Robust Indirect Adaptive Control of Time-Varying Plants with Unmodeled Dynamics and Disturbances.
It is shown that indirect pole-zero placement adaptive controllers are robust for systems with time-varying parameters as well as unmodeled dynamics and disturbances.  A parameter estimator with projection is used.  No special signal normalization is employed to ensure robustness.  The nominal system parameters need only be bounded, and their variations need only be small in an average sense.  This allows them to vary slowly with time, as well as to take large jumps occasionally. The adaptive controllers can also simultaneously tolerate small unmodeled dynamics, as well as bounded disturbances, with no restriction on the magnitude of the bound.
Introduction
In [1], Astrom and Wittenmark have proposed the use of indirect, certainty-equivalent self-tuning
controllers based on pole-zero placement for the servo-problem. Thus, if the goal is to enforce a
response
A   (q
to a command signal fr k g, then one first estimates system polynomials b
A
at each time instant k by fitting a model,
to the available data fy t ; u kg, at time k. Then, the control input u k is chosen such
that,
where the polynomials b
R k and b
S k are obtained by solving the Diophantine equation,
at each time instant. To estimate the parameter vector b
may use a recursive algorithm of the form
where
The choice of the "gain" matrix I yields a gradient update law. Control schemes of this
nature have proven popular in practice, and many successful implementations have been reported;
see [2, 3, 4, 5]. Also, the model reference adaptive control method, the backbone of continuous-time
adaptive control, is a "direct" version of such a pole-zero placement scheme.
Often, such adaptive control algorithms are employed to control plants which are subject to
time-variations, for which there is consequently an ever present need to adapt to the changing
system characteristics. It is therefore important to develop a theory of robustness for such adaptive
control algorithms for their use in the face of such system variations, as well as in the presence of
uncertainties such as unmodeled dynamics and disturbances.
In this paper we consider an indirect adaptive control law, as above, with a parameter estimation
scheme employing "projection" to keep the parameter estimates confined to a compact convex set.
We establish that this simple modification is powerful enough to provide robustness simultaneously
with respect to system time-variations, small unmodeled dynamics, and bounded disturbances. In
particular, no special signal normalization is used.
Some background on the robustness problem for adaptive systems is in order. Much attention
has been devoted over the past decade to the robustness problem caused by unmodeled dynamics
and bounded disturbances, and an excellent unification of work up till 1988 is provided by Ioannou
and Sun [6]. Recently, in an important paper, Ydstie [7] has shown that just the simple mechanism
of "projection," which confines the parameter estimates to a compact convex set, is sufficient for
robustness with respect to bounded disturbances and some unmodeled dynamics. This paper is
notable also for the introduction of a new proof technique involving a "switched signal." This work
has been extended to continuous-time, while also enlarging the class of unmodeled dynamics, by
Naik, Kumar and Ydstie [8]. The net effect of these investigations is to show that many of the
modifications, proposed in the 1980's to prove robustness with respect to unmodeled dynamics, are
not necessary. The original simple "projection" modification, established by Egardt [9] to be robust
with respect to bounded disturbances, is also robust with respect to small unmodeled dynamics.
Comparatively less attention has been devoted to the problem of robustness with respect to time-
variations. Solo [10] has established the boundedness of signals for a direct one-step ahead adaptive
control scheme using an a-posteriori estimate based gradient update law with leakage, when the
parameters are slowly varying, while Kreisselmeier [11] has done so for an indirect adaptive control
scheme with projection, for slow-in-the-mean parameter variations, which allows occasional
large parameter jumps. Tsakalis and Ioannou [12] have obtained similar results for continuous-time
