
x = Fx(t)x + fx(t,x) x(tQ) = xQ (4-6)
where
fi(t, 0) = 0 (4-7)
Assume that the following conditions are also satisfied
(1) F^(t) is such that the system
x = F^(t)x (4-8)
is exponentially asymptotically stable for the equilibrium point x =
0. In otherwords, the state transition matrix $(t, t0) associated with
(4-8) is such that
il$(t, tQ)II^ < me-3^"^ (4-9)
for some positive constants m and a.


62
(a) Design based on nonlinear servomechanism theory
3*
nj
(b) Design based on linear servomechanism theory
Figure 4-1. Tracking error: reference = 2sin(t), no disturbance