PII: S0005 –1098(98)00005 –3Automatica , Vol. 34, No. 5, pp. 631 —635, 1998
/p71998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
0005-1098/98 $19.00 #0.00
Brief Paper
On Plant and LQG Controller Continuity Questions *
F. DE BRUYNE -‡, B. D. O. ANDERSON‡ and M. GEVERS -
Key Words —LQG control; identification for control; continuity questions.
Abstract —Using the dual Youla parametrizations of controller-
based coprime factor plant perturbations and plant-based
coprime factor controller perturbations, we study the LQG
plant-controller continuity question. Indeed, we show that it is
possible to calculate a new optimal LQG controller from a pre-
vious one when the plant is slightly changed, and to quantify the
change in the controller as a function of the change in the plant.
In addition, we compute the degradation in the achieved LQG
cost when the LQG controller is computed on the basis of
a perturbation of the real plant. As a by-product, we characterize
the set of all plants that have the same optimal LQG controller.
/p71998 Elsevier Science Ltd. All rights reserved.
1.Introduction
Our motivation for this paper originates from the recent
schemes for iterative identification and control design, in which
models and model-based controllers are successively updated on
the basis of new data collected on the real plant operating in
feedback with the most recent controller: see Lee et al. (1992)
Schrama (1992) and Zang et al. (1995) for a representative
sample of these iterative design schemes and Gevers (1993) for
a tutorial presentation of the ideas. An implicit but unproven
assumption underlying these schemes is that a small change in
the plant model should result in a small change in the controller,
and hence a small change in the actual closed-loop system. This
in turn should result in a slightly modified identified plant model.
Our main contribution in this paper is to shed some light on
this continuity question in the case of a Linear Quadratic Gaus-
sian (LQG) control criterion. In this paper, we have opted for an
approach that uses the dual Youla parametrizations of control-
ler-based coprime factor plant perturbations and plant-based
coprime factor controller perturbations. This approach is moti-
vated by the fact that in many of the iterative identification and
control schemes presented in the literature, the identification
step is performed using the identification method developed in
(Hansen, 1989) in view of closed-loop experiment design. In that
method, the dual Youla parametrization is used to parametrize
the unknown plant, and the closed-loop identification is reduced
to an open loop identification of the Youla parameter. By using
a Youla parametrization-based approach for the minimization
of the control criterion, we embed the identification and control
steps in a uniform framework in which the controller and the
model are computed as a perturbation of, respectively, the
previous controller and the previous model.
In the sequel, the following concepts are used extensively.
*Received 17 June 1996; revised 21 November 1997. This
paper was recommended for publication in revised form by
Associate Editor Andre ´ L. Tits under the direction of Editor
Tamer Bas ,ar.Corresponding author Dr Franky De Bruyne.
Tel.##61 2 6279 8674; Fax ##61 2 6279 8688; E-mail
debruyne @syseng.anu.edu.au.
-CESAME, Universite ´ Catholique de Louvain, Ba ˆ timent Eu-
ler, Avenue Georges Lemaıˆtre 4-6, B-1348 Louvain-la-Neuve,
Belgium.
‡Department of Systems Engineering, Research School of
Information Sciences & Engineering, The Australian National
University, Canberra, ACT 0200, Australia.Proposition 1.1 (Vidyasagar, 1985). Suppose P/p15andC/p15have
fractional representations P/p15"N/p46D/p92/p16/p46andC/p15"N/p33D/p92/p16/p33,
where N/p46,D/p46,N/p33,D/p33belong to S, the ring of proper stable
transfer functions. Assume that the following Bezout equation
holds:
N/p33N/p46#D/p33D/p46"1. (1)
For any arbitrary stable (linear) operator S, define
N/p49"N/p33!D/p46S,D/p49"D/p33#N/p46S. (2)
1. Then C(S)"N/p49D/p92/p16/p49is a stabilizing controller for
P/p15"N/p46D/p92/p16/p46.
2.Anycontroller that stabilizes P/p15has a fractional representa-
tion (2) for some S3S. The dual result is as follows. For any
arbitrary stable (linear) operator Q, define
N/p47"N/p46!QD/p33,D/p47"D/p46#QN/p33. (3)
1. Then P(Q)"N/p47D/p92/p16/p47is stabilized by C/p15"N/p33D/p92/p16/p33.
2.Anyplant stabilized by C/p15has a fractional representation (3)
for some Q3S.
Our basic one-degree-of-freedom control loop is that of
Fig. 1 and our control design criterion is the following regula-
tion LQG index (expressed here in discrete time):
J/p42/p47/p37"lim
/p44/p29/p271
NE/p7/p44/p9
/p82/p14/p16(y/p17/p82#/afii9838u/p17/p82)/p8, (4)
where y/p82is the plant output, u/p82is the control signal. The distur-
bance signal v/p82is assumed zero mean stationary with spectral
density function /p10/p84. In the sequel, we consider a disturbance
rejection problem, i.e. r/p82"0.
