 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 

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Title: 
New results in indirect adaptive control 

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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. DAAG2984K0081.
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 ClosedLoop 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 multiinput multioutput 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
closedloop 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 discretetime 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 timevarying or nonlinear systems, most of the
stability results on adaptive controllers have been limited to linear
timeinvariant 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 prechosen reference model. Several individuals have developed
this approach, see for example [13]. 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 [412]. 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 [1314]
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 [1519] 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 leastsquares, 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 multiinput multi
output discretetime 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 closedloop
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 discretetime 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 wellknown 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 discretetime system described by the input/out
put difference equation:
q P
y(k) = j ajy(kj) + I bju(kj). (21)
jj=1
In (21), 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
(21) can be rewritten in the following form which is convenient for
parameter estimation:
y(k) = eT(kl)
where
'T = [a ..aan b1 ... bn] (22)
T (k1) = [y(kl) ... y(kn) u(k1) ... u(kn)].
The vector e consists of all the system parameters and ((kl) 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
leastsquares algorithm has the fastest convergence rate and is most
easily modified to handle output disturbances and slowly timevarying
systems. The algorithm results from minimizing the quadratic cost
function:
N
JN( 1/2 + (y(k)VT(kl1))2 +/2(e6(o))Tpl(o) (ee(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 leastsquares algorithm is
described by
e(k) = e(k1) + TP(k1)(k1) [y(k) eT(kl)(kl)]
1 + T(k1)P(k1)((k1)
P(k) = P(kl) P(k1)T(kl) T(kl)P(k1) P(O) = pT(o) > 0. (23)
1 + *T(k1)P(k1).(k1)
The key properties of the leastsquares scheme used for proving global
stability of an indirect adaptive controller are
i. ue(k)u
ii. ne(k)e(k1)n + 0 as k + =
iii. lim e (k = 0
k+ 1 + ,T(k1)P(kl),(k1)
where e(k) = y(k) oT(kl)((k1).
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 leastsquares 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
statespace observer realization of the system described by (21) is
given by
x(k+l) = Fx(k) + Gu(k) (24)
y(k) = Hx(k)
where
a 1 b1
a2 1 b2
F = G = H = [1 0 ... 0].
an1 1 bn1
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), (25)
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)) (26)
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) (27)
In (27), Q and the initial value R(O) are arbitrary positive
definite symmetric matrices.
In order to prove global stability of the closedloop 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(km)n 0 as k + for any finite m
iii. le(k) < a(k) n(kl) + B(k)
where a(k) and B(k) converge to zero.
An example of a parameter estimator possessing these properties is the
leastsquares estimator discussed previously. Clearly, properties (i)
and (ii) listed above are equivalent to properties (i) and (ii) given
for the leastsquares 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+r1
n n (F(k)G(k)L(k))ll
k=t
where
t+r1
n (F(k)G(k)L(k)) = (F(t+r) G(t+r)L(t+r))x
k=t
(F(t+r1) G(t+r1)L(t+r1)) ... (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 (25)(27) will result in a globally stable
closedloop 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 leastsquares 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 LozanoLeal 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 [49]. 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 [1516], indirect adaptive controllers
which require a persistently exciting input are not robust. Since the
adaptive closedloop system is inherently timevarying 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.
LozanoLeal and Goodwin [21] developed an estimation scheme which
gives nonsingular parameter estimates at each point in time and in the
limit for SISO linear discretetime systems. They modify (when neces
sary) the parameter estimates generated from the leastsquares 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 leastsquares 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 LozanoLeal 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 continuoustime 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 continuoustime adaptive controller.
CHAPTER III
SYSTEM DEFINITIONS AND ASSUMPTIONS
The system to be regulated is the multiinput multioutput linear
discretetime system described by
y(k) = A. y(kj) + B.u(kj) (31)
j=1 J j=1 J
In (31), 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 (31) can be rewritten in the following form
convenient for parameter estimation:
y(k) = PT4(kl) (32)
where
T = [A1 ... An B1 ... Bn
JT(k1) = [yT(k1) ... yT(kn) uT(k1) ... uT(kn)]
The n(m+r)xm matrix P consists of all the system parameters and the
n(m+r)xl vector *(kl) 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
(32) in the following form:
y(k) = eT a(k1) + Tb(kl) (33)
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 (k1) and *b(kl) are regression vectors whose components
come from <(kl). Clearly, the decomposition described in (33) is not
unique for multiinput multioutput systems. In order to minimize the
required computations for parameter estimation, the size of a (k1)
should be made as small as possible. An example will serve to
illustrate these concepts.
