Citation
New results in indirect adaptive control

Material Information

Title:
New results in indirect adaptive control
Creator:
Ossman, Kathleen A. K., 1959- ( Dissertant )
Kamen, Edward W. ( Thesis advisor )
Bullock, Thomas E. ( Reviewer )
Hearn, Donald W. ( Reviewer )
Peebles, Peyton Z. ( Reviewer )
Svoronos, Spyros ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1986
Language:
English

Subjects

Subjects / Keywords:
Adaptive control ( jstor )
Automats ( jstor )
Estimators ( jstor )
Governing laws clause ( jstor )
Matrices ( jstor )
Perceptron convergence procedure ( jstor )
Point estimators ( jstor )
Preliminary estimates ( jstor )
Signals ( jstor )
State estimation ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis, Ph.D.
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
An adaptive regulator which does not require a persistently exciting input is derived for multi-input multi-output linear discrete- time systems. The assumptions made on the unknown plant are (1) an upper bound on the system order is known, (2) the system parameters belong to known bounded intervals, and (3) the plant is stabilizable for all possible values of the unknown system parameters ranging over the known intervals. A recursive parameter estimator is described which forces the estimates of the system parameters to converge to the known intervals asymptotically. Using this parameter estimator, an adaptive LQ regulator is developed which results in a globally stable adaptive closed-loop system in the sense that the system inputs and outputs converge to zero asymptotically. The results are then extended to the case of tracking and/or deterministic disturbance rejection using the internal model principle. Simulations illustrating the performance of the adaptive controller for several discrete-time systems are included.
Thesis:
Thesis (Ph.D.)--University of Florida, 1986.
Bibliography:
Includes bibliographic references (leaves 100-102).
General Note:
Vita.
Statement of Responsibility:
by Kathleen A. K. Ossman.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Kathleen A. K. Ossman. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
030575333 ( ALEPH )
17647544 ( OCLC )
AEW2312 ( NOTIS )

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Full Text









NEW RESULTS IN INDIRECT ADAPTIVE CONTROL


By

KATHLEEN A. K. OSSMAN














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 acknowledge the following people for their contributions to the completion of this dissertation. Special thanks go to my advisor, Dr. Edward W. Kamen, for his numerous helpful suggestions

and creative ideas. Thanks also go to the members of my supervisory committee: Dr. T. E. Bullock, Dr. D. W. Hearn, Dr. P. Z. Peebles, and Dr. S. Svoronos.

This work was supported in part by the U. S-. Army Research Office, Research Triangle Park, N.C., under Contract No. DAAG29-84-K-0081.















TABLE OF CONTENTS




ACKNOWLEDGEMENTS .

ABSTRACT . CHAPTERS

I INTRODUCTION .
II BACKGROUND . * . . . . . . . . . . . . . .

III SYSTEM DEFINITIONS ANO ASSUMPTIONS . IV PARAMETER ESTIMATION .

V ADAPTIVE REGULATOR. . oo. .

Uniform Stabilizability of the Estimated System .
Adaptive Observer. .
Feedback Gain Sequence . . . . .
Stability of the Adaptive Closed-Loop System.

VI APPLICATION TO TRACKING AND DISTURBANCE REJECTIONo. VII SIMULATIONS. VIII DISCUSSION .o. REFERENCES. . BIOGRAPHICAL SKETCH .o. o. . . . . . .


ii iv



1
4

12 18 36 36 41 44 55 61 66 96 100

103

















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

NEW RESULTS IN INDIRECT ADAPTIVE CONTROL

By

KATHLEEN A. K. OSSMAN

August 1986

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

An adaptive regulator which does not require a persistently

exciting input is derived for multi-input multi-output linear discretetime systems. The assumptions made on the unknown plant are (1) an upper bound on the system order is known, (2) the system parameters belong to known bounded intervals, and (3) the plant is stabilizable for

all possible values of the unknown system parameters ranging over the known intervals. A recursive parameter estimator is described which forces the estimates of the system parameters to converge to the known intervals asymptotically. Using this parameter estimator, an adaptive LQ regulator is developed which results in a globally stable adaptive closed-loop system in the sense that the system inputs and outputs converge to zero asymptotically. The results are then extended to the case of tracking and/or deterministic disturbance rejection using the internal model principle. Simulations illustrating the performance of the adaptive controller for several discrete-time systems are included.















CHAPTER I
INTRODUCTION

Adaptive control, the problem, of controlling a system whose parameters are unknown prior to or changing during system operation, has been a major research topic during the past three decades. Although

originally intended for time-varying. or nonlinear systems, most of the stability results on adaptive controllers have been limited to linear time-invariant systems. The majority of adaptive controllers can be classified as either direct or indirect.

In direct adaptive control, no attempt is made to estimate the unknown system parameters. Instead, the controller parameters are updated directly using real time input/output information. One specific example of direct adaptive control is model reference adaptive control (MRAC) where the unknown plant is forced to behave asymptotically like some pre-chosen reference model. Several individuals have developed

this approach, see for example [1-3]. The assumptions needed to prove global stability of model reference adaptive controllers are (1) the

unknown plant is minimum phase, (2) the relative degree of the plant is known and (3) the sign of the plant gain is known.

Indirect adaptive controllers are applicable to nonminimum phase as well as minimum phase systems. In indirect adaptive control, the

unknown system parameters are estimated in real time using available input/output information then the control law is computed using the most

recent parameter estimates. A number of individuals have introduced indirect adaptive controllers, see for example [4-12]. The assumptions









used to prove global stability of indirect adaptive controllers are (1)

an upper bound on system order is known, (2) the estimated system is uniformly stabilizable, and (3) the parameter estimator possesses certain key properties. Since the adaptive controller developed in this

dissertation can be classified as indirect, each of these assumptions will be discussed in more detail.

The importance of the first assumption was illustrated in [13-141 by Rohrs and associates who investigated the behavior of adaptive controllers in the presence of unmodeled dynamics. Rohrs demonstrated through computer simulations that' all existing adaptive controllers could go unstable if the order of the system was underestimated. In

response to Rohrs' findings, several individuals [15-19] have achieved promising results in the development of robust adaptive controllers.

Verification of the second assumption, which requires the existence

of a stabilizing feedback control law for the estimated system, is difficult because it depends on. the parameter estimates which are generated in real time. Unless something more can be said about where the parameter estimates are going, the assumption cannot be verified a priori and, consequently, global stability cannot be ensured. The first approach towards satisfying this assumption was the use of a persistently exciting external input which allows perfect identification of the plant. This approach along with more recent results which do not

require persistent excitation are discussed in more detail in Chapter .Ii.

The third assumption requires that the parameter estimator possess certain key properties which are explained in detail in Chapter II. It will suffice to mention the existence of several parameter estimation









algorithms, .such as the recursive least-squares, projection, and orthogonalized projection algorithins discussed in �6], which do indeed have the required properties.

In this dissertation, an indirect adaptive regulator which does not

require a persistently exciting input is derived for multi-input multioutput discrete-time systems. In Chapter II, more explicit background material on parameter estimators and indirect adaptive controllers is presented. Chapter III contains the system definitions and the assumptions which include (1) an upper bound on system order is known,

(2) the unknown system parameters belong to known bounded intervals, and

(3) the plant is stabilizable for all possible values of the unknown parameters ranging over the known intervals. A parameter estimator is

derived in Chapter IV which has the required properties for proving global stability of indirect adaptil/e controllers and, in addition, forces the estimates of the parameters to converge to the known bounded intervals. In Chapter V, it is first shown that the assumptions listed in Chapter II ensure the parameter estimator described in Chapter IV will generate a uniformly stabilizable estimated system. Using this

parameter estimator, an adaptive LQ regulator is developed which, when applied to the unknown plant, results in a globally stable closed-loop system in the sense that the system inputs and outputs converge to zero asymptotically. The results are then extended in Chapter VI to the case of tracking and/or deterministic disturbance rejection using the internal model principle �20]. Chapter V.II includes simulations of the adaptive LQ controller for both SISO and MIMO discrete-time systems. A

discussion of the results and further work to be investigated is included in Chapter VII.















CHAPTER II
BACKGROUND

In indirect adaptive control, the system parameters are estimated in real time using available input/output information. The control law

is then computed for the estimated system using the most recent parameter estimates. A well-known problem in proving global stability of indirect adaptive control algorithms is the estimated system may not be controllable or even stabilizable for certain parameter estimates which are referred to as singular points. Global stability of the indirect

adaptive controller can only be ensured if there is some finite point in

time after which the parameter estimates are not arbitrarily close to a singular point. An example of a parameter estimator and an adaptive regulator will further illustrate the problem of singular points in indirect adaptive control.

Consider the SISO discrete-time system described by the input/output difference equation:


q p
y(k) =jll-ajy(k-j) + I b.u(k-j). (2-1)
j=1

In (2-1), y(k) is the system output and u(k) is the control input. It

is assumed that an upper bound, n, on q and p is known but some or all of the system parameters aj and b. are unknown. The system described by

(2-1) can be rewritten in the following form which is convenient for parameter estimation:

y(k) = eT ,(kl)









where
T -a . -an b1 . bn] (2-2)


T (k-1) = [y(k-1) . y(k-n) u(k-1) .u(k-n)].



The vector e consists of all the system parameters and *(k-l) is a regression vector of past inputs and outputs. Throughout the following discussion, u � o will designate the vector Euclidean norm defined by IlxH = (xTx)

Of all the parameter estimation schemes discussed in [6], the least-squares algorithm has the fastest convergence rate and is most easily modified to handle output disturbances and slowly time-varying systems. The algorithm results from minimizing the quadratic cost function:

N
JNO) = 1/2 1 +YR- V-Ie 2 2(0-6(0))Tp-(0) (6-6(O))o
k=1


The cost function consists of the sum of the squares of the prediction errors plus an additional term which takes initial conditions into account. The matrix P(O) can be interpreted as a measure of confidence in the initial parameter estimate e(O). Designating e(k) as the estimate of the system parameters at time k, the least-squares algorithm is described by


e(k) = e(k-1) + TP(k-1)0(k-1) y(k) - eT(k-1) (k-1)]
1 + * (k-1)P(k-1) (k-1)

P(k) = P(k-1) - P(k-1);(k-l) T(k'l)P(k-1)-, P(O) = pT(0) > 0. (2-3)
1 + T (k-1)P(k-1).(k-1)









The key properties of the least-squares scheme used for proving global stability of an indirect adaptive controller are



i. uiO(k)u
i i. iio(k)-e(k-1)u a 0 as k +


iii. lim e2(k) = 0
k+- 1 + ,T(k-1)P(k-1),(k-1)

where e(k) = y(k) - oT(k-I) (k-1).

The error e(k) is referred to as the prediction error since it is simply the difference between the actual output at time k and the predicted output at time k using the most recent parameter estimate. These properties do not depend on the type of control input chosen or on the boundedness of the system input and output. The least-squares algorithm

also has the property that the parameter estimates converge, although not necessarily to the true values. This property is not included among the key properties because it is not necessary for stability proofs.

An example of indirect adaptive control in the SISO case is Samson and Fuchs' LQ controller discussed in C12] and briefly outlined here. A

state-space observer realization of the system described by (2-1) is given by

x(k+l) = Fx(k) + Gu(k) (2-4)


y(k) = Hx(k)








where


-a, b
-a2 1b2

F . G . H= [1 0 . 0].

-anI 1 bn-I
-an 0 bn


Samson and Fuchs assume that (F,G) is stabilizable. This assumption
allows for nonminimum phase systems and systems with stable common poles and zeros. The adaptive LQ control law used by Samson and Fuchs [12] is

given by


u(k) = -L(k) (k), (2-5)


where u(k) is the input to the given plant, x(k) is the state estimate,

and L(k) is a stabilizing feedback gain for the estimated system (F(k), G(k)).

The state estimate (k) is generated from an adaptive observer. Letting F(k) and G(k) represent the current estimates for matrices F and G, the adaptive observer for the system is given by


X(k+l) = F(k)2(k) + G(k)u(k) + M(k) (y(k)-9(k)) (2-6)


AA
y(k) = Hx(k)









where

-al(k)
M(k) = . , ai(k) = estimate of ai at time k.



-an(k)

The feedback gain sequence is computed at each time point k by solving one step of a Riccati difference equation:


R(k+l) = Q + LT(k)L(k) + (F(k) - G(k)L(k))TR(k)(F(k) - G(k)L(k))


L(k) = [GT(k)R(k)G(k) + I-1 GT(k)R(k)F(k) (2-7)


In (2-7), Q and the initial value R(O) are arbitrary positive definite symmetric matrices.

In order to prove global stability of the closed-loop system, Samson and Fuchs make two assumptions in addition to the stabilizability assumption on (F,G) mentioned previously. First, the parameter estimator must possess the following three properties:


i. ue(k)o < M < - for all k

ii. ne(k) - e(k-m)n 0 as k + for any finite m iii. Ie(k) 4 c(k)ni@(k-l)ll + 8(k)

where a(k) and a(k) converge to zero.









An example of a parameter estimator possessing these properties is the

least-squares estimator discussed previously. Clearly, properties (i) and (ii) listed above are equivalent to properties (i) and (ii) given for the least-squares algorithm. It is not quite as obvious that Samson

and Fuchs' property (iii) is equivalent to property (iii) of the leastsquares algorithm. The proof is given in Chapter IV.

The second assumption for proving global stability is that the estimated system (F(k),G(k)) must be uniformly stabilizable.


Definition: The system (F(k),G(k)) is uniformly stabilizable if there exists an integer r > 1, a constant q and a uniformly bounded sequence L(k) such that


t+r-1
nII (F(k)-G(k)L(k))u1 where


t+r-1
I (F(k)-G(k)L(k)) = (F(t+r) - G(t+r)L(t+r))x
k=t

(F(t+r-1) - G(t+r-1)L(t+r-1)) . (F(t) - G(t)L(t)).



Theorem: Assuming the system (F,G) is stabilizable, the parameter estimator possesses the three required properties, and the estimated

system (F(k),G(k)) is uniformly stabilizable, the control law u(k) =

-L(k)^(k) described by (2-5)-(2-7) will result in a globally stable closed-loop system in the sense that the system input and output

converge to zero for any initial states in the plant and observer (Samson and Fuchs C12]).









In order to prove global stability, Samson and Fuchs must assume the estimated system (F(k),G(k)) is uniformly stabilizable. Unfortunately, this stabilizability condition cannot be checked a priori which

means global stability of the adaptive regulator cannot be guaranteed. If the least-squares estimator is used, the assumptio n that (F(k),G(k)) is uniformly stabilizable is equivalent to requiring that (F.,G.) is

stabilizable where F~ irn FMk and G. = lim GMk.

As previously mentioned, the problem of avoiding singular limit points is common to all indirect adaptive controllers. Until the recent work of Lozano-Leal and Goodwin �21], De Larminat �22] and Kreisselileier �23), the only way to avoid a singular limit point was to require a persistently exciting input which would force the parameter estimates to

converge 'to the true system parameters. This approach has been developed by a number of individuals, see for example [4-9]. The idea is to use an external input with many different frequencies which enriches the input/output information allowing perfect identification of the system parameters. As discussed in [15-16], indirect adaptive controllers which require a persistently exciting input are not robust. Since the

adaptive closed-loop system is inherently time-varying and nonlinear, any uncertainty in the plant could counteract the exciting input. The

problem of choosing a persistently exciting input in the presence of unmodeled plant dynamics has. not been completely resolved.

Lozano-Leal and Goodwin �21] developed an estimation scheme which gives nonsingular parameter estimates at each point in time and in the limit for SISO linear discrete-time systems. They modify (when necessary) the parameter estimates generated from the least-squares estimation algorithm with data normalization. This modification allows them









to prove global stability of an adaptive pole placement scheme without requiring a persistently exciting input. It is, however, possible for

the parameter estimates to converge to a point that is near a singular point, in which case the controller gains may be large. Also, a MIMO version of the algorithm is not availa ble at the present time.

De Larminat [22] has also proposed a parameter estimation scheme that does not yield singular points. He assumes a priori knowledge of a

space G which contains the actual system parameters and is devoid of singular points. He then introduces a modification to the estimates obtained from the standard least-squares algorithm. The modification is

only required for a finite period of time and produces estimates which belong to G for all time. Although his modification prevents convergence near a singular point, De Larminat's recursive procedure is not as explicit as the one proposed by Lozano-Leal and Goodwin [21].

As in the case of De Larminat, Kreisselmeier �23] assumes prior information on the system parameters is available. Specifically, he assumes that the components a. of the system parameter vector e lie within a known bounded interval [,min, mrax, liHe then modifies the
1 1
identification scheme for SISO continuous-time systems to force the parameter estimate of e. to converge to the set Cmm max formerr i. Kreisselmeier also assumes that for each e with e0 C min 0rnax I

there is a feedback control system with a prescribed degree of stability. This condition can be checked a priori since it does not involve the parameter estimates. With these assumptions, Kreisselmeier proves global stability of the SISO continuous-time adaptive controller.














CHAPTER III
SYSTEM DEFINITIONS AND ASSUMPTIONS
The system to be regulated is the multi-input multi-output linear discrete-time system described by



y(k) = - A.Y(k-j) + B.u(k-j) (3-1)
j=1 3 j=1 J

In (3-1), y(k) is the mx1 output vector and u(k) is the rxl control input vector. It is assumed that an upper bound, n, on p and q is known but all or some of the entries in the matrices A. and B. are unknown. The system described by (3-1) can be rewritten in the following form convenient for parameter estimation:


y(k) = pT4(k-l) (3-2)

where
pT = [ A, An B1 , . Bn]

T(k-1) = [yT(k-1) . yT(k-n) uT(k-1) . uT(k-n)]



The n(m+r)xm matrix P consists of all the system parameters and the n(m+r)xl vector *(k-l) is a regression vector of past inputs and outputs. In certain applications, some of the entries in the system matrix P will be known a priori. Since it is not necessary to estimate known parameters, a scheme for separating the known parameters from the unknown parameters is advantageous. This is accomplished by rewriting (3-2) in the following form:











y(k) = T a(k-1) + T(k-l) (3-3)


The Nxm matrix e contains all of the unknown entries in P while the matrix p contains only those entries of P which are known a priori. The vectors a (k-1) and *b(k-1) are regression vectors whose components come from (k-l). Clearly, the decomposition described in (3-3) is not unique for multi-input multi-output systems. In order to minimize the

required computations for parameter estimation, the size of a (k-1)

should be made as small as possible. An example will serve to

illustrate these concepts.

Example 1: Consider the linear discrete-time system with two outputs

and a single input described by



=k' yl(k) 1 a] y(k-1) + 3 u(k-)
Y2(k) - 2 y2(k-1) i


The parameter a is assumed to be unknown. The system can be rewritten

in the form


y(k) = PT(kR-1)

where

1 a 3-yl(k-1) 1 PT= (k-l) = yg(k-i)

2 [ u(k-i)


A decomposition for this system which minimizes the required computations for parameter estimation is given by










y(k) = Y2(k-1) + 2(k-)


OT a a(k1) : Y2(k-1) T 1 ] b(k-l) = (k-1).


Again, consider the linear discrete-time system described by (3-3). It is assumed that each component eij of the unknown system parameter matrix o belongs to a known bounded interval L m 'in max

This assumption is reasonable in those applications where some a priori information is available on the system. Clearly, the unknown system

parameter matrix e belongs to a subspace of ]RNxm which is defined by the known bounded intervals. In order to use the necessary concepts of boundedness, compactness and convergence, a suitable norm must be defined on IRNxm. The vector norm which will be used throughout this dissertation is the Euclidean norm defined by



Uxii : (xTx) 1/2


The matrix norm which is induced by the vector Euclidean norm is given

by


T 12
IMn = Ex (MTM)]


TT
where max (M M) is the eigenvalue of MTM with the greatest

magnitude. By the above assumption on the entries of e, the unknown system parameter matrix e belongs to a known compact subspace n of ]RNxm given by








Nxm min max
:O R Oijceoij ,6ijJ}


The concept of compactness (i.e., closed and bounded) is relative to the matrix norm previously defined.

The only additional assumption made on the system is a pointwise stabilizability assumption over 0. Specifically, the system described by (3-3) is assumed to be stabilizable for each eE0. This assumption can be verified a priori because it depends only on the known set a, not on the parameter estimates. The stabilizability condition can be tested using the transfer function matrix of the system. Let D(z-1) and N(z") denote the polynomial matrices defined by


n n
O(z"1) = I + I A.(z'J), N(z'1) : B(z-J).
j=1 j l

The coefficients of D(z-1) and N(z"I) depend on the system parameters and therefore can be viewed as functions of o. This dependancy is made explicit by denoting D(z-1) and N(z"1) as D(z-l,O) and N(z-1,o) respectively. The transfer function matrix of the system described by (3-1) is then given by


W(z1,6) = D-I(z-1,e)N(z-I,e).


The stability assumption is that the system described by (3-1) with transfer function matrix W(z"1, o) can be stabilized by dynamic output feedback for all oca. From well known results [24], this assumption is equivalent to the following rank condition:









rank [D(z,e) N(z,e)] = m for Izi < 1 and all eeo.


where m is the number of system outputs and


n n
D(z,e) = I + I Ajzj, N(z,o) = I B'zj j=1 J j=1 j

As previously mentioned, this stabilizability condition can be checked

since it does not depend on the parameter estimates. An example will

illustrate the test.


Example 2: Consider the system described in Example 1.


[1-z -az 3z1
ED(z,e) N(z,e)] = J
I -2z z


In order to satisfy the stabilizability assumption, the rank of [D(z,e) N(z,e)] must equal 2 for Izi 4 1 and ec. The determinants of
the 2x2 submatrices of [D(z,e) N(z,e)] are (2+a)z2 - 3z+1, z((6-a)z-3), z(1-4z). All three minors are zero only when z = 1/4 and a = -6. Hence the rank condition is violated only when z = 1/4 and a = -6. Therefore, the stabilizability assumption will be satisfied for any interval which does not contain the point I-6}.

The amount of effort involved in checking the rank condition in (3-4) will of course depend on the number of unknown parameters. It may

be possible to use a root-finding algorithm for polynomials in several variables (i.e., z and eij) to check the rank of [D(z,e) N(z,o)] for all GES. It should also be noted that by overestimating the max (p,q) in (3-1), the system cannot be controllable, but may be stabilizable. In








this case, adaptive pole placement would not be stabilizing feedback could be computed using the discussed in Chapter II.

To summarize, the assumptions on the r-input discrete-time system described by (3-1) - (3-3) are


possible, but a Riccati approach m-output linear


Al: An upper bound, n, on p and q in (3-1) is known; A2: the components oij of the unknown system parameter matrix e belong

to known bounded intervals [min ,max] ;

A3: the system described by (3-3) is stabilizable for all e belonging NxMmin _max.
to n where n = {e Nxm: Oij E Lij ,eij D.














CHAPTER IV
PARAMETER ESTIMATION Consider the r-input m-output linear discrete-time system described


y(k) = eT a (k-1) + T b(k-1).


(4-1)


As previously discussed in Chapter III, it is assumed that each component eij of the unknown system parameter matrix e belongs to a known bounded interval min max eij , eij.
The parameter estimation algorithm is given by


P(k-l) a (k-1)
e(k) = O(k-1) P(k-l)f(e(k-i)) + 2 T x (4-2)
nk-1 + a(k-l)P(k-l)-a (k-i)

Ey T(k) - *T (k-l)ip- T(k-i)e(k-1)]


P(k) = P(k-1) -


P(k-1)@a (k-l) T(k-1) P(k-1)
2 +T
nk.1 +a(k-1)P(k-1)a (k-1)


0 < P(O) = pT(0) < 21, (4-3)


0i(k-) - eTix when 6ij(k-I) > eiax fij(k-l) e i(k-1) - emn when e (k-)

0 when 0. .(k-1) E[emin 0maxI ij1 ij , ij


(4-4)










JI when the determinant of P(k) > e
nk- 1 =where e. is any small positive number. (4-5)
"max (l111 a (k-l)ii) otherwise

The parameter estimation algorithm described by (4-2) - (4-5) differs from the recursive least-squares estimator in three ways. First, the initial covariance matrix P(Q) in (4-3) must be less than 21,

whereas the least-squares algorithm allows for any positive definite initial covariance matrix. This condition is needed to ensure that the parameter estimator described by (4-2) - (4-5) possesses the properties listed in Theorem 4.1 which are necessary for proving global stability of the adaptive regulator discussed in Chapter V.

The second difference is the data normalization introduced through the term nk-1 defined in (4-5). In the least-squares algorithm,

T'k-1 is simply equal to one. Again, the data normalization is necessary for proving that the parameter estimator has the desirable properties given in Theorem 4.1. Simulations have shown that if e is not

sui tably small, the normal izati on has a detrimental ef fect on the transient response of the system. This effect can be minimized by

choosing e to be very small in (4-5). Simulations which show the effect of the data normalization and the choice of e on system transient response are discussed in Chapter VII.

