Deals with the problem of designing output tracking controllers for uncertain systems. The systems we consider may be non-minimum phase but are restricted to be linear. The problem is motivated by control applications where a desired output trajectory is specified, and the corresponding input to the system is to be found

Abdallah, Chaouki T.

Efromovich, S.

Heileman, G.L.

Koltchinski, V.

English

University of New Mexico

University of New Mexico
UNM Digital Repository
Electrical & Computer Engineering Faculty
Publications Engineering Publications
12-16-1998
Tracking control of uncertain systems
Chaouki T. Abdallah
V. Koltchinski
S. Efromovich
G.L. Heileman
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Recommended Citation
Abdallah, Chaouki T.; V. Koltchinski; S. Efromovich; and G.L. Heileman. "Tracking control of uncertain systems." Proceedings of the
37th IEEE Conference on Decision and Control (1998): 1867-1868. doi:10.1109/CDC.1998.758579.
Proceedings of the 37th IEEE 
Conference on Decision & Control 
Tampa, Florida USA December 1998 -_ Tracking Control of -- 
V. Koltchinski, S. Efromovich 
Department of Mathematics and Statistics 
University of New Mexico 
Albuquerque, NM 87131, USA 
{ vlad,efrom}@mat h.unm.edu 
Abstract 
This paper deals with the problem of designing output 
tracking controllers for uncertain systems. The sys- 
tems we consider may be non-minimum phase but are 
restricted to be linear. The problem is motivated by 
control applications where a desired output trajectory 
is specified, and the corresponding input to the system 
is to  be found. 
1 Introduction 
Mathematically, the output-tracking problem consid- 
ered may be studied as an inverse problem. The output 
tracking has a long history but more recently, Devasia 
et. al. [l] initiated a new line of research by studying 
stable inversion as a viable approach to exact-output 
tracking. This research has also benefited from con- 
tributions by Hunt et. al. [2, 31. At this stage, it is 
understood that the stable inversion of non-minimum 
phase systems (linear or nonlinear, time invariant or 
time varying) is possible at least for a large class of 
trajectories. The basic idea is that rather than finding 
an inverse for all trajectories, one only need be con- 
cerned with the inverse of the particular trajectory to 
be tracked. The resulting input then has a noncausal 
component and is of the feedforward variety. 
This paper uses a different approach to study a simi- 
lar but more general problem in the linear system case. 
The previously mentioned results assume given the sys- 
tem dynamics even though some robustness issues have 
been studied in [4,5]. In many cases, the system under 
study is highly uncertain and what we are given are 
samples of the desired output trajectory, rather than a 
closed-form time-dependent trajectory. Our approach 
relies on concepts from statistical learning theory and 
on the results of a recent paper by Efromovich and 
'The research of S. Efromovich is partially supported by NSF 
Grant DMS-9625412 and Sandia National laboratories Grant 
AE1679. The research of V. Koltchinski is partially supported 
by Sandia National laboratories Grant AU-4005. The research 
of C.T. Abdallah and G.L. Heileman is partially supported by 
Boeing Computer Services Grant 3-48181 
Uncertain Systems 
TA09 11:OO 
C.T. Abdallah, G.L. Heileman 
Department of EECE 
University of New Mexico 
Albuquerque, NM 87131, USA 
{ chaouki, heileman} @eece. unm. edu 
Koltchinski [6]. As such, our approach has much in 
common with the recent book by Vidyasagar [7], and 
other recent papers which attempt to  answer some con- 
trol questions statistically. 
2 Mathematical Set-Up 
The authors in [6] were interested in estimating a func- 
tion f E L2([0, 11) which is observed in Gaussian white 
noise modeled by the stochastic differential equation 
dY'(t) = ( H f ) ( t ) d t  + e d W ( t )  where H is a linear op- 
erator, e > 0 and W is a standard Brownian motion. 
This is again an inverse problem and applications of 
the approach to signal processing and communication 
theory is described by the authors of [6]. The novelty 
of their results lie in the facts that H may be unknown 
and that the problem may be ill-posed due to the non- 
invertibility of H .  In the setting of tracking control, 
the output is the desired trajectory to be tracked, H 
is an unknown system and f corresponds to the input 
to be found. The non-invertibility of H may be due 
to it being non-minimum phase. In order to  use the 
setup of [6], we first assume that samples of the desired 
output are given even though they may contain ran- 
dom errors, and that the system is unknown. We then 
use statistical learning theory [7] to generate training 
input functions along with the corresponding (possi- 
