• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Background
 System definitions and assumpt...
 Parameter estimation
 Adaptive regulator
 Application to tracking and disturbance...
 Simulations
 Discussion
 Reference
 Biographical sketch
 Copyright














Title: New results in indirect adaptive control
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00082425/00001
 Material Information
Title: New results in indirect adaptive control
Physical Description: Book
Language: English
Creator: Ossman, Kathleen A. K., 1959-
Publisher: s.n.
 Subjects
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Statement of Responsibility: by Kathleen A. K. Ossman.
 Record Information
Bibliographic ID: UF00082425
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000985894
oclc - 17647544
notis - AEW2312

Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
    Abstract
        Page iv
    Introduction
        Page 1
        Page 2
        Page 3
    Background
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
    System definitions and assumptions
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
    Parameter estimation
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
    Adaptive regulator
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
    Application to tracking and disturbance rejection
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
    Simulations
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
    Discussion
        Page 96
        Page 97
        Page 98
        Page 99
    Reference
        Page 100
        Page 101
        Page 102
    Biographical sketch
        Page 103
        Page 104
        Page 105
    Copyright
        Copyright
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 AND ASSUMPTIONS.........................

IV PARAMETER ESTIMATION........................................

V ADAPTIVE REGULATOR......................... .................

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

VI APPLICATION TO TRACKING AND DISTURBANCE REJECTION...........

VII SIMULATIONS.................................................

VIII DISCUSSION.................................................

REFERENCES....................... ... ...........................

BIOGRAPHICAL SKETCH.................. ... ............ ... ............


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 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.















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 [13-14]

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









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.














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
y(k) = j -ajy(k-j) + I bju(k-j). (2-1)
jj=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)









where

'T = [-a ..-aan b1 ... bn] (2-2)


T (k-1) = [y(k-l) ... 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 11 will designate the vector Euclidean norm defined by
lxi = (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
JN( 1/2 + (y(k)-VT(k-l1))2 +/2(-e6(o))Tp-l(o) (e-e(o)).
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(O) can be interpreted as a measure of confidence
in the initial parameter estimate e(0). Designating e(k) as the esti-
mate of the system parameters at time k, the least-squares algorithm is
described by


e(k) = e(k-1) + TP(k-1)(k-1) [y(k) eT(k-l)(k-l)]
1 + T(k-1)P(k-1)((k-1)

P(k) = P(k-l) P(k-1)T(k-l) T(k-l)P(k-1)- P(O) = pT(o) > 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. ue(k)u
ii. ne(k)-e(k-1)n + 0 as k + =

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

where e(k) = y(k) oT(k-l)((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 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)








where


-a 1 b1
-a2 1 b2

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


-an-1 1 bn-1
-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 2(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)R(k) + G(k)u(k) + M(k) (y(k)-9(k)) (2-6)


y(k) = H)(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. ne(k)n < M < for all k

ii. ne(k) e(k-m)n 0 as k + for any finite m

iii. le(k) < a(k) n(k-l) + B(k)

where a(k) and B(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 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
n n (F(k)-G(k)L(k))ll 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-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)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]).









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 (F ,G ) is

stabilizable where F = lim F(k) and G = lim G(k).

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 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









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 ei of the system parameter vector e lie

within a known bounded interval [emin ax He then modifies the
i i
identification scheme for SISO continuous-time systems to force the

parameter estimate of 0i to converge to the set [m, inmax] for every

i. Kreisselmeier also assumes that for each e with ie C [0in ', axI

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.














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 J j=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) = PT4(k-l) (3-2)
where

T = [-A1 ... -An B1 ... Bn

JT(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) = eT a(k-1) + Tb(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-l) 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



kl(k) 1 a yl(k-l1) 3
y(k) = = + u(k-l)
y?(k) -1 2 y2(k-1) 1


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


y(k) = PTJ(k-1)
where
1 a 3 yl(k-1)1
PT = c(k-l) = yp(k-l)
-1 2 1 u(k-1)


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








a 1 0 3
y(k) = Y 2(k-l) + (k-1)

a" 1 0 3'
T = a(k-1) = y2(k-1) T 1 ] b(k-l) = 4(k-1).
0 -1 2 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 e belongs to a known bounded interval e min emax
ij ij
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 ]RNxm. The vector norm which will be used throughout this

dissertation is the Euclidean norm defined by


xi = (xTx) 1/2


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

by


"T /2
IMn = Ex (MTM)] 1


where max (MTM) is the eigenvalue of MTM with the greatest
max
magnitude. By the above assumption on the entries of e, the unknown

system parameter matrix e belongs to a known compact subspace n of IRNxm

given by








Nxm min max
= {es P : e. e. ,e ]}.
:O R O ijceoij ,6ij 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 n. Specifically, the system described
by (3-3) is assumed to be stabilizable for each eEn. This assumption
can be verified a priori because it depends only on the known set n, 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-1)
denote the polynomial matrices defined by

n n
0(z-) = I + 1 A.(z'J), N(z-1) = BW(z").
j=1 J j=1 J

The coefficients of D(z-1) and N(z-1) depend on the system parameters

and therefore can be viewed as functions of e. This dependency is made

explicit by denoting D(z-1) and N(z"1) as D(z-l,e) and N(z-l,e)
respectively. The transfer function matrix of the system described by
(3-1) is then given by


