• TABLE OF CONTENTS
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 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Tracking and disturbance rejection...
 Application to linear systems
 Feedback control
 Practical considerations
 Application to the robotic...
 Simulation results
 Conclusions and open problems
 Reference
 Biographical sketch
 Copyright






Title: Tracking and disturbance rejection for nonlinear systems with applications to robotic manipulators
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00082430/00001
 Material Information
Title: Tracking and disturbance rejection for nonlinear systems with applications to robotic manipulators
Physical Description: v, 157 leaves : ill. ; 28 cm.
Language: English
Creator: Whitehead, Michael L., 1958- ( Dissertant )
Kamen, Edward W. ( Thesis advisor )
Lasiecka, Irena ( Reviewer )
Peebles, Peyton Z. ( Reviewer )
Sandor, George N. ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 1986
Copyright Date: 1986
 Subjects
Subject: Manipulators (Mechanism)   ( lcsh )
Servomechanisms   ( lcsh )
Feedback control systems   ( lcsh )
System analysis   ( lcsh )
Nonlinear theories   ( 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.
Statement of Responsibility: by Michael L. Whitehead.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082430
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000938559
oclc - 16568635
notis - AEP9804

Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    Abstract
        Page v
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
    Tracking and disturbance rejection for nonlinear systems
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
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        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
    Application to linear systems
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
    Feedback control
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
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        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
    Practical considerations
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
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        Page 87
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        Page 91
        Page 92
        Page 93
    Application to the robotic manipulator
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
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        Page 107
        Page 108
    Simulation results
        Page 109
        Page 110
        Page 111
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        Page 150
        Page 151
        Page 152
    Conclusions and open problems
        Page 153
        Page 154
    Reference
        Page 155
        Page 156
        Page 157
    Biographical sketch
        Page 158
        Page 159
        Page 160
    Copyright
        Copyright
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 2-

link 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 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 proceeding 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 recommended 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 system 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 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









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.














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

lAi will be the induced norm defined as
1

1/2
IlAII := sup nAxl = C[max (A'A)] (2-1)
II X1=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: 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









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

representative 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*(0) = x such that


S*(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- (r-1) + ... + a1.(1) + ao() = 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


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(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 initial 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(t) := u
1


0 < t< T

otherwise


(2-6)


Then let


u (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

00
Up(t) = a + E ckcos(kwt + k) (2-9)
k=1


Xp(t) = f(xp(t), Up(t), Wp(t))








where w is the fundamental frequency.

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


K
uK(t) = o + E kcos(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 Ilx (t) xK(t)i 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 lx (t) xK(t)l is for a given value of K. Also note that in
practice there is always some error, so that demanding

lx (t) xK(t)li = 0 is not reasonable.

We now mention an important practical point which was overlooked in

the proceeding discussion. For Ilx (t) xK(t)il 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 + E akcos(kit + k)
k=1


(2-12)


Taking Laplace transforms of both sides we get


UK(s) =


N(s)

s[ I (s2 + K2w)]


(2-13)


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


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


(2-14)


Next, by writing


K J
s[ (s+ k2)] = E as3 J = 2K + 1
k=1 j=0



SE a.s3] UK(S) = N(s)
j=0


(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


J dj
Sa. -- uK(t) = 0 t > 0 (2-18)
j=0 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


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) = alsinwt and the disturbance is

constant. Thus


x (t) = alsinwt

w*(t) = M2 (2-20)

where al and a2 are constants. Substituting (2-20) into (2-19) yields

the following

12 2 *
al cost = 3ausinwt + 1 (1-cos2wt) + [a (l-cos2wt) + 4]a2 + u (t)

(2-21)









Solving for u*(t) gives


u*(t) = aloCOSmt 3alsinwt + a,2( + a2)cos2wt a12(- + 2) 4a2

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


(s2 + 4w2)(s2+w2)(s) = s5 + 5w2s3 + 4w4s (2-23)


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


5 2 d3 4 d
(.) + 5w --3 (.) + 4wm (.) = 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 e Rrxr and T e Rr be defined as follows


0 1 0 . 0 0
0 0 1 . 0 0

C := T := (2-25)

0 0 0 .. 1 0
-aO -al -a2 . -ar- 1


with the coefficients aj, j = 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. [C, C, ..., C] T (2-27)

p blocks


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

p blocks


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.