We now summarize a solution of the minimization problem
(4) using the Youla parametrization. This solution borrows from
a collection of results from (Desoer et al., 1980; Francis, 1982;
Vidyasagar et al., 1982; Youla et al., 1976). Note that a more
elegant solution is obtained using a standard plant approach; we
refer to (Francis, 1982) for further details.
LetP/p15andC/p15be as described in Proposition 1.1. Replace
C/p15in Fig. 1 by an arbitrary controller C(S) defined in Proposi-
tion (1.1). For this ( P/p15,C(S)) configuration we have
y/p82"(D/p33#N/p46S)D/p46v/p82,u/p82"(N/p33!SD/p46)D/p46v/p82.
Using Parseval’s theorem, we obtain the following expression
for equation (4):
J/p42/p47/p37"1
2/afii9843/p16d/afii9853/p43/p34D/p33#N/p46S/p34/p17#/afii9838/p34N/p33!D/p46S/p34/p17/p44/p34D/p46/p34/p17/p10/p84. (5)
It is standard that a stable minimizing Scan be found analyti-
cally by means of spectral factorizations and projections, i.e. by
taking stable parts. Indeed, by completing the square, the LQG
control criterion can be rewritten as
J/p42/p47/p37"/p35AS#A/p92*B/p35/p17/p17#1
2/afii9843/p16d/afii9853/p43C!(A*A)/p92/p16B*B/p44(6)
631 Fig. 1. One degree of freedom control loop.
with
AA*"[/p34N/p46/p34/p17#/afii9838/p34D/p46/p34/p17]/p34D/p46/p34/p17/p10/p84, (7)
B"[N*/p46D/p33!/afii9838D*/p46N/p33]/p34D/p46/p34/p17/p10/p84, (8)
C"[/p34D/p33/p34/p17#/afii9838/p34N/p33/p34/p17]/p34D/p46/p34/p17/p10/p84, (9)
whereAis minimum phase, stable and of relative degree zero.
The minimizing Sis clearly given by S/p15/p16/p20"!A/p92/p16[A/p92*B]/p19/p20where [ ]/p19/p20denotes the stable part. Note that the constant term
in the partial fraction expansion of A/p92*Bmust be so par-
titioned between [ A/p92*B]/p21/p14/p19/p20and [A/p92*B]/p19/p20that [A/p92*B]/p21/p14/p19/p20hasz"0 as a zero. The preceding remark is used extensively in
Section 3. We refer the reader to Vidyasagar (1985) for more
details. The optimal control cost is
J/p15/p16/p20/p42/p47/p37"1
2/afii9843/p16d/afii9853/p7/p34[A/p92*B]/p21/p14/p19/p20/p34/p17#/afii9838
/p34N/p46/p34/p17#/afii9838/p34D/p46/p34/p17/p34D/p46/p34/p17/p10/p84/p8)
(10)
IfC/p15is the optimal controller for P/p15, then S/p15/p16/p20"0 minimizes
J/p42/p47/p37over all S3S.
Using the Youla parametrizations and the LQG control de-
sign criterion (4), we solve the following problems. Here,
C/p15denotes the optimal (and hence stabilizing) controller for P/p15.
1. Assume that the optimal LQG controller C/p15for a plant P/p15is
known and consider a new plant P/p16that is stabilized by
C/p15and that is obtained by a perturbation of size Qaway from
P/p15. We then compute the optimal LQG controller C/p16for
P/p16as a perturbation of size S/p16away from C/p15, where S/p16is
computed from P/p15,C/p15andQ. This allows us to relate the size
of a change Qin the plant to the size of the corresponding
change S/p16in the optimal LQG controller. We are especially
interested in the simple formula resulting when Qis small
where the size of Qis measured using either the H/p27or the
H/p17norm. Here, P/p15andP/p16could be seen as two successive
plant models in an iterative design scheme, with C/p15and
C/p16the corresponding optimal controllers. Alternatively,
P/p15could also be the true plant, with P/p16a model that is close
to it.
2. Under the same assumptions, we compute the increase in the
LQG cost (i.e. the performance degradation) that results from
applying the controller C/p16, optimal for P/p16, to the initial plant
P/p15. This increase is expressed as a function of the size of the
perturbation QofP/p16away from P/p15. Again, our main focus is
on small Q. The question addressed here is how much LQG
cost increase is incurred by applying to the real plant P/p15, say,
an optimal controller C/p16computed on the basis of a plant
model P/p16that is close to P/p15.
3. As a by-product, we characterize the set of all plants P/p16that
have the same optimal LQG controller, C/p15, as the original
plant P/p15, i.e. we characterize the set of perturbations Qthat
are such that the Youla parameter S/p16is zero over the fre-
quency axis.