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


The stability assumption is that the system described by (3-1) with
transfer function matrix W(z-1, e) can be stabilized by dynamic output
feedback for all esa. 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 een.


where m is the number of system outputs and


n n
D(z,e) = I + I A.z, N(z,e) = B.z.
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 3z
CD(z,e) N(z,e)] = z
z 1-2z z

In order to satisfy the stabilizability assumption, the rank of

[D(z,e) N(z,e)] must equal 2 for Izl < 1 and een. 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,9) N(z,o)] for all

een. 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 eij of the unknown system parameter matrix e belong

to known bounded intervals Ce in max];
i iij
A3: the system described by (3-3) is stabilizable for all e belonging
xm min max.
to a where a = {ee Nxm:ij E i j mi 1j .
Eemin ij













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


y(k) = eTa (k-1) + T b(k-1).


(4-1)


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

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

[yT (k) *b(k-l)*-aT(k-1)e(k-1)]


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


P(k-l)a (k-l) Ta(k-l)P(k-1)
2 +
nk-1 a+ (k-1)P(k-1) a(k-1)


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


eij(k-l) emx when eij(k-1) > emax

(mkl m<
fij(k-) = oe(k-l) eijn when e. (k-1) < 'ij
13 13 13ij

min max
0 when eij(k-1) me i eij a
13 ij ij


(4-4)









1 when the determinant of P(k) > e

nk-1 = where e is any small positive number. (4-5)
max (1,lIla(k-l)ll) 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(O) 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,

nk-1 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-1)f(e(k-1)) in (4-2). This term forces the estimates

eij(k) of the components of the system parameter matrix e to converge to
mim max
the sets CeOij ,ej ]. This property, when combined with the stabiliz-
ability assumption over n 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
m rn max
belong to the set Cee ,1 1'x] for every k but does converge to the set
ij 1ij
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
min max
[oi ex ]. The prediction error e(k) which appears in the theorem
ij 13
is defined as



e(k) = y(k) eT(k-l) a(k-1) %T(k-1).








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) < M < m for all k and i = 1,2,...,N, j = 1,2,...,m;


iii. 2 T k) + 0 as k + ;
nk-1 + a(k-1)P(k-l)a (k-1)

iv. ue(k)u < a(k)na(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 + w, which implies that oij(k) converges to the

set Cein'emax ] for i = 1,2,...,N, j = 1,2,...,m;
ij ij
vi. je.i(k) eij(k-p)I + 0 as k + for any integer p,
i = 1,2,...,N, j = 1,2,...,m;
min max
vii. If 6j emij or exij for every i and j, then e..(k) converges
ij 1j ij 13
min max
to a point in [e ,ei m
1j ij
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 < xTP(k)x all xe RN, and for all k > 0. Therefore, for each x IRN, the se-
quence xTP(k)x is monotone decreasing and must converge. Denoting

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

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








where

0


ei = 0
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(0) < 21, it then follows that P(k) must converge to a positive semi-
definite matrix P, < 21. To prove (ii), let '(k) = e(k)-e. Rewriting
(4-2) in terms of '(k) gives

-) P(k-1)a (k-1)Ta(k-1)6(k-1)
(k) = (k-1) a P(k-1)f(e(k-1)). (4-8)
nk-1 + (k-1)P(k-1)a(k-1)

Multiplying both sides of the equation (4-3) on the right by P-l(k-1)
gives

P(k-1)a (k-1)a (k-1)
P(k)P-l(k-1) = I 2 a (4-9)
nk-1 + a(k-1)P(k-1)a (k-1)

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


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








Now define the Lyapunov functional


V(k) = Tr(MT(k)P-'(k)e(k)) > 0,


where "Tr" denotes the trace operation.
Inserting equation (4-10) into the expression for V(k) gives


V(k) = Tr cTe(k-1)P (k-1)^'(k)] Tr[fT(e(k-1))P(k-l)P-'(k)Y(k)].

(4-11)


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


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


he(k) 12
nk- + (k-1)P(k-1) a(k-1)
nk. I+


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


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


Equation (4-10)
right hand side


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


for 1(k) is then inserted into the last term on the
of (4-12) giving


Ine(k) 12
2 Tak1)Pk1 a-
"ki_ + Oa(k-l)P(k-1)a(k-1)


-2Tr["T(k-1)f(e(k-1))]
(4-13)


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


It follows from the definition of f(e(k-1)) in equation (4-4) that

Tr[fT(e(k-1))'(k-l)] > Tr[fT(o(k-1))f(e(k-l))]. (4-14)


(4-12)








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


V(k) < V(k-) 2e(k)
nk-1 + a(k-l)P(k-l)a (k-1)
(4-15)
-Tr[fT ((k-1))C2e(k-1) -P( P(k)P(k-1)]f((k-1))].


