Robust Stochastic Performance Optimization

Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993

ROBUST STOCHASTIC PERFORMANCE OPTIMIZATION B. Bernbardsson Institute for Math. and its Appl., University of Minnesota, Minneapoiis, MN 55455, USA

Abstract The problem of finding the controller that optimizes the expected H 2 -norm of an uncertain system is solved in closed form for a class of problems including such signal processing applications as feedforward design, channel equalization, noise cancellation etc. The method uses covariance information on model uncertainty and does therefore match information obtainable from standard system identification. The optimal controller is found by using a spectral factorization to rewrite the problem as an H 2 -problem for an extended system. The article puts a restrictions on where uncertain parameters enter. The need for hard bounds on parameters can then be avoided. The method also avoids the conservativeness related to designing for worst cases. Keywords Robust Control, Identification, Optimal Control, Linear Systems, Robust Performance

1. INTRODUCTION

A natural approach is instead to try to use the information obtained from standard identification and to take the likelihood of different parameter variations into account. The controller design can then be made with respect to most probable parameter variations, instead of perhaps very rare worst cases.

The goal of robust controller design is to achieve robust performance, i. e. good performance in the face of plant uncertainty. This seems to be a harder problem than the robust stability problem which has been studied intensively during the last decade. No design method can tackle all the issues that have to be studied when making a real design. Typically a synthesis method will focus on one or two issues and leave the others to the designers common sense as hidden side conditions.

This paper presents results for a class of problems where there is a restriction on how uncertain elements enter, see assumptions 1 and 2 below. A short and instructive algorithm for synthesis of the optimal controller is presented. The current article hence describes a class of problems where classical identification methods suffice for designing robust performance controllers. If the assumptions on how the uncertain parameters enter are not met, other identification methods giving hard bound on parameters and/or other definitions of closed loop performance should be used.

Research in robust controller design has hitherto focused on stability. Most control systems can however be made useless for much smaller system variations than is needed for rendering the system unstable. This is of course well realized by people working in the robust stability area and it is a major goal of recent research to find synthesis methods for obtaining robust performance.

The method proposed should be seen as a complement to other design methods for robust performance synthesis, e. g. Quantitative Feedback Theory, [Horowitz and Sidi, 1972], LQGjLTR, p. [Maciejowski, 1989], or [Doyle et al., 1982], mixed H2/ Hoc [Ridgely, 1991], convex optimization [Boyd and Barratt, 1991] or adaptive control.

The formulation of uncertainty models is a fundamental issue. A common assumption is to assume hard bounds on the uncertainty, and design for worst cases. Since standard identification methods do not give hard bounds, new identification methods are being developed, see e. g. [Norton, 1987] or [Wahlberg, 1991]. It is hard to develop such methods that are not too conservative, i. e. give too large upper bounds on the model uncertainty. Too conservative estimates leads to conservative regulator designs resulting in low performance.

For a more detailed version this paper see [Bernhardsson, 1992].

279

2. PROBLEM FORMULATION 2.1 Uncertainty Models A general and useful way to describe an uncertain system is given in Figure 1. Here all uncertain elements have been collected in an upper loop in the matrix 6.. The uncertainty can be both parametric/structured and unparametric/unstructured. This is described in the structure of 6. = diag(hR' 15[, 6.[). For a description of the different blocks see [Doyle et al., 1982]. As a typical example an uncertain state space system

v w

r

z y

u

Fig. 1 General uncertainty model. It is assumed that Goo 0 and that either G lO 0 or G Ol O.

=

= (A + 6.A):l: + (B + 6.B)u y = (C + 6.C):l: + (D + 6.D)u

=

=

:i:

3. OPTIMAL CONTROLLER DESIGN

can be written in the form of Figure 1.

Although Figure 1 represents a very general way of representing robust performance problems we will now show that one actually can solve problem (3) in closed form for a class of interesting problems. Let the system be given by

2.2 Covariance Information Standard identification methods normally give covariance information on parameters. This is represented by a matrix

(4) = GoOv + G01w + G 02 u (5) z = GlOV + GUw + G 12 U (6) y = G 20 v + G 21 W + G 22 u where u = Ky, v = 6.r. Here Do is a real, constant but unknown matrix. We assume that E(6.) = 0

(1) where E denotes mathematical expectation and represents identification error.

r

6

2.3 Controller Design The goal is now to optimize expected controller performance measured from external signals w to outputs z. The H 2 -norm will be used to measure performance:

and that we are given covariance information on the elements of 6.. The special class of problems we can solve the problem for is given by Assumption 1

DEFINITION 1 For a stable system the H 2 -norm is defined by

IIGI12:= ( -1

21r

]00 tr G(jw)*G(jw)dw ) l -00

G OO

=0

(see Section 6 for a discussion on relaxation of this assumption).