plants, while Guo [13] and Meyn and Guo [14] have done so for discrete-time stochastic systems.
In these works, while the plant is allowed to vary with time, the effect of unmodeled dynamics is
not considered. In [15], de Larminat and Raynaud consider such robustness for a fairly general
indirect adaptive control law which uses two parallel estimators as well as a specially constructed
"normalization" signal. Middleton and Goodwin [16] also incorporate a normalization signal, and
additionally assume knowledge of the constant factor by which it overbounds the unmodeled dynam-
ics. This constant is then used to set up a normalized dead-zone, for which they establish robustness
to slow-in-the-mean parameter variations and small unmodeled dynamics. Giri et al. [17] prove the
robustness of an adaptive regulator for a plant with small-in-the-mean parameter variations and
unmodelled dynamics, using only the knowledge of the order of a nominal plant model. However,
this is done using a complicated control law involving an identification - stabilization time splitting,
utilizing a least squares based adaptation law that also employs a special normalization signal. Fur-
thermore, arbitrarily large bounded disturbances cannot be handled, and the regulation objective
is not achieved in the "ideal" case unless the algorithm is appropriately modified.
In all these works, the signals entering the adaptation law are normalizaed by a specially constructed
normalization signal, first proposed by Praly [18]. The effect of such normalization is
to ensure that the modelling error entering the adaptation law is bounded or small in the mean.
Construction of such a normalization signal requires a priori system knowledge and involves extra
computation. In addition to these practical considerations, it is of theoretical interest to see if
normalization is really necessary for robustness.
The key point of this paper is that such normalization is not required to ensure the robustness
of adaptive controllers to small unmodeled dynamics, bounded disturbances with arbitrarily large
bound, and slow-in-the-mean parameter variations. We show that one obtains robust adaptive
control by merely utilizing parameter projection, together with "extended regressor" normalization,
without recourse to any other modifications. An indirect adaptive pole-zero placement controller
is considered. The resulting necessity to cancel process zeros requires the nominal portion of the
time-varying plant to be minimum phase at every instant. On the other hand, the problem of
potential loss of estimated model controllability/stabilizability faced in adaptive pole placment is
not an issue here. This itself results in a simpler adaptive control scheme than those in [15], [17], and
[19], for instance. A result similar to ours has recently been reported in Wen [20], using a different
proof technique developed earlier in Wen and Hill [21]. They consider a unit delay plant with the
time variations restricted to be slow. The true nominal time varying parameter vector is assumed
to lie in a convex compact set, which is assumed to have the property that the nominal system
polynomials induced by any parameter vector in the set are uniformly coprime. This restrictive
assumption is the consequence of choosing an indirect pole-placement controller design. A final
point worth noting is that unlike the signal bounds derived in the present paper, those in Wen [20]
are not uniform in that they depend on the system initial condition.
Our main results are the following:
(i) A certainty equivalent adaptive controller, using a gradient based parameter estimator with
projection, and employing normalization based on an "extended regressor," ensures that all closed
loop signals are bounded, when applied to a nominally minimum phase discrete time plant with
slow-in-the-mean parameter variations, bounded disturbances, and small unmodelled dynamics
(Theorem 1).
(ii) In the absence of unmodelled dynamics and disturbances, and in case the parameter variations
asymptotically tend to zero, i.e., in the nominal case, the error in tracking a reference trajectory
converges to zero (Theorem 2). When unmodelled dynamics as well as bounded disturbances are
present, the mean-squared output prediction error is linear in the magnitude of the unmodelled