The major difference between the estimator given by (4-2) - (4-5) and the recursive least-squares estimator is the addition of a "correction term" -P(k-1)f(e(k-1)) in (4-2). This term forces the estimates

Oij(k) of the components of the system parameter matrix e to converge to the sets Lo j, o. ]. This property, when combined with the stabilizability assumption over Q discussed in Chapter III, eliminates the need








for a persistently exciting input. It is important to note that the
algorithm is not a projection algorithm. That is, eij(k) does not
rm eri maxl
belong to the set e ie'i. ] for every k but does converge to the set in the limit as k + -.

The idea for the correction term came from Kreisselmeier [23]. In

fact, f(e(k-1)) is defined exactly the same as in [23]; however, the parameter estimator described here for MIMO discrete-time systems is quite different from the adaptive observer Kreisselmeier uses for SISO continuous-time systems.

The following theorem shows the estimator described by (4-2) (4-5) has many of the same properties as recursive least-squares. In

addition, the estimates eij(k) converge to the known intervals ruin max
Oi , eT. ]. The prediction error e(k) which appears in the theorem ij 13
is defined as


e(k) = y(k) - eT (k-l)a(k-i) T T(k-).








Theorem 4.1: The estimation algorithm (4-2) - (4-5) has the following properties
i. P(k) converges to a positive semidefinite matrix P. < 21;

ii. I eij(k) i < M < - for all k and i = 1,2,.,N, j = 1,2,.,m;


iii. 2 T le~k) + 0 as k +

nk_1 + a (k-l)P(k-l)a (k-1)

iv. ue(k)u < a(k)# (k-1)ln + b(k), where a(k) and b(k) are positive scalar sequences which converge to zero;
v. f(e(k)) + 0 as k + w, which implies that oij(k) converges to the
set m ij 'max ] for i = 1,2,.,N, i = 1,2,.,m;

vi. iei(k) - eij(k-p)I + 0 as k + - for any integer p, i = 1,2,.,N, j = 1,2,004,m;
mi max
vii. If eij m 0i. or 0. for every i and j, then . .(k) converges Smin maxto a point in L ein mx.
Proof: Property (i) is a well-known property first proved by Samson [10) with nk_1 equal to one. The proof is included here for the sake of completeness. Equation (4-3) implies that 0 4 xTP(k)x~xTP(k-1)x for all xe RN, and for all k > 0. Therefore, for each x RN, the sequence xTP(k)x is monotone decreasing and must converge. Denoting

Pij(k) as the ijthm element of P(k), it follows that

(e. + e.)TP(k)(e. + e ) - eTP(k)ei eTP(k)e.
P.(k) = 1 i e{2 1 eL- T (4-7)








where

0


ei h0
1 ~ ith position.
0


0

Each of the terms on the right hand side of the equation (4-7) must converge which implies Pij(k) converges for every i and j. Noting that

P(O) < 21, it then follows that P(k) must converge to a positive semidefinite matrix P. < 21. To prove (ii), let '4(k) = o(k)-o. Rewriting

(4-2) in terms of "e(k) gives

'(k-1) - P(k-1) a(k'1),Ta(k'1)6(k'1)
2 T -P(k-1)f(6(k-1)). (4-8)
nk1 + a(k1)P(k1)a(k-i)


Multiplying both sides of the equation (4-3) on the right by P-1(k-1) gives
P(k-1)oa (k-1)T (k-1) P(k)P1l(k-1) =I-a a0(4-9)
k- + *T(k'l)P(k'1)a (k-i)


Combining (4-8) and (4-9) yields


e(k) = P(k)P' (k-1)O(k-1) - P(k-1)f(e(k-1)). (4-10)








Now define the Lyapunov functional V(k) = Tr( P(k)p -(k)'(k)) 0, where "Tr" denotes the trace operation. Inserting equation (4-10) into the expression for V(k) gives


IT -1 T1
V(k) = Tr ce (k-1)P (k-1)i'(k)] - Tr[f (e(k-1))P(k-l)p 1(k)(k)].

(4-11)


Inserting the expression (4-8) For "e(k) in the first term on the right hand side of (4-11) yields


V(k) = V(k-1) -


hne(k) 112
2- + T (kl)p(k-l) (k-i)
nk. I+


- Tr [VT(k-1)f(o(k-1))]


-Tr[fT(e(k-1))P(k-l)P' I (k)'(k)].


Equation (4-10) right hand side


V(k) = V(k-1) -


for 16(k) is then inserted into the last term on the
of (4-12) giving


ne(k) 112
2 T~k~~1
"ki_ + Oa(k-l)P(k-1) a(k-1)


2Tr[eT(k-1)f(4(k-1))] (4-13)


+ Tr[fT(o(k-1))P(k-1)p'I(k)P(k-1)f(o(k-1))]. It follows from the definition of f(O(k-1)) in equation (4-4) that

Tr[fT (o(k-1))"'(k-1)] > Tr[f T(o(k-1))f(e(k-1))1. (4-14)


(4-12)








Combining equations (4-13) and (4-14) gives


V(k) 4 V(k-1) - 2 T
nk-1 + a(k-l)P(k-l) a (k-1) (4-15)

-Tr[f T(e(k-1))C21-P(k-1)P- 1(k)P(k-1)]f(o(k-1))].


If the matrix 21 - P(k-1)P'1(k)P(k-) is positive definite for k sufficiently large, then V(k) will be a monotone decreasing sequence which must converge since V(k) is bounded below by zero. In order to show
that the matrix 21 - P(k-l)P-l(k)P(k-1) is positive definite for k sufficiently large, first assume that the determinant of P(k) does not converge to zero. In this case, P'1(k) converges to P-1 which implies that 21 - P(k-1)P'l(k)P(k-1) converges to 21 - P . The matrix 21 - P. is positive definite by property (i) in Theorem 4.1. Now suppose that
the determinant of P(k) does converge to zero. Using (4-3) and the

matrix inversion lemma gives

a(k-l) T(k-l)
-P-1(k) = . P-(k-1) a (
2
n k-i

Multiplying both sides of this equation on the left and right by P(k-1) and adding 21 to both sides give

P(k-l) a (k-l) a (k-l)P(k-1)
21 - P(k-I)P'I(k)P(k-1) 2 21-P(k-1)
2
n k -1(4-16)









Since P(k) converges, equation (4-3) implies that the matrix

T
P(k-l) a (k-1) a(k-l)P(k-1)
2 +T
k_1 a(k-l)P(k-l)a ( k-i)


converges to zero. For k sufficiently large, nk.1 = max (1,,ia (k-1)ii) which implies that
T T
P(R-I) � (k-1) Ta(k-1) P(R-1) P(R-I) a(k-l) Ta (k-l) P(R-1)

2 aT a >2
nk-1 + a(k-l)P(k-l) a(k-1) nk-1 + ,maxP(O)R4 a(k-I)i2


P(R-1 )a (k,-I )OT (k-1) P (k-1) m2ax a >0


nlk_1(1 + XmaxP(O))


P(k-1)a (k-1)0T(k-1)P(k-1)
It then follows that the matrix 2 converges
nk.1
to zero since it is bounded above by a matrix which is converging to zero. Thus by equation (4-16), the matrix 21 - P(k-1)P-1(k)P(k-1) also
converges to 21 - P. in the case when the determinant of P(k) converges to zero. Therefore, the matrix 21 - P(k-1)P-1(k)P(k-1) is positive definite for k sufficiently large since the matrix converges to the positive definite matrix 21 - P. By (4-15), this implies that V(k) is a monotone decreasing sequence for k sufficiently large and must therefore converge. It then follows that ij(k) is bounded (property (ii))

by noting that








V(k) = Tr(-T(k)P-l(k)e(k)) > Exmin P'l(k)] Tr( T(k)o6(k))


N m 2
V(k) > 1/2 Tr(CT(k)^d(k)) 1/2 1 1 [e .(k)- e 2 i=1 j=1 Jij

Now solving (4-15) recursively gives the relationship

q ne(j g )12
0 < V(q) 4 V(O) - 1 2(
q j=1 Tj1 + a a (4-17)

- I Tr[fT(e(j-l))C21 - P(j-1)P'I(j)P(j-1)]f(e(j-1))].
j=1

For sufficiently large k, the terms within the summations are non-negative and must converge to zero since V(k) is non-negative. Properties
(iii) and (v) then follow easily. Property (iv) follows immediately
from (iii) by first noting that


2le(k) II 1 b(k)

2 + oT(k.1)P(k-l)o (k-1)) 1/2 where b(k) is a positive scalar sequence which converges to zero. Multiplying both sides of the inequality by
2 1/2
(n2_1 + a(k-1)P(k-1)oa(k-1)) and noting that nk-1 < 1 + l0a (k-1)i gives


gIe(k)ii 4 b(k)(nk2_ + T 1/2

< b(k)(Nk_1 + (maxP(k-1))l/2 "ha(k.1)I)



4 a(k)ula(k-1)o + b(k)








where a(k) and b(k) are positive scalar sequences which converge to zero. To prove (vi), first subtract '(k-1) from both sides of equation (4-8). Pre-multiplying both sides of the resulting equation by its transpose and noting that e(k) - T(k-1)oa (k-1) gives


Tr[(6(k)-O(k-1)) T(e(k)-e(k-1))] = Tr[(A'(k) - eA(k-1))T(eI(k) -'(k-1))]: e(k)o T (k-l)P(k-l)2a (k-l) T (k) Tr [ 2a a TI
Tr kI+ ,Ta(k l)P(kl) a(k-i))2


2 T1
[fT((ki)) P(k- 2 (k) a
+2Tr T (k-l)P(k-l) (a
nk.1 + -a l

+ Tr[fT (e(k-1))P(k-1) 2f(o(k-1))]. (4-18)


Also

Ta (k-I)P(k-l) 2a (k-I) Ta (k-iP(k-)a (k-I)
2 T a(k-_) 4 [maxP(O)] 2 T
nk-I + a(kI)P(k) a nk-+a(k-)P(k-)a(k-1)

SIXmax P(O) (4-19)

Combining (4-18) and (4-19) gives Tr[(e(k)-e(k-1)) T(o(k)-e(k-1))] 4 [xmaxP(O)] l e(k)ii 2
max 1+ (k.1)P(k-l) a (k-i)


P(k-) 2 a(k-l) T(k'l)_'(k'l) + 2Tr [fT(o(k-1)) a a
+maOTr+fT (k-)P(k1) (k-i)


+ 2 (P(O))Tr~f T(o(k-i))f(e(k-i))]. (4-20)








By properties (iii) and (v), the first and third terms on the right side of the inequality (4-20) converge to zero. The second term on the right side of the inequality will also converge to zero by property (v) provided that the matrix

P(k-l) 2 a (k-1) T(k-1)
a a
2 7Ta -.1-1,kl)@
nk1 + ,T(k-I)P(k-l) (k-i)


is bounded. This matrix is shown to be bounded by first noting that

p2k1)@(k-1) T(k-1)
p 2 (k-l)a (k-1) T2k-k-12)a a (k-)2
(i)a 1a (k-1) (k-i ~(k-I)ii2
0 42 T(kil)P(k-1) (k-i) 2
nk-1 + @a '' ank-1
Ibk ki-I 2 + Xmin P(k-1)



Clearly, the numerator on the right hand side of the inequality is bounded above. The denominator is bounded below by one if the determinant of P(k) converges to zero, and by X min(P ) if the determinant

of P(k) does not converge to zero. Thus, all three terms on the right

hand side of the inequality (4-20) converge to zero. Since


N m
Tr[(e(k) - 6(k-1))T(o(k) - o(k-1))] =I X coij (k) - ai(k-1)]
i=1 j=i 1J
it follows that e ij (k) - i (k-1) must converge to zero which proves

property (vi) in the case where p = 1. The proof for arbitrary p follows from the Schwarz inequality. To prove (vii), equation (4-10) is

first solved recursively giving


n-1
e(n) = P(n)P1(o)'(0) - P(n) P' (i+l)P(i)f(e(i))
i=O









It then follows that



e(m) - e(n) = '(m) - 1'(n) = (P(m) - P(n))P1 (0)'(0)

m-1
-P(m) I P I(i + 1)P(i)f(e(i)) i=0

n-1 1
+P(n) P- (i +1)P(i)f(e(i)).
i =0


Without loss of generality, assume that m is greater than n. Replacing the matrix P(n) preceding the summation with P(m) + (P(n) - P(m)) gives



e(m) - e(n) = (P(m) - P(n))P'I(0)Y(O) - P(m) i P1(i + 1)P(i)f(O(i)) i~n


n-1
+ (P(n) - P(m)) I P-(i + 1)P(i)f(o(i)).
i=O
Using the properties of the induced matrix norm defined in Chapter II gives

le ijn(m) - 0i (n)l < 0le(m) - e(n)!




< ,P(m) - P(n)u oP-1 (0)"O'(O),


m-1 1
+ HP(m) ii 1 I1P (i+l)P(i)R n f(o(i))ll i=n
(4-21)
n-I1
+ ,P(n) - P(m)ll Z uip-l(i+1)P(i)i llf(e(i))1i.
i =0


If the right hand side of equation (4-21) can be made arbitrarily small for m and n sufficiently large, then each component 0i(k) of the matrix








sequence e(k) will be a Cauchy sequence. It will then follow using
property (v) that oij(k) converges to a point in the interval
mai omaxl
18 mi m " In order to prove that the right hand side of the inii '0 .1
equality (4-21) can be made arbitrarily small, two things must be
P1(
shown: lp (k+l)P(k)iu < v < - for all k, and I f(e(k))ii < L < .
k=1
In order to prove oP- (k+1)P(k)in < v < - for all k, first assume that
the determinant of P(k) does not converge to zero. In this case,

P-1(k+1) converges to P-1 which implies that P-1(k+1)P(k) converges to I. Therefore, P P'1(K+1)P(k) ii is bounded for all k when the determinant of P(k) does not converge to zero. Now suppose the determinant of P(k) does converge to zero. Applying the matrix inversion lemma to equation (4-3) gives

I + a(k) T(k)P(k)
P-l(k+l)P(k) a 2a
k


Using the properties of the induced matrix norm, it follows that

SNa(k) T(k) n
HP 1(k+l)P(k)H < iti + 2 aP(k)ii (4-22)
n'k

From (4-5), there exists a finite M such that nk = max (1,uca (k)it) for k > M. It then follows that all the components of the matrix

fa ( k) T(k)
2a have magnitude less than or equal to one for k > M which in
nlk
,a (k) a(k)
turn implies 2 is bounded for all k. Noting also that
nk
iiIii = 1 and P(k)n < 4, equation (4-22) implies P-1 (k+1)P(k)ii is








also bounded for all k in the case when the determinant of P(k) converges to zero. Therefore, there exists a finite v such that

iliP (k+i)P(k)il < v < for all k. Applying this result to the inequality (4-21) gives

mn-i

eij (m)-eij(n)i 4 iiP(m)-P(n)l IiP-1 (o)A'(0)II + 2v m Ilf(o(i))ii i=n

n-1
+ RP(m)-P(n)ov I uf(e(i))i. (4-23)
i=O

In order to complete the convergence proof, it must be shown that


Sf(O(k))n: < L < -. It follows from equation (4-13) that
k=O


V(k) .< V(k-i) -2 T e(k)a 2 - 2Tr[9'T(k-l)f(6(k-l))]
V1 k-I + a(k-1)P(k')O a(k-i)

+ 'max[P(k-l)P-l(k)P(k-l)]Tr[fT(e(k-l))f(O(k-l))]. (4-24)

Now assume < e Nowi asm j< ij j,.

6ij is an interior point of me min max
Lij ' i )" Using the definition of
f(e(k-1)) in (4-4), if f(ii (k-l)) * 0 then I"ii(k-1)I > q > 0 where q

m ,max_ , min ] for i=1, ., N,
= m n ij ij �ij ij "'"
j=1, ., m. Also from the definition of f(e(k-1)), it must be true that the sign of e'ij(k-1) is equal to the sign of fij(e(k-1)) for i=11 ., N and j=1, ., m. These facts when combined with equation (4-24), imply
i2 N m
V(k) < V(k-i) - 2 - 2q I I lfij(e(k-))l
"nk-1+ 1 T(k-1)P(k1)Oa (k-1) i= j=1

+max [P(k-I)P-1(k)P(k-l)Tr[fT (e(k-i))f(e(k-i))].
+a








Rearranging terms in this inequality results in


ie(k) u 2
2 + T(kl)P(kl)o a(k-1)
nk-1 a aa


N m
+ 2q I I fij(e(k-1))
i=1 j=1


V(k-1) - V(k) + Xmax[P(k-l)P- (k)P(k-1)]Tr[fT (e(k-1))f(e(k-1))].


Summing all the terms in this inequality from k equals one to M and N m
noting that nf(e(k-1))oi 4 1 I ifi(e(k-1))i gives
i=1 j=1


M le(k) 112
k=1 nk-12 + ,Ta(k.1)P(k.1)a(k-1)


M
+ 2q I 1lf(6(k-1))11
k=1


(4-25)


M
V(O) - V(M) + Z maxEP(k-1)P'l(k)P(k-1)]Tr[fT(e(k-1))f(e(k-1))]� k=1


If the right hand side of the inequality (4-25) is bounded in the limit as M approaches infinity, then the norm of f(e(k-1)) will be summable. The term V(M) is bounded since V(k) converges. Also


XmaxCP(k-I)P-I(k)P(k-1)]


is bounded since the matrix P(k-I)P-I(k)P(k-1)


converges to P. It remains to be shown that 00 T
I Tr[fT(e(k-l))f(6(k-1))] < � k=1

Equation (4-17) implies


k Tr[fT((k-1))E21-P(k-l)P-l(k)P(k-1)]f((k-1))] <
k=1








Also, there exists a finite N such that the matrix [21-P(k-1)P-1(k)P(k-1)] is positive definite for k > N because [21-P(k-1)P-1(k)P(k-1)] converges to the positive definite matrix 21P. Since


0 < min [21-P(k-1)P'l(k)P(k)P(k-1)]Tr[f T(o(k-1))f(o(k-1))]
k=N

' Z Tr[fT(e(k-1))[21-P(k-1)P' (k)P(k-1)]f(6(k-1))] ,
k=N


it follows that Z Tr[fT(o(k-1))f(o(k-1))] < =- Therefore, the right
k=l
hand side of the inequality (4-25) is bounded in the limit as M approaches infinity. This implies the existence of a finite L such that
I off(o(k-1))o < L < =. Returning to the inequality (4-23), let
k=l

e > 0 be given. The sequence oij(k) will be a Cauchy sequence if IOij(m)-eij(n)I is less than e for m and n sufficiently large. Since
P(k) converges to P=, there exists Nl(e) such that




iP(m)-P(n)o < el for m,n > NI(e)


where el = (1/3) min [e/(nP- (o)o(0)i1, e/(vL)].


Also, there exists N2(e) such that

m-1
I Of(e(i))II < e/6v for m,n > N2(E). i=n








It then follows from (4-23) that leij (m) - ij(n)I < e for m, n >

max (N1,N2). Combining this with property (v), eij(k) converges to a point in mn max f0o
[Lij ,oi I f i, ., N and j=l, ., m.
As discussed in Chapter II, properties (ii)-(iv) and (vi) are standard properties required for any parameter estimator used in indirect adaptive control. Property (v) forces the parameters to converge to a subset n of ]RNxm which contains no singular points. This additional property will ensure that the estimated system is uniformly stabilizable thus eliminating the need for a persistently exciting input. Property (vii), the convergence of eij(k) to a point in
omi n max i o rqie
ein'6 iJ' is not required for proving global stability of the
adaptive closed-loop system. Property (vii) will be used in Chapter VI for tracking and disturbance rejection based on the internal model principle. It is interesting that unlike the case of least squares where e(k) converges to 0 + PP'1(o)y'(O) as k + -, there is no closedform expression for the limit of e(k) as k + - when using the parameter estimator described by (4-2)-(4-5). However, if P(k) converges to zero, then e(k) converges to the true system parameter matrix 8 for both the parameter estimator discussed in this chapter and the recursive least-squares estimator. This is easily shown using the recursive solution of (4-10) which is given by


n-1
e(n) = P(n)PI(0)"(0) - P(n) p-1 (i+l)P(i)f(e(i)).
i=O

Taking the norm of both sides and using the properties of the induced matrix norm, it follows that






35


ii'(n) n < nP(n) ii nP-1 (0)"6(0) + iP(n) it n f P-1 (i+1)P(i )f(e(i )) .
i =0
n-r"
It was shown previously that ii I P-1(i+1)P(i)f(e(i))ii is bounded in i=O


the limit as n + =. Also P(n)i can be made arbitrarily small for n sufficiently large if P(k) converges to zero. Thus, 1'(n) converges to zero as n + - which implies O(k) converges to the true system parameter matrix e whenever P(k) converges to zero.














CHAPTER V
ADAPTIVE REGULATOR

The r-input m-output linear discrete-time system described by

(3-1)-(3-3) will be regulated using a MIMO version of Samson and Fuchs' adaptive LQ controller [12]. As discussed in Chapter Two, a necessary condition for the adaptive closed-loop system to be globally stable is a

uniformly stabilizable estimated system. It will be shown that the

parameter estimator described by equations (4-2)-(4-5) will give a

uniformly stabilizable estimated system provided that assumptions A1-A3 summarized at the end of Chapter Three are satisfied.






Uniform Stabilizability of the Estimated System


Using the parameter estimator described by (4-2)-(4-5), it will be shown that the estimated system can be stabilized using state feedback. As shown in [6], the state-space observer form realization for the system described by (3-1) is given by


x(k+1) = Fx(k) + Gu(k) (5-1)

y(k) = Hx(k)









where


-A1 I BI
~A2 B

F = G = . H [ [I 0 . 0]


-An1 I Bn-1
-An 0 Bn

The entries in F and G depend on the unknown system parameter matrix e. In fact, by the definition of e in (3-3), F and G can be viewed as continuous functions of e where e ranges over the parameter
Nxm: min max
space si = feP : ij c6eij ei .} This dependance can be made explicit by denoting F and G as F(e) and G(e) respectively. Assumption A3 in Chapter Three is equivalent to requiring that the system described by (F(e),G(e)) is pointwise stabilizable over a.


Definition: The pair (F(e),G(e)) is pointwise stabilizable over P. if for each o belonging to a, there exists a feedback matrix L(e) such that Ikmax (F(e)-G(e)L(e))I < q(e) < 1 where Xmax(.) is the eigenvalue of
(-) with the greatest magnitude.


It follows from known results C24], that pointwise stabilizability of (F(e),G(e)) over s is equivalent to the rank condition (3-4). Thus, stabilizability over a can be checked using the polynomial matrices D(z,e) and N(z,e) comprising the system transfer function. Given E > 0, let sie be an extension of the parameter space Q defined as


Nxm min max
n =fee~ IR I j eeo. -e' e. +ell. (5-2)
S13 13 1









Proposition 5.1: Suppose assumption A3 is satisfied so that (F(e),G(e)) is pointwise stabilizable over n. Then there exists an

e > 0 such that (F(e),G(e)) is also pointwise stabilizable over Q. Proof: Suppose (F(O),G(e)) is stabilizable over Q. For each e

belonging to n, there exists a bounded open neighborhood V, of e such that (F(e),G(O)) is stabilizable over V. This result is proved in

[25], by first showing that there exists a feedback matrix L(e) with entries which are continuous in a such that

Amax (F(e)-G(e)L(e))I < q(o) < 1. The existence of the open

neighborhood V6 of e then follows because the eigenvalues of (F(e)-G(o)L(e)) are continuous functions of 0. Since a is a compact

subset of Nxm and U V8 is an open cover of Q, there exists a
r

finite subcover U V0 of n. For k=1,2,., let ek (1/2 )k. For
j=l j

every k > 1, a - U V is a compact subset of RNxm and
�k j=1

r r
-UV C2 -UV0
k+1 j=1 j k j=1 .

r
Suppose a - U Va is nonempty for all k > 1. By the properties of
ek j=1 j


nested compact sets (see [26, p.38]) there is at least one point p which


r
belongs to nCk - U V for all k > 1. Therefore, pen for all
�k j=1 j

r
k > 1 which implies that pe. However, pj U V which forms an
j=l 8j









open cover of s and hence we have a contradiction. Thus,


r
k j=1 Ve must be empty for k > k which implies (F(a),G(e))

will be pointwise stabilizable over Q k for any k ) k1.



Let e(k) be defined as the estimate at time k of the system parameter matrix 0, and let (F(k),G(k)) denote the time-varying estimated system (F(e(k)),G(e(k))).


Definition: The estimated system (F(k),G(k)) is uniformly stabilizable as a time-varying system [27], if there exists a bounded feedback matrix L(k) such that (F(k)-G(k)L(k)) is uniformly asympototically stable as a

time-varying system matrix.


The following theorem gives conditions under which the estimated system is uniformly stabilizable.