If the matrix 21 P(k-1)P'1(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-1)P-l(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-1(k) converges to P-1 which implies
that 21 P(k-1)P-1(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


ia(k-l) -P-l1(k) = P-l(k-1) a a
2
n k-1

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

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








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

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

converges to zero. For k sufficiently large, nk-1 = max (1,11a(k-1)II)
which implies that

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

nk-1 + a(k-1)P(k-l)a(k-1) nk-1 + maxP(O)na(k-1)i2


P(k-l)a (k-1)OTa(k-1)P(k-l)

k1a max-l


P(k-1)a (k-1)aT(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-l) 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-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 eij(k) is bounded (property (ii))
by noting that








V(k) = Tr('T (k)p-l(k)(k)) > [EminP-1(k)] Tr(/T(k)6(k))


N m
V(k) > 1/2 Tr(TkT(k)(k)) = 1/2 I I
i=1 j=1


e ij(k) ei ]2.


Now solving (4-15) recursively gives the relationship

q iie(j)112
0 < V(q) < V(O) I 2 T
j=1 nj1 + a(J-1)P(j-l) a(j-l)


(4-17)


I Tr[fT(e(j-1))[2I P(j-1)P1(j)P(j-1)]f(o(j-1))].
j=1

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


2e(k) II < b(k)
2( + a(k-l)P(k-l) where b(k) is a positive scalar sequence which converges to zero.
Multiplying both sides of the inequality by
2 1/2
(nk-+1 a(k-1)P(k-l)a(k-1)) 'and noting that nk-1 < 1 + Ia(k-1)il
gives


2 T 1/2
iie(k)ii < b(k)(nk_1 + Ta(k-1)P(k-1) (k-1))

< b(k)(nk1 + (axP(k-1))1/2 i"a(k-1)II)



< a(k)u4a(k-1)o + b(k)








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


Tr[(o(k)-O(k-1))T(e(k)-e(k-1))] = Tr[(A(k) e(k-1))T (e(k) 6(k-1))]=

e(k)Ta (k-)(k-) (k-)eT(k)



2 T
[(-i+ T)P(k-1) a(k-lk-
+2Tr fT(e(k-1)) -(k--)a
nk.1 + (-)a

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


Also


Ta (k-l)P(k-I)2 a (k-1) Ta(k-1)P(k-)a (k-1)
2 Tk(k-i) < 4 maxP(O)] 2 T
k-1 + a (k-1)P(k1)a-) -+a(k-)P(k-1)a(k-1)
< max P(O) (4-19)

Combining (4-18) and (4-19) gives

Tr[(e(k)-e(k-1))T(e(k)-e(k-1))] < [CxmaxP(O)] l2e(k) 2
nk2 1 +a(k-l)P(k-l)a (k-1)

P(k-l)2 a(k-l)Ta(k-l)'(k-l)
+ 2Tr fT(o(k-1)) 2 a
k-1 + a (k-1)P(k-1))] (k-1)

+ 2m (P(O))TrCfT(e(k-1))f(e(k-1))]. (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) pro-

vided that the matrix

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

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

T
ST P ( k-1 -1)a(k-1)
P2 (k-l)a (k-l) a(k-1) a (k-) ll 2
( a a(k-1) (ki (k-I)ii
0 42 TJki1P ~k1O (k-i) 2
k- a2+ minP(k-1)
Ina(k-1)n2 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 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) e(k-1))T(o(k) o(k-1))] = I CX.ij(k) e.i(k-1)]
i=1 j=1

it follows that eij(k) e.i(k-1) must converge to zero which proves

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

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








It then follows that


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

m-1
-P(m) P1(i + 1)P(i)f(e(i))
i=0

n-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-1(0)d(0) P(m) Z P1'(i + 1)P(i)f(e(i))
i=n

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

gives


leij(m) eij(n)|l < e1(m) e(n)n


< P(m) P(n)u BP-O1()''(O)I


m-1 1
+ IP(m)l I l P '(i+l)P(i)in if(e(i))nI
i=n
(4-21)
n-1
+ ,P(n) P(m)n H uP (i+1)P(i)i llf(e(i))n.
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 eij(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
min max
Ce ie.m]. In order to prove that the right hand side of the in-
ij J1
equality (4-21) can be made arbitrarily small, two things must be
-1
shown: lip (k+l)P(k)iu < v < for all k, and IIf(e(k))ii < L < ".
k=l
-1
In order to prove nP (k+1)P(k)n < v < o 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 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 ) I + ka(k)a(k)P(k)
P-l(k+l)P(k) = 2 a
nk