bly noisy) outputs. Based on this data, a data-driven 
learning machine is obtained to estimate the desired 
input for the original tracking problem. 
More specifically, we review the mathematical setup of 
[6]: Let f E F 7-1 where 7-1 is a Hilbert space, and 
let H be a linear operator from 7-1 into another Hilbert 
space R such that our unknown system satisfies the 
linear operator equation H f  = g. We do not assume 
that the system H is known but rather that the func- 
tion class F is given. The problem we study is then to 
find the input f such that the output g tracks a given 
trajectory, when we only have available noisy samples 
of the desired trajectory. 
Suppose an orthonormal basis is given in 31 by {ei; i 2 
1) and an orthonormal basis of R is { $ j ; j  2 1). 
Both bases will be assumed known. We denote in- 
ner products over a Hilbert space by (.,.). The op- 
erator H defines the infinite matrix (hij)&, where 
hij = (Hej,qbi). Note that one can approximate the 
possibly infinite-dimecsional H by the n x n matrixes 
Hn = (hij):j=l. Suppose that we are given a noisy se- 
quence of the desired outputs Yj = ( H  f ,  q j )  + q j ;  j = 
1,2 , .  . . and that we can generate a training set of data 
X ( j i )  = hji + U C j i ;  i ,  j = 1,2, .  . *, where the random 
variables {q j ,  Cji; i ,  j = 1,2, .  . .} are i.i.d and Normal, 
and where E ,  u are nonnegative real numbers. The re- 
sults of [6] then place conditions on the different math- 
ematical objects so that we can obtain an asymptotic 
or a minimax estimate of the desired tracking input as 
E and u tend to zero. The studied risk is the minimax 
integrated squared error 
We know define a few more terms to facilitate the state- 
ment of the main results: Let ?in denote the subspace 
of ‘?i spanned by {e l , . . . en}  and R, be the subspace 
of R spanned by {$l,..*qn}. Let Pn and IIn de- 
note the orthogonal projections on an and R,, and 
f n  = Pnf .  Let A I be the operator norm of A and 
IIAI(H = + Tr(ATA) be the Hilbert norm of A. We let 
Hn = IInHPn be an operator from 31 into R. If we re- 
strict Hn to be an operator from ?in to R,, we obtain 
a matrix [HIn which is the n x n principal submatrix 
of H,. We assume throughout that for a large enough 
n, [HIn is invertible. Then, we let Tn = H i l H  be a 
projection operator from ‘l-t onto ?in. Next, let 
so that X, = [HIn + uZn. The projection estimate is 
then given by f n  = H;,;Y(n) where 
(3) 
j=1 
and 
Xi1  
0 otherwise 
if X ,  is nonsingular and IJX;’l( 5 y H;; = 
In particular, we consider the following form for y, y = 
2 3  where a ( € )  is a nondecreasing function of E > 0 
such that a(€)  = o(log-1’2(c-1)). Finally, we denote 
An = IIHL’II; Bn = IlH;’ll~; S(o,n) = u f i A n  (4) 
The following theorem is then obtained: 
Theorem 1 [6] Suppose that 
6 ( U J , n ) 9  
sup log(u-’B,1 = o(6-2) 
sup logy = o(6-2) 
Y 2 2An 
6 ( ~ , n ) S 5  
Then, for some p > 0 and for any  C > 1, there exists 
60 > 0 such that for all u , ~ ,  and n satisfying 
the following holds 
1EIfn - TnflI 5 CBi(u211T,f1)2 + e 2 )  + e-fi62(uin) 
Moreover, if u 5 C E ,  the following holds 
E l l f n  - T n f l l  I: CBX(u211T,fl12 + E ~ )  
The above theorem illustrates the fact that as n in- 
creases, one is able to obtain a-more accurate estimate 
of the desired tracking input f .  In fact, one can find 
the minimum number of samples necded to  achieve a 
certain accuracy in the estimate of f .  Note also that 
if E and u are zero (i.e. deterministic case), one can 
guarantee the convergence of the estimate to zero as 
n increases. In the full version of the paper, we illus- 
trate the main theorem using numerical examples and 
present other relevant results from paper [6] which may 
be applied to  control problems. 
References 
[l] S. Devasia, D. Chen, and B. Paden, “Nonlinear 
inversion-based output tracking,” IEEE Trans Auto. 
Control, vol. AC-41, pp. 930-943, 1996. 
[2] L. Hunt, G. Meyer, and R. Su, “Noncausal in- 
verses for linear systems,” IEEE Trans Auto. Control, 
[3] L. Hunt, G. Meyer, and R. Su, “Driven dynam- 
ics for time-varying linear systems,” IEEE Trans Auto. 
[4] 3. Huang and W. rugh, “On a nonlinear 
multivariable servomechanism problem,” Automatica, 
[5] J. Huang and C. Lin, “Internal model principle 
and robust control of nonlinear systems,” in Proceed- 
ings 32nd Conf. Decision Contr., (San Antonio, TX), 
[6] S. Efromovich and V. Koltichinskii, “Projection 
estimation for inverse problems with unknown opera- 
tor,” Preprint, 1997. 
[7] M. Vidyasagar, A Theory of Learning and Gen- 
eralization. Berlin: Springer-Verlag, 1996. 
vol. AC-41, pp. 608-611, 1996. 
Contd ,  vol. AC-42, pp. 1313-1317, 1997. 
vol. 26, pp. 963-972, 1990. 
pp. 1501-1506,1993. 
1868 


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