Example 1: Consider the linear discretetime system with two outputs
and a single input described by
kl(k) 1 a yl(kl1) 3
y(k) = = + u(kl)
y?(k) 1 2 y2(k1) 1
The parameter a is assumed to be unknown. The system can be rewritten
in the form
y(k) = PTJ(k1)
where
1 a 3 yl(k1)1
PT = c(kl) = yp(kl)
1 2 1 u(k1)
A decomposition for this system which minimizes the required
computations for parameter estimation is given by
a 1 0 3
y(k) = Y 2(kl) + (k1)
a" 1 0 3'
T = a(k1) = y2(k1) T 1 ] b(kl) = 4(k1).
0 1 2 1
Again, consider the linear discretetime system described by
(33). 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 (33) 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(z1) and N(z1)
denote the polynomial matrices defined by
n n
0(z) = I + 1 A.(z'J), N(z1) = BW(z").
j=1 J j=1 J
The coefficients of D(z1) and N(z1) depend on the system parameters
and therefore can be viewed as functions of e. This dependency is made
explicit by denoting D(z1) and N(z"1) as D(zl,e) and N(zl,e)
respectively. The transfer function matrix of the system described by
(31) is then given by
W(z1,6) = D1(z1,e)N(z1,).
The stability assumption is that the system described by (31) with
transfer function matrix W(z1, 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.
1z az 3z
CD(z,e) N(z,e)] = z
z 12z 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((6a)z3),
z(14z). 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 I6}.
The amount of effort involved in checking the rank condition in
(34) will of course depend on the number of unknown parameters. It may
be possible to use a rootfinding 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
(31), 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 rinput
discretetime system described by (31) (33) are
possible, but a
Riccati approach
moutput linear
Al: An upper bound, n, on p and q in (31) 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 (33) 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 rinput moutput linear discretetime system described
y(k) = eTa (k1) + T b(k1).
(41)
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(kl)a (k1)
e(k) = e(kl) P(kl)f(e(kl)) + 2 T x (42)
k1 + a(kl)P(kl)a (k1)
[yT (k) *b(kl)*aT(k1)e(k1)]
P(k) = P(k1) 
P(kl)a (kl) Ta(kl)P(k1)
2 +
nk1 a+ (k1)P(k1) a(k1)
0 < P(O) = pT(0) < 21,
(43)
eij(kl) emx when eij(k1) > emax
(mkl m<
fij(k) = oe(kl) eijn when e. (k1) < 'ij
13 13 13ij
min max
0 when eij(k1) me i eij a
13 ij ij
(44)
1 when the determinant of P(k) > e
nk1 = where e is any small positive number. (45)
max (1,lIla(kl)ll) otherwise
The parameter estimation algorithm described by (42) (45)
differs from the recursive leastsquares estimator in three ways.
First, the initial covariance matrix P(O) in (43) must be less than 21,
whereas the leastsquares algorithm allows for any positive definite
initial covariance matrix. This condition is needed to ensure that the
parameter estimator described by (42) (45) 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 nk1 defined in (45). In the leastsquares algorithm,
nk1 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 (45). 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 (42) (45)
and the recursive leastsquares estimator is the addition of a "correc
tion term" P(k1)f(e(k1)) in (42). 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(k1)) is defined exactly the same as in [23]; however, the
parameter estimator described here for MIMO discretetime systems is
quite different from the adaptive observer Kreisselmeier uses for SISO
continuoustime systems.
The following theorem shows the estimator described by (42) 
(45) has many of the same properties as recursive leastsquares. 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(kl) a(k1) %T(k1).
Theorem 4.1: The estimation algorithm (42) (45) 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 + ;
nk1 + a(k1)P(kl)a (k1)
iv. ue(k)u < a(k)na(k1)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(kp)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 wellknown property first proved by Samson
[10] with nk1 equal to one. The proof is included here for the sake of
completeness. Equation (43) 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 (47)
where
0
ei = 0
1 ith position.
0
0
Each of the terms on the right hand side of the equation (47) 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
(42) in terms of '(k) gives
) P(k1)a (k1)Ta(k1)6(k1)
(k) = (k1) a P(k1)f(e(k1)). (48)
nk1 + (k1)P(k1)a(k1)
Multiplying both sides of the equation (43) on the right by Pl(k1)
gives
P(k1)a (k1)a (k1)
P(k)Pl(k1) = I 2 a (49)
nk1 + a(k1)P(k1)a (k1)
Combining (48) and (49) yields
e(k) = P(k)P (kl)O(k1) P(k1)f(e(k1)). (410)
Now define the Lyapunov functional
V(k) = Tr(MT(k)P'(k)e(k)) > 0,
where "Tr" denotes the trace operation.