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


1 Na(k)Ta(k) n
IP l(k+l)P(k)u < IIll + a 2 nP(k)ll (4-22)
nk

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

a(k)Ta(k)
S(k)(k have magnitude less than or equal to one for k > M which in
2
nk
a (k)T a(k)
turn implies 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) con-

verges to zero. Therefore, there exists a finite v such that

iP -(k+l)P(k)n < v < for all k. Applying this result to the in-
equality (4-21) gives

m-1
leij(m)- ij(n)i < IIP(m)-P(n)ll IP'-1(0)e'(0) + 2v m Ilf(o(i))il
i=n

n-1
+ iP(m)-P(n)nv auf(e(i))i. (4-23)
i=O

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


C nf(e(k))i < L < -. It follows from equation (4-13) that
k=0

V(k) < V(k-1) 2- 2 e(k)n2 2Tr['T(k-l)f(e(k-1))]
nk- + a(k-1)P(k-1)a(k-1)

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

min max
Now assume < ei oj ij
eij is an interior point of em n m x]). Using the definition of
ij ) Using the definition of
f(e(k-1)) in (4-4), if fij(e(k-l)) 0 then |i.j(k-1)l > q > 0 where q
max min
Smin [e e. e -ei ] for i=1, ..., N,
= 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 'ij(k-1) is equal to the sign of fij(e(k-1)) for i=l,

..., N and j=1, ..., m. These facts when combined with equation (4-24),

imply
it2 N m
V(k) < V(k-1) 2 T k ---- 2q I |fi..(e(k-1))l
nk-1+ aT(k-1)P(k-1)a (k-1) i=1 j=1

+ max [P(k-1)P1(k)P(k-l)Tr[f (e(k-))f(e(k-1))].
max








Rearranging terms in this inequality results in


Ine(k) Iu2
k-+ Ta(k-1)P(k-l)Ya(k-1)
nk-1 a aa


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




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


M e(k)
0 < 1 2 T---(k)
k=1 n + a(k-1)P(k-l)(a(k-1)


M
+ 2q I If(e(k-1))n
k=l


(4-25)


M
< V(O) V(M) + "max[P(k-l)P-l(k)P(k-1)]Tr[fT(e(k-1))f(e(k-1))].
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


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

is bounded since the matrix

P(k-1)P-l(k)P(k-1)

converges to P,. It remains to be shown that

I Tr[fT(e(k-1))f(e(k-1))] < .
k=l
Equation (4-17) implies

STr[fT(e(k-1))2-P(k-)-(k)P(k-1)]f(e(k-1))] < .
k=l








Also, there exists a finite N such that the matrix
[2I-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 21-
P,. Since


0 < min[21-P(k-1)Pl(k)P(k)P(k-1)]Tr[fT(o(k-1))f(o(k-1))]
k=N

< Tr[fT(e(k-1))[2I1-P1)p)P(l)P(k)P(k-1)]f((k-1))] < =,
k=N

it follows that 1 Tr[fT(e(k-1))f(e(k-1))] < =. 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

I of(e(k-1))n < L < =. Returning to the inequality (4-23), let
k=l
e > 0 be given. The sequence oij(k) will be a Cauchy sequence if

eOij(m)-eij(n)j is less than e for m and n sufficiently large. Since
P(k) converges to P=, there exists N1(e) such that




lP(m)-P(n)i < e1 for m,n > Nj(e)


where e1 = (1/3) min Ce/(nP (O0)(0)) e/(vL)].


Also, there exists N2(e) such that

m-1
Z nf(e(i))nI < /6v for m,n > N2().
i=n








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

max (N1,N2). Combining this with property (v), oij(k) converges to a
Smin max
point in [ ,o ] for i=l, ..., N and j=l, ..., m.
ij ij
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 n of IRNxm 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 eij(k) to a point in
min max
ei 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 e + P.P'1(0)'(O) as k + -, there is no closed-

form expression for the limit of e(k) as k + O 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-1
e(n) = P(n)P-1(0)(0) P(n) p-1 (i+l)P(i)f(e(i)).
i=0

Taking the norm of both sides and using the properties of the induced

matrix norm, it follows that






35

n-1
I < (n) < n(n) I P-1(0)" (0) + IIP(n)ii n I P (i+ )P(i )f(e(i ))i.
i=O

n-1
It was shown previously that Ii P-(i+1)P(i)f(e(i))n is bounded in
i=O


the limit as n + . Also IIP(n)uI 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 e(k) converges to the true system param-

eter 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 B
-A2 2

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


-An 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 i = {ee R : e.jeeiji ]}. 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 9 belonging to a, there exists a feedback matrix L(e) such that

Xmax(F(e)-G(6)L(e))l < q(e) < 1 where max(*) is the eigenvalue of
(*) with the greatest magnitude.