dynamics, bounded disturbances, and average rate of parameter variations (Theorem 3). Thus the
adaptive controller provides robust performance in addition to robust boundedness.
System Description
Consider a plant
a
where u and y denote the input and output, while v represents the cumulative effect of unmodelled
dynamics and disturbances. The coefficients fa i;k ; b i;k g may vary with time, and so the system
is allowed to be time-varying. We wish to investigate the robustness of indirect adaptive control
schemes for such systems which simultaneously consist of time-variations, unmodelled dynamics
and disturbances.
Let us suppose that the goal of adaptive control is to generate a closed-loop response to a
command signal r k , which satisfies (1), where q \Gamma1 denotes the usual backward-shift operator,
. If the plant in (5) were time-invariant and "ideal" without any unmodelled
dynamics or disturbances, i.e., given by A(q then the control law,
the Diophantine equation,
results in the cancellation of all original process zeroes (hence necessitating some sort of minimum-phase
condition), and a closed-loop transfer-function q \Gammad B   (q
A   (q \Gamma1 ) from r k to y k , as desired.
We will consider here a certainty-equivalent indirect adaptive control scheme, which first forms
parameter estimates
at each time-step k, and then uses the resulting estimated polynomials b
A
b a
to solve the Diophantine equation,
for b
R k and b
We will consider the minimum degree solution for b
R so that it is monic, and
of degree (d \Gamma 1). We note that such a solution can be found through a (d \Gamma 1)-step long division
of the polynomial A   by -
A. Then we apply the input u k given by,
It is well known in adaptive control, see Egardt [9], that if one simply uses a pure gradient
based parameter estimator, then the resulting adaptive system is destabilized by even a small
bounded disturbance. It is therefore necessary to somehow modify the parameter estimator to
secure robustness with respect to even just bounded disturbances.
We will consider a parameter update law using parameter projection. This modification, motivated
by the seminal work of Egardt [9], simply projects the parameter estimate vector onto a
compact convex set C at each time step k. The set C is chosen such that
th component of ') - b min ? 0 for every ' 2 C:
(By thus ensuring that the estimated leading coefficient of b
positive, we also ensure that
the control law (9) is well defined, since it does not involve division by zero). The parameter
estimates are recursively specified by
and
where the regressor OE is given by (4). Here Proj C [\Delta] denotes "orthogonal" projection onto the
compact convex set C, defined uniquely by Proj C [x
The vector / k is an "extension" of the regression vector OE k defined in (4),
Above, the constant - in the step-size is chosen such that
2, and the algorithm is initialized with -
shall refer to this as the Parameter
Estimator with Projection (PEP).
Since we are dealing with time-varying polynomials in a shift-operator, we distinguish between the notations,
multiplying
. For a signal xk , Ck (q
as is usual.
2 When the delay is unity, i.e., i.e., the extended regressor is identical to the regressor.
Remarks: All the results of this paper can be extended to least-squares type parameter esti-
mators, if the condition number of the covariance matrix is kept bounded, and also to "direct"
pole-zero placement schemes. In fact, though we do not show it here, if one uses a direct one-step
ahead adaptive control law, then robustness can be established easily through a similar analysis.
3 The Assumptions
Our goal is to analyze the behavior of these adaptive controllers when they are applied to plants of
the form shown in (5), which consist simultaneously of time-variations, unmodelled dynamics, and
disturbances. We shall make the following assumptions on the plant (5) and the reference model
(1).
where C is a compact convex set, such that the (p 1)-th component of
every vector in C is larger than or equal to b min ? 0. Let K ' be a constant which bounds k'k
for all ' 2 C.
(A2) The zeros of B
lie in the open disk
(a is the time-varying parameter vector.
oem
and oe is as in (A2).
Xa
q \Gammai has all its zeros in the open disk jqj 2 ! oe 000 ! 1. 3 For later use,
define oe 0 := maxfoe 00 ; oe 000 g.