Inserting equation (410) into the expression for V(k) gives
V(k) = Tr cTe(k1)P (k1)^'(k)] Tr[fT(e(k1))P(kl)P'(k)Y(k)].
(411)
Inserting the expression (48) for "(k) in the first term on the right
hand side of (411) yields
V(k) = V(k1) 
he(k) 12
nk + (k1)P(k1) a(k1)
nk. I+
 Tr [VT(k1)f(o(k1))]
Tr[fT(e(k1))P(k1)P'1(k)'(k)].
Equation (410)
right hand side
V(k) = V(k1) 
for 1(k) is then inserted into the last term on the
of (412) giving
Ine(k) 12
2 Tak1)Pk1 a
"ki_ + Oa(kl)P(k1)a(k1)
2Tr["T(k1)f(e(k1))]
(413)
+ Tr[fT(e(k1))P(k1)((k)P(k1)f(e(k1))].
It follows from the definition of f(e(k1)) in equation (44) that
Tr[fT(e(k1))'(kl)] > Tr[fT(o(k1))f(e(kl))]. (414)
(412)
Combining equations (413) and (414) gives
V(k) < V(k) 2e(k)
nk1 + a(kl)P(kl)a (k1)
(415)
Tr[fT ((k1))C2e(k1) P( P(k)P(k1)]f((k1))].
If the matrix 21 P(k1)P'1(k)P(kl) 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(k1)Pl(k)P(kl) is positive definite for k
sufficiently large, first assume that the determinant of P(k) does not
converge to zero. In this case, P1(k) converges to P1 which implies
that 21 P(k1)P1(k)P(k1) 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 (43) and the
matrix inversion lemma gives
ia(kl)
Pl1(k) = Pl(k1) a a
2
n k1
Multiplying both sides of this equation on the left and right by P(kl)
and adding 21 to both sides give
P(kl)a (kl) Ta (kl)P(k1)
21 P(k)Pl(k)P(k1) = 2IP(k1) 2 
n k1 (41
(416)
Since P(k) converges, equation (43) implies that the matrix
P(kl) a(kl)Ta(kl)P(k1)
2 T
nk1 + a(k)P(kl)a( k1)
converges to zero. For k sufficiently large, nk1 = max (1,11a(k1)II)
which implies that
P(kl)a (kl) a(k1)P(k1) P(kl) a(kl) a (kl)P(k1)
nk1 + a(k1)P(kl)a(k1) nk1 + maxP(O)na(k1)i2
P(kl)a (k1)OTa(k1)P(kl)
k1a maxl
P(k1)a (k1)aT(k1)P(k1)
It then follows that the matrix 2 converges
nk1
to zero since it is bounded above by a matrix which is converging to
zero. Thus by equation (416), the matrix 21 P(k1)P1(k)P(kl) also
converges to 21 P. in the case when the determinant of P(k) converges
to zero. Therefore, the matrix 21 P(k1)P1(k)P(kl) is positive
definite for k sufficiently large since the matrix converges to the
positive definite matrix 21 P.. By (415), 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)pl(k)(k)) > [EminP1(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 (415) recursively gives the relationship
q iie(j)112
0 < V(q) < V(O) I 2 T
j=1 nj1 + a(J1)P(jl) a(jl)
(417)
I Tr[fT(e(j1))[2I P(j1)P1(j)P(j1)]f(o(j1))].