Theorem 5.1: Assuming A1-A3 are satisfied, the parameter estimator described by (4-2)-(4-5) will generate an estimated system (F(k),G(k)) which is uniformly stabilizable as a time-varying system.


Proof: By Proposition 5.1, there exists e > 0 such that (F(e),G(e)) is pointwise stabilizable for all e belonging to si. Using the results in Kamen and Khargonekar [25], since a is compact, there exists a feedback matrix L(e) with entries which are continuous in e and a positive constant q < 1 such that


Imax (F(o)-G(e)L(e))I < q < 1 for all eEn








Property (v) in Theorem 4.1 implies that there exists a finite N such that e(k) belongs to Q, for all k > N. It then follows from the
result in Kamen and Khargonekar given above that there exists a feedback gain sequence L(k) continuous in e such that


I max (F(k)-G(k)L(k))I < q < I for all k > N.


Since the entries in (F(k)-G(k)L(k)) are continuous functions of O, property (vi) in Theorem 4.1 implies [F(k)-G(k)L(k)][F(k-1)-G(k-1)L(k-1)] converges to zero as k goes to infinity. Desoer [28] proves A(k) will be uniformly asymptotically stable as a system matrix if (i) all eigenvalues of A(k) have magnitude less than some constant q < 1 for all k and (ii) A(k)-A(k-1) converges to zero as k goes to infinity. Using Desoer's results on stability of slowly timevarying systems, (F(k)-G(k)L(k)) is uniformly asymptotically stable as a system matrix. Thus the estimated system (F(k),G(k)) is uniformly stabilizable as a time-varying system.


The adaptive control law for regulating the system described by (3-1) is of the form u(k) = -L(k)^(k). The state estimate X(k) is
generated from an adaptive observer which will be discussed in the next section. The feedback gain matrix L(k) is chosen to stabilize the estimated system (F(k),G(k)). The existence of a stabilizing feedback
gain L(k) is guaranteed by Theorem 5.1. Several possible options for L(k) are given in the section following the adaptive observer section.






41


Adaptive Observer


The adaptive observer for the estimated system is given by


x(k+l) = F(k)2(K) + G(k)u(k) + M(k)(y(k)-9(k))

9(k) = H(k)


-A,(k) -A2(k) F(k)=

-An (k)


M(k)


7-Al(k)"
-A2(k)




-An(k)


G(k) =


Bl (k) B2 (k)




Bn(k)


(5-3)


H = [1 0 . . . 0].


The adaptive observer defined by (5-3) is a dead-beat observer. The

output 9(k) of the adaptive observer can be expressed in the following form:


A
y(k) = [-Al(k-1) . -An(k-n) Bl(k-1) . Bn(k-n)] (k-l)


(5-4)


where








This expression is easily derived by first rewriting (5-3) as


A


0 I


Bl(k-1) B2(k-1)




Bn(k-l)"


u(k-1)


"Al(k-1)
-A2(k-1)




-An(k-1)


y(k-1).


(5-5)


Multiplying equation (5-5) by H gives


9(k) = Hx(k) = E0 1 0 . O]X(k-1) + Bl(k-1)u(k-1) - Al(k-1)y(k-1) Substituting into this equation for X(k-1) using (5-5) yields


-(k) = CQ 0 I . O]R(k-2) + B1(k-1)u(k-1) + B2(k-l)u(k-2)

- Al(k-1)y(k-I) - A2(k-2)y(k-2)


Continuing the substitution using (5-5) gives


y(k) = [0 . 0 I] (k-n+l) +


n-1
SBj (k-j)u(k-j) j=l


I Z A.(k-j)y(k-j) j=l J

n n
I B (k-j)u(k-j) - I A (k-j)y(k-j) j=1 j=1


which is equivalent to (5-4).








The error between the output of the adaptive observer (5-3) and the output of the system (3-1) as defined as e(k) = :(k) - y(k).


The observer error e(k) has the same properties as the prediction error e(k) given in Theorem 4.1; that is, lle(k)n < a'(k)Hj(k-1)o + b'(k) (5-6)


where a'(k) and b'(k) are positive scalar sequences which converge to zero. This is shown by first rewriting e(k) as


e(k) = A(k) - ceT (k-l)a (k-i) T(k-1)]


+ [OT(k-l) a(k-i) + T % (k-1)] - y(k)


'(k)(k) (k) - e T(k-l) a(k-1) + T (k-1)] - e(k).


Taking the norm of both sides and using the triangle inequality gives


ll(k)i 4 OM(k) - (T (k-l)a (k-1) + (k-l))il + ile(k)ll


Noting from (3-1)-(3-3) that e T(k-1)a (k-1) + T b(k-1) is equal to [-AI(k-1) . -An(k-1) B1(k-1) . 9 n(k-i)](k-i) and using expression (5-4) gives








l 1(k)ii 4 ii[J (k) . J2n(k)] (k-l)iu + iie(k)ii (5-7)


< 1[ 1 (k) . 2n]J2n ii i (k-1)ii + iie(k)ii

-Ai (k-i) + Ai (k-i) li~n
where Ji(k) =

Bi (k-i) - Bi (k-i) n+1i<2n.




It follows from property (vi) in Theorem 4.1 that aJi(k) converges to zero as k goes to infinity for i=1,.,2n. Applying this property and property (iv) of Theorem 4.1 to (5-7) gives the desired inequality (5-6). Property (5-6) is very important in the proof of global stability of the adaptive closed-loop system.


Feedback Gain Sequence


The feedback gain sequence L(k) must be chosen so that (F(k)G(k)L(k)) is exponentially stable as a time-varying system matrix. As shown in Theorem 5.1, a stabilizing feedback does exist because the estimated system (F(k),G(k)) generated from the parameter estimator (4-2)-(4-5) is uniformly stabilizable under assumptions A1-A3. There are several options in the literature for choosing L(k), some of which will be discussed in this section.
One approach for computing L(k) is to stabilize the estimated system (F(k),G(k)) pointwise in time using control law strategies for time-invariant systems such as pole placement or LQ control. Of course, in the case of pole placement, the stabilizability assumption over the parameter space Q (assumption A3) must be strengthened to reachability









over n. This approach has been developed by a number of individuals, see for example [4-8]. A drawback to this approach is that (F(k),G(k))

may not be reachable or stabilizable at certain points in time (i.e., 0(k) may be a singular point for some values of k). Recall that

Proposition 5.1 and property (v) of Theorem 4.1 only guarantee that (F(k),G(k)) is stabilizable for all k exceeding some finite N. Thus,

for time points prior to N, the chosen control law may not have a solution. Therefore, in this approach it is necessary to check whether

or not (F(k),G(k)) is reachable or stabilizable at each time point k, which adds to the number of on-line calculations. Also, a decision must

be made on how to modify the control law when singular points are encountered. A common choice in the literature is to update the control law only when (F(k),G(k)) is stabilizable.

Another approach first introduced by Kreisselmeier for continuoustime systems is to compute the feedback gain asymptotically [29]. This

approach was developed for SISO discrete-time systems by Samson and Fuchs [10-12] using an LQ control strategy, and later extended to MIMO discete-time systems by Ossman and Kamen [30-32].

In [30-32], the stabilizing feedback L(k) is computed at each time point k by solving in real time one step of a Riccati difference equation. More precisely, let


L(k) = [GT(k)RkG(k) + I]-IGT(k)RkF(k) (5-8)


where R(k) is the solution to the Riccati difference equation


Rk+1 = Q + LT(k)L(k) + (F(k) - G(k)L(k))TRk(F(k) - G(k)L(k)).


(5-9)








In (5-9), both Q and the initial value Ro are positive definite symmetric matrices. In contrast to the first approach, where the algebraic Riccati equation would be solved for each k, this asymptotic approach requires the solution of only one iteration of the Riccati difference equation at each point in time. Obviously, this approach

offers considerable savings in on-line calculations. Also, it is not

necessary to check whether or not (F(k),G(k)) is stabilizable at each point k because L(k) given by (5-8)-(5-9) is well-defined even for isolated singular points. In order to prove that (F(k)-G(k)L(k)) is exponentially stable when L(k) is computed using (5-8)-(5-9), it is first necessary to show u(L(k)-L(k-1))u converges to zero as k + -.


Theorem 5.2: Suppose the parameter estimates are generated using the algorithm (4-2)-(4-5). Then subject to assumptions A1-A3, the feedback gain sequence given by (5-8)-(5-9) has the property

HL(k)-L(k-I)i + 0 as k + -.
Proof: The proof is rather lengthy and will therefore be divided into steps. Throughout the proof, Pk() will refer to the sequence generated from the time-invariant Riccati difference equation (RDE) given by



Pk+l(0): Q + FT (6)P(O)F(8) - FT(e)Pk(o)G()[I + GT (M)Pk()G(o)]-I


x GT (O)Pk(e)F(e), PO > 0.




The matrix P(e) will denote the solution to the algebraic Riccati equation (ARE) given by










P.(8) = Q + FT(e)P.(e)F(8) - FT ()P (e)G(o)[I + GT (6)P (e)G(e)]-1


x G T(e)P(e)F(e).


Let Q denote the compact parameter space defined in (5-2) such that
(F(e),G(e)) is pointwise stabilizable for all a belonging to Q. Let

> 0 be given. It will first be shown that HRk - Rklii < � for k

sufficiently large where Rk is computed using (5-9). Step 1: The sequence of matrices Rk given by equation (5-9) is

bounded. That is, Rk belongs to a compact subset D of NxN.
Assumption A3 implies (FT (e),G ()) is detectable for all e belonging

to the parameter space a. Using arguments similar to those in Theorem 5.1, it follows easily that (FT(k),GT(k)) is uniformly detectable.
Anderson and Moore [27], prove that Rk and L(k) given by (5-8)-(5-9) will be bounded if (FT(k),GT(k)) is uniformly detectable.
1 2
Step 2: Let Pk(e) and Pk(e) denote the sequences generated from the RDE using initial conditions P0 and P0 respectively. Defining

APk(e) : P1() - P2(e), it then follows that


APT 1 T 1 -1 T
APk+l(0) = F (0)(I - Pk(e)G(B)[G (e)Pk(e)G(O) + I] G (e))


x(APk(e) + APk(O)G(B)G T(O)P2 (e)G(e) + I]-I G ()APk())




x (I P1(e)G(e)G T(6)P1()G(e) + I1 G T()) TF(e).
I- keGe[T Ik-









This relationship was introduced by Samson [10] in the single-input
case. It can also be verified in the multi-input case, but due to the large amount of algebra involved the derivation will not be included here.

Step 3: For any PO > 0 there exists an N(e,Po) such that


liPk(e) - P=(e)il < E/6 for all Oe n and for all k > N.

Since (F(e),G(e)) is stabilizable for each o belonging to Q, it follows from well-known results that Pk(e) converges to P.(e) pointwise over

It was shown in Green and Kamen ([33], Theorem 1) that "the

convergence is uniform over a. if n. is compact. Step 4: For all PO belonging to a compact subset D of NxN, there

exists an N(s) such that


OPN() - P=(e)u < e/6 for all Oet^.

� m M
The compactness of 0 implies there exist matrices P0 and P0 such that
pm < p0 . M for all PO belonging to D. It follows from the
0 P0
relationship given in step two that

pm PM fralPO
Pk (e) < Pk(e) < Pk(e) for all PonD.


From step 3, there exist integers NI(EP ) and N2(e,PM) such that


IPk(e) - P.(e)ii < s/6 for all k > N1 and for all oe^;


lip (e) - P=(e)u < e/6 for all k > N2 and for all Bel .









Thus,


llPk(Q) - P(o)i < E/6 for all e�12, for all Po O, and for all k > max(N1,N2).


Step 5:

for all


Given a > 0, there exists a 6(a) such that for all PkD and

A


lipk+1(8) - Pk+1( )Q < a


whenever lie - ^en < 6


This property follows easily from the fact that Pk+1(6) is a continuous function of e and the sets D and n, are compact.
e
Step 6: Let N be a fixed finite integer such that the condition in step four is satisfied. There exists a finite integer Ml(e) such that



nRk+N - Pkii < e/3 for all k > M


where Pk is the solution to the ARE when the matrices are evaluated at S(k). In order to prove this result, a family of sequences Pt follows


F(.) and G(.) is defined as


t
Pk = Rk for all k 4 t. Pk+1 = Q+FT ((t))P tF((t))FT ((t))pt G((t))[I + GT (e(t))P G(O(t))] 1


x GT(e(t))PtF(e(t)) for all k > t.








In other words, P is equal to Rk for all k up to time t. After time

t P is generated from the time-invariant Riccati difference equation
,Pk

initialized by Rt and using the constant matrices F(.) and G(.) evaluated at e(t). For every 6(t) belonging to n , the sequence Pt tk
will be bounded for all k. In fact, Pk converges to P t whenever
k Ck
O(t) en,. It will first be shown that iR -Pk+Ni < k /6 for k
Sk+N -k+N< 1 fo k
sufficiently large. The matrix P k is simply the Nth step of a timek+N
invariant RDE intitialized at time k by Rk and using constant matrices F(.) and G(.) evaluated at e(k). The interpretation for

uRk+N -Pku being small for k sufficiently large is that the timevarying RDE given by (5-9) will not vary much from the time-invariant RDE over a finite interval of time N as long as e(k) does not change

much over the time interval N. Equation (5-9) can be rewritten as



R k+i+1 =Q+ F T ((k))Rk+iF(o(k)) - F T(o(k))Rk+iG(e(k))


x [I + G T(o(k))Rk+iG(o(k))]"1 GT ((k))Rk+i F(e(k)) + Eki; Rk

for all k and for all ie[O,N].


The matrix Ek+i is given by


Ek+i FT (O(k+i))Rk+iF(e(k+i)) - FT (e(k+i))Rk+i G(o(k+i)) x [I + GT (O(k+i))Rk+iG(o(k+i))]-I GT ((k+i))Rk+iF(O(k+i))


- FT(o(k))Rk+iF(O(k)) + FT(o(k))Rk+iG(o(k))

x [I + GT(e(k))Rk+iG(e(k))]"1 GT(O(k))Rk+iF(o(k)) for all k and for all iECO,N].








Since Rk belongs to the compact set D for all k, it follows from step five that given any a > 0 there exists s(a) such that


,1Zk+i o < a whenever ie(k+i) - o(k)ii < 6 for ie[O,N].


k Qk Let Q"k =Rk - k


k
Qk+i +1


Using the equation introduced in step two gives


F T(e(k))(I - Rk+iG(e(k))[G T(e(k))Rk+iG(e(k)) + I] - 16(k)))


k k T k
x (Qk+i + Qk+i G(o(k))[GT(o(k))Pk+iG(e(R)) + I]
T kT

x G T(e(k))Qk+i) (I - Rk+iG(e(k))G T(e(k))Rk+iG(e(k)) + I]T T kT k Q O
x GT(e(k) ))T + Zki ~k = 0 ie[0,N].


It follows from the boundedness of e(k), Rk, and sufficiently large) that


(for t


iF T(e(k))(I -.Rk+iG(e(k))G T(e(k).)Rk+iG(e(k)) + 1]-iGT(e(k)))II < KI< O and IG(e(k))[GT(e(k))Pk+i G(O(k)) + ]- G T(e(k))N < K2 < -. Thus for all k such that e(k)en,, we have

k K12 k i+ K2 k 0 2 + 1 k , ie0,N].
nQk+i+lu < K(Qk+i 2Qk+i k+i k








Clearly, there exists a al(e) such that if oE k+i 11 < 1 for all ie[O,N] then


k
Qk+N < C/6.



Let 6(a1) be chosen such that



HEk+iU < a 1for all ie[O,N] whenever ne(k+i) - e(k): < 6.


It follows from properties (v) and (vi) of Theorem 4.1 that there exists a finite integer M1 such that



el(k)gen for all k > M1


and ne(k+i) - e(k)n < 6(a ) for all ic [0,N] and for all k > M .

Therefore
k K
Qk+N =nRk+N - Pk+Nn < e/6 for all k > M1.


Using the results in step four, we obtain


k p
HPk+N k < e/6 for all k > M1. Thus

uRk+N - k IRk+N - Pk N + iP k k 1 < C/3 for all k > M1.
00k+N k+N - P0 10


which proves the result.









Step 7: There exists a finite integer M2(e) such that


0pk+l- k H < E/3 for all k > M



The result follows immediately from the uniform continuity of P (.) over ^ and the convergence of Iie(k+i) - e(k)n to zero. Step 8: There exists a finite integer M(e) such that



nRk - R k-1 < e for all k > M.


Applying the triangle inequality and using the results in steps six and seven gives

IRk+N+I - Rk+Nn 4 oRk+N+1 - pk+lI + npk+li k 11 + lipk Rk+N1



for all k > max (M1, M2).


The result then follows easily by setting M = max(M1, M2) + N + 1. Step 9: If L(k) is computed using (5-8)-(5-9), then HL(k) -L(k-l)ll converge to zero as k goes to infinity. This result follows easily

since fIRk - Rk-l1l, IF(k) - F(k-1) ii, and iG(k) - G(k-)i all converge to zero.

Using the results of Theorem 5.2, it is now possible to prove that (F(k)-G(k)L(k)) is exponentially stable as a time-varying system matrix. The exponential stability of (F(k)-G(k)L(k)) is very important

for proving global stability of the adaptive closed-loop system.








Theorem 5.3: Suppose e(k) is generated using the parameter estimator (4-2)-(4-5) and the feedback gain sequence L(k) is computed using (5-8)(5-9). Then subject to assumptions A1-A3, (F(k)-G(k)L(k)) is

exponentially stable as a time-varying system matrix. Proof: Assumption A3 implies that (FT (),G T(e)) is detectable for each e belonging to Q. Using arguments similar to those in
Proposition 5.1 ind Theorem 5.1, it follows that (FT(k),GT(k)) is

uniformly detectable. In [27], Anderson and Moore prove that if

(FT(k),GT(k)) is uniformly detectable then (F(k)-G(k)L(k))T is exponentially stable as a time-varying system matrix where L(k) is given by (5-8)-(5-9). Since matrix products do not commute, it does not immediately follow that (F(k)-G(k)L(k)) is exponentially stable. However, Samson and Fuchs [12] show A(k) will be exponentially stable as a time-varying system matrix if AT(k) is exponentially stable and if IA(k)-A(k-1)n converges to zero as k + . Since (F(k)-G(k)L(k))T is exponentially stable, it then follows that (F(k)-G(k)L(k)) will also be exponentially stable if u(F(k)-G(k)L(k)) - (F(k-1)-G(k-1)L(k-1)), converges to zero as k + . Property (vi) of Theorem 4.1 implies both IF(k)-F(k-1)ii and oG(k)-G(k-1)II converge to zero. It was shown in Theorem 5.2 that llL(k)-L(k-1)n also converges to zero. Since

F(k)-G(k)L(k)-(F(k-1)-G(k-1)L(k-1)) =
(F(k)-F(k-1)) - G(k)(L(k)-L(k-1)) - (G(k)-(G(k-1))L(k-1) and both G(k) and L(k) are bounded, it follows that I(F(k)-G(k)L(k))-(F(k-1)-G(k-1)L(k-1))n does indeed converge to zero. Thus the matrix (F(k)-G(k)L(k)) is exponentially stable as a time-varying system matrix.








Stability of the Adaptive Closed-Loop System


The control law chosen to regulate the system is given by


u(k) = -L(k) (k) (5-10)


where x(k) is generated from the adaptive observer (5-3) and L(K) is a stabilizing feedback for the estimated system (F(k),G(k)). The following theorem shows that the adaptive regulator consisting of the

observer (5-3) and the control law (5-10) results in a globally stable closed-loop system; that is, for any initial states in the plant and the observer, the input u(k) and the output y(k) converge to zero.


Theorem 5.4: Suppose that the parameter estimator (4-2)-(4-5) is used so that there is a stabilizing feedback L(k) for the estimated system (F(k),G(k)) subject to assumptions A1-A3. Then with the adaptive

regulator defined by (5-3) and (5-10), the resulting closed-loop system is globally stable.


Proof: The proof is based on a MIMO extension of the results in [121. Letting (k) = (k)-y(k) and using equations (5-3) and (5-10) gives




A(k+1) = (F(k)-G(k)L(k)) (k) - M(k)A(k) y(k) = H%(k) - e(k).









Also, the regression vector 4(k) defined in (3-2) can be written in the form


(k) = S4(k-1)


where


0 . 0 0 . 0
Im(n-1)
S = 0 . 0 0 . 0
0

Ir(n-1) :

0
Defining z(k+l) =A(kl gives


+ D(k) (k) + V(k)


H 0
D(R) = -L(k)


, v(k) =


z(k+l) = A(k)z(k) + w(k)


-e(k)
0 0

0


(5-11)


where
S D(k) v 1
A(k) = w(k) ) j
0 F(k)-G(R)L(K) -M(R)e(R)


The matrix S is a stable matrix since it is lower-block triangular. The

matrix F(k)-G(k)L(k) is stable by construction; therefore, A(k) in (5-11) is exponentially stable as a time-varying system matrix. In

addition, it follows easily from (5-6) and the boundedness of M(k) that


Iw(k)n < a(k)uz(k)n +b(k)


(5-12)


where a(k) and b(k) are positive scalar sequences which converge to zero as k + -. Thus, (5-11) can be viewed as an exponentially stable timevarying system driven by an input w(k) which can grow no faster than linearly with the state, z(k). Following the proof in [12], it will be shown that Iiz(k) a converges to zero. Let A(k+N,k) denote the state transition matrix for (5-11) defined by










A(k+M,k) =


M-1 11 A(k+i) i =0*


Since A(k) is an exponentially stable time-varying system matrix, it follows that


iiA(k+M,k)t < R 1 < - for all k,M > 0.


(5-13)


Also, given 0<6<1 there exists a finite p such that


nA(k+p,k)i < 6 for all k.


(5-14)


For the remainder of the proof, p will be a fixed constant such that(5-14) is satisfied. Taking the norm of both sides of (5-11) and using (5-12) gives


liz(k+1) e ItA(k)aItz(k) ii + nw(k) ii


(5-15)


(A(k)u + a(k))uz(k)ii + b(k)


Noting that HA(k)n, i(k), and E(k) are all bounded and p is fixed, it follows from equation (5-15) that there exist finite constants R2 and R3 such that


ttz(k+i)ii < R2z(k)ti + R3 for all is [O,p] and for all k. Solving equation (5-11) recursively gives


(5-16)








z(k+p) = A(k+p,k)z(k) +


p-1 I A(k+p,k+j+i)w(k+j). j=0


Taking the norm of both sides and using (5-12) results in


iiz(k+p)i 4 1IA(k+p,k)lh iz(k)3


4 nA(k+p,k) li iiz(k)ii


p-1
+ I fIA(k+p,k+j+1)nI iw(k+j)1u j=0

p-1 + I nA(k+p,k+j+1)ii(a(k+j)ifz(k+j)ii + b(kj)). j=0


It follows from equations (5-13), (5-14), and (5-16) that

P p-1
Iiz(k+p)ii .4 az(k)ii + max(1,Rj) I a(k+j)(R2 iz(k)ii + R3) + b(k+j) for j=0


all k.
Since a(k) converges to zero, there exists a finite N such that


(5-17)


i(k) < I p for all k > N.
6 + PR2max(1,R,)

Therefore, for all k > N,

~,p-i
uz(k+p)i < roz(k)i + max(1,R1) P a(k+j)R3 +b(k+j) or equivalently, j=0


iz(k+p)ii 4 riiz(k)n + c(k)


(5-18)


where r < 1 and c(k) converges to zero as k + o. In order to show
(5-18) implies liz(k)o converges to zero, let e>O be given. Solving (5-18) recursively and using (5-16) gives










n-1 -j
iiz(k+i+np)ii rn iz(k+i)i + I rn c(k+i+jp) j=0

n-1 n-1-j
iz(k+i+np)ii 4 r (R2iz(k)ii + R3) + I r c(k+i+jp) j=0


for all k > N, iE[O,p]. (5-19)


Since c(k) converges to zero, there exists a finite N, > N such that c(k) < for all k > N1.
(177


Applying this inequality to (5-19) gives nz(Nl+i+np)i; < rn(R2iz(N1)i + R3) + e/2, ie[O,p]. Also, since r < 1 there exists a finite N2 such that



r (R nz(N) + R) for all n > N2.


Thus


Hz(N1+i+np)ii < e for all n > N2, i�[O,p]


which implies tz(k)ii < e for all k > N1 + N2P. By the definition of z(k), both the system input u(k) = -L(k) (k) and the system output y(k) must converge to zero. Therefore the adaptive






60


closed-loop system is globally stable using the adaptive regulator defined by (5-3) and (5-10).













CHAPTER VI
APPLICATION TO TRACKING AND DISTURBANCE REJECTION


The adaptive controller derived in Chapter V can. be applied to the problem of tracking with disturbance rejection using the internal model principle. The internal model principle has been discussed by a number of individuals, see for example [20].