The assumptions (A1) and (A2) require that the time-varying parameter vector ' k be bounded,
have leading b 1 -coefficient uniformly bounded away from zero, and be strictly minimum-phase,
at every time-instant k. This latter restriction is necessitated by the requirement of cancelling
all process zeroes. Assumption (A3) allows slow-in-the-mean parameter time-variations, and thus
occasional jumps too. Assumption (A4) allows small unmodelled dynamics, and bounded disturbances
with arbitrarily large bound. Finally, (A5) simply requires the reference model to be stable
with a prescribed margin of stability.
Remark: It is worthwhile to note that assumption (A4) includes the case of an incorrect assumption
on the delay, since m k\Gamma1 includes input terms upto time k \Gamma 1. Furthermore, the class of
unmodelled dynamics covered by (A4) allows the true plant to be non-minimum phase [22, 23].
3 Without loss of generality, if deg(A
In order to implement the above adaptive control schemes, one thus needs to know upper bounds
and ' on the orders of the nominal time-varying portion of the plant, the nominal delay d, and
the sign (say, positive) and a lower bound b min ? 0 on the leading term in the polynomial B
modeling the numerator of the nominal portion of the plant, for every k. In addition, since one
has to project the parameter estimates onto a compact convex set guaranteed to contain the true
parameter ' k for all k, one has to know the bound K ' on the norms of the time-varying parameters
Finally, one also needs to know the reference model given by A   (q
r k .
Our central result in this paper is that under the above assumptions, the adaptive control laws
are stable for all K v and k ffi small enough, i.e., whenever the unmodelled dynamics is small enough,
the parameter variations are small enough on the average, and the disturbances are bounded,
though without any restriction on the magnitude of the bound.
4 Properties of the Parameter Estimator
We now derive some important properties of the parameter estimator PEP, which are independent
of the control laws used. Let us denote by e
, the parameter estimation error. Using the
definitions of OE k and ' k in (4) and (A3), we can express the plant (5) as
Further, substituting (15) in (11) gives
e
Lemma 1. The following properties hold for the parameter estimator PEP.
ii) The parameter estimation errors are uniformly bounded, i.e., k e
iii) The parameter estimates f b
are uniformly bounded also, with k b
ae
ae
The quantity ae k is defined as,
ae
and the constants are given by, K ev := (1+ 1
-(2\Gamma-\Gammaffl) .
Proof
i) This is immediate from the projection (12), and the property (A1) of C.
ii) This is immediate from (A1) and the projection (12).
iii) This follows from (ii) by (A1).
iv) Let us define . By (10),
e
Using kOE
e
Using this inequality in (19) and noting that k e
due to projection, we get
Thus, using (16) we get,
e
Substituting this into inequality (20) gives
Summing from t using (A3) as well as k e
, gives the
desired result. 2
The boundedness of parameter estimates yields the following important result that the controller
parameters are Lipschitz functions of the plant parameters.
Lemma 2. Let ' c :=(coefficients of R(q \Gamma1 ), coefficients of S(q denote the "controller" parameter
vector, where R and S are functions of (coefficients of A(q coefficients of B(q
through (6). Let ' c;1 and ' c;2 denote the controller parameter vectors corresponding to two different
plant parameter vectors ' 1 and ' 2 respectively. Then
is a constant that only depends on C.
Proof The relation (6) can be re-written as a system of linear equations,
denotes the Sylvester matrix corresponding to the polynomials A(q \Gamma1 ) and q \Gammad , and
a   is formed from the coefficients of A   (q being monic implies that there exists a
positive ffl such that, for all ' in LC , j det(M(' j ))j - ffl. We therefore get
thereby concluding the proof. 2
5 The Switched System
We now introduce, for purposes of analysis only, the "switched" system,
1, and the indicator function I k\Gamma1 is defined as,
I
Lemma 3. For all k,
Above, the constants K vz ; K uz , and K aez can all be made as small as desired by choosing K v small
enough.
4 In the remainder of the paper, we will use the symbol K to generically denote any positive constant that does