j=1
For sufficiently large k, the terms within the summations are nonnega
tive and must converge to zero since V(k) is nonnegative. Properties
(iii) and (v) then follow easily. Property (iv) follows immediately
from (iii) by first noting that
2e(k) II < b(k)
2( + a(kl)P(kl)
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(k1)P(kl)a(k1)) 'and noting that nk1 < 1 + Ia(k1)il
gives
2 T 1/2
iie(k)ii < b(k)(nk_1 + Ta(k1)P(k1) (k1))
< b(k)(nk1 + (axP(k1))1/2 i"a(k1)II)
< a(k)u4a(k1)o + b(k)
where a(k) and b(k) are positive scalar sequences which converge to
zero. To prove (vi), first subtract 'e(k1) from both sides of equa
tion (48). Premultiplying both sides of the resulting equation by its
transpose and noting that e(k) = T(k1)a (k1) gives
Tr[(o(k)O(k1))T(e(k)e(k1))] = Tr[(A(k) e(k1))T (e(k) 6(k1))]=
e(k)Ta (k)(k) (k)eT(k)
2 T
[(i+ T)P(k1) a(klk
+2Tr fT(e(k1)) (k)a
nk.1 + ()a
+ Tr[fT(e(k1))P(k1) 2f((k1))]. (418)
Also
Ta (kl)P(kI)2 a (k1) Ta(k1)P(k)a (k1)
2 Tk(ki) < 4 maxP(O)] 2 T
k1 + a (k1)P(k1)a) +a(k)P(k1)a(k1)
< max P(O) (419)
Combining (418) and (419) gives
Tr[(e(k)e(k1))T(e(k)e(k1))] < [CxmaxP(O)] l2e(k) 2
nk2 1 +a(kl)P(kl)a (k1)
P(kl)2 a(kl)Ta(kl)'(kl)
+ 2Tr fT(o(k1)) 2 a
k1 + a (k1)P(k1))] (k1)
+ 2m (P(O))TrCfT(e(k1))f(e(k1))]. (420)
By properties (iii) and (v), the first and third terms on the right side
of the inequality (420) 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)(kl)
2 Taa
nk + a(k1)P(k) a(k1)
is bounded. This matrix is shown to be bounded by first noting that
T
ST P ( k1 1)a(k1)
P2 (kl)a (kl) a(k1) a (k) ll 2
( a a(k1) (ki (kI)ii
0 42 TJki1P ~k1O (ki) 2
k a2+ minP(k1)
Ina(k1)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 (420) converge to zero. Since
N m
Tr[(e(k) e(k1))T(o(k) o(k1))] = I CX.ij(k) e.i(k1)]
i=1 j=1
it follows that eij(k) e.i(k1) 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 (410) is
first solved recursively giving
n1
(n) = P(n)P1(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)
m1
P(m) P1(i + 1)P(i)f(e(i))
i=0
n1
+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))P1(0)d(0) P(m) Z P1'(i + 1)P(i)f(e(i))
i=n
n1
+ (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 BPO1()''(O)I
m1 1
+ IP(m)l I l P '(i+l)P(i)in if(e(i))nI
i=n
(421)
n1
+ ,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 (421) 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 (421) 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,
P1(k+1) converges to P 1 which implies that P1(k+1)P(k) converges to
I. Therefore, P P1(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 (43) gives
P ) I + ka(k)a(k)P(k)
Pl(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 (422)
nk
From (45), 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 (422) 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 (421) gives
m1
leij(m) ij(n)i < IIP(m)P(n)ll IP'1(0)e'(0) + 2v m Ilf(o(i))il
i=n
n1
+ iP(m)P(n)nv auf(e(i))i. (423)
i=O
In order to complete the convergence proof, it must be shown that
C nf(e(k))i < L < . It follows from equation (413) that
k=0
V(k) < V(k1) 2 2 e(k)n2 2Tr['T(kl)f(e(k1))]
nk + a(k1)P(k1)a(k1)
+ xmax[P(kl)Pl(k)P(kl)]Tr[fT(e(kl))f(e(kl))]. (424)
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(k1)) in (44), if fij(e(kl)) 0 then i.j(k1)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(k1)), it must be true
that the sign of 'ij(k1) is equal to the sign of fij(e(k1)) for i=l,
..., N and j=1, ..., m. These facts when combined with equation (424),
imply
it2 N m
V(k) < V(k1) 2 T k  2q I fi..(e(k1))l
nk1+ aT(k1)P(k1)a (k1) i=1 j=1
+ max [P(k1)P1(k)P(kl)Tr[f (e(k))f(e(k1))].
max
Rearranging terms in this inequality results in
Ine(k) Iu2
k+ Ta(k1)P(kl)Ya(k1)
nk1 a aa
N m
+ 2q I I f..(e(k1))
i=1 j=1
Summing all the terms in this inequality from k equals one to M and
N m
noting that nf(e(k1))I < 1 If..(e(k1))i gives
i=1 j=1
M e(k)
0 < 1 2 T(k)
k=1 n + a(k1)P(kl)(a(k1)
M
+ 2q I If(e(k1))n
k=l
(425)
M
< V(O) V(M) + "max[P(kl)Pl(k)P(k1)]Tr[fT(e(k1))f(e(k1))].
k=l
If the right hand side of the inequality (425) is bounded in the limit
as M approaches infinity, then the norm of f(e(k1)) will be summable.