It follows from known results [24], that pointwise stabilizability

of (F(9),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 be an extension of the parameter space a defined as


S{ ]RNxm min max
n = {Be R : .ele.. -e, 6. +e]}. (5-2)
J 13 13 ij









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 Q. For each e

belonging to n, there exists a bounded open neighborhood V, of e such

that (F(e),G(e)) 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

Smx(F(e)-G(e)L(e))| < q(e) < 1. The existence of the open

neighborhood V6 of a then follows because the eigenvalues of

(F(6)-G(o)L(e)) are continuous functions of e. Since a is a compact

subset of Nxm and U Ve is an open cover of Q, there exists a
Sen

r
finite subcover U Ve of n. For k=1,2,..., let ek (1/2 )k. For
j=1 J

r
every k > 1, n U V is a compact subset of RNxm and
Sk j=1 j

r r
S UV C n -U V
Sk+1 j=1 j k j=1 j

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

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


r
belongs to n U V for all k > 1. Therefore, pen for all
k j=l k

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








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

r
S- jU Ve must be empty for k > k1 which implies (F(e),G(e))

will be pointwise stabilizable over Qk for any k > k1.


Let e(k) be defined as the estimate at time k of the system

parameter matrix e, 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 s Using the results in

Kamen and Khargonekar [25], since n 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(e)-G(e)L(e))| < 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


max (F(k)-G(k)L(k))l < q < 1 for all k > N.


Since the entries in (F(k)-G(k)L(k)) are continuous functions of e,
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 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.





41


Adaptive Observer


The adaptive observer for the estimated system is given by


x(k+1) = F(k)2(K) + G(k)u(k) + M(k)(y(k)-9(k))

y(k) = HM(k)


-A1(k)
-A2(k)

F(k) =

-An (k)


M(k) =


7-Al(k)
-A2(k)




-An(k)


G(k) =


B (k)
B2(k)




Bn(k)


(5-3)


H = [I 0 . 0].


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) = [-Al(k-1) ... -An(k-n) B1(k-1) ... Bn(k-n)](k-1)


(5-4)


where








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


X(k) =


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


y(k) = Hx(k) = [0 I 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


y(k) = [0 0 I ... 0]x(k-2) + B1(k-1)u(k-1) + B2(k-l)u(k-2)
Al(k-1)y(k-1) A2(k-2)y(k-2)


Continuing the substitution using (5-5) gives


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


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


I A.(k-j)y(k-j)
j=1 J
n n
= 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) = 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,


ln(k)n < a'(k)nH(k-1)n + 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) = y(k) ceT(k-l)Da(k-l) + T1b(k-l)]


+ [eT(k-l) a(k-1) + T b(k-1)] y(k)


'(k) = y(k) [eT(k-l)a(k-1) + TTb(k-1)] e(k).


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


nl(k)n < l (k) (eT(k-l) (k-1) + (k-l))+ + ne(k)n


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








4iU(k)i < 1[J1 (k) ... J2n(k)](k-l)I + lie(k)n (5-7)


< I[J1 (k) ... J2n(k)] ii iu (k-1)ii + Ile(k)ii
-Ai (k-i) + Ai (k-1) 1 where Ji(k) =
Bi (k-i) Bi (k-1) n+1



It follows from property (vi) in Theorem 4.1 that Ji.(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 n (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.,

6(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)








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 n(L(k)-L(k-1))a 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
L(k)-L(k-l1) + 0 as k + -.
Proof: The proof is rather lengthy and will therefore be divided into

steps. Throughout the proof, Pk(e) will refer to the sequence generated
from the time-invariant Riccati difference equation (RDE) given by



Pk+l(e) = Q + FT()Pk()F() FT()Pk(e)G(e)[I + GT ()Pk(6)G(e)]-1


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 (e) = Q + FT(e)P(e)F(e ) FT(e)P (e)G(e)[I + GT (e)P(e)G(e)-1


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


Let g denote the compact parameter space defined in (5-2) such that
(F(e),G(e)) is pointwise stabilizable for all a belonging to nQ. Let
S> 0 be given. It will first be shown that IRk Rk-,l < e 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),GT(e)) is detectable for all e belonging
to the parameter space n. 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 Po and P2 respectively. Defining
APk(e) = P(6) P2(e), it then follows that


APk+l(e) = FT(e)(I Pk(e)G(B)[GT(e)Pk(e)G(O) + I]-GT(e))


x(APk(e) + APk(e)G(B)CGT ()P (e)G(e) + I]-1 GT()APk())



x (I Pk(e)G()[GT ()Pk(6)G() + I]-1 GT())TF(e).
k k








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)i < E/6 for all eOe and for all k > N.

Since (F(e),G(e)) is stabilizable for each e belonging to Qn, it follows

from well-known results that Pk(e) converges to P,(e) pointwise over

ng. It was shown in Green and Kamen ([33], Theorem 1) that the
convergence is uniform over a if a is compact.
E E
Step 4: For all PO belonging to a compact subset D of NxN, there

exists an N(e) such that


nPN(e) P (e)n < e/6 for all OeS^.