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))

A(t) = An(t) + BH[x*(t) -x(t)]
(2-29)
u(t) = -Klx(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) + ar-l (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

e RPxpr and A e 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 E Rpr, z(t) can be generated as follows


z(t) = K2 (t)


S(t) = An (t) n (0) = no (2-32)


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 no 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 no spans pr dimensional space, the initial condition vector
defining K2n (t) will span pr dimensional space.
1
To prove uniqueness, let no be another initial state such that


z(t) = K2n(t) (t) = An (t) n (0) = no (2-33)








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


1 .
and the vector [n n ] 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.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(0) = xo and n(0) = no 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, 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) = -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 Be 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*(0) = 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*(0) = no 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) 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(0), n(O)] differs
*
from [x no], the resulting state trajectory [x(t), n(t)] may not
*
converge to [x (t), n (t)] as t + m. To achieve (asymptotic)

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.








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)
(2-37)
W(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 [x (t), n (t)] is the trajectory which
*
gives e(t) = 0, t > 0 and results from the initial state [x n ].

Since it is our goal to have the trajectory [x(t), n(t)] converge,
*
eventually, to the trajectory [x (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)+x(t), u*(t)+u(t), w*(t)) f(x*(t), u*(t), w*(t))

f(t) = Ai(t) BHx(t) (2-38)

u(t) = -Kl^(t) K2"(t)









It is seen that the system NCT has an equilibrium point at

W(t) = '(t) = 0


Theorem 2.2: Suppose that the hypotheses of Theorem 2.1 are satisfied

and that for some choice of K1 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 '(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

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 accommodate 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


P1AP = 1 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 tket 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 D(t,O) are made up of modes of
the form t e 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 '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 '(t) depends on '(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 2'(t) = 0,

?(t) = 0 is


(t) F*(t) G*(t)K1 -G*(t)K2 (t)240)
S(2-40)
t -BH A ?(t)









where



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



and



G(t) = af(x,u,w)
Gu x = x (t)
u 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:


"X(t)
xA(t) (2-43)



SF*(t) -G* (t)K1 -G*(t)K2
FA(t) := B A (2-44)
-BH A

f(x (t)+x, u (t)+u, w*(t)) f(x*(t), u*(t), w*(t))
A? BH'

(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 sup fA(t, x A)l
11m sup ) = 0 (2-47)
IlxAI+O t>0 IlxA



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









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 Rqrxqr, B e Rqrxp and K2 e 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 e Rqrxqr defined by

(2-49), B e Rqrxp arbitrary, and K2 e 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'[XI 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'[EI C] = 0



Now define a matrix Q e Rqxqr to be the following


wQ
WI
Q 0


. O]
0.


It can be readily seen that


Q [XI A] = 0


Now let D e 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
correspondence with the eigenvalues of the C matrices which make 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:

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) = -Kp,1x(t) Kp,2n(t) (2-57)


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


u(t) = -Klx(t) K2n(t) (2-58)
where


K1 = 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, assump-

tion (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 assump-

tion, assumption (A.2). Here, the requirement was made that the input

and state trajectories which occurred 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 necessary 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) + r (r1) + **. + y(.)(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 < 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.









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

(B.2) rank = n + p i=1,2,...,F
-H 0

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) = x 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 Rmxn and u(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.

1* -1
X*(s) = [sI-F] x + [sI-F]-U(s) + [sI-F]- EW (s) (3-4)
and

R*(s) = H[sI-F]Ix + H[sI-FI GU(s) + H[sI-FJ EW*(s) (3-5)


where

F:= F GK (3-6)


Now consider the following conditions


(1) m > p
(2) rank [ xiI-F] = n for all xi i = 1,2, ...,
(3) rank H[AiI-F]G = p for all i 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), 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.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 Xi, 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



irn F G a Xi F + GK G
rank = rank (3-7)
-H 0 0 H[Eil F + GK]-1G

This is accomplished by premultiplying and postmultiplying the left-hand

side of (3-7) by



nxn and by xn 0
H[XI F + GK]-1 Ipxp nd K 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, ..., F. 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

R s)1 a + 1 b + 1 c (3-8)
(s-x1) 2 c (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

hold true, tracking will occur for some initial state xo and input

U(s). The input U(s) can be chosen as


1 1 1
(s) w + 2 v + r (3-10)
1 (s-X1) s-2

where w, v, r e Rm are vectors given by


v = {H[1 I-F-1G}- b

w = {H[X1I-F]-1G} {a + H[X1I-]2Gv} (3-11)

r = {H[X2I-FT-G}-1c


The initial state xo should be taken as


x= -{ [F-xllI Gw + [F-Xll] Gv + [F-x21] Gr } (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 H[EI F]-1 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


sI-F-1 N-- = { (-1)j-l[cI-l-j 1
(s-X) j=1 (s-X) -

+ { [sI-F]-1 [-I]-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* + [T-XlI'Gw + [F-XlI-2 Gv + [F-X21]-Gr }

1 1 1
+ HEX1 I-rF]-Gws1 + HCx II-F]-Gv 1 2 H[Xl I-F]-2Gv 1
S1(s-x1)


+ HE I-F]" Gr 1 (3-14)
2 (s-X2)


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) = [xII-F1 GwT- + X1I--Gv 1
Xl11 ll' (s-x1r
(S-X)2 (3-15)

-2 1 [ 2 1 1
[Ax1-F] Gv1- 1 + 2I-F] Gr1s-
1 (s-X1j) 2 X2








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


(3) + [-x2-2~] ()(2) + [21X2 + X2 () (1) + [- 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:


F(t) F 0 o (t) G1
+ G(t)
S-BH A (t)


W(t) = -K1 (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 i., i = 1, 2, ..., F given by condition (B.2)








then the pair

F 0 G
(3-18)
-BH A O


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)
= 0 (3-20)
BH XI A v









This means that x is also an eigenvalue of the matrix


F GK1 -GK2

-BH A


Since the system LCT can be written as


x(t F GK1 -GK2 (t)
1 2(3-21)
(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.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)+(t), u*(t)+Z(t), w*(t)) f(x*(t), u*(t), w*(t))

n(t) = A (t) BHx(t)
'i(t) = -Kl (t) K2r(t) (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 x:(t)
XA(t) := (4-2)
XA M (t








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

approximation to NCT as


XA(t) FA(t)xA(t) (4-3)
where



FAt) F*(t) G*(t)K1 -G*(t)K2
FA(t) := (4-4)
-BH A

and




F*(t = af(x,u,w) *G(t)= f(xuw)
F(t ax x = x (t) u x = x (t)
u = (t) u = u(t) (4-5
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 compli-
cations. 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


S= 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


S= 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


lla(t, t )ll < me-a(t-to) (4-9)


for some positive constants m and a.









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


sup lifl(t,x)l < LIIxIl L > 0
t>t
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,t )xo + fJ (t, T)fl(T, x(T))dT
t


(4-12)


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

follows that


Ilx(t) i < me-a(t-to)l1x o + mea(t-)L Ilx(T) idT


(4-13)


Multiplying through by eat gives


at t
eatllx(t)ll < me llx ll + mL f eaT lIx(T)IldT
t
0


(4-14)


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









eat ix(t) II


< me


nx(t)n < m 1:


ato mL(t-t )



(mL-a)(t-t )
xo lie


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


RA(t) F (xA(t) + [FA(t) FA]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 GK1

-BH


-GoK2

A


(4-19)


GO = af(x,u,w)
au


x= 0
u= 0
w= 0


(4-20)









We see that FA is simply the matrix resulting from a linearization of

the system NCT about the origin. In addition, because the original
system is time-invariant Fo, Go, and FA 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.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 IF*(t) Fni =
t>O

(ii) sup iG (t) GI = 2
t>O

where e1 and e2 are positive constants.


(iii) The pair [Fo, Go] is stabilizable



(iv) rank -Fo = 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 FA take the role of

Fl(t) and [FA(t) FA]XA(t) take the role of fl(t,x). Condition (1)

of the Poincare-Liapunov theorem then requires exponential stability of

the system


XA = FAxA (4-21)


In order to meet this stability requirement, the pair


Fo 0 Go
S ) (4-22)
-BH A 0


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 o(t,t ) associated with (4-21) will satisfy the

inequality










llo(t,to )li < mea(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 IFA(t) Flli < L (4-24)
t>0

where L is small enough to insure that


(mL a) < 0 (4-25)


Equation (4-24) can be verified by using the definitions for

FA(t) and FA to obtain the relationship


** *0 *
llFA(t) FIli < IF (t) Fol + llG(t) Go .l[K1, K2]li (4-26)


Using (i) and (ii) then gives


sup llFA(t) FAl< + 211[K1, K2]li (4-27)
t>~

If el 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









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 iF (t) Fon and

nG (t) Golli will be small whenever IIr (t)I and ilw (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), CI and 62 will be,

respectively, the maximum values that IIF (t) Foll. and
o
IIG (t) G ll 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 IK1, K2 1]i 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, 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 per-

formance of the control algorithm. Also, simulations are provided which

show the consequence of using a controller based on linear servo-

mechanism theory.











Example 4.1

Consider again the system of Example 2.2


i(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 ia 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() () ) + 4t- (.) = 0
dt dt


(4-30)


The internal model system is thus


0
0
(t = 0
0
0


0 0
0 0
1 0 n(t)
0 1
-5 0


0
0
+ 0 e(t)
0
1


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=alsint


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


= (3+2x+4wx)
x=alsint

w=a:2


= 3+(2a1+4ala2)sint

(4-33)


x=alsint
w= a2


(4-34)


The linearization of NCT may be written as


xA(t)


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


1
0
0
0
-4


* -K2
0
1
0
0
0


0
0
1
0
-5


xA(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 Go = 1


(4-32)


(4-37)









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


sup IIF(t) FIi = (2a1 + 4a 12) =1 (4-38)
t>0
and

sup IG (t) GI = 0 = e2 (4-39)
t>0



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

have


em 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 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, c1 =1/2









and a2 = 1. Then, from (4-38), we get e1 = 3 so that elm-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 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.

































0 00-2-0-----0----0-- --- ---- 0-------------
0 I

i I I





Si i










Cu I
i i i I


n nn nn U i i nn 1i
nil U t~in if Rn' f ii I U mI


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


i.D


.l U


-.I


".


.


t.


."


AL


















-------------4------ 4--------I4-----I----- ---S-------------
i i i



1 .0 2 i i i









0 I I I

0.0D 2.00 ,.00 6.00 8.00 LD.0D 120.0 14.00 1C


TIME (SECONDS)


(a) Design based on nonlinear servomechanism theory


(b) Design


6.00 8.00 L. 00 12.00
TIME (SECONDS)


based on linear servomechanism theory


16.00


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


W
CI-



I-


i.00












. . . . . ; - - - -












C i I I

(1
0 ilI


2. 00


4.00


6.00 8.00
TIME (SECONDS)


LO. 00


12.00


1Q.oo


(a) Design based on nonlinear servomechanism theory


(b) Design based on linear servomechanism theory


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


.00


16.00


Figure 4-3.



































C. UU


4. UU


(a) Design based


8I


















tI i i i ,

o,-- i--i....-- .-i-. ._i.--- i- -


.uu 0.00 LD.DO 12.00 La
TIME (SECONDS)



on nonlinear servomechanism theory


(b) Design based on linear servomechanism theory






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


cr:
0
C)
-,J
Qc

IU
i-i
(-


. uu


qU.0


16.00


--








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


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:

(t 0 x(t) G E
= + u(t) + w(t) + r(t)
n(t) -BH A n(t) 00


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) 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) s 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 servo-

mechanism 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 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) = xo and n(0) = 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


- (t F 0 x (t) G oE 0
= + ss(t) + w(t) + r(t)
L (t) -BH A ns(t) 0 0


uss(t) = -K1Xss(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 F(t) G
i = H' + (t) (4-46)
(t) -BH A LE(t) 0
where


x(t) = x(t) xss(t)
(t) = n(t) nss(t) (4-47)
i(t) = u(t) uss(t) = -Kl1(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 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 := [K1, K2] is found

by solving the algebraic Riccati equation for optimal control. That is,

the positive semidefinite P satisfying


Q + PFA + FAP PGAR-1G P = 0 (4-48)


is obtained and K is selected as


K = R-1GP (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


F 0 G
FA -H GA = (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
























x(t)


Figure 4-5. Actual and steady-state trajectories










J = I [A(t)Q A(t) + u'(t)Ru(t)]dt (4-51)
0
where
x(t) x (t)
XA(t) := ss (4-52)
and n(t) n(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 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









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



0
J = e2tCx'(t)Qx(t) + u'(t)Ru(t)]dt (4-55)



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) = f^(t) + GQ(t) (4-56)

where


F = F + ca G = G


(4-57)









A new performance index is taken as


00
J= f [' (t)Q (t) + '(t)Rt(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 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.








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

y*(t) = Hx (t) (5-2)
S= (t) y (t)


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 assump-
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








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) F*(t)
(t) := w*(t) (t)
S*(t) := x*(t) *(t) (5-3)
u(t) := u*(t) u(t)


where ?*(t, t, )*(t),(t), and ^*(t) denote the deviations of the true
signals from the nominal signals. Note that because the system under
consideration is nonlinear, x (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


x(t) := x(t) x*(t)
(5-4)
(t) :t ) Ut) 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) = -Kl(t)R*(t) *(t) (5-5)


The above equation is a mere consequence of definition (5-3) and the
structure of the controller.
Although Kl(t) and K2(t) are shown to be functions of time, for the

present, assume that they are constant. Also, assume that 2 (t) and

d*(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, K2(t)) is observable, tracking will occur for some
initial state [x*(O), n (0)]. Consequently, we shall require that

W*(t) 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 C*(t) superimposed on some nominal
trajectory x (t). The trajectory x (t) is not restricted by assumption
(A.2).





79




















/ I
Sx (t)
x (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 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 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) n(t) (5-7)

and

FA) F*(t) G*(t)Kl(t) -G*(t)K2(t (5-8)
FA(t) := (5-8)
-BH A


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


XA(t) FA(t)xA(t) + [FA(t) FA(t)]xA(t) (5-10)









where

F*( (t) (t)Kl(t) -G(t)K2(t)
FA(t) := (5-11)
S-BH A

and





F*(t) Df(x,u,w) *(t) af(x,u,w)
ax x = (t) au x = (t)
(t -* (5-12)
u = *(t) u = u*(t)
w = w (t) w (t)

Now (see Theorem 4.1) if the system


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


is the exponentially asymptotically stable and sup ilF*(t) FA(t)ii. is
t>O

suitably small then (5-10) is exponentially asymptotically stable as

desired.

First consider conditions under which the quantity

sup llFA(t) FA(t)ii is sufficiently small. When the true reference
t>0

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 IIFA(t) -FA(t)i. is
t>0

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

necessary that the pair


F (t) 0 G*(t)

L -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 w (t) are themselves, slowly

varying.

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 -*
lII r-F(t)II 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.

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.









Robustness with Respect to Generation of the Nominal Signals

It was previously indicated that, in practice, the nominal signals

x (t) and u (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
-a
input actually supplied to the system is ua(t) while, as before, the

input needed to obtain the nominal trajectory is U (t). Now make the

definition



udt) = a(*(t) (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 ud(t) acting in the input channel so that, effectively,

G (t) ud(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 perspec-

tive it is evident that the method described in the previous section can
-d
still be applied by simply modeling u (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



















a(t)
a (t)


-dt)
. (t


(t)--ud(t)=Tia(t)


Modeling incorrect nominal inputs


Figure 5-3.









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


_d a (5-16)
x (t) = x (t) x(t) (5-16)


-a
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 3d(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)
-d
it is apparent that d (t) can be modeled as an input disturbance.
-d
Hence, we may conclude that the dynamics associated with x (t) must

also be included in the internal model system.

To summarize, we have shown that robustness with respect to the
_* -
open-loop signals x (t) and u (t) is obtained provided that any

deviations from these signals are successfully 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 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

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) + dr-ls(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, ...,] (5-20)


where






90



0 1 0 ... 0 0
0 0 1 ... 0 0
Cd T"[ ] (521)
Cd = : : : : ,T= : (5-21)
0 0 0 ... 1 06
-do -dl -do ... -dr-1 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 ca and a2 are constants. The sampled signals are


x (k) = al

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) = 2 2
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+2coswT)


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


0C =
Cd = 0
1


(5-27)










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


To be consistent 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) = -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) FA,d(k)xA(k) (5-30)


where _
whe Fd(k) Gd(k)K1(k) -Gd(k)K2(k)
FA,d(k) = (5-31)
-BdH Ad








-* -*
The discrete-time matrices Fd(k) and Gd(k) corresponding to the plant

are obtained [26] by the relationships

-*
-* F (kT)T
Fd(k) = e(k (5-32)

T -*
Gd(k) = [ fe (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 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.










J(6)'' + Tv(o,) = TA(t) + Tg(e) + Td(w(t),e) (6-1)


For a manipulator having N links, e(t), 6(t) and *'(t) are the vectors of
length N defining angular positions, velocities, and accelerations of
the actuator joints. The matrix J(e) RNxN 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 e (see
[24]) and is thus always invertible. The inertia torque vector
TV (,6) e 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)j we have the following


TV(e,6), = 9'PJ(e)9 (6-2)


where PJ(e) e RNxN is a purely configuration dependent matrix referred
to in [28] as the intertia power modeling matrix. The term TA(t) e 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(0)w(t) (6-3)


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




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