not depend on either Kv or k ffi , and whose exact value is unimportant for the proofs which follow.
i) These are obvious from (4), (13) and (14), where we also note that by choosing m 0 large, the
constant K can be chosen independently of the initial conditions.
ii) From
oe k\Gammaj [K y y 2
oe k\Gammaj [K y e 2
it is clear that it suffices to show that,
oe k\Gammaj y 2
oe k\Gammaj e 2
oe k\Gammaj u 2
and
oe k\Gammaj u 2
oe k\Gammaj y 2
oe k\Gammaj u 2
where K 0 denotes a constant that may depend on the initial conditions, since the effects of the
initial conditions can then be accounted for by making z 0 appropriately large, thus ensuring
that Kmz and kmz do not depend on the initial conditions.
For simplicity, let us first consider the case d = 1. In that case b
R
A dropping q \Gamma1 for brevity, and recalling the
notation for multiplying time-varying polynomials in the shift operator, we obtain
A   y
R
A
R
(from
A
From (A5) it follows that there exist constants
for every root q of A   (q Using the boundedness of fr k g it follows from (24) that
oe k\Gammaj y 2
oe
thus proving (22). For (23), recall from (9) that
' k is bounded, and the coefficients of b
continuously on
, the result (23) follows from (26).
Now let us turn to the case d ? 1. In that case,
A
e
Hence operating on both sides by b
R keeping in mind the consequences of multiplying
time-varying polynomials in the shift-operator, and the associated notation, we have,
R
A
R
A
R
R
A
R
R k\Gammad
R k\Gammad
R
R
A
R
R k\Gammad
R
R
A
R
(using
Hence,
A   y
R
R
A
R
R
R k\Gammad
Now note that
R
A
R
R
R k\Gammad
(b a
R k\Gammai
R
jb a j;k\Gamman \Gamma b a
R
deg(b S)
(b s
s
(b r
deg(b S)
deg(b S)
je k\Gamman j;
where we have used Lemma 2. Now employing this bound in (27), we can establish (22) just
as in the case of d = 1. The result (23) again follows from (22) by (26).
iii) From (21), z k - gz
. Thus we only need to bound e 2
k and u 2
in
terms of z k\Gamma1 . For the first, by (A4) and (i), it follows from (16), that
Also, from (14), u 2
Ku .
iv) From (A4) and (ii), v 2
v) From the control law (26), and the boundedness of fr k g, u 2
Hence,
and the required bound follows.
k . Hence, from (29) and (v), ae k - Kae
vii) Using i), ii) of Lemma 3 gives k/
Now we examine the implications of the switching mechanism in more detail.
Lemma 4.
There exists a constant ffl aez ? 0, such that ae k - ffl aez z
ii) For every positive integer N , there exist positive constants L(N); - k(N), and K v max (N) with
the following property. If K v 2 [0; K vmax
Proof
1. Then, from the definition of I
Hence, we have
2Ku , then the claim is true with ffl aez ? 0 chosen smaller than (g \Gamma
oe)=2K u . So let us only consider the interesting case, ae
. From (30) we
then have e 2
2Ky
, while from (16), e 2
k , and so
, the required result follows by choosing K vz small enough, and ffl aez !
ii) We will bound the growth rate of ae k =z k , and then use this in a reversed time argument. Consider
ae k+1
z k+1
z k+1
by Lemma 3(vi), where we set K aez1 := 0 in the case of PEP. Hence,
ae k+1
z k+1
oe
ae k
z k
oe
ae k
z k
where K a and K b are appropriately chosen. So,
a
z t
- K N
a
z t
Hence,
ae t
z t
a
a
Now we simply define
a
noting that by choosing large enough, and K aez1 ; K aez small enough (by choosing
enough), we can ensure that - k(N) ?
6 A Representation of the Closed-Loop System
In this section, we obtain a non-minimal state space description of the closed-loop system, consisting
of a stable state-transition matrix, and driven by a composite input consisting of the output
prediction error, the unmodelled dynamics and disturbances, the filtered reference input, and in
the case of d ? 1, also of "small" fractions of the input and output.
Consider first, for simplicity, the case d = 1. Then, using (24),
Hence,
[A
(b
are defined appropriately. Hence, defining the "state-vector"
we obtain the closed-loop system representation,
where
\Gammaa
a 0
We note, for future use, that the eigenvalues of F k , at each instant k, are the roots of A   (q
and B by (A2) and (A5), F k is a stable matrix with all eigenvalues - i lying in the
disk
Now let us turn to the case of d ? 1. First, recall that (27) gives
A   y
R
where
R
R
R k\Gammad
R
A
S k\Gammad .
Next, operating by b
A k on (9), adding q \Gammad b
to both sides, and then using (8) gives
A   b
This implies
A   B k u
A
A
where e