The compactness of 0 implies there exist matrices P0 and P0 such that

Pm < P P M for all PO belonging to 0. It follows from the
relationship given in step two that


Pk (e) < Pk(e) < P (e) for all PgeD.


From step 3, there exist integers N(ePm) and N2(e,PM) such that


IPm(e) P (e)i < e/6 for all k > N1 and for all eso ;


i (e) P (e)n < e/6 for all k > N2 and for all BEn .
k 2.








Thus,


lPk(f ) P (k)i < E/6 for all een8 for all P0OD,

and for all k > max(N1,N2).


Step 5:

for all


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

S


lPk+1() k+1( )" < a


whenever ne ^n < 6


This property follows easily from the fact that Pk+1(e) is a continuous

function of e and the sets D and n, 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 M1(s) such that



nRk+N P k. < e/3 for all k > M
ccN 1


where P is the solution to the ARE when the matrices
are evaluated at 8(k).

In order to prove this result, a family of sequences Pk

follows


F(.) and G(.)


is defined as


Pk = Rk for all k < t.



Pk+l = Q+FT(e(t))P F(e(t))-FT ((t))P G(e(t))I[ + GT(e(t))PtG(e(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, Pk is generated from the time-invariant Riccati difference equation
t, Pk
initialized by Rt and using the constant matrices F(.) and G(.)
evaluated at e(t). For every e(t) belonging to n the sequence p
t k
will be bounded for all k. In fact, Pk converges to Pt whenever
k
O(t)en,. It will first be shown that IlR, P k i < e/6 for k
S k+N k+N
sufficiently large. The matrix Pkk is simply the Nth step of a time-
k+N
invariant RDE intitialized at time k by Rk and using constant matrices
F(.) and G(.) evaluated at e(k). The interpretation for
k
IRk+N -Pk+N being small for k sufficiently large is that the time-
varying 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


Rk+i+1 = Q + FT(e(k))Rk+iF(o(k)) FT ((k))Rk+iG(e(k))


x [I + GT(o(k))Rk+iG(o(k))]-1 GT(e(k))Rk+iF(e(k)) + Ek+i; 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(e(k+i))

x [I + GT(O(k+i))Rk+iG(o(k+i))]-1 GT(e(k+i))Rk+iF(O(k+i))

FT(o(k))Rk+iF(e(k)) + FT(o(k))Rk+iG(e(k))

x [I + GT(e(k))Rk+iG(e(k))]"- GT(e(k))Rk+iF(e(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 ((a) such that


.Zk+ill < a whenever lie(k+i) e(k)ll < ( for ie[O,N].


k k
Let Qk = Rk P


k+
k+i +1


Using the equation introduced in step two gives


= FT(e(k))(I Rk+iG(e(k))[GT(e(k))Rk+iG(e(k)) + I] 1e(k)))


k k T k -1
x (Qki + k+i G(e(k))[GT(e(k))Pk+iG(e(k)) + I]-1


x GT(e(k))Qk+i) (I Rk+iG(e(k))[GT(e(k))Rk+iG(e(k)) + I]-


x GT(e(k)))T + k+i; Q = 0, iCO.,N].


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


(for t


lFT(e(k))(I -.Rk+iG(e(k))[GT(e(k))Rk+iG(e(k)) + I]-lGT(e(k)))l < K1<


and IIG(e(k))CGT(e(k))Pk+ G(O(k)) + I]-GT(e(k))n < K2 <


Thus for all k such that e(k)en,, we have


k 2 k + K 2 + Q k = ieN
"Qk+i+lI < 1 (Qk+i + K2 Qk+i ) + k+i k = O, i'[O,N].








Clearly, there exists a c1(e) such that if lk+i ll < I1 for all ie[O,N]
then


k
,Qk+N[ < e/6.


Let 6(al) be chosen such that



Hsk+iU < a1 for all ie[O,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 M1 such that


e(k)sng for all k > M1


and ne(k+i) e(k)n < 6(al) for all ic [0,N] and for all k > M1.

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


Using the results in step four, we obtain


k k
nPk+N P < e/6 for all k > M1.
Thus

kRk+N Ik+N k + iPkN kl < s/3 for all k > M1.
00 k+N k+N P 10


which proves the result.








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


k+1 k
HP P k < e/3 for all k > M2.


The result follows immediately from the uniform continuity of

P (.) over nG and the convergence of Ile(k+l) e(k)u to zero.
Step 8: There exists a finite integer M(e) such that


nRk Rk-, < e for all k > M.