The term V(M) is bounded since V(k) converges. Also
XmaxCP(k1)P1(k)P(k1)]
is bounded since the matrix
P(k1)Pl(k)P(k1)
converges to P,. It remains to be shown that
I Tr[fT(e(k1))f(e(k1))] < .
k=l
Equation (417) implies
STr[fT(e(k1))2P(k)(k)P(k1)]f(e(k1))] < .
k=l
Also, there exists a finite N such that the matrix
[2IP(k1)P1(k)P(k1)] is positive definite for k > N because
[21P(k1)P1(k)P(k1)] converges to the positive definite matrix 21
P,. Since
0 < min[21P(k1)Pl(k)P(k)P(k1)]Tr[fT(o(k1))f(o(k1))]
k=N
< Tr[fT(e(k1))[2I1P1)p)P(l)P(k)P(k1)]f((k1))] < =,
k=N
it follows that 1 Tr[fT(e(k1))f(e(k1))] < =. Therefore, the right
k=l
hand side of the inequality (425) is bounded in the limit as M ap
proaches infinity. This implies the existence of a finite L such that
I of(e(k1))n < L < =. Returning to the inequality (423), 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
m1
Z nf(e(i))nI < /6v for m,n > N2().
i=n
It then follows from (423) 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 closedloop 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 (42)(45). 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
leastsquares estimator. This is easily shown using the recursive
solution of (410) which is given by
n1
e(n) = P(n)P1(0)(0) P(n) p1 (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
n1
I < (n) < n(n) I P1(0)" (0) + IIP(n)ii n I P (i+ )P(i )f(e(i ))i.
i=O
n1
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 rinput moutput linear discretetime system described by
(31)(33) 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 closedloop system to be globally stable is a
uniformly stabilizable estimated system. It will be shown that the
parameter estimator described by equations (42)(45) will give a
uniformly stabilizable estimated system provided that assumptions A1A3
summarized at the end of Chapter Three are satisfied.
Uniform Stabilizability of the Estimated System
Using the parameter estimator described by (42)(45), it will be
shown that the estimated system can be stabilized using state
feedback. As shown in [6], the statespace observer form realization
for the system described by (31) is given by
x(k+1) = Fx(k) + Gu(k) (51)
y(k) = Hx(k)
where
A1 I B
A2 2
F = G = H = [I 0 ... 0]
An I Bn1
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 (33), 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 (34). 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]}. (52)
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 timevarying
estimated system (F(e(k)),G(e(k))).
Definition: The estimated system (F(k),G(k)) is uniformly stabilizable
as a timevarying 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
timevarying system matrix.
The following theorem gives conditions under which the estimated system
is uniformly stabilizable.
Theorem 5.1: Assuming A1A3 are satisfied, the parameter estimator
described by (42)(45) will generate an estimated system (F(k),G(k))
which is uniformly stabilizable as a timevarying 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(k1)G(k1)L(k1)] 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(k1) 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 timevarying system.
The adaptive control law for regulating the system described by
(31) 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) =
7Al(k)
A2(k)
An(k)
G(k) =
B (k)
B2(k)
Bn(k)
(53)
H = [I 0 . 0].
The adaptive observer defined by (53) is a deadbeat observer. The
output y(k) of the adaptive observer can be expressed in the following
form:
y(k) = [Al(k1) ... An(kn) B1(k1) ... Bn(kn)](k1)
(54)
where
This expression is easily derived by first rewriting (53) as
X(k) =
0 I
Bl(k1)
B2(k1)
Bn(kl)
u(k1)
Al(k1)
A2(k1)
An(k1)
y(k1).
(55)
Multiplying equation (55) by H gives
y(k) = Hx(k) = [0 I 0 ... O]x(k1) + Bl(k1)u(k1) Al(k1)y(k1)
Substituting into this equation for x(k1) using (55) yields
y(k) = [0 0 I ... 0]x(k2) + B1(k1)u(k1) + B2(kl)u(k2)
Al(k1)y(k1) A2(k2)y(k2)
Continuing the substitution using (55) gives
y(k) = [0 ... 0 I] '(kn+l) +
n1
I B (kj)u(kj)
j=1
I A.(kj)y(kj)
j=1 J
n n
= B (kj)u(kj) I A (kj)y(kj)
j=1 j=1
which is equivalent to (54).
The error between the output of the adaptive observer (53) and the
output of the system (31) 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(k1)n + b'(k) (56)
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(kl)Da(kl) + T1b(kl)]
+ [eT(kl) a(k1) + T b(k1)] y(k)
'(k) = y(k) [eT(kl)a(k1) + TTb(k1)] e(k).