Consider the r-input m-output plant described by



x(k+l) = Fx(k) + Gu(k) + Dv(k) (6-1)

y(k) = Hx(k)

where

-A1 I B1
- A 2 18
A2I B2


F . . G: . H rI 0 . 0].


-An_1 I Bn_1
-An 0 Bn


The vector v(k) consists of the exogenous disturbance signals. As in

previous chapters, it is assumed that some or all of the entries in the matrices A. and B. are unknown and will be estimated using the parameter

estimator described by (4-2)-(4-5). Letting r(k) denote an m-vector

reference signal, the objective is to design an adaptive controller which stabilizes the system described by (6-1) and forces the tracking error e(k) = y(k)-r(k) to converge to zero as k + -.








As mentioned, the controller design will be based on the internal model principle. Assume that the disturbance vector v(k) and the reference signal r(k) both satisfy the difference equation:


z(k+q) + aq-1z(k+q-1) + . + aIZ(k+l) + a0z(k) = 0.


(6-2)


Let r be defined as


-al


-aq.1


The internal model can then be realized as


Xc(k+l) = AcXc(k) + Bce(k)


(6-3)


where Ac = block diagonal {r,


r *., r} m-tupl e


Bc block diagonal {T, T tuple T




The following theorem gives the conditions under which it is possible to

design an adaptive controller which will stabilize the plant described








by (6-1) and will drive the tracking error e(k) to zero asymptotically.


Theorem 6.1: Suppose assumptions A1-A3 are satisfied and in addition


zIF(e) 0 G(o)
rank = n+mq for all een and
-BCH zI-Ac 0 for all z: Izi > 1

Nxmmi n max
where { = {s R e j ,6i j. Let xc(k) be generated from the
error driven system described in (6-3) and X(k) be generated from the adaptive observer given by (5-3). Then there exists a control law u(k)

-Llx(k)-L2xc(k) which when applied to the plant (6-1) results in a stable closed-loop system and drives the tracking error e(k) = y(k) r(k) to zero.
Proof: Using equations (5-3) and (6-3) and setting v(k) and r(k) equal to zero, we obtain


[x(k+1) 1 F(k) [ x(k) 1 G(k)] [-M(k)]
[c(k+) L BcH A [xc(kJ+ u(k) + LBcje(k) (6-4)


where e(k) = H'(k) - y(k). Assuming the given rank condition holds, it follows from property (v) of Theorem 4.1 that the system described by (6-4) is stabilizable. The stabilizing control law is of the form


u(k) = -Ll(k)A(k) -L2Xc(k). (6-5)



Let F(k) : and G(k) = [
BcH Ac r
As in the case of the adaptive regulator derived in Chapter V,a









stabilizing feedback gain L(k) = [L1(k) L2(k)] can be computed from one step of the Riccati difference equation


L(k) = [ T(k)Rk'G(k) + I"1 GT(k)RkF(k)


(6-6)


Rk+1 = Q + LT(k)L(k) + (F(k)-6(k)L(k))TR k((k)-r^(k)L(k)).


Again, Ro and Q are arbitrary positive definite symmetric matrices.

In order to use the existing results on tracking and disturbance rejection, the closed-loop system consisting of (6-4) and the control law (6-5) must converge to a time-invariant system. Properties (v) and (vii) of Theorem 4.1 imply that the pair (F(k),G(k)) converges to a stabilizable pair (F.,G). Using the results of Samson [o], L(k)

converges to L., the optimal LQ feedback for the pair (F.,G) given by L = [GTR G. + I]" GT RF




where the matrix R. is the solution to the ARE for (Fw,G.) . More

precisely,


R = Q + LTL + (FTG L )TR.(F- .)


Applying the control law (6-5) to the system described results in an exponentially stable system driven by e(k).

results in Chapter Five, the observer error e(k) converges Also, it follows from property (ii) of Theorem 4.1 that bounded. Thus, the closed-loop system consisting of (6-4)


by (6-4) From the to zero. M(k) is and the









control law u(k) = -Llx(k)-L2xc(k) is exponentially stable and converges to a time-invariant system. Finally, from the results in Chen [20], if the control law (6-5) is applied to the plant (6-1), the resulting closed-loop system is still stable (not including the exogenous disturbance v(k)) and the system output y(k) converges to the reference signal r(k) as k +The adaptive controller described by (6-3) and (6-5)-(6-6) causes the plant (6-1) to reject deterministic disturbances and to track a given reference signal r(k). Simulations of the adaptive controller for various types of external disturbance and reference signals are included in Chapter VII.













CHAPTER VII
SIMULATIONS


This chapter contains several simulations of both the adaptive regulator derived in Chapter V and the adaptive controller discussed in Chapter Six applied to S150 as well as MIMO discrete-time systems.



Example 7.1: Consider the non-minimum phase discrete-time system described by the following difference equation



y(k) = 2y(k-1)-O.99y(k-2)+O.5u(k-1)+3u(k-2).


The adaptive regulator defined by the observer (5-3) and the control law (5-10) with L(k) computed using (5-8)-(5-9) was implemented. Three

different algorithms were used to estimate the system parameters: the

estimation scheme described by (4-2)-(4-5), the recursive least-squares algorithm defined by (2-3), and the estimation scheme given by (4-2)(4-5) with data normalization (i.e., ok-1 = max(l, i, (k-l) Hi) for all k). In all three cases, the initial "covariance" P(O) was chosen to be I and the initial state of the plant was [I 0jT. Also, Q and Roin

(5-9) were chosen to be I. The parameter ranges, initial parameter

estimates and steady state estimates for all three estimation schemes are displayed in Table 7-1. As seen from Table 7-1, the estimation

scheme described by (4-2)-(4-5) forces the estimates of the parameters to converge to the given ranges; whereas, recursive least squares does


































-a, -a2 bi b2

Actual Parameters 2.0 -0.99 0.5 3.0

Initial Estimates 1.95 1.0 0.75 6.0

Parameter Ranges [1.9, 2.0] C-2.0, 4.0] C-0.5, 2.01 C2.5, 9.5]1

Estimation Scheme 2.00 -1.00 .0.530 3.04
(4-2)-(4-5)

Recursive Least 2.11 -1.05 0.594 3.19
Squares

Estimation Scheme 2.00 -0.991 0.571 3.14
(4-2)-(4-5) with
ax(1,ie(k)ii)
nk = m


not. Figures 7-1-7-3 contain plots of the output response, control input, and estimated parameters for each of the three cases. A

comparison of Figures 7-1 and 7-2 show the estimation algorithm defined by (4-2)-(4-5) has a slightly better transient response than least squares. Figure 7-3 illustrates the detrimental effect of the data normalization on the transient response of the system. As previously

mentioned, in (4-5) should be chosen very small to minimize the effect of data normalization on the system transient response.




Table 7-1

DATA





















~I P.
- i i'i. *1


20 30 40 50 60 70 ' k


It iiIiI I.'


I'
- ,***~~ ~ ii


ii III II


20 30 40 50 60 7o0 {k-


Figure 7-1
Estimation Scheme (4-2)-(4-5)


y(k)


20 15 10
5


-5
-10
-15
-20


u(k)


20 15 10
5


-5
-10
-15


-20




















-al(k)


b2 (k)

b1 (k)


20 30 40 50 60 70 '(k(d)

Figure 7-1 continued
Estimation Scheme (4-2)-(4-5)
















20
15 10
5


-5
-10
-15
-20







u(k)

20 15
10
5


-5
-10
-15
-20


30 4om 5o0o -T -(b)

Figure 7-2
Recursive Least Squares


I i I I


11 ' j,, , . . I I lI ,,


Jill 'i





















-al(k)


_ 20 30 40 50 _a2(k o


70 (k)


b2(k) bl(k)


20 30 40 50 6070(k)


Figure 7-2 continued Recursive Least Squares





















20 30 40 50 60 70 (k)





(a)


1114 It *-.


Ii' ~ IJ'~~20


30 40 50 60 70 (k)


Figure 7-3
Estimation Scheme (4-2)-(4-5)
with nk= max(I, IVP(k)II )


y(k)


15
10
5


-5

-10
-15
-20


u(k)

I








Example 7.2: Consider the non-minimum phase stabilizable but not

reachable discrete-time system described by


y(k) = 2.6y(k-1)-2.13y(k-2)+0.54y(k-3)+u(k-l)+1.5u(k-2)-u(k-3).


The adaptive controller defined by the adaptive observer (5-3) and the control law (6-5) with L(k) computed from (6-6) was implemented to force the output to track the reference signal r(k) = 10. No disturbance was introduced in this example. The internal model was given by


Xc(k+l) = Xc(k) 4. e(k)


with e(k) = y(k)-10. The parameters were estimated using (4-2)-(4-5). The initial conditions were chosen to be: P(O) = I, x(O) = [0 1 O]T, xc(O)=O, and Ro=I. The initial parameter estimates, parameter ranges,

and steady state values of the estimates are displayed in Table 7-2. Plots of the output response, control input, and parameter estimates are shown in Figure 7-4.

Equation (4-4) indicates that the initial "covariance" matrix P(O) should satisfy O
y(k)=6.5xlO8 and the control input was u(k)=-2.6x1O11. The parameter

estimates at k=20 were -al(k)=451, -a2(k)=-943, -a3(k)=484, bl(k)=1.0, b2(k)=-447, and b3(k)=-898. Thus, choosing P(O) smaller than 21 is not

merely a technicality in the proof of Theorem 4.1, but is a necessary condition for preserving stability of the adaptive closed-loop system.















Tabl e 7-2 DATA


_ _ _ _ _- -a, -a2 -a3J b, b2 j b3

Actual Parameters 2.6 -2.13 0.54f . 1.5 -1.0

Initial Estimates 2.5 -1.0 -0.5f 2.0 1.6 r -1.5

Parameter Ranges [2.5,2.8] C-5,31 C-2,1J C-4,8] C1.5,1.74 C-3,01
Estimation Scheme 2.5 -1.92 0.429 0.9991 1.60 -0.795
(4-2)-(4-5) ______ ___________













y(k)


40 30 [ I Il



11101 20 30 40 50 60 70 (k)
IOi
-20

-30
-40
(a)


20 30 40 50 60 70 (k)





(b)


Figure 7-4
Estimation Scheme (4-2)-(4-5)


40 30
20 10


-10
-20
-30
-40




















-a.,(k)
-a 3 ( k)


20 30 40 50 60 70


-a 2(k)


b 2(Ck)


U ir A


20 30 40 50 60 70 b 3(k)





(d)

Figure 7-4 continued
Estimation Scheme (4-2)-(4-5)









Example 7.3: Consider the discrete-time system described by


y(k) = -2y(k-1)-5y(k-2)-u(k-1)-0.5u(k-2)


The adaptive controller defined by the observer (5-3) and the control law (6-5) with L(k) computed from (6-6) was implemented to force the output to track the reference signal r(k)=5. A step disturbance defined

by

[ O]T K < 40

V(k) [3 O]T K > 40


was introduced. The internal model was given by


Xc(k+l) = xc(k) + e(k)


with e(k)=y(k)-5. The parameters were estimated using the algorithm defined by (4-2)-(4-5). The initial conditions were chosen to be: P(O)=I, x(O)=[1 O]T, Xc(O)=O, Ro=I. The initial parameter estimates, parameter ranges, and steady state estimated values are displayed in Table 7-3. The output response, control input, and parameter estimates are shown in Figure 7-5. For comparison purposes, plots of the output responses and control input when the parameters are known exactly are shown in Figure 7-6. As expected, the transient response is considerably worse when the system parameters must be estimated.

However, once the system is in steady state, the adaptive system seems to respond equally well to a sudden step disturbance as the system using exact parameters.















Table 7-3 DATA


-a, -a2 bI b2

Actual Parameters -2.0 -5.0 -1.0 -0.5

Initial Estimates -0.5 -6.0 -1.0 0.0

Parameter Ranges [-3, 21 [-8, -4] C-1.1, -0.9] [-1, 1.0]

Estimation Scheme -1.64 -4.3 -0.90 -0.478
(4-2)-(4-5)






















20 40 60 80 100 120


140 (k)


u(k

80 60
40 20

- fj ' :"I .



-60 .
-80
(b)

Figure 7-5 Estimation Scheme (4-2)-(4-5)


20 15 10
5


-5
-10
-15
-20























,. 20 40


.-.


60 80 100 120


z- -4U 60 80 100 120


I 4 b 2(k) 140- bl(k)


(d)

Figure 7-5 continued
Estimation Scheme (4-2)-(4-5)


14n


-al(k)

-a2(k)


. l


140














y(k)

20 15 10

.,


_________________________ II
-~ I,
is


20 40 60 80 100


140 (k)


(a).


uk


. 20 40'.60


100


140 (k)


Figure 7-6
Exact Parameters


-5
-10
-15
-20


-20
-40
-60
-80









Example 7.4: Consider the non-minimum phase discrete-time system

described by


y(k)=2y(k-1)-0.99y(k-2)+0.5u(k-1)+3u(k-2)


The adaptive controller defined by the observer (5-3) and the control

law (6-5) with L(k) computed using (6-6) was implemented to force the system output to track the reference signal r(k)=5 and reject the sinusoidal disturbance v(k)=2sin (7rk/2). The internal model was given by






Xc(k+1) 0 0 J Xc(k) + 0 e(k)






with e(k) = y(k)-5. The system parameters were estimated using the

algorithm (4-2)-(4-5). The initial conditions were chosen to be P(O)=I, x(O)= [1 O]T, xc(O)=[O 0 O]T, and Ro=I. The initial parameter estimates, parameter ranges, and steady state estimated values are displayed in Table 7-4. Figure 7-7 includes plots of the system output response, control input, and estimated system parameters. For comparison purposes, both the system output and control input are shown in Figure 7-8 for the case when the parameters are known exactly.














Table 7-4 DATA


-a, -a2 b1 b2

Actual Parameters 2 -0.99 0.5 3.0

Initial Estimates 1.95 1.0 0.75 5.0

Parameter Ranges �1.9, 2.0] C-2.0, 4.0] C-0.5, 2.0] [2.5, 7.5]

Estimation Scheme 1.91 -0.920 0.406 2.83
(4-2)-(4-5)





















I,,. -


30 40 50 60 70 (k)


40 50 60 70 (k)


(b)

Figure 7-7
Estimation Scheme (4-2)-(4-5)


A~k

A^


I I. III. I
ii I.


I i
I.,,


- .i I .
10 11'I t20
Iit


-30 -


-40


u (I )


-10
-20
-30
-40


. i ll ', ". 2 ' 3. . 0o


-t- 1






















-~ U1


20 3 4n0 50 60 7 0


b.,(k)


h. (ic


10 20 30 40 50 60 70






(d)

Figure 7-7 continued
Estimation Scheme (4-2)-(4-5)


F



*1


- Ii .


I


f f, I


3


--I.U-


b W






















20 30 40 50 60 1 70 (k)


20 30 40 50 60 70 (k)


Figure 7-8
Exact Parameters


y(k)


40 30 20 10


-10
-20
-30
-40


I ~


" LU


u(k)


10


i








Example 7.5: Consider the single input two output discrete-time system described by


y(k) al y (k)1 + 3u(k-1)

_y2(k) -1 2 Y2 (k-1) bj\


where a and b are unknown parameters. As discussed in Chapter Three,

the required computations for parameter estimation can be reduced by separating the known parameters from the unknown parameters. This is

accomplished by rewriting the system equations as


y(k) = T, a(k-i) + *T b(k-1)

with




T 'Y k1 * 1 0 3' [y1(k-1)'1
F (k-1) = 9 T % (k-1) y2(k-1)I.
ab) Lu(k-1) ] 2 0 L u(k-1)j






The adaptive regulator defined by the observer (5-3) and the control law (5-10) with L(k) computed using (5-8)-(5-9) was implemented. The system parameter matrix e was estimated using the algorithm (4-2)-(4-5). The

initial conditions were chosen to be: P(O)=I, x(O)=[1 1]T, and Ro=I.

The initial estimates, parameter ranges, and steady state estimates are

given in Table 7-5. Plots of the output response, control input, and parameter estimates are shown in Figure 7-9.





















a b

Actual Parameters 4.0 1.0

Initial Estimates 6.0 -1.0

Parameter Ranges [2.0, 10.01 E-4.0, 2.01

Estimation Scheme 3.981 1.001
(4-2)-(4-5)


Table 7-5

DATA













yl(k)


40 30
20 L 10L


-10
-20
-30
-40






Y2 (k)

40 30 20 10


-10
-20
-30
-40


10 20 30 40 50 60 70 (k)





(b)


Figure 7-9
Estimation Scheme (4-2)-(4-5)


10 20 30 40 50 60 70 (k)





(a)

























20 30 40 50 60 70 (k)


80 60 40 20


-20
-40
-60
-80









8
6
4
2


-2
-4
-6
-8


-" 10 20 30 40 50 60 70 (k)






(d)

Figure 7-9 continued Estimation Scheme (4-2)-(4-5)


u(k)


a(k)

b(k)



-









Example 7.6: Consider the multi-input multi-output discrete-time system described by




E k)1 [I -a,21[y1 (k-i) +1 [ b12 1 ul (k-i)1

Y2(k) L1 2 J Y2 (k-i) b21 b22 u2 (k-i)

The order of required computations for parameter estimation is reduced by rewriting the system in the form


y(k) T Ta (k-i) + T b(k-I)

with

OT [a12 0 b12 , iaT(k.1) = Y2(k-i) ul(k-1) u2(k-l)],
0 b21 b22


= [ i 1 1bT(k-1) : Eyl(k-1) Y2(k-1) ul(k-1)].
1 2 0

The adaptive regulator defined by the observer (5-3) and the control law

(5-10) with L(k) computed using (5-8)-(5-9) was applied to the system. The algorithm given by (4-2)-(4-5) was used to estimate the system parameter matrix e. The initial conditions for the simulation were: P(O)=I, x(O)=[1 O]T, and Ro=I. The initial parameter estimates,

parameter ranges, and steady state estimates are displayed in Table 7-6. Figure 7-10 contains plots of the output response, control input, and parameter estimates.














Tabl e 7-6

DATA


____________12_ -a2121 2
Actual Parameters 0.75 2.0 3.0 -0.5

Initial Estimates 0.65 -1.0 2.5 1.0

Parameter Ranges �0.5, 0.8] [-5.0, 3.0] [1.0, 4.0] [-1.0, 3.0)

Estimation Scheme 0.785 1.58 2.97 -0.320
(4-2)-(4-5) 1













y1 (k)


r
10


5





-5


-101


20 30 40 50 60 70 (k)





(a)


Y2 (k)


10


5





-5


-10


1'i 10 j i


20 30 40 50 60 70 (k)





(b)


Figure 7-10
Estimation Scheme (4-2)-(4-5)













Ul(k) 10








-5


-10 u2 (k)


20 30 40 50 60 70 (k)





(d)

Figure 7-10 continued
Estimation Scheme (4-2)-(4-5)


20 30 40 50 60 70 (k)





(c)


,., "10


-5


-10


IIm .





(k)

77=
10 20 30 40 50 60 70 b 22 (k)





M


10 20 30 40 50 60 70





(e)


Figure 7-10 continued
Estimation Scheme (4-2)-(4-5)














CHAPTER VIII
OI.SCUSSION

A globally stable adaptive LQ controller which does not require persistent excitation was introduced for multi-input multi-output linear

discrete-time systems. The assumptions made on the plant were (1) an upper bound on system order is known, (2) the unknown system parameters belong to known bounded intervals, and (3) the plant is stabilizable for

all values of the unknown parameters ranging over the known bounded intervals. When applied to the unknown plant using this parameter estimator, the adaptive LQ controller ensures the system inputs and outputs will remain bounded, and forces the output to track a given reference signal in the presence of a deterministic external disturbance. Some remaining considerations include robustness,

application to time-varying or nonlinear systems, and analysis of transient response.

In many control applications, the order of the model will be lower than that of the plant. As mentioned in the introduction, Rohrs et al. [13-141 demonstrated that most adaptive controllers could go unstable if the order of the system was underestimated, even if the modelling errors were small. Since knowledge of an upper bound on system order is one of

the assumptions made on the adaptive controller presented in this dissertation, it is likely that this controller will also go unstable for systems with unmodelled dynamics. There have been several

approaches in the literature towards development of robust adaptive controllers, some of which will be briefly discussed here.




Full Text
91
Example 7.6: Consider the multi-input multi-output discrete-time system
described by
Vl(k)'
'l
"a12
*1
(k-1)*
1
b12
ux (k-1)
y2 -1
2
/2
(k-1)
b2i
b22 _
u2 (k-1)
The order of required computations for parameter estimation is reduced
by rewriting the system in the form
y(k) eTa(k-l) + i|T<(b( k-1)
with
9
T =
1
-1
0
2
>12
)22
1
0
, <|>aT(k-l) = [y2(k-l) u1(k-l) u2(k-l)],
, 4>bT(k-1) = Cyx(k-l) y2(k-1) u^k-1)].
The adaptive regulator defined by the observer (5-3) and the control law
(5-10) with L(k) computed using (5-8)-(5-9) was applied to the system.
The algorithm given by (4-2)-(4-5) was used to estimate the system
parameter matrix e. The initial conditions for the simulation were:
P(0)=I, x(0)=[l 0]^, and RQ=I. The initial parameter estimates,
parameter ranges, and steady state estimates are displayed in Table
7-6. Figure 7-10 contains plots of the output response, control input,
and parameter estimates.


58
P-1
z(k+p) = A(k+p,k)z(k) + J A(k+p,k+j+l)w(k+j).
j=0
Taking the norm of both sides and using (5-12) results in
P-1
nz(k+p)n < iiA(k+p,k)n nz(k) n + £ nA(k+p,k+j+l) n nw(k+j)n
j=0
P1
< nA(k+p,k)n tiz(k) it + l nA(k+p,k+j+l) n(a(k+j) nz(k+j) 11 +b(k+j)).
j=0
It follows from equations (5-13), (5-14), and (5-16) that
p P-1 _
I!z(k+p) ii < 6iiz(k) ii + max(l,R!j l a(k+j)(R9iiz(k) II + R~) + b(k+j) for
1 j=0
all k. (5-17)
Since I(k) converges to zero, there exists a finite N such that
a(k) <
1
6 + pRgdiaxil ,R^)
for all k > N.
Therefore, for all k > N,
p P-1 _
iz(k+p)n < rnz(k)ii + max(l,R¡|) l a(k+j)R, + b(k+j) or equivalently.
1 j=0 J
iz(k+p)ii < rnz(k)n + c(k)
(5-18)
where r < 1 and c(k) converges to zero as k - . In order to show
(5-18) implies uz(k)n converges to zero, let e>0 be given. Solving
(5-18) recursively and using (5-16) gives


7
where
F =
"al
1
bl
-a2
1
b2


G =


"an-l

1

bn-l
"an
0
bn
H = [1 0
0].
Samson and Fuchs assume that (F,G) is stabilizable. This assumption
allows for nonminimum phase systems and systems with stable common poles
and zeros. The adaptive IQ control law used by Samson and Fuchs [12] is
given by
u(k) = -L(k)x(k), (2-5)
where u(k) is the input to the given plant, x(k) is the state estimate,
and L(k) is a stabilizing feedback gain for the estimated system (F(k),
fi(k)).
The state estimate x(k) is generated from an adaptive observer.
Letting F(k) and G(k) represent the current estimates for matrices F and
G, the adaptive observer for the system is given by
x(k+l) = F(k)$(k) + G(k)u(k) + M(k) (y(k)-y(k)) (2-6)
y(k) = H?(k)


82
Example 7.4: Consider the non-minimum phase discrete-time system
described by
y(k)=2y(k-l)-0.99y(k-2)+0.5u(k-l)+3u(k-2)
The adaptive controller defined by the observer (5-3) and the control
law (6-5) with L(k) computed using (6-6) was implemented to force the
system output to track the reference signal r(k)=5 and reject the
sinusoidal disturbance v(k)=2sin (irk/2). The internal model was given
by
xc(k+D -
1
o
1
o'
1
o
1
0
0
1
xe00 +
0
1
-1
4
1
1
l*
1
e(k)
with e(k) = y(k)-5. The system parameters were estimated using the
algorithm (4-2)-(4-5). The initial conditions were chosen to be P(0)=I,
x(0)= [1 Of. xc(0)=[0 0 0]^, and R0=I. The initial parameter
estimates, parameter ranges, and steady state estimated values are
displayed in Table 7-4. Figure 7-7 includes plots of the system output
response, control input, and estimated system parameters. For
comparison purposes, both the system output and control input are shown
in Figure 7-8 for the case when the parameters are known exactly.