(2)

Assumption 2

The optimal robust performance controller is given by the linear time invariant controller K(s) that stabilizes the loop under all uncertainties 6. (if possible) and minimizes the expected H 2-norm:

G 02

=0

or

It is easy to see that if assumptions 1 and 2 are satisfied the stability of the closed loop system will not depends on 6.. If we do not want to assume hard bounds on parameters this is crucial. If there is feedback around an uncertain element without a hard bound, there is always a possibility for the system becoming unstable for large parameter errors.

(3) where E denotes mathematical expectation with respect to the distribution of 6. and Tm (K) denotes the closed loop system from w to z. Problems like (3) have been proposed previously and are generally quite hard. Note that (3) is related to the stochastic embedding method used by [Goodwin and Salgado, 1989]. Different criteria that take the likelihood of different modeling errors into account have been used bye. g. [Chung and Belanger, 1976], [Grimble, 1984], or [Kassam and Poor, 1985, Nahi and Knobbe, 1976, Stengel and Ray, 1991]. A scalar input estimation problem of the form (3) is solved in [Sternad and Ahlen, 1993], we refer to this paper for more history. The results therein also follow from Theorem 1 below.

THEOREM

1

Assume that G oo = 0 and G 02 spectral factorization

= O. Introduce the

P(s)F*(s) = E(6.G01(S)G~l(S)6.*)

(7)

I:;.

The optimal controller solving (3) is then given by the solution ofthe following standard H 2 -problem.

~nll

(GlOP

G 12 K(I 280

-

G

u) +

G 22 K)-1 (G 20 P

G 21

)

11:

(8)

Using u = Ky gives

PROOF

w

r = [Goo + G02 K(I [G01 + G 02 K(I z = [GlO + G 12 K(I [G l l + Gl'JK(I -

G 22 K)-1G 20 ]v + G22 K )-1G 2l ]w G 22 K)-1G 20 ]v + G 22 K)-1G 21 ]w

z

Assuming G oo = 0 and G 02 = 0 and using v = D.r the problem is reduced to minimizing (3) where

Fig. :I Block Diagram for the Feedforward Example

= G l l + G 12 K(I -

G 22 K)-1G 2l + +[GlO + G12K(I - G22K)-1G20]D.G01

T"..,(K)

Note that the information needed to perform the expectation in (3) is exactly the covariance between different elements in D.. Performing the expectations in (3) and noting that the terms linear in D. disappear due to E(D.) = 0 we get

liT".., II~ =

(GlOP

11

G12 K(I - G22 Kr

Gl l

1

)

I

+

(G 20 P

G 2l

)

11:

10·' '--~~~~-~~~

tot

10-'

10'

FtequellC)'(radI'l

This is a standard LQG-problem and can be solved as usual. For P = 0 the design reduces to the nominal design.

Fig. S Solution for the Feedforward Example, CT = 0.1 0.2 (dashed) and CT 0.4 (dotted) (solid), CT

=

The result when G 20 = 0 follows by the dual argument. THEOREM 2 Assume that G oo 0 and G 20 spectral factorization

=

Note that G oo = G 20 = 0, we can therefore use Theorem 2. Since GlO = 1/(6 + 1), the spectral factorization will in fact be scalar

= O. Introduce the

P(s)

The optimal controller solving (3) is then given by the solution of

The following example is made very simple to illustrate ideas. Successful feedforward control crucially depends on a good model of the system. To exemplify this we consider the feedforward problem in Figure 2, where

=

0.1, E(9) = 0, and E(92 ) Assume that 90 Written in the form (4)-(6) we have

o G= [.+1 :

o

1

.+1 1

(~) + [ _2~1 1K(s) _+1 (.+1)"

The optimal controller is given by K(s) -5(s)/ R(s) where

4. A SIMPLE EXAMPLE

G _ 6 +96+ 1 1(6+1)2

(T -s+l

The problem can therefore be solved as a standard H 2 problem for the system G=

1 G2 = - 6+1

= (TGlO(s) =

With more than one uncertain element a method for doing MIMO spectral factorization must be used.

(9)

2

=

R(s)R( -s) = (s2 5(s)

=

90 s + 1) - (T26 2

= _(2+90 )(s+1)2 R(l)

Figure 3 shows the amplitude of K (iw) for (T = 0.1,0.2,0.4. Note that when the uncertainty in 9 is increased the robust controller decreases the maximum amplitude, which agrees with intuition.