. Next, note that by (4), (7), (15), and (16),
A
and that by (34) and (5),
A k\Gamma1+d y k+d \Gamma e
A k\Gamma1+d y k+d
A k\Gamma1+d y k+d ; (35)
where e
. Using (34) and (35) in (33), and defining \Delta 5;k := A   ( e
A k\Gammad
R k\Gammad
R k\Gammad
A k\Gammad
A
A   B k\Gammad u
A
We can now use (36) and (32) to get the required closed-loop system representation. Define
l 1;k := q \Gammad B   r k
R
l 2;k := A
and C
Note that by assumptions (A2) and (A5), all roots of C k (q \Gamma1 ) lie in the open disk jqj
for all k - 1.
Using the above definitions, we get the representation,
where
\Gammaa
and the ?'s represent non-zero scalars whose exact values are unimportant. Note that the eigen-values
of F k are the zeros of A   (q \Gamma1 ) and the zeros of B One difference from (31) is that
for the case d ? 1, the system (37) is driven by terms involving the input and output. However,
these occur only in products with the \Delta i terms, which are either parameter or parameter estimate
differences or "swapping" terms.
7 The Contraction Property
We will consider the "composite" Lyapunov function,
the discrete-time Lyapunov equation,
We note that since the F k 's lie in a compact set, and each F k has all its eigenvalues inside the
disk of radius oe 01
2 , we have
F
In what follows, we will first show that W k has a bounded growth rate, and then that W k has
a certain "contraction" property, viz. W k+T ! W k for a certain T whenever W k is large enough.
These will then be used to prove the boundedness of W k , and hence of all signals in the closed-loop
system, thus establishing "robust boundedness" of the adaptive system.
Lemma (Bounded Growth Rate of W ). There exist constants Kw and kw such that,
Proof First,
Hence, for some Kwz and kwz ,
The Key Lemma. For every constant L large enough, whenever there is an interval [a; b] with
the following properties hold.
iii) If I
1). Suppose either that I
then suppose there exists an N such that I there
exists
Proof
We already have z k - W k , by the definition of W k . To show that W k - 2Kwz z k for L large
enough, we note that because W k - Kwz z k
Kwz
Kwz
Hence if we choose L large enough that kwz - Kwz [2L \Gamma kwz
Kwz ], then kwz - Kwz z k , thus
yielding W k - 2Kwz z k .
ii) This follows easily from the lemma above.
since 2Kwz - 1 implies W a - 2KwzL - L. Hence W b - 2Kwz z b - 2Kwz [g b\Gammaa
]W a .
First, let us consider the case simplicity. We note that
Now defining
since
I , we have the following inequalities,
and
min
(from (26)). (40)
Hence,
K u
min
min
choose
large enough, and ffl 5 enough so that,
K
and define K ee :=
min
min
. This gives,
However,
and so,
Now focusing on x T
Now note that,
F
oe (i\Gamma1\Gammaj)=2 oe j=2 Kkffi k kkx
Hence
Thus we have,
z
z
k is the term in the square brackets above.
From this recursive bound, we obtain,
Y
Y
and so,
W a
Y
W a
Y
KwzL
Now, since log x
F
ee
Using vii) of Lemma 3 and the fact that z j \Gammai - L, we get z
noting that
min
min L
we obtain
min
min L
From (17), using Lemma 4(ii), we get
K ev
Hence,
K ee
K ev
K ee
K ee K e
min
min L
K ee
K ev
min
min L
K ee
K ev
min
\GammaL
K ee
K ev
2k vz K l
min
By choosing K v , k ffi small enough, and L large enough, we get
for some
We thus obtain,
W a
e
KeKee
Now we turn to the case d ? 1. The essential difference between the cases
is that we replace b e;k e k by b 1 l 1;k , and replace l k by b 1;k\Gammad in (39). Hence we need to bound
l 2
1;k , and l 2
2;k in terms of e 2
(\Delta) , and kffi (\Delta) k. This is done as in the proof of Lemma 3, except
that A   (B which is part of \Delta 5;k u k\Gammad has to be handled differently. It gives
rise to terms involving kffi j k.
Let us consider how one overbounds l 2
1;k . As in the proof of Lemma 3(ii), we obtain,
and
Combining these yields l 2
Next, consider l 2
2;k . First, note that we have the following inequality, whose counterpart in
the case is (40).
This is obtained by applying the control law (9), using the boundedness of the parameter
estimates, and the boundedness from below, by b min ? 0, of - b 1;k\Gammad .
For the term A   (B
(b
where the last two inequalities follow from (44), and the definition of z(\Delta). The remaining
terms can be handled similarly. 2
The Contraction Lemma
Consider 1. Then there exist N , L large enough, and K v
then
Proof There are four cases to consider.
Figure