29
It then follows that
e(m) e(n) = '(m) een) = (P(m) P(n))P*(O)e,(0)
m-1 1
-P(m) l P'Mi + l)P(i)f(0(i))
1=0
+P(n) Y P-1(1 +l)P(1)f(0(D).
1=0
Without loss of generality, assume that m is greater than n. Replacing
the matrix P(n) preceding the summation with P(m) + (P(n) P(m)) gives
i m-1 1
0(m) 0(n) = (P(m) P(n))P"1(O)0(O) P(m) Y P-1(i + 1)P(i)f(0(i))
i=n
+ (P(n) P(m)) l P-1(i + l)P(i)f(0(i)).
1=0
Using the properties of the induced matrix norm defined in Chapter II
gives
0ij(m) 0ij(n)| < lo(nn) 0(n) i
< iP(m) P(n) ii nP"1(0)?(0)n
. m-1 n
+ nP(m> n l Pi(i+l)P(i) i iif(e(i)) ii
i=n
(4-21)
n-l ,
+ nP(n) P(m) ii l nP (i+l)P(i)n If (0(i)) ii .
i=0
If the right hand side of equation (4-21) can be made arbitrarily small
for m and n sufficiently large, then each component e.^(k) of the matrix
* J


90
u(k)
Figure 7-9 continued
Estimation Scheme (4-2)-(4-5)


64
stabilizing feedback gain L(k) = [L^(k) L?(k)] can be computed from one
step of the Riccati difference equation
L(k) = [GT(k)RkG(k) + I]"1 GT(k)RkF(k) (6-6)
Rk+1 = Q + LT(k)L(k) + (F(k)4(k)L(k))TRk(F(k)-G(k)L(k)).
Again, RQ and Q are arbitrary positive definite symmetric matrices.
In order to use the existing results on tracking and disturbance
rejection, the closed-loop system consisting of (6-4) and the control
law (6-5) must converge to a time-invariant system. Properties (v) and
(vii) of Theorem 4.1 imply that the pair (F(k),G(k)) converges to a
stabilizable pair (F ,G ). Using the results of Samson CLO], L(k)
00 00
converges to L the optimal LQ feedback for the pair (F ,G ) given by
00 00
L = [GTR G + I]-1 GT R F
00 00 00 CO 00 00 00
where the matrix R is the solution to the ARE for (F ,G ) More
00 00
precisely,
R = Q + LTL + (F -G L )TR (F -G L ).
Applying the control law (6-5) to the system described by (6-4)
results in an exponentially stable system driven by e(k). From the
results in Chapter Five, the observer error e(k) converges to zero.
Also, it follows from property (ii) of Theorem 4.1 that M(k) is
bounded. Thus, the closed-loop system consisting of (6-4) and the


TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT iv
CHAPTERS
I INTRODUCTION 1
II BACKGROUND 4
III SYSTEM DEFINITIONS AND ASSUMPTIONS 12
IV PARAMETER ESTIMATION 18
V ADAPTIVE REGULATOR 36
Uniform Stabilizability of the Estimated System 36
Adaptive Observer 41
Feedback Gain Sequence 44
Stability of the Adaptive Closed-Loop System 55
VI APPLICATION TO TRACKING AND DISTURBANCE REJECTION 61
VII SIMULATIONS 66
VIII DISCUSSION 96
REFERENCES 100
BIOGRAPHICAL SKETCH 103
i i i


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
f t'}'c~> *- J
S. /Svoronos
Professor of Chemical Engineering
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
August 1986
Dean, College of Engineering
Dean, Graduate School


45
over fl. This approach has been developed by a number of individuals,
see for example [4-8]. A drawback to this approach is that (F(k),G(k))
may not be reachable or stabilizable at certain points in time (i.e.,
0(k) may be a singular point for some values of k). Recall that
Proposition 5.1 and property (v) of Theorem 4.1 only guarantee that
(F(k),G(k)) is stabilizable for all k exceeding some finite N. Thus,
for time points prior to N, the chosen control law may not have a
solution. Therefore, in this approach it is necessary to check whether
or not (F(k),G(k)) is reachable or stabilizable at each time point k,
which adds to the number of on-line calculations. Also, a decision must
be made on how to modify the control law when singular points are
encountered. A common choice in the literature is to update the control
law only when (F(k),G(k)) is stabilizable.
Another approach first introduced by Kreisselmeier for continuous
time systems is to compute the feedback gain asymptotically [29]. This
approach was developed for SISO discrete-time systems by Samson and
Fuchs [10-12] using an LQ control strategy, and later extended to MIMO
discete-time systems by Ossman and Kamen [30-32].
In [30-32], the stabilizing feedback L(k) is computed at each time
point k by solving in real time one step of a Riccati difference
equation. More precisely, let
L(k) = [GT(k)RkG(k) + I]_1GT(k)RkF(k) (5-8)
where R(k) is the solution to the Riccati difference equation
Rk+1 = Q + LT(k)L(k) + (F(k) G(k)L(k))TRk(F(k) G(k)L(k)).
(5-9)


57
M-l
A(k+M,k) = n A(k+i)
i=0'
Since A(k) is an exponentially stable time-varying system matrix, it
follows that
nA(k+M,k)I < R1 < for all k,M > 0. (5-13)
Also, given 0<5<1 there exists a finite p such that
nA(k+p,k)ii < 6 for all k. (5-14)
For the remainder of the proof, p will be a fixed constant such that
(5-14) is satisfied. Taking the norm of both sides of (5-11) and using
(5-12) gives
nz(k+l)n < A(k) iinz(k) n + nw(k)ii (5-15)
< ( h A( k) d + i(k)) nz(k) ii + b(k).
Noting that nA(k)n, a(k), and b(k) are all bounded and p is fixed, it
follows from equation (5-15) that there exist finite constants R2 and Rg
such that
11 z(k+i) 11 < R^t!z(k) 11 + Rg for all ie [0,p] and for all k. (5-16)
Solving equation (5-11) recursively gives


37
where
*
-A1 1
_A2
32


G =




-Vi
I
Bn-1
An
0
Bn
H = [I 0 ... 0]
The entries in F and G depend on the unknown system parameter matrix
6. In fact, by the definition of e in (3-3), F and G can be viewed as
continuous functions of 9 where 9 ranges over the parameter
space n = {9e ]R^xm: 9..e[9?!n 9*il?x]}. This dependance can be made
1J T J T 3
explicit by denoting F and G as F(e) and G(e) respectively. Assumption
A3 in Chapter Three is equivalent to requiring that the system described
by (F(9),G(9)) is pointwise stabilizable over n.
Definition: The pair (F(g),G(9)) is pointwise stabilizable over fi if
for each 9 belonging to n, there exists a feedback matrix L(e) such that
Uma(F(9)-G(9)L(9))I < q(9) < 1 where x^ () is the eigenvalue of
maX max
() with the greatest magnitude.
It follows from known results [24], that pointwise stabilizability
of (F(0) ,G(e)) over n is equivalent to the rank condition (3-4). Thus,
stabilizability over a can be checked using the polynomial matrices
D(z,9) and N(z,0) comprising the system transfer function. Given e > 0,
let be an extension of the parameter space n defined as
Q
e
rn mNxrn rAmin AmaxL i,
{0e F : 9..e[9.. -e, 9.. +e]}.
(5-2)


3
algorithms, such as the recursive least-squares, projection, and
orthogonalized projection algorithms discussed in [6], which do indeed
have the required properties.
In this dissertation, an indirect adaptive regulator which does not
require a persistently exciting input is derived for multi-input multi-
output discrete-time systems. In Chapter II, more explicit background
material on parameter estimators and indirect adaptive controllers is
presented. Chapter III contains the system definitions and the
assumptions which include (1) an upper bound on system order is known,
(2) the unknown system parameters belong to known bounded intervals, and
(3) the plant is stabilizable for all possible values of the unknown
parameters ranging over the known intervals. A parameter estimator is
derived in Chapter IV which has the required properties for proving
global stability of indirect adaptive controllers and, in addition,
forces the estimates of the parameters to converge to the known bounded
intervals. In Chapter V, it is first shown that the assumptions listed
in Chapter II ensure the parameter estimator described in Chapter IV
will generate a uniformly stabilizable estimated system. Using this
parameter estimator, an adaptive LQ regulator is developed which, when
applied to the unknown plant, results in a globally stable closed-loop
system in the sense that the system inputs and outputs converge to zero
asymptotically. The results are then extended in Chapter VI to the case
of tracking and/or deterministic disturbance rejection using the
internal model principle [20]. Chapter VII includes simulations of the
adaptive LQ controller for both SISO and MIMO discrete-time systems. A
discussion of the results and further work to be investigated is
included in Chapter VII.


101
[13] C. Rohrs, "Adaptive Control in the Presence of Unmodeled
Dynamics," Ph.D. dissertation, Dept. Elec. Eng. Comput. Sci.,
Mass. Inst. Techno!., Aug. 1982.
[14] C. Rohrs, L. Valavani, M. Athans, G. Stein, "Robustness of
Continuous-Time Adaptive Control Algorithms in the Presence of
Unmodeled Dynamics," IEEE Trans. Automat. Contr., Vol. AC-30, No.
9, pp. 881-889, Sept. 1985.
[15] G. Kreisselmeier, "A Robust Indirect Adaptive-Control Approach,"
Int. J. Control, Vol. 43, No. 1, pp. 161-175, 1986.
[16] G. Kreisselmeier and B.D.O. Anderson, "Robust Model Reference
Adaptive Control," IEEE Trans Automat. Contr., Vol. AC-31, No. 2,
Feb. 1986.
[17] K.S. Narendra and A.M. Annaswamy, "Persistent Excitation and
Robust Adaptive Algorithms", in Proc. 3rd Workshop on Applications
of Adaptive System Theory, Yale University, New Haven, CT, pp. Il
ls, 1983.
[18] B.D.O. Anderson, "Exponential Convergence and Persistent
Excitation," in Proc. 21st IEEE Conf. Decision Contr., Orlando,
FL, pp. 12-17, 1982.
[19] P.A. Ioannou and P.V. Kokotovic, "Singular Perturbations and
Robust Redesign of Adaptive Control," in Proc. 21st Conf. Decision
and Contr., Orlando, FL, 1982.
[20] C. Chen, Linear System Theory and Design, CBS College Publishing,
New York, N.Y., 1984.
[21] R. Lozano-Leal and G.C. Goodwin, "A Globally Convergent Adaptive
Pole Placement Algorithm Without a Persistency of Excitation
Requirement," IEEE Trans. Automat. Contr., Vol. AC-30, No. 8, pp.
795-798, Aug. 1985.
[22] P. De Larminat, "On the Stability Condition in Indirect Adaptive
Control," Automtica, Vol. 20, pp. 793-795, 1984.
[23] G. Kreisselmeier, "An Approach to Stable Indirect Adaptive
Control," Automtica, Vol. 21, pp. 425-431, July 1985.
[24] P.P. Khargonekar and E.D. Sontag, "On the Relation Between Stable
Matrix Fraction Factorizations and Regulable Realizations of
Linear Systems over Rings," IEEE Trans. Automatic Control, Vol.
AC-27, pp. 627-638, 1982.
[25] E.W. Kamen and P.P. Khargonekar, "On the Control of Linear Systems
Whose Coefficients Are Functions of Parameters," IEEE Trans.
Automat. Contr., Vol. AC-29, No. 1, pp. 25-33, Jan. 1984.
[26] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New
York, 1976.


95
8
6
4
2
-2
-4
-6
-8
10
20
30
40
50
60
b
-a
-12
12
(k)
XE)
70
(e)
8
6
4
2
-2
-4
-6
-8
UmXf.
10
20
30 40 50
(f)
60
b21(k)
70 b22(k)
Figure 7-10 continued
Estimation Scheme (4-2)-(4-5)


85
Figure 7-7 continued
Estimation Scheme (4-2)-(4-5)


41
Adaptive Observer
The adaptive observer for the estimated system is given by
x(k+l) = F(k)x(K) + G(k)u(k) + M(k)(y(k)-y(k)) (5-3)
y(k) = H>t (k)
where
*A1(k) I
A2(k)

\w
B2(k)

F(k) =

I
_-An(k) 0
G(k) =


Bn(k)
M(k)
-Ax(k)
-A2(k)
H = [I 0 .... 03.
The adaptive observer defined by (5-3) is a dead-beat observer. The
output y(k) of the adaptive observer can be expressed in the following
form:
y(k) = C-AjU-1)

-An(k-n) B1(k-1)
Bn(k-n)](j>(k-l)
(5-4)


30
sequence e(k) will be a Cauchy sequence. It will then follow using
property (v) that e^. (k) converges to a point in the interval
[e?1n>e?!x]. In order to prove that the right hand side of the in-
T J 1J
equality (4-21) can be made arbitrarily small, two things must be
-1
shown: nP (k+l)P(k)ii < v < for all k, and £ nf(e(k))ii < L < .
-1 k=1
In order to prove nP (k+l)P(k)n < v < for all k, first assume that
the determinant of P(k) does not converge to zero. In this case,
P_1(k+1) converges to P-1 which implies that P^(k+l)P(k) converges to
oo
I. Therefore, i P"*(K+l)P(k) n is bounded for all k when the deter
minant of P(k) does not converge to zero. Now suppose the determinant
of P(k) does converge to zero. Applying the matrix inversion lemma to
equation (4-3) gives
P-1(k+l)P(k) =
I + *a(k)^(k)P(k)
2
Using the properties of the induced matrix norm, it follows that
i a(k) J(k)n
iPA(k+l)P(k) i < iiln + 2 J iiP(k) I, (4-22)
nk
From (4-5), there exists a finite M such that nk = max (1,na(k) ii) for
k > M. It then follows that all the components of the matrix
+a(k)*a(k)
2
nk
have magnitude less than or equal to one for k > M which in
turn implies
^aOO^k)
2
nk
Hill = 1 and nP(k)n < 4,
is bounded for all k.
equation (4-22) implies
Noting also that
HP'^k+ljPk) ii is


8
where
M(k)
-a^k)
, a^(k) = estimate of a^ at time k.
-an(k)
The feedback gain sequence is computed at each time point k by
solving one step of a Riccati difference equation:
R(k+1) = Q + LT(k)L(k) + (F(k) G(k)L(k))TR(k)(F(k) G(k)L(k))
L(k) = [GT(k)R(k)G(k) + I]1 GT(k)R(k)F(k) (2-7)
In (2-7), Q and the initial value R(0) are arbitrary positive
definite symmetric matrices.
In order to prove global stability of the closed-loop system,
Samson and Fuchs make two assumptions in addition to the stabilizability
assumption on (F,G) mentioned previously. First, the parameter
estimator must possess the following three properties:
i. ne(k)n < M < for all k
ii. ne(k) e(k-m) n + 0 as k > for any finite m
iii.|e(k)| < a(k) n<|>(k-l) n + $(k)
where a(k) and g(k) converge to zero.


92
Table 7-6
DATA
"a12
b12
b21
b22
Actual Parameters
0.75
2.0
3.0
-0.5
Initial Estimates
0.65
-1.0
2.5
1.0
Parameter Ranges
I1
00

o
A
in

o
i i
[-5.0, 3.0]
[1.0, 4.0]
[-1.0, 3.0)
Estimation Scheme
(42)(45)
0.785
1.58
2.97
-0.320


BIOGRAPHICAL SKETCH
The author was born in Cincinnati, Ohio, on July 19, 1959. She
received a BSEE and a MSEE degree from the Georgia Institute of
Technology in March 1982 and December 1982 respectively. She expects to
receive a Ph.D. in electrical engineering from the University of Florida
in August 1986.
103


50
In other words, Pk is equal to Rk for all k up to time t. After time
t, p£ is generated from the time-invariant Riccati difference equation
initialized by Rt and using the constant matrices F(*) and G()
evaluated at 9(t). For every e(t) belonging to n the sequence p£
will be bounded for all k. In fact, P^ converges to P^ whenever
K oo
9(t)eiU. It will first be shown that nRk+N P^r < e/6 for k
sufficiently large. The matrix p^ is simply the Nth step of a time-
invariant ROE initialized at time k by Rk and using constant matrices
F() and G() evaluated at 9(k). The interpretation for
llRk+N Pk+N11 bei'ng sn,a",1 For k sufficiently large is that the time-
varying RDE given by (5-9) will not vary much from the time-invariant
ROE over a finite interval of time N as long as 9(k) does not change
much over the time interval N. Equation (5-9) can be rewritten as
Rk+i+i 55 Q + FT(0(k))Rk+iF(e(k)) FT(e(k))Rk+iG(e(k))
X [I + GT(9(k))Rk+.G(9(k))]1 GT(9(k))Rk+.F(9(k)) + Zk+.; RR
for all k and for all ie[0,N].
The matrix zk+i is given by
Ek+i = FT(0(k+i))Rk+iF(9(k+i)) PT(e(k+i))Rk+-j G(9(k+i))
x [I + GT(9(k+i))Rk+.G(9(k+i))]"1 GT(9(k+i))Rk+.F(9(k+i))
FT(9(k))Rk+.F(9(k)) + FT(9(k))Rk+.G(9(k))
x [I + GT(9(k))Rk+.G(9(k))]'1 GT(9(k))Rk+.F(9(k)) for all k and
for all ie[0,N].



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CHAPTER V
ADAPTIVE REGULATOR
The r-input m-output linear discrete-time system described by
(3-1)-(3-3) will be regulated using a MIMO version of Samson and Fuchs'
adaptive LQ controller [12]. As discussed in Chapter Two, a necessary
condition for the adaptive closed-loop system to be globally stable is a
uniformly stabilizable estimated system. It will be shown that the
parameter estimator described by equations (4-2)-(4-5) will give a
uniformly stabilizable estimated system provided that assumptions A1-A3
summarized at the end of Chapter Three are satisfied.
Uniform Stabilizability of the Estimated System
Using the parameter estimator described by (4-2)-(4-5), it will be
shown that the estimated system can be stabilized using state
feedback. As shown in [6], the state-space observer form realization
for the system described by (3-1) is given by
x(k+l) = Fx(k) + Gu(k) (5-1)
y(k) = Hx(k)
36


21
Theorem 4.1: The estimation algorithm (4-2) (4-5) has the following
properties
i. P(k) converges to a positive semidefinite matrix P^ < 21;
ii. | 9..(k)| < M < for all k and i = 1,2,...,N, j = l,2,...,m;
J
m
ie(k)
2
nk-l
4>^(k-l) P( k-1) <(>a(k-l)
+ 0 as k + ;
iv. iie(k)n < a(k)a(k-1)n + b(k), where a(k) and b(k) are positive
scalar sequences which converge to zero;
v. f(e(k)) + 0 as k + , which implies that j(k) converges to the
set [9,I'1n>0^x] for i = 1,2 N, j = 1,2,...,m;
' J \J
VI .
|0. .(k) 0..(k-p) j -* 0 as k for any integer p,
' J J
i 1 y2y t yNy j ^92^ ^
vii. If 0.. 0^" or 0^x for every i and j, then 0..(k) converges
' J J U ^ J
j, f-jroi n .max-,
to a point in [0.. ,0.. ]*
* j j
Proof: Property (i) is a well-known property first proved by Samson
[10] with ti^.i equal to one. The proof is included here for the sake of
completeness. Equation (4-3) implies that 0 < x^P(k)x all xe RN, and for all k > 0. Therefore, for each x eEN, the se
quence x^P(k)x is monotone decreasing and must converge. Denoting
P.jj(k) as the ij-^L element of P(k), it follows that
(e. + eJ.)TP(k)(ei + e^.) elp(k)ei eTp(k)e..
PijOO =
(4-7)


68
y(k)
u(k)


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
l/l f
E. W. Kamen, Chairman
Professor of Electrical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
T. E. Bullock
Professor of Electrical Engineri
ng
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
_
Professor of Industrial and Systems
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
P. Z. Peebles '
Professor of Electrical Engineering


59
nz(k+i+np) ii < rnnz(k+i)ii + l rn*^c(k+i+jp)
j=0
n-1 .
nz(k+i+np) ii < rn(R9nz(k) ii + R-) + £ rn Jc(k+i+jp)
d j=0
for all k > N, ie[0,p]. (5-19)
Since c(k) converges to zero, there exists a finite > N such that
c(k) < f for all k > N,.
1
Applying this inequality to (5-19) gives
Hz(N^+i+np) ll < rn(R2Hz(N^) It + Rg) + e/2, ie[0,p].
Also, since r < 1 there exists a finite N2 such that
r < I(R2hz(N^)ii + R3) for a11 n > V
Thus
Hz(N^+i+np) 11 < e for all n > N2, ie[0,p]
which implies nz(k) 11 < e for all k > + N2p.
By the definition of z(k), both the system input u(k) = -L(k)x(k) and
the system output y(k) must converge to zero. Therefore the adaptive


27
where a(k) and b(k) are positive scalar sequences which converge to
zero. To prove (vi), first subtract e^k-l) from both sides of equa
tion (4-8). Pra-multiplying both sides of the resulting equation by its
transpose and noting that e(k) = -/eT(k-l)<¡> (k-1) gives
Tr[(e(k)-9(k-l))T(0(k)-9(k-l))] = Tr[(e(k) -'9(k-l))T(0(k) -?(k-l))>
e(k)J(k-l)P(k-l)2* (k-l)eT(k)
y p ~ q ~ ~ ~
(Vl + ^(k-l)P(k-l)^a(k-l))2 _
+2Tr
t P(k-l)2* (k-l)J(k-l)9(k-l)
fT( 9(k-l)) 2 2
Vl + a(k1)p(kl)*a(,c_1)
(4-13)
+ Tr[fT(e(k-l))P(k-l)2f(e(k-l))].
Also
I(k-l)P(k-l)2* (k-1) a(k-l)P(k-lH (k-1)
3 3 < CW>()3 T-TT
n2_! + ^(k-l)P(k-l)*a(k-l)
nk-l++aik"1,p(k'1)*a(k1)
< X P(0)
max '
(4-19)
Combining (4-18) and (4-19) gives
TrC(0(k)-e(k-l))1(0(k)-e(k-l))3 < CxmaxP(0)]
ie(k) ii'
Vl+Vk1)P(k_1)+a(k1)
+ 2Tr
T P(k-l)2 (k-l)/(k-l)0(k-l)
fT(0(k-l))^- a a
nk-l + a (k-DP(k-l)^(k-l)
+ *2ax (P(O))Tr[fT(0(k-l))f(0(k-l))]. (4-20)


77
Example 7.3: Consider the discrete-time system described by
y(k) = -2y(k-l)-5y(k-2)-u(k-l)-0.5u(k-2)
The adaptive controller defined by the observer (5-3) and the control
law (6-5) with L(k) computed from (6-6) was implemented to force the
output to track the reference signal r(k)=5. A step disturbance defined
by
Of
of
K < 40
K > 40
was introduced. The internal model was given by
xc(k+l) = xc(k) + e(k)
with e(k)=y(k)-5. The parameters were estimated using the algorithm
defined by (4-2)-(4-5). The initial conditions were chosen to be:
P(0)=I, x(0)=[l of, xc(0)=0, R0=I. The initial parameter estimates,
parameter ranges, and steady state estimated values are displayed in
Table 7-3. The output response, control input, and parameter estimates
are shown in Figure 7-5. For comparison purposes, plots of the output
responses and control input when the parameters are known exactly are
shown in Figure 7-6. As expected, the transient response is
considerably worse when the system parameters must be estimated.
However, once the system is in steady state, the adaptive system seems
to respond equally well to a sudden step disturbance as the system using
exact parameters.


35
ne(n)n < iP(n) 11 nP"1(0)'e(0) n + nP(n) u n l P-1(i+l)P(i )f (e(i)) n.
i =0
nr -1
It was shown previously that ii l P (i+l)P(i )f (e(i)) n is bounded in
i=Q
the limit as n - . Also iiP(n)n can be made arbitrarily small for n
sufficiently large if P(k) converges to zero. Thus, e^n) converges to
zero as n which implies e(k) converges to the true system param
eter matrix e whenever P(k) converges to zero.