Taking the norm of both sides and using the triangle inequality gives
nl(k)n < l (k) (eT(kl) (k1) + (kl))+ + ne(k)n
Noting from (31)(33) that e (k1)a (k1) + T b(k1) is equal to
[Al(k1) ... A (k1) B1(k1) ... n(k1)]T(k1) and using expression
(54) gives
4iU(k)i < 1[J1 (k) ... J2n(k)](kl)I + lie(k)n (57)
< I[J1 (k) ... J2n(k)] ii iu (k1)ii + Ile(k)ii
Ai (ki) + Ai (k1) 1
where Ji(k) =
Bi (ki) Bi (k1) 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 (57) gives the desired inequality
(56). Property (56) is very important in the proof of global
stability of the adaptive closedloop 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 timevarying 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
(42)(45) is uniformly stabilizable under assumptions A1A3. 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
timeinvariant 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 [48]. 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 online 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 discretetime systems by Samson and
Fuchs [1012] using an LQ control strategy, and later extended to MIMO
discetetime systems by Ossman and Kamen [3032].
In [3032], 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) (58)
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)).
(59)
In (59), 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 online 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 (58)(59) is welldefined 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 (58)(59), it is
first necessary to show n(L(k)L(k1))a converges to zero as k .
Theorem 5.2: Suppose the parameter estimates are generated using the
algorithm (42)(45). Then subject to assumptions A1A3, the feedback
gain sequence given by (58)(59) has the property
L(k)L(kl1) + 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 timeinvariant 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 (52) 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 (59).
Step 1: The sequence of matrices Rk given by equation (59) 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 (58)(59)
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 singleinput
case. It can also be verified in the multiinput 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 wellknown 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 timeinvariant 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 (59) will not vary much from the timeinvariant
RDE over a finite interval of time N as long as e(k) does not change
much over the time interval N. Equation (59) 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 (58)(59), then nL(k) L(kl)l
converge to zero as k goes to infinity. This result follows easily
since IRk Rkl11, IF(k) F(k1) i, and IG(k) G(k1)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 timevarying system
matrix. The exponential stability of (F(k)G(k)L(k)) is very important
for proving global stability of the adaptive closedloop system.
Theorem 5.3: Suppose e(k) is generated using the parameter estimator
(42)(45) and the feedback gain sequence L(k) is computed using (58)
(59). Then subject to assumptions A1A3, (F(k)G(k)L(k)) is
exponentially stable as a timevarying 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 timevarying system matrix where L(k) is given
by (58)(59). 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 timevarying system matrix if AT(k) is exponentially stable and if
IA(k)A(k1)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(k1)G(k1)L(k1))I
converges to zero as k + .. Property (vi) of Theorem 4.1 implies both
IF(k)F(k1)n and nG(k)G(k1)II converge to zero. It was shown in
Theorem 5.2 that IL(k)L(k1)I also converges to zero. Since
F(k)G(k)L(k)(F(k1)G(k1)L(k1)) =
(F(k)F(k1)) G(k)(L(k)L(k1)) (G(k)(G(k1))L(k1)
and both G(k) and L(k) are bounded, it follows that
ll(F(k)G(k)L(k))(F(k1)G(k1)L(k1))n does indeed converge to
zero. Thus the matrix (F(k)G(k)L(k)) is exponentially stable as a
timevarying system matrix.
Stability of the Adaptive ClosedLoop System
The control law chosen to regulate the system is given by
u(k) = L(k)2(k) (510)
where x(k) is generated from the adaptive observer (53) 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 (53) and the control law (510) results in a globally stable
closedloop 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 (42)(45) is used
so that there is a stabilizing feedback L(k) for the estimated system
(F(k),G(k)) subject to assumptions A1A3. Then with the adaptive
regulator defined by (53) and (510), the resulting closedloop 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 (53) and (510) 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 (32) can be written in
the form
(k) = S4(k1)
where
0 ... O 0 ... 0
Im(n1)
S = 0 ... 0 0 ... 0 ,
0
Ir(n1)
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
(511)
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 lowerblock triangular. The
matrix F(k)G(k)L(k) is stable by construction; therefore, A(k) in
(511) is exponentially stable as a timevarying system matrix. In
addition, it follows easily from (56) and the boundedness of M(k) that
nw(k)B < a(k)nz(k)n + b(k)
(512)
where a(k) and b(k) are positive scalar sequences which converge to zero
as k + . Thus, (511) 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 (511) defined by
A(k+M,k) =
M1
1 A(k+i)
i=0*
Since A(k) is an exponentially stable timevarying system matrix, it
follows that
iiA(k+M,k)Il < R1 < for all k,M > 0.
(513)
Also, given 0<5<1 there exists a finite p such that
nA(k+p,k)ii < 6 for all k.
(514)
For the remainder of the proof, p will be a fixed constant such that
(514) is satisfied. Taking the norm of both sides of (511) and using
(512) gives
Iiz(k+1) e < IiA(k) Iiz(k) I + nw(k) I
(515)
< (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 (515) 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 (511) recursively gives
(516)
z(k+p) = A(k+p,k)z(k) +
p1
I A(k+p,k+j+l)w(k+j).
j=0
Taking the norm of both sides and using (512) results in
iz(k+p)a < IA(k+p,k) niz(k)n
< nA(k+p,k)n iz(k)ni
p1
+ fIA(k+p,k+j+l)n nw(k+j)n
j=0
p1
+ I nA(k+p,k+j+1) i(a(k+j)iiz(k+j)ii + b(k+j)).
j=0
It follows from equations (513), (514), and (516) that
P p1
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
(517)
i(k) < I for all k > N.
6 + pR2max(1,R,)
Therefore, for all k > N,
pl
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)
(518)
where r < 1 and c(k) converges to zero as k + o. In order to show
(518) implies uiz(k)n converges to zero, let e>0 be given. Solving
(518) recursively and using (516) gives
n1
n1
j=0
n1 n1j
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]. (519)
Since c(k) converges to zero, there exists a finite N1 > N such that
c(k) < for all k > N1.
1r
Applying this inequality to (519) 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
closedloop system is globally stable using the adaptive regulator
defined by (53) and (510).
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 rinput moutput plant described by
x(k+l) = Fx(k) + Gu(k) + Dv(k) (61)
y(k) = Hx(k)
where
A1 I B1
A2 2
A2 I B2
F = G = H = [I 0 ... 0].
An I Bn1
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 (42)(45). Letting r(k) denote an mvector
reference signal, the objective is to design an adaptive controller
which stabilizes the system described by (61) 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+q1) + ... + alz(k+l) + a0z(k) = 0.
(62)
Let r be defined as
al
aq1
The internal model can then be realized as
Xc(k+l) = AcXc(k) + Bce(k)
(63)
where Ac = block diagonal {r,
r, ..., r}
mtuple
0
c = block diagonal {T, T, ..., T with T =
mtuple 1
The following theorem gives the conditions under which it is possible to
design an adaptive controller which will stabilize the plant described
by (61) and will drive the tracking error e(k) to zero asymptotically.
Theorem 6.1: Suppose assumptions A1A3 are satisfied and in addition
zIF(e) 0 G(e)
rank = n+mq for all een and
BcH zIAc 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 (63) and x(k) be generated from the
adaptive observer given by (53). Then there exists a control law u(k)
SL~x(k)L2xc(k) which when applied to the plant (61) results in a
stable closedloop system and drives the tracking error e(k) = y(k) 
r(k) to zero.
Proof: Using equations (53) and (63) 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) (64)
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
(64) is stabilizable. The stabilizing control law is of the form
u(k) = Ll(k)A(k) L2xc(k). (65)
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) + ]11 ^T(k)Rk (k)
(66)
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 closedloop system consisting of (64) and the control
law (65) must converge to a timeinvariant 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 (65) 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 closedloop system consisting of (64)
by (64)
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 timeinvariant system. Finally, from the results in Chen [20], if
the control law (65) is applied to the plant (61), the resulting
closedloop 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 (63) and (65)(66) causes
the plant (61) 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 discretetime systems.
Example 7.1: Consider the nonminimum phase discretetime system
described by the following difference equation
y(k) = 2y(k1)0.99y(k2)+0.5u(k1)+3u(k2).
The adaptive regulator defined by the observer (53) and the control law
(510) with L(k) computed using (58)(59) was implemented. Three
different algorithms were used to estimate the system parameters: the
estimation scheme described by (42)(45), the recursive leastsquares
algorithm defined by (23), and the estimation scheme given by (42)
(45) with data normalization (i.e., nk1 = max(l, Hi(kl) 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
(59) were chosen to be I. The parameter ranges, initial parameter
estimates and steady state estimates for all three estimation schemes
are displayed in Table 71. As seen from Table 71, the estimation
scheme described by (42)(45) forces the estimates of the parameters
to converge to the given ranges; whereas, recursive least squares does
not. Figures 7173 contain plots of the output response, control
input, and estimated parameters for each of the three cases. A
comparison of Figures 71 and 72 show the estimation algorithm defined
by (42)(45) has a slightly better transient response than least
squares. Figure 73 illustrates the detrimental effect of the data
normalization on the transient response of the system. As previously
mentioned, in (45) should be chosen very small to minimize the effect
of data normalization on the system transient response.
Table 71
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
(42)(45)
Recursive Least 2.11 1.05 0.594 3.19
Squares
Estimation Scheme 2.00 0.991 0.571 3.14
(42)(45) 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 71
Estimation Scheme (42)(45)
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 71 continued
Estimation Scheme (42)(45)
',:
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 72
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 72 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 73
Estimation Scheme (42)(45)
with nk= max(l, ll(k)jI )
y(k)
15
10
5
5
10
15
20
u(k)
I
Example 7.2: Consider the nonminimum phase stabilizable but not
reachable discretetime system described by
y(k) = 2.6y(k1)2.13y(k2)+0.54y(k3)+u(kl)+1.5u(k2)u(k3).
The adaptive controller defined by the adaptive observer (53) and the
control law (65) with L(k) computed from (66) 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 (42)(45).
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 72.
Plots of the output response, control input, and parameter estimates are
shown in Figure 74.
Equation (44) 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 closedloop system.
Table 72
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
(42)(45)
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 74
Estimation Scheme (42)(45)
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 74 continued
Estimation Scheme (42)(45)
,.., '
10
Example 7.3: Consider the discretetime system described by
y(k) = 2y(k1)5y(k2)u(k1)0.5u(k2)
The adaptive controller defined by the observer (53) and the control
law (65) with L(k) computed from (66) 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 (42)(45). 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 73. The output response, control input, and parameter estimates
are shown in Figure 75. For comparison purposes, plots of the output
responses and control input when the parameters are known exactly are
shown in Figure 76. 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 73
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
(42)(45)
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 75
Estimation Scheme (42)(45)
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 75 continued
Estimation Scheme (42)(45)
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 76
Exact Parameters
5
10
15
20
20
40
60
80
) I I I
Example 7.4: Consider the nonminimum phase discretetime system
described by
y(k)=2y(k1)0.99y(k2)+0.5u(k1)+3u(k2)
The adaptive controller defined by the observer (53) and the control
law (65) with L(k) computed using (66) 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 (42)(45). 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 74. Figure 77 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 78 for the case when the parameters are known exactly.
Table 74
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
(42)(45)
I,,.~  
30 40 50 60 70 (k)
 40 50 60 70 (k)   _ . ... ,.
40 50 60 70 (k)
(b)
Figure 77
Estimation Scheme (42)(45)
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 77 continued
Estimation Scheme (42)(45)
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 78
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 discretetime system
described by
yl(k) 1 al y1 (k1) + 3 'u(k1)
= + u(kl)
y2(k) 1 2 y2 (k1) 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(k1) + *Tb(k1)
with
T a 0 y(k1) 1 0 3' y1(k1)
S= a(k1) = = b(k1) = y2(kl) .
0 b Lu(k1) 1 2 0 u(k1)J
The adaptive regulator defined by the observer (53) and the control law
(510) with L(k) computed using (58)(59) was implemented. The system
parameter matrix e was estimated using the algorithm (42)(45). 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 75. Plots of the output response, control input, and
parameter estimates are shown in Figure 79.
Table 75
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
(42)(45)
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 79
Estimation Scheme (42)(45)
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 79 continued
Estimation Scheme (42)(45)
u(k)
 a(k)
b(k)

Example 7.6: Consider the multiinput multioutput discretetime system
described by
yl(k)1 [i a,2 [y1 (k1) 1 1 b12 u (k1)
= II+ .
2(k) 1 2 Y2 (k1) b21 b22 u2 (k1)
The order of required computations for parameter estimation is reduced
by rewriting the system in the form
y(k) = Ta (k1) + T% b(kl)
with
OT a12 0 b12 %aT(k1)= [y2(kl) ul(k1) u2(kl)],
0 b21 b22
S1 0 1
T = i bT(k1) = Cyl(k1) y2(k1) ul(k1)].
1 2 0
The adaptive regulator defined by the observer (53) and the control law
(510) with L(k) computed using (58)(59) was applied to the system.
The algorithm given by (42)(45) 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
76. Figure 710 contains plots of the output response, control input,
and parameter estimates.
Table 76
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
(42)(45)
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 710
Estimation Scheme (42)(45)
ul(k)
10
5
5
10
u2(k)
20 30 40 50 60 70 (k)
(d)
Figure 710 continued
Estimation Scheme (42)(45)
20 30 40 50 60 70 (k)
(c)
,,i "10
5
10
rLSt;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 710 continued
Estimation Scheme (42)(45)
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 multiinput multioutput linear
discretetime 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 timevarying 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.
[1314] 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.