1: Illustration of Case 2
Case 1: Suppose I From part (iii) of the Key Lemma,
Hence, for N , L large enough, and K v correspondingly small, we have (45).
Case 2: Suppose
1g.
Note that this implies that I
First suppose Using parts (ii), (iii) and (iv) of the Key Lemma, we have
ww
ww
ww K
yielding (45) when N; L are large enough, and K v max and k ffi max appropriately small. If
then by parts (ii) and (iii) of the Key Lemma,
ww
hence giving (45).
Case 3: Suppose First suppose
Using parts (ii), (iii) and (iv) of the Key Lemma, we have
Figure

2: Illustration of Case 3
I
I
I
Figure

3: Illustration of Case 4
and using parts (ii) and (iii) of the Key Lemma gives
This yields,
wz K 3
ww
!/
and hence (45). If t the first two inequalities get replaced by
This gives
ww
hence giving (45).
Case 4: Suppose
Note that this implies that I
Case 4a: Suppose First suppose Using parts (ii), (iii) and (iv) of the Key
Lemma, we have
and
which gives
ww
exp(\Gamma-N
and hence (45). If t the first two inequalities get replaced by
This gives
and hence (45).
Case 4b: Suppose first suppose that t 2 ! 2N . Using parts (ii), (iii) and (iv)
of the Key Lemma, we have
and
which gives
wz K 4
ww
and hence (45). If t the first two inequalities get replaced by
This gives
ww
thus establishing (45) in all cases. 2
Theorem 1 (Robust Boundedness Theorem). Consider the adaptive control system with the
plant satisfying assumptions (A1-A5). Then, all signals in the closed-loop adaptive system are
bounded, whenever K v and k ffi are small enough.
Proof As a consequence of the Bounded Growth Rate Lemma and the Contraction Lemma, W
is bounded. To see this, note first that the Contraction Lemma shows that if W stays above a
certain value (2Kwz L) continuously for a certain time interval (2N must contract.
That is, W cannot continue to grow for more than 2N consecutive samples at a time, once it
has grown larger that 2KwzL. Since W has a bounded growth rate, it follows that it is bounded.
Since W bounds all other signals through z and x, we conclude that all closed-loop signals are
Remark: These results extend easily to recursive least-squares based schemes which keep the
condition number of the covariance matrix bounded.
8 Performance for a Nominal System
In this section, we consider the performance that can be achieved in the absence of unmodelled
dynamics, and when the parameter variations go to zero asymptotically, specifically when kffi
We show that the desired performance objective is met.
Theorem 2. If K
(i) lim
(ii) lim
Remark: Note that we do not need the nominal plant parameters to be time-invariant; we only
require them to satisfy kffi k k 2 ' 1 . That is,X
Proof
(i) Recall from Lemma 1(iii) that
ae
ae
gives
ae
Recalling that ae (\Delta) is uniformly bounded by Theorem 1, and letting ae max := max k-0 fae k g, we
get
Fixing t and letting T go to infinity gives e
(ii) Recalling (10), the fact that k b
due to parameter projection, and using
the Cauchy-Schwarz inequality, we get,
Using this in (27), the closed-loop boundedness of all signals, and using (1)(a) yields the
desired result. 2
9 Robust Performance
We now show that the performance of the adaptive controller, as measured by the mean square
output prediction error, is robust in that it is linear (hence also continuous) in the magnitude of
the unmodelled dynamics, bounded disturbances, and average rate of parameter variations.
Theorem 3.
lim sup
are generic constants which can only decrease (or remain constant) as K v , k v , k ffi
decrease.
Proof Using (A4) in Lemma 1(iii) and recalling that fm k g and fae k g are uniformly bounded by
Theorem 1, we get,
ae
ae
This implies
lim sup
which yields the desired result after using Lemma 1(iii). 2
Simulation Example
We now demonstrate the advantage of the suggested adaptive control algorithm through a simulation
example. The system is modelled as
where a k , and b k are unknown, possibly time-varying parameters. The true unknown system is,
however, given by the following.
where d k denotes a discrete square wave disturbance of period 100 and amplitude 0.15. The adaptive
control is designed to track the following model reference trajectory,
y
As Fig. 4 shows, using an unmodified LMS-type gradient estimator or an LMS-type gradient estimator
with parameter projection, causes the output to blow up. Fig. 5 illustrates the undesirable
behavior that results if we use a normalized gradient estimator, as used in the ideal case. Finally,
Fig. 6 illustrates the results obtained if we use the adaptive control algorithm with parameter
projection, as proposed in this paper. Fig. 6(b) also exhibits the nice parameter tracking that is
achieved.

Figure

4: (a) Unmodified (LMS-type) gradient estimator: y blows up (b) Gradient estimator with
parameter projection: y blows up

Figure

5: Normalized gradient estimator: y is not well-behaved

Figure

Proposed adaptive control algorithm: (a)(i) System output, y (ii) Model reference output,
y   (b)(i) True and estimated a k (ii) True and estimated b k
Concluding Remarks
We have presented an indirect adaptive pole-zero placement control law using a simple parameter
estimator employing projection. We have shown that this is robust for plants which simultaneously
feature unknown slow-in-the-mean time-variations of the nominal parameters, as well as small
unmodeled dynamics, and bounded disturbances, without any restriction on the magnitude of the
bound. The plant parameters may even make occasional jumps. No special normalization is used.
Instead, the signals entering the parameter update law are normalized by the squared norm of
an "extended" regressor, which requires neither any a priori system knowledge nor any additional
computation.
It is straightforward to extend this analysis to recursive least-squares based update laws which
monitor, and keep bounded, the condition number of the covariance matrix.
Several issues still need to be explored. A major restriction is that we require the frozen
nominal plant to be minimum phase at every instant. Transient performance, and the precise sizes
of unmodeled dynamics and parameter variations tolerated, are issues which require deeper study.

Acknowledgements

: The writing of this paper was completed while one of the authors was
visiting the Department of Computer Science and Automation, Indian Institute of Science, Banga-
lore. The author is grateful to Professor N. Viswanadham for the warm hospitality and facilities
provided.



--R

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--TR
robust performance;robustness;adaptive systems;indirect adaptivecontrol;unmodeled dynamics;time-varyingplants;stability;disturbances