67
not. Figures 7-1-7-3 contain plots of the output response, control
input, and estimated parameters for each of the three cases. A
comparison of Figures 7-1 and 7-2 show the estimation algorithm defined
by (4-2)-(4-5) has a slightly better transient response than least
squares. Figure 7-3 illustrates the detrimental effect of the data
normalization on the transient response of the system. As previously
mentioned, e in (4-5) should be chosen very small to minimize the effect
of data normalization on the system transient response.
Table 7-1
DATA
"al
~a2
bl
b2
Actual Parameters
2.0
-0.99
0.5
3.0
Initial Estimates
1.95
1.0
0.75
6.0
Parameter Ranges
[1.9, 2.0]
[-2.0, 4.0]
[-0.5, 2.0]
[2.5, 9.5]
Estimation Scheme
(4-2)-(4-5)
2.00
-1.00
0.530
3.04
Recursive Least
Squares
2.11
-1.05
0.594
3.19
Estimation Scheme
(4-2)-(4-5) with
= max(l,ne(k)n)
2.00
-0.991
0.571
3.14


38
Proposition 5.1: Suppose assumption A3 is satisfied so that
(F(e) ,G(e)) is pointwise stabilizable over n. Then there exists an
e > 0 such that (F(e),G(e)) is also pointwise stabilizable over n .
Proof: Suppose (F(e)G(e)) is stabilizable over ft. For each 9
belonging to ft, there exists a bounded open neighborhood VQ of 9 such
that (F(e),G(9)) is stabilizable over V0. This result is proved in
[25], by first showing that there exists a feedback matrix L(e) with
entries which are continuous in 9 such that
|x (F(9)-G(e)L(0))| < q(9) < 1. The existence of the open
neighborhood V0 of 9 then follows because the eigenvalues of
(F(e)-G(e)L(9)) are continuous functions of 9. Since ft is a compact
subset of I'*xm and U V0 is an open cover of n, there exists a
9 eft
finite subcover U V of ft. For k=l,2,..., let ek = ( V2 )k. For
j=l 9j
r
every k > 1, n U Va isa compact subset of FNxm and
ek j=l 9j
r r
ft U VQ C ft U Vfl
ek+l j=l 9j ek j=l 9j
Suppose ft U V. is nonempty for all k > 1. By the properties of
ek j=l 9j
nested compact sets (see [26, p.38]) there is at least one point p which
belongs
k > 1
r
to ft U for all k > 1.
ek j-i ej
Therefore, nen
ek
which implies that pen.
However, p^
r
U V
j=l
for all
which forms an


40
Property (v) in Theorem 4.1 implies that there exists a finite N such
that e(k) belongs to ne for all k > N. It then follows from the
result in Kamen and Khargonekar given above that there exists a feedback
gain sequence L(k) continuous in 9 such that
< q < 1 for all k > N.
fnaX
Since the entries in (F(k)-G(k)L(k)) are continuous functions of 9,
property (vi) in Theorem 4.1 implies [F(k)-G(k)L(k)]-
[F(k-l)-G(k-l)L(k-l)] converges to zero as k goes to infinity. Oesoer
[28] proves A(k) will be uniformly asymptotically stable as a system
matrix if (i) all eigenvalues of A(k) have magnitude less than some
constant q < 1 for all k and (ii) A(k)-A(k-1) converges to zero as k
goes to infinity. Using Desoer's results on stability of slowly time-
varying systems, (F(k)-G(k)L(k)) is uniformly asymptotically stable as a
system matrix. Thus the estimated system (F(k),G(k)) is uniformly
stabilizable as a time-varying system.
The adaptive control law for regulating the system described by
(3-1) is of the form u(k) = -L(k)x(k). The state estimate x(k) is
generated from an adaptive observer which will be discussed in the next
section. The feedback gain matrix L(k) is chosen to stabilize the
estimated system (F(k),G(k)). The existence of a stabilizing feedback
gain L(k) is guaranteed by Theorem 5.1. Several possible options for
L(k) are given in the section following the adaptive observer section.


CHAPTER VI
APPLICATION TO TRACKING AND DISTURBANCE REJECTION
The adaptive controller derived in Chapter V can be applied to the
problem of tracking with disturbance rejection using the internal model
principle. The internal model principle has been discussed by a number
of individuals, see for example [20].
Consider the r-input m-output plant described by
where
F =
x(k+l) = Fx(k) + Gu(k) + Dv(k)
y(k) = Hx(k)
(6-1)

'-Ai I
B1
A?
2 I

b2


G =


-Vi I

Bn-1
-An 0
. Bn .
H = [I 0 ... 0].
The vector v(k) consists of the exogenous disturbance signals. As in
previous chapters, it is assumed that some or all of the entries in the
matrices Aj and Bj are unknown and will be estimated using the parameter
estimator described by (4-2)-(4-5). Letting r(k) denote an m-vector
reference signal, the objective is to design an adaptive controller
which stabilizes the system described by (6-1) and forces the tracking
error e(k) = y(k)-r(k) to converge to zero as k -> .
61


15
ft {0s IR
Nxm
V
r imn inax-,..
L0.. ,0.. j}.
The concept of compactness (i.e., closed and bounded) is relative to the
matrix norm previously defined.
The only additional assumption made on the system is a pointwise
stabilizability assumption over ft. Specifically, the system described
by (3-3) is assumed to be stabilizable for each 0eft. This assumption
can be verified a priori because it depends only on the known set ft, not
on the parameter estimates. The stabilizability condition can be tested
using the transfer function matrix of the system. Let D(z"*) and N(z"*)
denote the polynomial matrices defined by
OU1) = 1+1 A. (z~J), NU"1) = l B.(zj).
j=l J j=l J
The coefficients of D(z"l) and N(z"^) depend on the system parameters
and therefore can be viewed as functions of 0. This dependancy is made
explicit by denoting Diz1) and N(z-1) as D(z-1,0) and N(zl,e)
respectively. The transfer function matrix of the system described by
(3-1) is then given by
W(z-1,0) = D1(z"1,8)N(z1,9).
The stability assumption is that the system described by (3-1) with
transfer function matrix W(z-1, 0) can be stabilized by dynamic output
feedback for all 0eft. From well known results [24], this assumption
is equivalent to the following rank condition:


31
also bounded for all k in the case when the determinant of P(k) con
verges to zero. Therefore, there exists a finite v such that
nP*(k+l)P(k) ii < v < for all k. Applying this result to the in
equality (4-21) gives
1 a/ m-l
|9i (m)-e. .(n) | < nP(m)-P(n) n nP"i(0)9(0)ii + 2v l nf(e(i))n
J i=n
n-1
+ nP(m)-P(n)nv £ nf(e(i))a. (4-23)
i=0
In order to complete the convergence proof, it must be shown that
l ilf(e(k)) n < L < . It follows from equation (4-13) that
k=0
V(k) < V(k-l) -
iie(k)
Vi + +¡(k-l)P(k-D*a(k-l)
- 2Tr[e (k-l)f(e(k-l))]
+ xmax[P(k-l)P"1(k)P(k-l)]Tr[fT(0(k-l))f(9(k-l))]. (4-24)
Now assume < 9.. <0^X for i = 1, ..., N and j = 1, ..., m (i.e.,
0i is an interior point of [el1!11, 0n!'?x]). Using the definition of
'J ij ij
f(0(k-l)) in (4-4), if fij(0(k-l)) 0 then |eij-(k-l)| > q > 0 where q
n -IT13X mirin a i,
= [(j ij. 'ij-,] N.
J
j=l, ..., m. Also from the definition of f(e(k-l)), it must be true
that the sign of e^k-l) is equal to the sign of fin-(9(k-l)) for i=l,
ij
..., N and j=l, ..., m. These facts when combined with equation (4-24),
imply
V(k) < V(k-l) -
N m
2 -y- 2q l £ |fii (0(k-l))
- ,ln' 1' i=l j=l J
ne(k) ii'
nk-l+ i>;(k-DP(k-l) + ^xCPk-nP'^kPk-mrC^eU-Untetk-1))].


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
NEW RESULTS IN INDIRECT ADAPTIVE CONTROL
By
KATHLEEN A. K. OSSMAN
August 1986
Chairman: Dr. Edward W. Kamen
Major Department: Electrical Engineering
An adaptive regulator which does not require a persistently
exciting input is derived for multi-input multi-output linear discrete
time systems. The assumptions made on the unknown plant are (1) an
upper bound on the system order is known, (2) the system parameters
belong to known bounded intervals, and (3) the plant is stabilizable for
all possible values of the unknown system parameters ranging over the
known intervals. A recursive parameter estimator is described which
forces the estimates of the system parameters to converge to the known
intervals asymptotically. Using this parameter estimator, an adaptive
LQ regulator is developed which results in a globally stable adaptive
closed-loop system in the sense that the system inputs and outputs
converge to zero asymptotically. The results are then extended to the
case of tracking and/or deterministic disturbance rejection using the
internal model principle. Simulations illustrating the performance of
the adaptive controller for several discrete-time systems are included.
iv


56
Also, the regression vector <{>(k) defined in (3-2) can be written in
the form
4>( k) = S(j>(k-1) + D(k)x(k) + V(k)
where
"o ... 0 0 ... 0
H
1
CD
1
yn-D
0
0
0 ... 0 0 ... 0
, D(k) =
-L(k)
, v(k) =
0
0
Ip(n-l) :

0
0
0
-
flO 1
Defining z(k+l) =
x(k+l)
gives
where
A(k) =
S
0
z(k+l) = A(k)z(k) + w(k)
(5-11)
D(k)
F(k)-G(k)L(k)
w(k) =
v(k)
-M(k)e(k)
The matrix S is a stable matrix since it is lower-block triangular. The
matrix F(k)-G(k)L(k) is stable by construction; therefore, A(k) in
(5-11) is exponentially stable as a time-varying system matrix. In
addition, it follows easily from (5-6) and the boundedness of M(k) that
iiw(k)n < a(k)nz(k)n + b(k)
(5-12)
where a(k) and b(k) are positive scalar sequences which converge to zero
as k . Thus, (5-11) can be viewed as an exponentially stable time-
varying system driven by an input w(k) which can grow no faster than
linearly with the state, z(k). Following the proof in [12], it will be
shown that iiz(k) t converges to zero. Let A(k+N,k) denote the state
transition matrix for (5-11) defined by


42
This expression is easily derived by first rewriting (5-3) as
*00 =
0 I



+
Bx(k-1)
B2(k-1)


u(k-l) +
-A1(k-1)
-A2(k-1)


y(k-i).
i
0 0

Bn(k-1)

_An(k-l)_
(5-5)
Multiplying equation (5-5) by H gives
y(k) = Hx(k) =[010 ... 0]x(k-l) + B^k-DuU-l) A^k-Dyik-l)
Substituting into this equation for x(k-l) using (5-5) yields
y(k) =[001 ... 0]x(k-2) + B1(k-l)u(k-l) + B2(k-l)u(k-2)
- Ax(k-l)y(k1) A2(k-2)y(k-2)
Continuing the substitution using (5-5) gives
* n-1
y(k) = [0 ... 0 I] x(k-n+l) + l B. (k-j)u(k-j)
j=l J
n-1
- I A.(k-j)y(k-j)
j=l J
= l B (k-j)u(k-j) l A (k-j)y(k-j)
j=l J=1
which is equivalent to (5-4).


REFERENCES
[1] K.E. Narendra and L.S. Valavani, "Stable Adaptive Controller
Design-Direct Control," IEEE Trans. Automat. Contr., Vol. AC-23,
No. 4, pp. 570-583, Aug. 1978.
[2] K.S. Narendra, Y.H. Lin, and L.S. Valavani, "Stable Adaptive
Controller Design, Part II: Proof of Stability," IEEE Trans.
Automat. Contr., Vol. AC-25, No. 3, pp. 440-448, June 1980.
[3] G.C. Goodwin, P.J. Ramadge, and P.E. Caines, "Discrete-Time Multi-
variable Adaptive Control," IEEE Trans. Automat. Contr., Vol. AC-
25, No. 3, pp. 449-456, June 1980.
[4] G.C. Goodwin and K.S. Sin, Adaptive Control of Nonminimum Phase
Systems," IEEE Trans. Automat. Contr., Vol. AC-26, No. 2, pp. 478-
483, April 1981.
[5] G.C. Goodwin, D.J. Hill, and M. Palaniswami, "A Perspective on
Convergence of Adaptive Control Algorithms," Automtica, Vol. 20,
pp. 519-531, 1984.
[6] G.C. Goodwin and K.S. Sin, Adaptive Filtering, Prediction and
Control, Prentice Hall, New York, 1984.
[7] H. Elliot, R. Cristi, and M. Das, "Global Stability of Adaptive
Pole Placement Algorithms," IEEE Trans. Automat. Contr., Vol. AC-
30, No. 4, pp. 348-356, April 1985.
[8] 8.0.0. Anderson and R.M. Johnstone, "Global Adaptive Pole
Positioning," IEEE Trans. Automat. Contr., Vol. AC-30, No. 1, pp.
11-22, Jan. 1985.
[9] G. Kreisselmeier, "On Adaptive State Regulation," IEEE Trans.
Automat. Contr., Vol. AC-27, No. 1, pp. 3-16, Feb. 1982.
[10] C. Samson, These de Docteur-Ingenieur, Laboratoire d'Automatique
de 1'IRISA, Campue de Beaulieu, 35042 Rennes Cedex.
[11] C. Samson, "An adaptive LQ Controller for Nonmiminum Phase
Systems," Int. J. Control, Vol. 35, pp. 1-28, 1982.
[12] C. Samson and J.J. Fuchs, "Discrete Adaptive Regulation of Not-
Necessarily Minimum-Phase Systems," Proc. IEE, Vol. 128, Pt. D,
No. 3, pp. 102-108, May 1981.
100


63
by (6-1) and will drive the tracking error e(k) to zero asymptotically.
Theorem 6.1: Suppose assumptions A1-A3 are satisfied and in addition
rank
zI-F(e) 0 G(e)
-BCH zI-Ac 0
n+mq for all eft and
for all z: |z| > 1
where n = {0e ffTxm: 0. .e [o'!1!n,0^x]- Let x (k) be generated from the
ij ij ij
error driven system described in (6-3) and x(k) be generated from the
adaptive observer given by (5-3). Then there exists a control law u(k)
= -Lj£(k)-L2Xc(k) which when applied to the plant (6-1) results in a
stable closed-loop system and drives the tracking error e(k) = y(k) -
r(k) to zero.
Proof: Using equations (5-3) and (6-3) and setting v(k) and r(k) equal
to zero, we obtain
x(k+l)
xc(k+l)
F(k)
BCH
0
x(k)
xc(k)
G(k)
0
u(k) +
(6-4)
where e(k) = Hx(k) y(k). Assuming the given rank condition holds, it
follows from property (v) of Theorem 4.1 that the system described by
(6-4) is stabilizable. The stabilizing control law is of the form
u(k) = -L^k^k) -L2xc(k).
(6-5)
A
F(k)
0
A
G('kf
Let F(k) =
_BCH
Ac_
and G(k) =
0
As in the case of
the
adaptive regulator d


6
The key properties of the least-squares scheme used for proving global
stability of an indirect adaptive controller are
i.ne(k)n ii.ii9(kj-e(k-l) h > 0 as k +
iii.lim =¡= 6 =0
k+ 1 + <(> (k-l)P(k-l)<(>(k-1)
where e(k) = y(k) e^(k-l)(k-l).
The error e(k) is referred to as the prediction error since it is simply
the difference between the actual output at time k and the predicted
output at time k using the most recent parameter estimate. These pro
perties do not depend on the type of control input chosen or on the
boundedness of the system input and output. The least-squares algorithm
also has the property that the parameter estimates converge, although
not necessarily to the true values. This property is not included among
the key properties because it is not necessary for stability proofs.
An example of indirect adaptive control in the SISO case is Samson
and Fuchs' LQ controller discussed in [12] and briefly outlined here. A
state-space observer realization of the system described by (2-1) is
given by
x(k+l) = Fx(k) + Gu(k) (2-4)
y(k) = Hx(k)


43
The error between the output of the adaptive observer (5-3) and the
output of the system (3-1) as defined as
e(k) y(k) y(k).
The observer error e(k) has the same properties as the prediction error
e(k) given in Theorem 4.1; that is,
iie(k)n < a'(k)i*(k-l)i + b'(k) (5-6)
where a1(k) and b'(k) are positive scalar sequences which converge to
zero. This is shown by first rewriting e(k) as
e(k) = y(k) [0T(k-lHa(k-l) + *\(k-l)]
+ [eT(k-l)a(k-l) + .\(k-l)] y(k)
e(k) = y(k) [eT(k-lHa(k-l) + /ib(k-l)] e(k).
Taking the norm of both sides and using the triangle inequality gives
iie(k)ii < ny(k) (9T(k-l)b(k-l))ii + ne(k)n
Noting from (3-1)-(3-3) that eT(k-1)4>a(k-1) + ^^(k-l) is equal to
[-A^(k-l) ... -An(k-1) B1(k-1) ... 3n(k-1)]<¡>(k-l) and using expression
(5-4) gives


87
Example 7.5: Consider the single input two output discrete-time system
described by
y^k)"
l
a
*1
(k-1)'
'3 *
y2(k)
-l
2
*2
(k-1)
b
u(k-l)
where a and b are unknown parameters. As discussed in Chapter Three,
the required computations for parameter estimation can be reduced by
separating the known parameters from the unknown parameters. This is
accomplished by rewriting the system equations as
y(k) = eT4>a(k-l) + ipT4>5 (k-1)
with
T
0
a 0'
.a(k-i) -
rH
1
CM
L>>
,/ =
.0 b.
u(k-l).
t
0 3
2 0
b(k-l)
y1(k-D'
y2(k-i)
u(k-l)
The adaptive regulator defined by the observer (5-3) and the control law
(5-10) with L(k) computed using (5-8)-(5-9) was implemented. The system
parameter matrix 9 was estimated using the algorithm (4-2)-(4-5). The
initial conditions were chosen to be: P(0)=I, x(0)=[l 1]T, and RQ=I.
The initial estimates, parameter ranges, and steady state estimates are
given in Table 7-5. Plots of the output response, control input, and
parameter estimates are shown in Figure 7-9.


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TITLE: New Results.in Indirect Adaptive Control (record number: 985894)
PUBLICATION DATE! 1986
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79
yflO
Figure 7-5
Estimation Scheme (4-2)-(4-5)


99
estimation to cease. Most likely, these modifications could be used in
the estimation scheme introduced in Chapter Four since it is similar to
the least-squares algorithm. Thus far, the only results on stability of
adaptive controllers applied to time-varying systems require precise
assumptions on the parameter variations, see for example Goodwin and
Teoh [37].
Due to the difficulty of rigorous analysis, the transient response
of systems using adaptive controllers has received very little
attention. Much of the literature on adaptive control contains only
algorithms not simulations. The simulations included in Chapter Seven
show wide variations in the system output response and control input
initially. Once the identification scheme begins to converge, the
system response becomes well-behaved.
There are many open questions in the area of adaptive control. As
indicated in the discussion above, it may be possible to modify the
adaptive LQ controller developed in this dissertation for time-varying
systems or for systems with unmodelled dynamics.


76
Figure 7-4 continued
Estimation Scheme (4-2)-(4-5)


14
y(k) =
y2(k-i)
eT =
+a(ic-i) = y2(k-i)
1 0 3
-1 2 1
1 0 3
-1 2 1
+(k-l)
b(k-l) = (k-1).
Again, consider the linear discrete-time system described by
(3-3). It is assumed that each component e^. of the unknown system
parameter matrix 9 belongs to a known bounded interval [0i!*1.n, a1??*].
This assumption is reasonable in those applications where some a priori
information is available on the system. Clearly, the unknown system
parameter matrix 9 belongs to a subspace of ]R^xm which is defined by
the known bounded intervals. In order to use the necessary concepts of
boundedness, compactness and convergence, a suitable norm must be
defined on jRNxm. The vector norm which will be used throughout this
dissertation is the Euclidean norm defined by
IX l = (x x)
V,
The matrix norm which is induced by the vector Euclidean norm is given
by
T ^2
* CWM M)]
where ^max^1^ is the ei9envalue of MTM with the greatest
magnitude. By the above assumption on the entries of 0, the unknown
system parameter matrix 0 belongs to a known compact subspace n of ]RNxm
given by


78
Table 7-3
DATA
"al
"a2
bl
b2
Actual Parameters
-2.0
-5.0
-1.0
-0.5
Initial Estimates
-0.5 '
-6.0
-1.0
0.0
Parameter Ranges
[-3, 2]
I1
1

CO
1
1 1
[-1.1, -0.9]
[-1, i.o]
Estimation Scheme
(42)(45)
-1.64
-4.3
-0.90
-0.478


73
Example 7.2: Consider the non-minimum phase stabilizable but not
reachable discrete-time system described by
y(k) = 2.6y(k-l)-2.13y(k-2)+0.54y(k-3)+u(k-l)+1.5u(k-2)-u(k-3).
The adaptive controller defined by the adaptive observer (5-3) and the
control law (6-5) with L(k) computed from (6-6) was implemented to force
the output to track the reference signal r(k) =10. No disturbance was
introduced in this example. The internal model was given by
xc(k+l) = xc(k) + e(k)
with e(k) = y(k)-10. The parameters were estimated using (4-2)-(4-5).
The initial conditions were chosen to be: P(0) = I, x(0) = [0 1 0]T,
xc(0)=0, and R0=I. The initial parameter estimates, parameter ranges,
and steady state values of the estimates are displayed in Table 7-2.
Plots of the output response, control input, and parameter estimates are
shown in Figure 7-4.
Equation (4-4) indicates that the initial "covariance" matrix P(0)
should satisfy 0 the system rapidly went unstable. At time k=20, the output was
y(k)=6.5xl0 and the control input was u(k)=-2.6xl01^. The parameter
estimates at k=20 were -a1(k)=451, -a2(k)=-943, -a3(k)=484, bj(k)=1.0,
b2(k)=-447, and b2(k)=-898. Thus, choosing P(0) smaller than 21 is not
merely a technicality in the proof of Theorem 4.1, but is a necessary
condition for preserving stability of the adaptive closed-loop system.


25
Since P(k) converges, equation (4-3) implies that the matrix
P(k-l)a(k-l)<}>g(k-l)P{k-l)
Vi + k1}
converges to zero. For k sufficiently large, n. 1 = max (1, n<(> (k-l) n)
K-l a
which implies that
P(k-lHa(k-l)^(k-l)P(k-l) P(k-lHa(k-l)<(,J(k-l)P(k-l)
nk-l + ia(k-1)P(k-1)<(,a(k"1) \-l + xmaxP(0),+a(k"1)|Z
P(k-l) (k-l) > a a > o
Vl^1 +
P(k-lU (k-l) It then follows that the matrix ^ converges
Vl
to zero since it is bounded above by a matrix which is converging to
zero. Thus by equation (4-16), the matrix 21 P(k-l)P"l(k)P(k-l) also
converges to 21 PM in the case when the determinant of P(k) converges
to zero. Therefore, the matrix 21 P(k-l)P"^(k)P(k-l) is positive
definite for k sufficiently large since the matrix converges to the
positive definite matrix 21 P^. By (4-15), this implies that V(k) is
a monotone decreasing sequence for k sufficiently large and must there
fore converge. It then follows that 9. .(k) is bounded (property (ii))
' J
by noting that


75
y(k)
U(k)


24
Combining equations (4-13) and (4-14) gives
V(k) < V(k-l) -
iie(k) ii^
nk-l + ¡.(k-l)P(K-lHa(k-l)
(4-15)
-TrCfT(e(k-l))C2I-P(k-l)P-1(k)P(k-l)]f(0(k-l))].
If the matrix 21 P(k-l)P"*(k)P(k-l) is positive definite for k suffi
ciently large, then V(k) will be a monotone decreasing sequence which
must converge since V(k) is bounded below by zero. In order to show
that the matrix 21 P(k-l)P-1(k)P(k-l) is positive definite for k
sufficiently large, first assume that the determinant of P(k) does not
converge to zero. In this case, P"^(k) converges to p~* which implies
00
that 21 P(k-l)P"l(k)P(k-l) converges to 21 P^. The matrix 21 P^
is positive definite by property (i) in Theorem 4.1. Mow suppose that
the determinant of P(k) does converge to zero. Using (4-3) and the
matrix inversion lemma gives
^(k-iH^k-D
-P1(k) = p-^k-l) -
2
11 k-1
Multiplying both sides of this equation on the left and right by P(k-l)
and adding 21 to both sides give
21 P(k-l)P-1(k)P(k-l) = 2I-P(k-1)
P(k-lHa(k-lH¡(k-l)P(k-l)
2
n
k-1
(4-16)


102
[27] B.D.O. Anderson and J.B. Moore, "Detectability and Stabilizability
of Time-Varying Discrete-Time Linear Systems," SIAM J. Control,
Vol. 19, No. 1, pp. 20-32, Jan. 1981.
[28] C.A. Desoer, "Slowly Varing Discrete System xt+1 = Atxt,"
Electron. Lett., No. 6, pp. 339-340, 1970.
[29] G. Kreisselmeier, "Adaptive Control via Adaptive Observation and
Asymptotic Feedback Matrix Synthesis," IEEE Trans. Automat.
Contr., AC-25, No. 4, pp. 717-722, Aug. 1980.
[30] K.A. Ossman and E.W. Kamen, "An Adaptive LQ Controller for MIMO
Linear Discrete-Time Systems," in Proc. 24th IEEE Conf. Decision
Contr., Ft. Lauderdale, FL, pp. 843-844, 1985.
[31] K.A. Ossman and E.W. Kamen, "A Parameter Estimator with
Convergence to Pre-Specified Intervals for Discrete-Time Systems,"
in Proc. Amer.- Contr. Conf., Seattle, Wash., 1986.
[32] K.A. Ossman and E.W. Kamen, "Adaptive Regulation of MIMO Linear
Discrete-Time Systems Without Requiring a Persistent Excitation,"
submitted to IEEE Trans. Automat. Contr., Jan. 1986.
[33] W.L. Green and E.W. Kamen, "Stabilizability of Linear Systems Over
a Commutative Normed Algebra with Applications to Spatially-
Distributed and Parameter-Dependent Systems," J. Contr. and
Optimization, pp. 1-18, Jan. 1985.
[34] G. Kreisselmeier and K.S. Narendra, "Stable Model Reference
Adaptive Control in the Presence of Bounded Disturbances," IEEE
Trans. Automat. Contr. Vol. AC-27, No. 6, pp. 1169-1168, Dec.
1982.
[35] B.B. Peterson and K.S. Narendra, "Bounded Error Adaptive Control,"
IEEE Trans. Automat. Contr., Vol., AC-27, No. 6, pp. 1161-1168,
Dec. 1982.
[36] C. Samson, "Stability Analysis of Adaptively Controlled Systems
Subject to Bounded Disturbances," Automtica, Vol. 19, pp. 81-86,
1983.
[37] G.C. Goodwin and E.K. Teoh, "Adaptive Control of a Class of Linear
Time-Varying Systems," IFAC Workshop on Adaptive Systems in
Control and Signal Processing, San Francisco, June 1983.


39
open cover of ti and hence we have a contradiction. Thus,
r
fl_ 1) V. must be empty for
k ji ej
will be pointwise stabilizable over
k > kj which implies (F(9),G(e))
n for any k > k..
ek 1
Let 9(k) be defined as the estimate at time k of the system
parameter matrix 9, and let (F(k),G(k)) denote the time-varying
estimated system (F(9(k)) ,G(9(k))).
Definition: The estimated system (F(k),G(k)) is uniformly stabilizable
as a time-varying system [27], if there exists a bounded feedback matrix
l(k) such that (F(k)-G(k)L(k)) is uniformly asympototically stable as a
time-varying system matrix.
The following theorem gives conditions under which the estimated system
is uniformly stabilizable.
Theorem 5.1: Assuming A1-A3 are satisfied, the parameter estimator
described by (4-2)-(4-5) will generate an estimated system (F(k),G(k))
which is uniformly stabilizable as a time-varying system.
Proof: By Proposition 5.1, there exists e > 0 such that (F(9),G(e)) is
pointwise stabilizable for all 9 belonging to n£. Using the results in
Kamen and Khargonekar [25], since is compact, there exists a feedback
matrix L(9) with entries which are continuous in 9 and a positive
constant q < 1 such that
|x (F(g)-G(e)L(e))I < q < 1 for all 9efi
i max\ e


51
Since Rk belongs to the compact set D for all k, it follows from step
five that given any a > 0 there exists 6(a) such that
ns^n < a whenever ue(k+i) 0(k) 11 < 5 for ie[0,N].
k k
Let Qk = Pk# Using the equation introduced in step two gives
Qk+i+l = Rkt1G(0(k))[GT(e(|O>Rk+iG> + n ie(k)))
X (Qfc+i + s(9(k))CGT'(8(k))Pki,s(e(k)) + I]'1
X GT(8(k))Q| X GT(e(k)))T + Ik+i; Qk = 0, ieCO.N].
It follows from the boundedness of e(k), Rk, and pjj (for t
sufficiently large) that
llFT(e(k))(I Rk+iG(Q(k))CGT(8(k))Rk+iG(0(k)) + I]V(0(k))) 11 < -
and iiG(e(k))[GT(e(k))PkJ\ G(e(k)) + I]"1GT(8(k)) 1 < K2 < .
Thus for all k such that e(k)ei2^, we have
"Vi+l1 < Kl2('Cl' + K2,GHi'2> + 'W* Qk = - eC0-N:>-


19
1 when the determinant of P(k) > e
nk_1 = / where s is any small positive number. (4-5)
max (1, ii<|> (k-1) ii) otherwise
k a
The parameter estimation algorithm described by (4-2) (4-5)
differs from the recursive least-squares estimator in three ways.
First, the initial covariance matrix P(0) in (4-3) must be less than 21,
whereas the least-squares algorithm allows for any positive definite
initial covariance matrix. This condition is needed to ensure that the
parameter estimator described by (4-2) (4-5) possesses the properties
listed in Theorem 4.1 which are necessary for proving global stability
of the adaptive regulator discussed in Chapter V.
The second difference is the data normalization introduced through
the term ^ defined in (4-5). In the least-squares algorithm,
nk i is simply equal to one. Again, the data normalization is neces
sary for proving that the parameter estimator has the desirable proper
ties given in Theorem 4.1. Simulations have shown that if e is not
suitably small, the normalization has a detrimental effect on the
transient response of the system. This effect can be minimized by
choosing e to be very small in (4-5). Simulations which show the effect
of the data normalization and the choice of e on system transient re
sponse are discussed in Chapter VII.
The major difference between the estimator given by (4-2) (4-5)
and the recursive least-squares estimator is the addition of a "correc
tion term" -P(k-l)f(e(k-l)) in (4-2). This term forces the estimates
8.jj(k) of the components of the system parameter matrix 0 to converge to
the sets Ce?lnT^XU. This property, when combined with the stabiliz-
ability assumption over n discussed in Chapter III, eliminates the need


CHAPTER II
BACKGROUND
In indirect adaptive control, the system parameters are estimated
in real time using available input/output information. The control law
is then computed for the estimated system using the most recent param
eter estimates. A well-known problem in proving global stability of
indirect adaptive control algorithms is the estimated system may not be
controllable or even stabilizable for certain parameter estimates which
are referred to as singular points. Global stability of the indirect
adaptive controller can only be ensured if there is some finite point in
time after which the parameter estimates are not arbitrarily close to a
singular point. An example of a parameter estimator and an adaptive
regulator will further illustrate the problem of singular points in
indirect adaptive control.
Consider the SISO discrete-time system described by the input/out
put difference equation:
q
P
(2-1)
In (2-1), y(k) is the system output and u(k) is the control input. It
is assumed that an upper bound, n, on q and p is known but some or all
of the system parameters aj and bj are unknown. The system described by
(2-1) can be rewritten in the following form which is convenient for
parameter estimation:
y(k) = eT(k-l)
4


20
for a persistently exciting input. It is important to note that the
algorithm is not a projection algorithm. That is, 0.¡j(k) does not
belong to the set [0l.n1.n,et!?x] for every k but does converge to the set
T J 1J
in the limit as k - .
The idea for the correction term came from Kreisselmeier [23]. In
fact, f(e(k-l)) is defined exactly the same as in [23]; however, the
parameter estimator described here for MIMO discrete-time systems is
quite different from the adaptive observer Kreisselmeier uses for SISO
continuous-time systems.
The following theorem shows the estimator described by (4-2) -
(4-5) has many of the same properties as recursive least-squares. In
addition, the estimates 0.jj(k) converge to the known intervals
[9?]n9^x]. The prediction error e(k) which appears in the theorem
is defined as
e(k) = y(k) eT(k-l)a(k-l) /^(k-l).


26
V(k) Tr(eT(k)p-1(k)'e(k)) > [A^P'V)] Tr(eT(k)'e(k))
T N m ,
v(k) > V2 Tr(T(k>r(k)) = V2 ^ ^ Ce1j(k) 0,j].
Now solving (4-15) recursively gives the relationship
q ie(j)i
0 < V(q) < V(0) .I 2 Y77
j=l nj-l +
(4-17)
- I Tr[fT(0(j-l))[2I P(j-l)P"1(j)P(j-l)]f(e(j-l))].
j=l
For sufficiently large k, the terms within the summations are non-nega
tive and must converge to zero since V(k) is non-negative. Properties
(iii) and (v) then follow easily. Property (iv) follows immediately
from (iii) by first noting that
iie(k)
(nk-i+ *I < b(k)
where b(k) is a positive scalar sequence which converges to zero,
Multiplying both sides of the inequality by
2
(n£_i + a(k-l)P(k-l)a(k-l)) and noting that < 1 + iia(k-1) ii
gives
ne(k) < b(k)(n^ + ¡(k-l)P(k-l)*a(k-l))
< b(k)(Vl + UmaxP(><-l))1/2 *+a(k-l)i)
< a(k)

71
Figure 7-2 continued
Recursive Least Squares


10
In order to prove global stability, Samson and Fuchs must assume
the estimated system (F(k),G(k)) is uniformly stabilizable. Unfortu
nately, this stabilizability condition cannot be checked a priori which
means global stability of the adaptive regulator cannot be guaranteed.
If the least-squares estimator is used, the assumption that (F(k),G(k))
is uniformly stabilizable is equivalent to requiring that is
stabilizable where F = lim F(k) and G = lim G(k).
oo K-M 00 K-H
As previously mentioned, the problem of avoiding singular limit
points is common to all indirect adaptive controllers. Until the recent
work of Lozano-Leal and Goodwin [21], Oe Larminat [22] and Kreisselmeier
[23], the only way to avoid a singular limit point was to require a
persistently exciting input which would force the parameter estimates to
converge to the true system parameters. This approach has been devel
oped by a number of individuals, see for example [4-9]. The idea is to
use an external input with many different frequencies which enriches the
input/output information allowing perfect identification of the system
parameters. As discussed in [15-16], indirect adaptive controllers
which require a persistently exciting input are not robust. Since the
adaptive closed-loop system is inherently time-varying and nonlinear,
any uncertainty in the plant could counteract the exciting input. The
problem of choosing a persistently exciting input in the presence of
unmodeled plant dynamics has not been completely resolved.
Lozano-Leal and Goodwin [21] developed an estimation scheme which
gives nonsingular parameter estimates at each point in time and in the
limit for SISO linear discrete-time systems. They modify (when neces
sary) the parameter estimates generated from the least-squares estima
tion algorithm with data normalization. This modification allows them


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44
n(k)ii < nC^ (k) ... J2n(k)] (5-7)
where
It follows from property (vi) in Theorem 4.1 that nd,.(k)it converges to
zero as k goes to infinity for i=l 2n. Applying this property and
property (iv) of Theorem 4.1 to (5-7) gives the desired inequality
(5-6). Property (5-6) is very important in the proof of global
stability of the adaptive closed-loop system.
Feedback Gain Sequence
The feedback gain sequence L(k) must be chosen so that (F(k)-
G(k)L(k)) is exponentially stable as a time-varying system matrix. As
shown in Theorem 5.1, a stabilizing feedback does exist because the
estimated system (F(k),G(k)) generated from the parameter estimator
(4-2)-(4-5) is uniformly stabilizable under assumptions A1-A3. There
are several options in the literature for choosing L(k), some of which
will be discussed in this section.
One approach for computing L(k) is to stabilize the estimated
system (F(k),G(k)) pointwise in time using control law strategies for
time-invariant systems such as pole placement or LQ control. Of course,
in the case of pole placement, the stabilizability assumption over the
parameter space n (assumption A3) must be strengthened to reachability


5
where
(2-2)
T(k-l) = [y(k-l) ... y(k-n) u(k-l) ... u(k-n)].
The vector 0 consists of all the system parameters and (k-l) is a
regression vector of past inputs and outputs. Throughout the following
discussion, n 11 will designate the vector Euclidean norm defined by
T ^2
11x11 = (x x) .
Of all the parameter estimation schemes discussed in [6], the
least-squares algorithm has the fastest convergence rate and is most
easily modified to handle output disturbances and slowly time-varying
systems. The algorithm results from minimizing the quadratic cost
function:
k=l
The cost function consists of the sum of the squares of the prediction
errors plus an additional term which takes initial conditions into
account. The matrix P(0) can be interpreted as a measure of confidence
in the initial parameter estimate 0(0). Designating 0(k) as the esti
mate of the system parameters at time k, the least-squares algorithm is
described by
Cy(k) eT(k-l)(t>(k-l)]
P(k) = P(k-i) P(k-l)4>(k-l)/(k-l)P(k-ll.
1 + (k-l)
; P(0) = PT(0) > 0.
(2-3)


69
Figure 7-1 continued
Estimation Scheme (4-2)-(4-5)


13
y(k) 0T<>a(k-l) + /^(k-l)
(3-3)
The Nxm matrix e contains all of the unknown entries in P while the
matrix tp contains only those entries of P which are known a priori. The
vectors (k-1) and 4>. (k-1) are regression vectors whose components
come from (k-l). Clearly, the decomposition described in (3-3) is not
unique for multi-input multi-output systems. In order to minimize the
required computations for parameter estimation, the size of (k-1)
St
should be made as small as possible. An example will serve to
illustrate these concepts.
Example 1: Consider the linear discrete-time system with two outputs
and a single input described by
y(k) =
yi(k)
1 a
'y^k-D
+
3
y?(k)
-1 2
y2(k-i)
1
u(k-l)
The parameter a is assumed to be unknown. The system can be rewritten
in the form
y(k) = PT <(>( k-1)
where
' 1 a 3'
y^k-i)'
4>(k-l) =
y2(k-D
-1 2 1
u(k-l)_
A decomposition for this system which minimizes the required
computations for parameter estimation is given by


47
PJe) = Q + ft(0)poo(8)f(0) ft(0)poo(0)g(0)[i + gt(0)pw(0)g(0)]1
X GT(0)Poo(0)F(0).
Let ftg denote the compact parameter space defined in (5-2) such that
(F(e),G(0)) is pointwise stabilizable for all 0 belonging to ftg. Let
e > 0 be given. It will first be shown that nR^ R^ ^ii < e for k
sufficiently large where R^ is computed using (5-9).
Step 1: The sequence of matrices R^ given by equation (5-9) is
bounded. That is, R^ belongs to a compact subset D of ^xN.
T T
Assumption A3 implies (F (8),G ()) is detectable for all 0 belonging
to the parameter space ft. Using arguments similar to those in Theorem
5.1, it follows easily that (FT(k),GT(k)) is uniformly detectable.
Anderson and Moore [27], prove that R^ and L(k) given by (5-8)-(5-9)
will be bounded if (FT(k) ,G^(k)) is uniformly detectable.
1 2
Step 2: Let Pk(@) and P£(0) denote the sequences generated from the
. 1 2
RDE using initial conditions PQ and PQ respectively. Defining
APk(0) = P^(0) P^(0), it then follows that
Pk+1(e) ft(e) x(iPk(e) + iPk(0)G(e)[GT(e)Pk (e)G(e) + I]"1 GT(8)aPk(e))
x (I pJ(e)G(9)[GT(8)pJ(6)G(e) + I]'1 GT(0))TF(e).


97
Many individuals, including Kreisselmeier and Narendra [34],
Peterson and Narendra [35], Goodwin and Sin [6], and Samson [36],
consider the problem of maintaining stability of adaptive controllers in
the presence of bounded disturbances. A dead zone is introduced into
either the identification scheme or the adaptive law to "shut off" the
adaptive algorithm whenever the magnitude of the identification error
falls below a certain bound related to the upper bound on the
disturbance. Global stability of the adaptive system can be ensured
only if an upper bound on the disturbance is known.
Narendra and Annaswamy [17] and Anderson [18] propose using an
external input to ensure persistent excitation of the adaptive system.
Robustness will then follow as a consequence of exponential stability of
the closed-loop system. However, there is currently no method for
ensuring persistent excitation in the presence of unmodeled plant
dynamics unless the disturbance due to plant uncertainty is bounded.
Ioannou and Kokotovic [19] introduce a a-modification (i.e., the
addition of a linear feedback term) to the adaptive law for a model
reference adaptive controller. As in the case of the persistent
excitation approach, stability can be guaranteed only if the disturbance
due to plant uncertainty are bounded.
All of the approaches to robustness discussed so far require the
disturbance to be bounded. A disturbance due to unmodelled plant
dynamics is internally generated and therefore dependant on the system
input and output. Thus, boundedness of the disturbance can only be
assumed if stability of the adaptive closed-loop system has already been
established. Kreisselmeier [15] introduces a relative dead zone into
the estimation scheme which depends not on the identification error


53
Step 7: There exists a finite integer Mgte) such that
nPk+1 pkn < e/3 for all k > M0.
The result follows immediately from the uniform continuity of
P () over si* and the convergence of H8(k+1) e(k)u to zero.
00 £
Step 8; There exists a finite integer M(e) such that
nRk Rk 1 < e for a11 k >
Applying the triangle inequality and using the results in steps six and
seven gives
k+1 k+1 k k
llRk+N+l Rk+NB < BRk+N+l P + i,p o PJ + "P* Rk+NB < e
for all k > max (M^, M2).
The result then follows easily by setting M = max(M^, M2) + N + 1.
Step 9: If L(k) is computed using (5-8)-(5-9), then nL(k) L(k-l)n
converge to zero as k goes to infinity. This result follows easily
since nR^ R^ ^il, nF(k) F(k-l)ll, and nG(k) G(k-1) 11 all converge
to zero.
Using the results of Theorem 5.2, it is now possible to prove that
(F(k)-G(k)L(k)) is exponentially stable as a time-varying system
matrix. The exponential stability of (F(k)-G(k)L(k)) is very important
for proving global stability of the adaptive closed-loop system.


11
to prove global stability of an adaptive pole placement scheme without
requiring a persistently exciting input. It is, however, possible for
the parameter estimates to converge to a point that is near a singular
point, in which case the controller gains may be large. Also, a MIMO
version of the algorithm is not available at the present time.
De Larminat [22] has also proposed a parameter estimation scheme
that does not yield singular points. He assumes a priori knowledge of a
space G which contains the actual system parameters and is devoid of
singular points. He then introduces a modification to the estimates
obtained from the standard least-squares algorithm. The modification is
only required for a finite period of time and produces estimates which
belong to G for all time. Although his modification prevents conver
gence near a singular point, De Larminat's recursive procedure is not as
explicit as the one proposed by Lozano-Leal and Goodwin [21].
As in the case of De Larminat, Kreisselmeier [23] assumes prior
information on the system parameters is available. Specifically, he
assumes that the components 9. of the system parameter vector 9 lie
within a known bounded interval [emln, eTax]. He then modifies the
identification scheme for SISO continuous-time systems to force the
parameter estimate of e.. to converge to the set [e?1 n,em^x] for every
i. Kreisselmeier also assumes that for each e with 9.e [e1?1 n,er[,ax]
there is a feedback control system with a prescribed degree of stabil
ity. This condition can be checked a priori since it does not involve
the parameter estimates. With these assumptions, Kreisselmeier proves
global stability of the SISO continuous-time adaptive controller.


54
Theorem 5.3: Suppose e(k) is generated using the parameter estimator
(4-2)-(4-5) and the feedback gain sequence L(k) is computed using (5-8)-
(5-9). Then subject to assumptions A1-A3, (F(k)-G(k)L(k)) is
exponentially stable as a time-varying system matrix.
T T
Proof: Assumption A3 implies that (F (@),G (e)) is detectable for
each 9 belonging to n. Using arguments similar to those in
Proposition 5.1 and Theorem 5.1, it follows that (F^(k),GT(k)) is
uniformly detectable. In [27], Anderson and Moore prove that if
(F^(k),G^(k)) is uniformly detectable then (F(k)-G(k)L(k) )^ is
exponentially stable as a time-varying system matrix where L(k) is given
by (5-8)-(5-9). Since matrix products do not commute, it does not
immediately follow that (F(k)-G(k)L(k)) is exponentially stable.
However, Samson and Fuchs [12] show A(k) will be exponentially stable as
a time-varying system matrix if AT(k) is exponentially stable and if
iiA(k)-A(k-1) n converges to zero as k - . Since (F(k)-G(k)L(k))^ is
exponentially stable, it then follows that (F(k)-G(k)L(k)) will also be
exponentially stable if u(F(k)-G(k)L(k)) (F(k-l)-G(k-l)L(k-l))n
converges to zero as k - . Property (vi) of Theorem 4.1 implies both
iiF(k)-F(k-1) n and nG(k)-G(k-l) n converge to zero. It was shown in
Theorem 5.2 that nL(k)-L(k-l)n also converges to zero. Since
F(k)-G(k)L(k)-(F(k-l)-G(k-l)l(k-l)) =
(F(k)-F(k-l)) G(k)(L(k)-L(k-l)) (G(k)-(G(k-l))L(k-l)
and both G(k) and L(k) are bounded, it follows that
n(F(k)-G(k)L(k))-(F(k-l)-G(k-l)L(k-l))n does indeed converge to
zero. Thus the matrix (F(k)-G(k)L(k)) is exponentially stable as a
time-varying system matrix.


94
u1(k)
: A A
!.*.! . 1 .
V L-
'j |10 20
30
40
50
60
70 (k)
1 i
: ¥
(c)
u2(k)
10:
5
* i\
1j* * ir |Kit y r- tL iw 1 ,L liMimim 1 1 1 i -
V V0 20 30 40 50 60 70 (k)
-5 ;
-10
(d)
Figure 7-10 continued
Estimation Scheme (4-2)-(4-5)


83
Table 7-4
DATA
"al
"a2
bl
b2
Actual Parameters
2
-0.99
0.5
3.0
Initial Estimates
1.95
1.0
0.75
5.0
Parameter Ranges
[1.9, 2.0]
11
o

*
o

CM
1
1 1
[-0.5, 2.0]
[2.5, 7.5]
Estimation Scheme
(4-2)-(4-5)
1.91
-0.920
0.406
2.83


52
Clearly, there exists a cr^e) such that if nzk+. II < ^ for all ie[0,N]
then
nQ
k
k+N
11 < e/6.
Let S(o^) be chosen such that
nsk+i-H < Oj for all ie[0,N] whenever ne(k+i) e(k) n < 6.
It follows from properties (v) and (vi) of Theorem 4.1 that there exists
a finite integer such that
e(k)ei2g for all k >
and ne(k+i) e(k)n < 6(a^) for all ie [0,N] and for all k > M^.
Therefore
"Qk+N" = Rk+N pk+N11 < e/6 for 311 k > 'V
Using the results in step four, we obtain
aPk+N pn < fr a^ k > Mi*
Thus
Rk+N P11 < 11Rk+N Pk+N11 + 11Pk+N P11 < e^3 for a11 k > Mi*
which proves the result.


65
control law u(k) = -L-jX(k)-L2xc(k) is exponentially stable and converges
to a time-invariant system. Finally, from the results in Chen [20], if
the control law (6-5) is applied to the plant (6-1), the resulting
closed-loop system is still stable (not including the exogenous
disturbance v(k)) and the system output y(k) converges to the reference
signal r(k) as k -* .
The adaptive controller described by (6-3) and (6-5)-(6-6) causes
the plant (6-1) to reject deterministic disturbances and to track a
given reference signal r(k). Simulations of the adaptive controller for
various types of external disturbance and reference signals are included
in Chapter VII.