The outline of our paper is as follows. In Section 2 we
compute how much change is induced in a controller by
a change in a plant model, while in Section 3 we characterize the
set of all plants that have the same optimal LQG controller. In
Section 4 we express the degradation in the LQG cost that
results from computing the LQG controller on the basis of
a model that is a perturbed version of the actual plant. The
validity of the theoretical results is checked in Section 5. We
conclude in Section 6.2.Plant and corresponding controller perturbations
In this section we examine the change that results in an
optimal LQG controller when a plant model is changed from
some initial model P/p15to a model P/p16that is expressed as a con-
troller-based perturbation of P/p15. Consider first a plant model
P/p15and its corresponding optimal controller C/p15, both factorized
as before. Let now P/p16be some plant that is stabilized by C/p15.I ti s
obvious that P/p16is contained in the set
P"/p43P/p16(Q)"(N/p46!QD/p33)(D/p46#QN/p33)/p92/p16with Q 3S/p44, (11)
of all models stabilized by C/p15. The set of all controllers stabiliz-
ing a given P/p16(Q)3Pis then given by
CM"/p43C/p16(S/p16,Q)"[N/p33/p16(S/p16,Q)][D/p33/p16(S/p16,Q)]/p92/p16withS/p16,Q3S/p44
(12)
where N/p33/p16(S/p16,Q)"N/p33!S/p16(D/p46#QN/p33) and D/p33/p16(S/p16,Q)"D/p33#
S/p16(N/p46!QD/p33). Let C/p16be any controller in the set CM. Using
Parseval’s relation, we get the following LQG index:
J/p42/p47/p37(P/p16,/p10/p84/p16,C/p16)"1
2/afii9843/p16d/afii9853/p43/p34D/p33#(N/p46!D/p33Q)S/p16/p34/p17.
#/afii9838/p34N/p33!(D/p46#N/p33Q)S/p16/p34/p17/p44/p34D/p46#QN/p33/p34/p17/p10/p84/p16/p13
(13)
2.1.Computation of S /p16/p15/p16/p20as a function of Q . In this subsection, we
characterize the optimal controller C/p15/p16/p20/p16, i.e. we compute S/p16/p15/p16/p20that
minimizes J/p42/p47/p37and express it as a function of Qand the coprime
factorizations of the plant P/p15and its corresponding optimal
controller C/p15. Thus, S/p16/p15/p16/p20, which expresses C/p15/p16/p20/p16as a perturbation
ofC/p15, is to be defined as a function of Q, which expresses P/p16as
a perturbation of P/p15.
Recall that AandB, related to the plant P/p15and its optimal
controller C/p15, are given by
AA*"[/p34N/p46/p34/p17#/afii9838/p34D/p46/p34/p17]/p34D/p46/p34/p17/p10/p84and
B"[N*/p46D/p33!/afii9838D*/p46N/p33]/p34D/p46/p34/p17/p10/p84.
Two situations can occur when the system is perturbed: either
the perturbation influences only the plant model, and the noise
model remains unchanged or both the plant model and the noise
model are influenced. We consider the case where /p10/p84/p16varies with
Qin such a way that /p34D/p46#QN/p33/p34/p17/p10/p84/p16(Q) is independent of Q, i.e.
/p34D/p46#QN/p33/p34/p17/p10/p84/p16(Q)"/p34D/p46/p34/p17/p10/p84/p16(0)"/p34D/p46/p34/p17/p10/p84. (14)
This is typical of an ARMAX model structure, i.e. equation (14)
makes sure that the perturbed system ( P/p16,/p10/p84/p16) remains an
ARMAX system if the original system ( P/p15,/p10/p84) is an ARMAX
system. By an ARMAX model structure, we mean an auto-
regressive moving-average model structure with exogeneous in-
put. Note that the case /p10/p84/p16(Q)"/p10/p84which is typical of an output
error (OE) model structure leads to derivations that are more
involved. We are now in a position to calculate the perturbed
version of B,
BM"[(N*/p46!Q*D*/p33)D/p33!/afii9838(D*/p46#Q*N*/p33)N/p33]/p34D/p46/p34/p17/p10/p84.
"B!Q*[/p34D/p33/p34/p17#/afii9838/p34N/p33/p34/p17]/p34D/p46/p34/p17/p10/p84. (15)
There will be a corresponding change from AtoAM, i.e.
AMAM*"[/p34N/p46!QD/p33/p34/p17#/afii9838/p34D/p46#QN/p33/p34/p17]/p34D/p46/p34/p17/p10/p84. (16)
The minimizing Youla parameter S/p16/p15/p16/p20is given by
S/p16/p15/p16/p20"!AM/p92/p16[AM/p92*BM]/p19/p20
"AM/p92/p16[AM/p92*Q*[/p34D/p33/p34/p17#/afii9838/p34N/p33/p34/p17]/p34D/p46/p34/p17/p10/p84]/p19/p20(17)
because Bis unstable by optimality of C/p15andAM/p92*is unstable
by definition.
2.2.A continuity question . In this subsection, we investigate the
continuity properties of LQG controllers with respect to plant632 Brief Papers