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

seven gives


k+1 k+1 k k
k+N+ Rk+N < Rk+N+ p k+ + p~P P k + llP R II < E
k+N+1 CkN k+N+1 C C = P k+N


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 nL(k) L(k-l)l

converge to zero as k goes to infinity. This result follows easily

since IRk Rkl11, IF(k) F(k-1) i, and IG(k) G(k-1)uI 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(B),GT e)) is detectable for
each e belonging to Q. Using arguments similar to those in
Proposition 5.1 and 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))I
converges to zero as k + .. Property (vi) of Theorem 4.1 implies both
IF(k)-F(k-1)n and nG(k)-G(k-1)II converge to zero. It was shown in
Theorem 5.2 that IL(k)-L(k-1)I 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
ll(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)2(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 e(k) = y(k)-y(k) and using equations (5-3) and (5-10) gives




A(k+1) = (F(k)-G(k)L(k))1(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 ... O 0 ... 0
Im(n-1)
S = 0 ... 0 0 ... 0 ,
0

Ir(n-1)

0

Defining z(k+l) = A(k) gives
X(k+l)


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

H
0
D(k) = -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(k)
A(k) = w(k) = .
0 F(k)-G(k)L(k) -M(k) (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


nw(k)B < 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) converges to zero. Let A(k+N,k) denote the state
transition matrix for (5-11) defined by










A(k+M,k) =


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


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

follows that


iiA(k+M,k)Il < 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


Iiz(k+1) e < IiA(k) Iiz(k) I + nw(k) I


(5-15)


< (iA(k)u + a(k))nz(k)ii + b(k).


Noting that HA(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 R3

such that


Iiz(k+i)i1 < 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+l)w(k+j).
j=0


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


iz(k+p)a < IA(k+p,k) niz(k)n


< nA(k+p,k)n iz(k)ni


p-1
+ fIA(k+p,k+j+l)n nw(k+j)n
j=0
p-1
+ I nA(k+p,k+j+1) i(a(k+j)iiz(k+j)ii + b(k+j)).
j=0


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


P p-1
liz(k+p)n < 5 lz(k)n + max(1,R1) I a(k+j)(R2inz(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 for all k > N.
6 + pR2max(1,R,)

Therefore, for all k > N,


p-l
az(k+p)ii < rnz(k)ii + max(1,R1) Z a(k+j)R3 + b(k+j) or equivalently,
j=0


nz(k+p)n < riz(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 uiz(k)n converges to zero, let e>0 be given. Solving
(5-18) recursively and using (5-16) gives










n-1

n-1
j=0

n-1 n-1-j
Iz(k+i+np)i < r (R2iiz(k)ii + R3) + I r '1jc(k+i+jp)
j=0


for all k > N, ie[0,p]. (5-19)


Since c(k) converges to zero, there exists a finite N1 > N such that


c(k) < for all k > N1.
1-r

Applying this inequality to (5-19) gives


nz(N1+i+np)il < rn(R2nz(N1)i + R3) + e/2, ie[0,p].


Also, since r < 1 there exists a finite N2 such that


n 2
2(R2rnz(N)n + R3) for all n > N2.


Thus


nz(N1+i+np) i < e for all n > N2, ie[O,p]


which implies tz(k)in < 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
-A2 2
A2 I B2

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


-An 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 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 + -.









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) + aqq1z(k+q-1) + ... + alz(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-tuple


0

c = block diagonal {T, T, ..., T with T =
m-tuple 1



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


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

Nxmmi n max
where n = {ee Rm: eje [e 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)
S-L~x(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) F(k) 0 x(k) G(k) -M(k)
+ u(k) + e(k) (6-4)
xc(k+1) BcH Ac xc(k) u0 -Bc

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) = -Ll(k)A(k) -L2xc(k). (6-5)


F(k) 0 A G('k)
Let F(k) = and G(k) =
BcH Ac 0
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) = C^T(k)R 'G(k) + ]1-1 ^T(k)Rk (k)


(6-6)


Rk+1 = Q + LT(k)L(k) + (F(k)-6(k)L(k))TRk(F(k)-G(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 [10], L(k)

converges to L., the optimal LQ feedback for the pair (F ,G ) given by


L = CGTR G + I"1 GT RF
00 00 CO 0000


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

precisely,


R = Q + LTL + (F -GL )TR (F -GL ).
W O Go .


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) = -L$x(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 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-1)-0.99y(k-2)+0.5u(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., nk-1 = max(l, Hi(k-l) H) for all

k). In all three cases, the initial covariancee" P(O) was chosen to be

I and the initial state of the plant was [1 O]T. Also, Q and Ro 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








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


-aI -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] [-2.0, 4.0] [-0.5, 2.0] [2.5, 9.5]

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
nk = max(l,ie(k)li)





















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


20 30 40 50 60 70 '(k


I.'