86
y(k)
40
30
20
10
-10
-20
-30
-40
20 30 40 50 60 TOO TO
(a)
u(k)
Figure 7-8
Exact Parameters


NEW RESULTS IN INDIRECT ADAPTIVE CONTROL
By
KATHLEEN A. K. OSSMAN
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


93
yx(k)
y200
Figure 7-10
Estimation Scheme (4-2)-(4-5)


CHAPTER I
INTRODUCTION
Adaptive control, the problem of controlling a system whose
parameters are unknown prior to or changing during system operation, has
been a major research topic during the past three decades. Although
originally intended for time-varying or nonlinear systems, most of the
stability results on adaptive controllers have been limited to linear
time-invariant systems. The majority of adaptive controllers can be
classified as either direct or indirect.
In direct adaptive control, no attempt is made to estimate the
unknown system parameters. Instead, the controller parameters are
updated directly using real time input/output information. One specific
example of direct adaptive control is model reference adaptive control
(MRAC) where the unknown plant is forced to behave asymptotically like
some pre-chosen reference model. Several individuals have developed
this approach, see for example [1-3]. The assumptions needed to prove
global stability of model reference adaptive controllers are (1) the
unknown plant is minimum phase, (2) the relative degree of the plant is
known and (3) the sign of the plant gain is known.
Indirect adaptive controllers are applicable to nonminimum phase as
well as minimum phase systems. In indirect adaptive control, the
unknown system parameters are estimated in real time using available
input/output information then the control law is computed using the most
recent parameter estimates. A number of individuals have introduced
indirect adaptive controllers, see for example [4-12]. The assumptions
1


CHAPTER IV
PARAMETER ESTIMATION
Consider the r-input m-output linear discrete-time system described
by
y(k) = 9%a(k-l) + /^(k-l)
(4-1)
As previously discussed in Chapter III, it is assumed that each compo
nent 8.. of the unknown system parameter matrix 9 belongs to a known
bounded interval [9*1n, 9II,^X].
L ij 5 ij J
The parameter estimation algorithm is given by
P(k-lU (k-1)
9(k) = 9(k-1) P(k-l)f(9(k-l)) +-= = ^ x (4-2)
\ 1 + ^(k-l)P(k-l)* (k-1)
[yT(k) <(>J(k-l)^-^(k-l)e(k-l)]
P(k) = P(k-l) -
P(k-lHa(k-l)^(k-l)P(k-l)
\-l + ¡(k-lWk-D^k-l)'
0 < P(0) = PT(0) < 21,
(4-3)
'ij(k-
max
ij
when 9. .(k-1) > 9^x
fij(k-l) = ^
utki)
- 0I!'in
IJ
when 9 - (k-1) < g?!0
* \J V
(4-4)
0
when 9. .(k-1) e[9^-n, 9x]
18


CHAPTER VIII
DISCUSSION
A globally stable adaptive LQ controller which does not require
persistent excitation was introduced for multi-input multi-output linear
discrete-time systems. The assumptions made on the plant were (1) an
upper bound on system order is known, (2) the unknown system parameters
belong to known bounded intervals, and (3) the plant is stabilizable for
all values of the unknown parameters ranging over the known bounded
intervals. When applied to the unknown plant using this parameter
estimator, the adaptive LQ controller ensures the system inputs and
outputs will remain bounded, and forces the output to track a given
reference signal in the presence of a deterministic external
disturbance. Some remaining considerations include robustness,
application to time-varying or nonlinear systems, and analysis of
transient response.
In many control applications, the order of the model will be lower
than that of the plant. As mentioned in the introduction, Rohrs et al.
[13-14] demonstrated that most adaptive controllers could go unstable if
the order of the system was underestimated, even if the modelling errors
were small. Since knowledge of an upper bound on system order is one of
the assumptions made on the adaptive controller presented in this
dissertation, it is likely that this controller will also go unstable
for systems with unmodelled dynamics. There have been several
approaches in the literature towards development of robust adaptive
controllers, some of which will be briefly discussed here.
96


84
y(k)
40
30
20
10
-10
-20
-30
-40
I I
'i 11
I It
\ .10 l i 20
30 40 50 60 70 (k)
(a)
Figure 7-7
Estimation Scheme (4-2)-(4-5)


74
Table 7-2
DATA
"al
-a2
-a3
bl
J
b2
b3
Actual Parameters
2.6
-2.13
0.54
1.0
1.5
-1.0
Initial Estimates
2.65
-1.0
-0.5
2.0
1.6
-1.5
Parameter Ranges
[2.5,2.8]
[-5,3]
[-2,1]
[-4,8]
[1.5,1.7]
[-3,0]
Estimation Scheme
(42)(45)
2.5
-1.92
0.429
0.999
i
!
1.60
-0.795


9
An example of a parameter estimator possessing these properties is the
least-squares estimator discussed previously. Clearly, properties (i)
and (ii) listed above are equivalent to properties (i) and (ii) given
for the least-squares algorithm. It is not quite as obvious that Samson
and Fuchs' property (iii) is equivalent to property (iii) of the least-
squares algorithm. The proof is given in Chapter IV.
The second assumption for proving global stability is that the
estimated system (F(k),G(k)) must be uniformly stabilizable.
Definition: The system (F(k),G(k)) is uniformly stabilizable if there
exists an integer r > 1, a constant q and a uniformly bounded sequence
L(k) such that
t+r-1
l H (F(k)-G(k)L(k))I k=t
where
t+r-1
n (F(k)-G(k)L(k)) = (F(t+r) G(t+r)L(t+r))x
k=t
(F(t+r-1) G(t+r-l)L(t+r-l)) ... (F(t) -G(t)L(t)).
Theorem: Assuming the system (F,G) is stabilizable, the parameter
estimator possesses the three required properties, and the estimated
system (F(k),G(k)) is uniformly stabilizable, the control law u(k) =
-L(k)x(k) described by (2-5)-(2-7) will result in a globally stable
closed-loop system in the sense that the system input and output
converge to zero for any initial states in the plant and observer
(Samson and Fuchs [12]).


46
In (5-9), both 0 and the initial value R0 are positive definite
symmetric matrices. In contrast to the first approach, where the
algebraic Riccati equation would be solved for each k, this asymptotic
approach requires the solution of only one iteration of the Riccati
difference equation at each point in time. Obviously, this approach
offers considerable savings in on-line calculations. Also, it is not
necessary to check whether or not (F(k),G(k)) is stabilizable at each
point k because L(k) given by (5-8)-(5-9) is well-defined even for
isolated singular points. In order to prove that (F(k)-G(k)L(k)) is
exponentially stable when L(k) is computed using (5-8)-(5-9), it is
first necessary to show n(L(k)-L(k-l))n converges to zero as k + <.
Theorem 5,2: Suppose the parameter estimates are generated using the
algorithm (4-2)-(4-5). Then subject to assumptions A1-A3, the feedback
gain sequence given by (5-8)-(5-9) has the property
nL(k)-L(k-l)n + 0 as k + .
Proof: The proof is rather lengthy and will therefore be divided into
steps. Throughout the proof, Pk(9) will refer to the sequence generated
from the time-invariant Riccati difference equation (RDE) given by
Pk+1(e) = Q + FT(e)Pk(e)F(e) FT(e)Pk(9)G(e)[i + GT(9)Pk(e)G(0)]1
x GT(9)Pk(0)F(9), PQ > o.
The matrix P^e) will denote the solution to the algebraic Riccati
equation (ARE) given by


ACKNOWLEDGEMENTS
I would like to acknowledge the following people for their
contributions to the completion of this dissertation. Special thanks go
to my advisor, Dr. Edward W. Kamen, for his numerous helpful suggestions
and creative ideas. Thanks also go to the members of my supervisory
committee: Dr. T. E. Bullock, Dr. D. W. Hearn, Dr. P. Z. Peebles, and
Dr. S. Svoronos.
This work was supported in part by the U. S. Army Research Office,
Research Triangle Park, N.C., under Contract No. DAAG29-84-K-0081.
ii


33
Also, there exists a finite N such that the matrix
[2I-P(k-l)P"*(k)P(k-l)] is positive definite for k > N because
[2I-P(k-l)P"l(k)P(k-l)] converges to the positive definite matrix 21-
PM. Since
00
00
0 < l xmin[2I-P(k-l)P"1(k)P(k)P(k-l)]Tr[fT(e(k-l))f(0(k-l))]
k=N min
00
< I Tr[fT(0(k-l))[2I-P(k-l)P1(k)P(k-l)]f(0(k-l))] <
k=N
00
it follows that l Tr[f^(0(k-l))f(0(k-l))] < . Therefore, the right
k=l
hand side of the inequality (4-25) is bounded in the limit as M ap
proaches infinity. This implies the existence of a finite L such that
00
l Ilf(0(k-1)) ii < L < . Returning to the inequality (4-23), let
k=l
e > 0 be given. The sequence 0^j(k) will be a Cauchy sequence if
10^j(m)-0-¡j(n) | is less than e for m and n sufficiently large. Since
P(k) converges to P^, there exists N^(e) such that
nP(m)-P(n)n < for m,n > N^(e)
where £l = (1/3) min [e/(nP"1(O)0(O) ii e/(vL)].
Also, there exists N2U) such that
m-1
l nf(e(i))n < e/6v for m,n > N?(e).
i=n


CHAPTER III
SYSTEM DEFINITIONS AND ASSUMPTIONS
The system to be regulated is the multi-input multi-output linear
discrete-time system described by
yOO -? A.y(k-j) + ? B,u(k-j) (3-1)
3=1 J 3=1 J
In (3-1), y(k) is the mxl output vector and u(k) is the rxl control
input vector. It is assumed that an upper bound, n, on p and q is known
but all or some of the entries in the matrices Aj and Bj are unknown.
The system described by (3-1) can be rewritten in the following form
convenient for parameter estimation:
y(k) = PT$(k-l) (3-2)
where
PT = C-A]_ ... -An ... Bn]
4>T(k-1) = [yT(k-l) ... yT(k-n) uT(k-l) ... uT(k-n)]
The n(m+r)xm matrix P consists of all the system parameters and the
n(m+r)xl vector (k-l) is a regression vector of past inputs and
outputs. In certain applications, some of the entries in the system
matrix P will be known a priori. Since it is not necessary to estimate
known parameters, a scheme for separating the known parameters from the
unknown parameters is advantageous. This is accomplished by rewriting
(3-2) in the following form:
12


22
where
O
O
1
O
- ith position.
0
Each of the terms on the right hand side of the equation (4-7) must
converge which implies P-jj(k) converges for every i and j. Noting that
P(0) < 21, it then follows that P(k) must converge to a positive semi-
definite matrix P^ < 21. To prove (ii), let 'e(k) = 0(k)-e. Rewriting
(4-2) in terms of e(k) gives
p(k-lHa(k-l)^(k-l)e(k-l)
ei(k) = 'e(k-l) -
P(k-l)f(e(k-l)). (4-8)
\-l + ¡(k-DP(k-l)*a(k-l)
Multiplying both sides of the equation (4-3) on the right by P-1(k-l)
gives
P(k)P1(k-1) = I -
P(k-l) (4-9)
\-l + 4(k~1)P{k'1K(k'1)
Combining (4-8) and (4-9) yields
e(k) = P(k)P-1(k-l)e(k-l) p(k-l)f(e(k-l))
(4-10)


81
yflO
u(k)
Figure 7-6
Exact Parameters


32
Rearranging terms in this inequality results in
II c \ Pit / u gq
Vl + ^(k-l)P(k-l)+a(k-l)
< V(k-l) V(k) + X^C^-DP'1
N m
I l
i=l j=l
fij(0(k-D)
(k)P(k-l)]Tr[fT(0(k-l))f(9(k-l))].
Summing all the terms in this inequality from k equals one to M and
N m
noting that nf(e(k-l))n < l l |f.¡_.(e(k-l))| gives
1=1 j=l J
M
0 < I
ne(k) ii*
M
2 t + 2q l llf(e(k-l))
k=1 nk-l + *;(k-1)P(k-1)*a(|c-1) k=1
M
(4-25)
< V(0) V(M) + l AmaxCP(k-l)P1(k)P(k-l)]TrCfT(0(k-l))f(9(k-l))].
k=l
If the right hand side of the inequality (4-25) is bounded in the limit
as M approaches infinity, then the norm of f(e(k-1)) will be summable.
The term V(M) is bounded since V(k) converges. Also
AmaxCPik-DP^OOPU-l)]
is bounded since the matrix
P(k-l)P1(k)P(k-l)
converges to P^. It remains to be shown that
l Tr[fT(e(k-l))f(e(k-l))] < ..
k=l
Equation (4-17) implies
l Tr[fT(0(k-l))[2I-P(k-l)P'1(k)P(k-l)]f(e(k-l))] <
k=l


80
8
6
4
2
-2
-4
-6
-8
. i
2U 40"
- > 1 .
60 bo too no
(d)
t£{k)
bj(k)
w
Figure 7-5 continued
Estimation Scheme (4-2)-(4-5)


60
closed-loop system is globally stable using the adaptive regulator
defined by (5-3) and (5-10).


28
By properties (iii) and (v), the first and third terms on the right side
of the inequality (4-20) converge to zero. The second term on the right
side of the inequality will also converge to zero by property (v) pro
vided that the matrix
P(k-l)2*a(k-l)^(k-l)
Vl + *¡(k-l)P(k-lHa(k-l)
is bounded. This matrix is shown to be bounded by first noting that
0 <
P2(k-l)a(k-l)^(k-l)
\-l + +¡(k-l)P(k-lHa(k-l)
2 Uk-lH'ik-l)
P(k-1) a a
iia(k-l)ii<
'k-1
iia(k-l)
-y+ AminP(k-l)
,2 min
Clearly, the numerator on the right hand side of the inequality is
bounded above. The denominator is bounded below by one if the deter
minant of P(k) converges to zero, and by x (P ) if the determinant
min o
of P(k) does not converge to zero. Thus, all three terms on the right
hand side of the inequality (4-20) converge to zero. Since
Tr[(e(k) e(k-l))T(0(k) e(k-l))] = J J [e..(k) e^k-l)]2,
1=1 j=l u ,J
it follows that 0. .(k) e..(k-1) must converge to zero which proves
J 1 J
property (vi) in the case where p =1. The proof for arbitrary p fol
lows from the Schwarz inequality. To prove (vii), equation (4-10) is
first solved recursively giving
e(n) = ?(n)p-l(0)Q(0) P(n)
l P"1(i+l)P(i)f(0(i))
i=0


72
y(k)
20;
15-
lo
st
-5
-10
-15
-20
tit
4 n-JJ-
iV'20
30
40
50
60
70 (k)
(a)
Figure 7-3
Estimation Scheme (4-2)-(4-5)
with nk= max(1, ||(k)|| )


89
yx(k)
y2(k)


88
Table 7-5
DATA
a
b
Actual Parameters
4.0
1.0
Initial Estimates
6.0
-1.0
Parameter Ranges
[2.0, 10.0]
11
o

CM
v
o

I
1 I
Estimation Scheme
(4-2)-(4-5)
3.981
1.001


34
It then follows from (4-23) that |e.^(m) 0^(n)| < s for m, n >
max (Nj,^). Combining this with property (v), e^j(k) converges to a
point in [9.. ,0.. ] for i=l, .... N and j=l, ..., m.
I J 1 J
As discussed in Chapter II, properties (ii)-(iv) and (vi) are
standard properties required for any parameter estimator used in in
direct adaptive control. Property (v) forces the parameters to converge
to a subset 0 of ]R^xm which contains no singular points. This addi
tional property will ensure that the estimated system is uniformly
stabilizable thus eliminating the need for a persistently exciting
input. Property (vii), the convergence of 0-¡j(k) to a point in
[9?]n9^X] is not required for proving global stability of the
adaptive closed-loop system. Property (vii) will be used in Chapter VI
for tracking and disturbance rejection based on the internal model
principle. It is interesting that unlike the case of least squares
where e(k) converges to 0 + PooP"1(O)'0(O) as k - , there is no closed-
form expression for the limit of 0(k) as k -> when using the param
eter estimator described by (4-2)-(4-5). However, if P(k) converges to
zero, then e(k) converges to the true system parameter matrix e for both
the parameter estimator discussed in this chapter and the recursive
least-squares estimator. This is easily shown using the recursive
solution of (4-10) which is given by
?(n) P(n>p-1{0)'e(0) P(n) J P'1 (1+l)P(l)f(e(1)).
i=Q
Taking the norm of both sides and using the properties of the induced
matrix norm, it follows that


2'
used to prove global stability of indirect adaptive controllers are (1)
an upper bound on system order is known, (2) the estimated system is
uniformly stabilizable, and (3) the parameter estimator possesses
certain key properties. Since the adaptive controller developed in this
dissertation can be classified as indirect, each of these assumptions
will be discussed in more detail.
The importance of the first assumption was illustrated in [13-14]
by Rohrs and associates who investigated the behavior of adaptive
controllers in the presence of unmodeled dynamics. Rohrs demonstrated
through computer simulations that all existing adaptive controllers
could go unstable if the order of the system was underestimated. In
response to Rohrs' findings, several individuals [15-19] have achieved
promising results in the development of robust adaptive controllers.
Verification of the second assumption, which requires the existence
of a stabilizing feedback control law for the estimated system, is
difficult because it depends on the parameter estimates which are
generated in real time. Unless something more can be said about where
the parameter estimates are going, the assumption cannot be verified a
priori and, consequently, global stability cannot be ensured. The first
approach towards satisfying this assumption was the use of a
persistently exciting external input which allows perfect identification
of the plant. This approach along with more recent results which do not
require persistent excitation are discussed in more detail in Chapter
II.
The third assumption requires that the parameter estimator possess
certain key properties which are explained in detail in Chapter II. It
will suffice to mention the existence of several parameter estimation


55
Stability of the Adaptive Closed-Loop System
The control law chosen to regulate the system is given by
u(k) = -L(k)x(k) (5-10)
where x(k) is generated from the adaptive observer (5-3) and L(K) is a
stabilizing feedback for the estimated system (F(k),G(k)). The
following theorem shows that the adaptive regulator consisting of the
observer (5-3) and the control law (5-10) results in a globally stable
closed-loop system; that is, for any initial states in the plant and the
observer, the input u(k) and the output y(k) converge to zero.
Theorem 5.4: Suppose that the parameter estimator (4-2)-(4-5) is used
so that there is a stabilizing feedback L(k) for the estimated system
(F(k),G(k)) subject to assumptions A1-A3. Then with the adaptive
regulator defined by (5-3) and (5-10), the resulting closed-loop system
is globally stable.
Proof: The proof is based on a MIMO extension of the results in [12].
Letting £(k) = y(k)-y(k) and using equations (5-3) and (5-10) gives
x(k+l) = (F(k)-G(k)L(k))x(k) M(k)e(k)
y(k) = Hx(k) e(k).


48
This relationship was introduced by Samson [10] in the single-input
case. It can also be verified in the multi-input case, but due to the
large amount of algebra involved the derivation will not be included
here.
Step 3: For any Pg > 0 there exists an N(e,Pg) such that
nPu(0) P ()H < e/6 for all eeiU and for all k > N.
Since (F(0),G(0)) is stabilizable for each 0 belonging to flg, it follows
from well-known results that P^(e) converges to P^e) pointwise over
£1a. It was shown in Green and Kamen ([33], Theorem 1) that the
convergence is uniform over Qg if is compact.
Step 4: For all Pg belonging to a compact subset D of there
exists an N(e) such that
iPM(e) P (0) n < e/6 for all eesu.
n <*> e
The compactness of 0 implies there exist matrices P and Pg such that
Pg < Pg < Pg for all Pg belonging to 0. It follows from the
relationship given in step two that
Pk (9) < Pk(9) < Pk(e) for all PQeD.
...
From step 3, there exist integers N1(e,Pg) and N2(e,Pg) such that
iiP1^() P (0)n < e/6 for all k > N. and for all 0efiA;
K 00 i e
nPk(e) Pje) 11 < e/6 for all k > N2 and for all OeSU.


62
As mentioned, the controller design will be based on the internal
model principle. Assume that the disturbance vector v(k) and the
reference signal r(k) both satisfy the difference equation:
z(k+q) + <* iZ(k+q-l) + ... + ^(k+l) + a^z(k) = 0. (6-2)
Let r be defined as
0 1
ao ~al ~aq-l
The internal model can then be realized as
xc(k+l) = Acxc(k) + Bce(k)
where Ac = block diagonal {r, r, ..., r}
m-tuple
0

0
1
Bc = block diagonal {T, T, ..., T} with t =
m-tuple
(6-3)
The following theorem gives the conditions under which it is possible to
design an adaptive controller which will stabilize the plant described


16
rank [D(z,e) N(z,0)] = m for |z| < 1 and all 0efl.
where m is the number of system outputs and
n n .
D(z,e) = I + l A.zJ, N(z,0) = l B.zJ.
j-1 J j=l J
As previously mentioned, this stabilizability condition can be checked
since it does not depend on the parameter estimates. An example will
illustrate the test.
Example 2: Consider the system described in Example 1.
[D(z,e) N(z,e)] =
1-z
z
-az 3z
l-2z z
In order to satisfy the stabilizability assumption, the rank of
[D(z,0) N(z,0)] must equal 2 for |z| < 1 and 0en. The determinants of
the 2x2 submatrices of [D(z,0) N(z,0)] are (2+a)z2 3z+l, z((6-a)z-3),
z(l-4z). All three minors are zero only when z = V4 and a = -6. Hence
the rank condition is violated only when z = V4 and a = -6. Therefore,
the stabilizability assumption will be satisfied for any interval which
does not contain the point {-6}.
The amount of effort involved in checking the rank condition in
(3-4) will of course depend on the number of unknown parameters. It may
be possible to use a root-finding algorithm for polynomials in several
variables (i.e., z and 0ij) to check the rank of [D(z,0) N(z,e)] for all
0eG. It should also be noted that by overestimating the max (p,q) in
(3-1), the system cannot be controllable, but may be stabilizable. In


17
this case, adaptive pole placement would not be possible, but a
stabilizing feedback could be computed using the Riccati approach
discussed in Chapter II.
To summarize, the assumptions on the r-input m-output linear
discrete-time system described by (3-1) (3-3) are
Al:
A2:
A3:
An upper bound, n, on p and q in (3-1) is known;
the components eij of the unknown system parameter matrix e belong
to known bounded intervals [e^Q^];
the system described by (3-3) is stabilizable for all e belonging
to n where n = {ee RNxm: e [e^.n ,9^x]}.
* J J J


49
Thus,
llP.(0) P (9)11 < e/6 for all QeSU, for all PfteD,
K E U
and for all k > maxCN^.Ng).
Step 5: Given a > 0, there exists a 6(a) such that for all P^eD and
for all e,0e£U
e
Pk+1(9) Pk+1^^11 < 0 whenever ne en < 6
This property follows easily from the fact that Pk+1(@) is a continuous
function of 8 and the sets D and fi* are compact.
Step 6: Let N be a fixed finite integer such that the condition in step
four is satisfied. There exists a finite integer Mj(e) such that
nRk+N P11 < for all k >
1/
where P is the solution to the ARE when the matrices F() and G(*)
00
are evaluated at e(k).
In order to prove this result, a family of sequences p£ is defined as
fol1ows
Pk = Rk ^or k < t.
Pk+1 = Q+FT(o(t))PkF(e(t:))-FT(0(t))P^ G(0(t))[I + ST(0(t))P¡jG(0(t))]1
x GT(9(t))P^F(9(t)) for all k > t.


23
Now define the Lyapunov functional
V(k) = TrCeT(k)P1(k)^(k)) > 0,
where "Tr" denotes the trace operation.
Inserting equation (4-10) into the expression for V(k) gives
V(k) = Tr [0'T(k-l)P"1(k-l)/e(k)] Tr[fT(e(k-l))P(k-l)P1(k)e(k)].
(4-11)
Inserting the expression (4-8) for e(k) in the first term on the right
hand side of (4-11) yields
V(k) = V(k-l) -5 TSSUilJ
Vl + +a(k-1)p(k-1)*a(k1)
Tr [0T(k-l)f(e(k-l))]
-Tr[fT(e(k-l))P(k-l)P1(k)e'(k)]. (4-12)
Equation (4-10) for e(k) is then inserted into the last term on the
right hand side of (4-12) giving
V(k) = V(k-l) -
ne(k) ii2
Vl + ¡(k-l)P(k-l)+a(k-l)
-2Tr[0T(k-l)f(0(k-l))]
(4-13)
It follows from the definition of f(0(k-l)) in equation (4-4) that
Tr[fT(0(k-l))ff(k-l)] > Tr[fT(9(k-l))f(0(k-l))].
(4-14)


70
y(k)
U(k)
20
15
10
5
-5
-10
-15
-20
:
1
1
i A
: "'V
20
30
40
(b)
Q
gg
ft VT
Figure 7-2
Recursive Least Squares


98
alone but on the identification error normalized by the signals in the
adaptive system. In order to prove global stability of the adaptive
system, Kreisselmeier assumes that the unknown system parameters e.
belong to known bounded intervals ["T11 n,9r?axl. He also assumes that
the modelling error meets a certain growth criterion which will not be
discussed here. It will suffice to mention that Kreisselmeier gives
some good examples of types of plant uncertainties which meet his
criterion. By incorporating the relative dead zone into his
identification scheme, Kreisselmeier is able to prove global stability
of an indirect adaptive controller for SISO continuous-time systems with
modelling errors. Although Kreisselmeier's identification scheme
differs from the parameter estimator introduced in Chapter Four, it may
be possible to introduce a relative dead zone into the estimation scheme
in order to develop a robust adaptive controller for MIMO discrete-time
systems.
Adaptive control was originally intended for time-varying and/or
nonlinear systems; however, most of the results have been restricted to
linear time-invariant systems. Goodwin and Sin [6] make several
suggestions on modifying the least-squares algorithm for time-varying
systems: (1) addition of a positive term to the covariance P(k), (2) a
finite data window, and (3) exponential data weighting. In the finite
data window, the old data are discarded periodically by some chosen
pattern. In exponential data weighting, a forgetting factor is
introduced into the covariance equation which has the effect of
weighting the most recent data heavier than older data. All three
methods preserve the fast initial convergence of least squares while
preventing the covariance from converging to zero which would cause


CHAPTER VII
SIMULATIONS
This chapter contains several simulations of both the adaptive
regulator derived in Chapter V and the adaptive controller discussed in
Chapter Six applied to SISO as well as MIMO discrete-time systems.
Example 7.1: Consider "the non-minimum phase discrete-time system
described by the following difference equation
y(k) = 2y(k-l)-0.99y(k-2)+0.5u(k-l)+3u(k-2).
The adaptive regulator defined by the observer (5-3) and the control law
(5-10) with L(k) computed using (5-8)-(5-9) was implemented. Three
different algorithms were used to estimate the system parameters: the
estimation scheme described by (4-2)-(4-5), the recursive least-squares
algorithm defined by (2-3), and the estimation scheme given by (4-2)-
(4-5) with data normalization (i.e., = max(l, H(j>(k-1) n) for all
k). In all three cases, the initial "covariance" P(0) was chosen to be
I and the initial state of the plant was [1 0]T. Also, Q and RQ in
(5-9) were chosen to be I. The parameter ranges, initial parameter
estimates and steady state estimates for all three estimation schemes
are displayed in Table 7-1. As seen from Table 7-1, the estimation
scheme described by (4-2)-(4-5) forces the estimates of the parameters
to converge to the given ranges; whereas, recursive least squares does
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