(10)

=

+ 90 s + 1)(s2 -

(11)

(T2.

5. CONCLUSIONS (.+1)' .'+9 p .+1

A natural performance robustness problem has been formulated . The formulation uses information obtainable from standard identification in the

· 1

(.+1)"

o

281

form of covariances of parameters. In this way we avoid finding hard bounds on parameters.

DOYLE, J. C., J . E. WALL, and G . STEIN (1982): "Performance and robustness analysis for structured uncertainty." In IEEE Proceedings on Decision and Control, Orlando, volume 2, pp. 629636.

We have shown how the problem can be analytically solved for a class of open-loop type problems. This class includes interesting applications in signal processing such as feedforward design, channel equalization, noise cancellation, optimal different iation[Carlsson et al., 1991]. The optimal controller can be found using standard LQG-software and it is easy to see how uncertainty in the elements influence the resulting controller, see (8) and (9). We have implicitly assumed that the resulting H 2 control problem is nonsingular. This is the case for most applications, but it should be possible to describe the conditions for this more exactly.

GOODWIN, G. and M . SALGADO (1989): "A stochastic embedding approach for quantifying uncertainty in the estimation of restricted complexity models." International Journal of Adaptive Control and Signal Processing, 3, pp. 333356. GRIMBLE, M . (1984) : "Wiener and Kalman filters for systems with random parameters." IEEE Transactions on Automatic Control, AC-29:6, pp. 552-554.

The most severe restriction is of course the open loop way uncertain elements are supposed to enter. The assumption G oo = 0 can be relaxed to GooLl nilpotent. This means that there is no signal loop around an uncertain parameter in the open loop system. Higher order moments are then however needed to calculate the expected value of the H 2 -norm. If Gaussianity is assumed the higher order moments can be calculated from the mean and variances. The problem can then be solved analogously to above.

HOROWITZ, I. and M. SIDI (1972): "Synthesis of feedback systems with large plant ignorance for prescribed time domain tolerances." International Journal of Control, pp. 287-309. KASSAM, S. and V. POOR (1985): "Robust techniques for signal processing: A survey." Proceedings of the IEEE, 73:3, pp. 433-48l. MACIEJOWSKI, J. M. (1989): Multivariable Feedback Design. Addison Wesley.

The problem has been described in continuous time. The results carry over directly to discrete time systems.

NAHI, N. and E . KNOBBE (1976): "Optimallinear recursive estimation with uncertain system parameters." IEEE Transactions on Automatic Control, AC-21:2, pp. 263- 266.

The solution should be interesting to use in connection with adaptive control and adaptive signal processing. We now have a method were identification and controller design match each other. Applications of this is left for future research.

NORTON, J . P. (1987): "Identification of parameter bounds of armax models from records with bounded noises." International Journal of Control, 45:2, pp. 375-390.

Acknowledgement The author is grateful to M. Sternad and A. Ahlim for interesting discussions and for providing early versions of [Sternad and Ahlen, 1993] .

RIDGELY, B. (1991): A Non-conservative Solution to the General Mixed H 2/ H 00 Optimization Problem. PhD thesis, Massachusetts Institute of Technology, Cambridge.

6. REFERENCES

STENGEL, R. and L. RAY (1991): "Stochastic robustness of linear time-invariant control systems." IEEE Transactions on Automatic Control, 36:1, pp. 82-87.

BERNHARDSSON, B . (1992) : Theory for Digital and Robust Control of Linear Systems. PhD thesis, Lund Institute of Technology.

STERNAD, M. and A. AHLEN (1993): "Robust filtering and feedforward control based on pro babilistic descriptions of model errors." Automatica. To appear, also available as Report UPTEC 91071R Dept. of Technology, Uppsala University, Sweden.

BOYD, S. and C. BARRATT (1991): Linear Controller Design - Limits of Performance. Prentice Hall. CARLSSON, B., A. AHLEN, and M . STERNAD (1991): "Optimal differentiation based on stochastic signal models." IEEE Transactions on Signal Processing, SP-39, pp. 341-353.

WAHLBERG, B. (1991): "System identification using laguerre models." IEEE Transactions on Automatic Control, 36:5, pp. 551-562.

CHUNG, R. and P. BELANGER (1976): "Minimum sensitivity filter for linear time-invariant stochastic systems with uncertain parameters." IEEE Transactions on Automatic Control, AC21:1, pp. 98-100. 282