I11'i


'IIi ii i


20 30 40 50 60 70-- 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


--


-- -1-



















-al(k)


b2(k)
b (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 40 50 60 ---T





(b)

Figure 7-2
Recursive Least Squares


I I I


Ii 20


Jl i

























-al(k)


- 20 30 40 50 _2(ko


70 (k)


b2(k)

bl(k)


20 30 40 50 60 70 (k)


Figure 7-2 continued
Recursive Least Squares


____


I I I I _I t I I L


L-


'I




















20 30 40 50 60 70 (k)





(a)


1114 It *-.


SII 'J "' 20


30 40 50 60 70 (k)


Figure 7-3
Estimation Scheme (4-2)-(4-5)
with nk= max(l, |ll(k)jI )


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) + 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 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 covariancee" matrix P(0)
should satisfy 0 the system rapidly went unstable. At time k=20, the output was

y(k)=6.5x108 and the control input was u(k)=-2.6x1011. 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(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.














Table 7-2

DATA


-al -a2 -a3 b I 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 2.5 -1.92 0.429 0.9991 1.60 -0.795
(4-2)-(4-5)












y(k)

40
30
30 l II
10 I !
20.[ I, I--______________________________


-10 0 20 30 40 50 60 70 (k)

-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)
-a3 (k)


20 30 40 50 60 70


-a2(k)


b2(k)
I..- f l,


U ir A


20 30 40 50 60 70 b3(k)





(d)

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


,.., '


10









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


[0 O]T K < 40
V() [3 0]T K > 40

was introduced. The internal model was given by


xc(k+1) = 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)=[1 O]T, c(0O)=0, 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


-al -a2 b1 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] [-8, -4] [-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


-20 tI 20 40 '- 60 80 100 1
-40 '
-60 I
-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


- -4U 60 80 100 12
- ZU 4U 60 80 100 120


t 4 b 2(k)
140 b (k)


(d)

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


S14n


-al(k)

-a2(k)


......... l


140















y(k)

20
15
10
5


is


20 40 60 80 100


140 (k)


(a).


u(k)


.- 20 40'-... 60
1 '* --.


100


140 (k)


Figure 7-6
Exact Parameters


-5
-10
-15
-20


-20
-40
-60
-80


) I I I








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 (wk/2). The internal model was given

by





0 1 0 0
Xc(k+1) = 0 0 1 xc(k) + 0 e(k)
1 -1 1 1




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 ]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


-ai -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] [-2.0, 4.0] [-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) - - _- ---. ... ,.
40 50 60 70 (k)


(b)

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


y(k)
^


i i
' I.


l i
I.,,


10 20

I-..


-30


-40


u(I )


-10
-20
-30
-40


LLi U,4e
ii
I '


. .. I il 20"'2 "' 30

.


-I -


- -:---























-> l )k


n2 30n 4 60 g7 0


bo(k)


h )k(


10 20 30 40 50 60 70






(d)

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


F


*1


,- I


I


I


I-- -~


I -


3


I


10 _


--*'***'~~ ~ ~ ~ ~ W ---


























20 30 40 50 60 70 (k)


_ '._-._, -_--; _- -_ ---_ --_, .- __- _- 'k -_- -'_- ._____- _-
20 30 40 50 60 70 (k)






(b)


Figure 7-8
Exact Parameters


y(k)
I


40
30
20
10



-10
-20
-30
-40


_1 I"


S 1U


u(k)


10


-,~----------,--,


II ' I I I I I


I


'"


!








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


yl(k) 1 al y1 (k-1) + 3 'u(k-1)
= + u(k-l)
y2(k) -1 2 y2 (k-1) b


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) = eTa(k-1) + *Tb(k-1)
with



T a 0 y(k-1) 1 0 3' y1(k-1)
S= a(k-1) = = b(k-1) = y2(k-l) .

0 b Lu(k-1) 1 2 0 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.














Table 7-5

DATA


a b

Actual Parameters 4.0 1.0

Initial Estimates 6.0 -1.0

Parameter Ranges [2.0, 10.0] [-4.0, 2.0]

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













Yl(k)

40
30
20 L
10 L


-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


S 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




yl(k)1 [i -a,2 [y1 (k-1) 1 1 b12 u (k-1)
= II+ .
2(k) -1 2 Y2 (k-1) b21 b22 u2 (k-1)

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


y(k) = Ta (k-1) + T% b(k-l)
with

OT a12 0 b12 %aT(k-1)= [y2(k-l) ul(k-1) u2(k-l)],
0 b21 b22

S1 0 1
T = i bT(k-1) = Cyl(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(0)=I, x(0)=[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.














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 [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)













yl(k)


I0
10

5



II

-5


-10:
I


20 30 40 50 60 70 (k)





(a)


y2(k)


10


5





-5


-10


I t 1
j
* [ : .


20 30 40 50 60 70 (k)





(b)


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














ul(k)

10


5





-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)


,,i "10


-5


-10


r--LSt;re I L i r I i I


,*1

























,~tV11


I----


10 20 30 40 50 60 70







(e)


b21(k)



10 20 30 40 50 60 70 b22(k)






(f)


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


iI


I I I I -r I__rr_____


---


l


J














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.

[13-14] 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.




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs