Systems & Control Letters 4 (1984) 79-83 North-Holland

April 1984

Another balanced controller reduction algorithm

J.A. DAVIS and R.E. SKELTON u=GZ, School of Aeronazitics an Astronautics, Purdue University, West Lafayette, IN 47907, USA

Received 28 August 1983 Revised 12 December 1983

The problem of controller reduction has been addressed by a number of authors [l-4] using the concept of balanced coordinates originally proposed by Moore [5]. This note estah- lishes a more direct link between the Riccati balancing methods in [2-41 and Moore’s work and proposes yet another candidate method as a logical extension of (2-41.

Keywords: Controller reduction, Model reduction, Optimal control, Stochastic processes.

1. Introduction

Consider the standard Linear Quadratic Gaus- sian (LQG) design S,(n) for the following system (S(n)): s(n): i=Ax+Bu+Dw,

XE.%?n,t4E9m,WE@,

y=cx, ye?‘k,

z=Mx+u, Z EL&,

E{w(t)}=O, E{u(t)}=O,

+(o)} =o,

E{ w(t)wT(7)} = Wli(t-T),

E{ U(f)UT(T)} = vi3(t --7),

E{w(t)UT(T)}=O, f#T,

E{x(0)wT(t)} =o, t20,

E{ x(o)UT(t)} = 0, t 2 0,

v= Ko{ Idyll;+ Il4M)~ Em{ .} p ,p/{ .},

Q>O, R > 0,

where S,(n) is

S,(n): i=A,i+-t~, Ivan,

(14 (lb) (lc)

(24 (2b) (2c) (2d) (24 (20 (34

(3b)

(44

(4b) A,=A+BG-FM,

G= -Rp’BTK,

F= PMTV-‘,

(4c)

(54

(5b)

with K and P satisfying the algebraic Riccati equa- tions

KA + ATK - KBR-‘BTK + C’QC = 0,

PAT •t AP - PMTV-‘MP •t D WDT = 0.

(5c)

(5d)

If S(n) is a minimal realization then the solutions to the Riccati equations are guaranteed positive definite and unique [6].

2. Review of balancing algorithms

The controller reduction algorithm proposed by Jonckheere and Silverman [2] and Verriest [3,4] balances the algebraic Riccati equations (5c), (Sd) such that

P=K= [of,a;,...,a,Z]

and deletes the states associated with the smallest u: ‘s. We shall label this method of Balanced Ric- cati Reduction Algorithm (BRRA).

The BRRA balances Riccati solutions whereas [5] balances Liapunov solutions. To see the con- nection, use (5a) and (5b) and write (5~) and (5d) as follows: BRRA: K(A+BG)+(A+BG)TK

+ [ C'T GIT] ;; =O, [ 1 C’ = \/Q C, G’ = &G, (64

P(A - FM)T + (A - FM)P

=O,

D’=D&?, F’=F@. (6b)

The interpretation of the BRRA from the point of view of Moore’s work is to balance the observabil-

0167-6911/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland) 79

Volume 4, Number 2 SYSTEMS & CONTROL LE’ITERS April 1984

ity grammian associated with the matrix pair

[A +BG [ $11 and the controllability grammian associated with the matrix pair

[A-FM [D’ F’]].

Instead, we propose in what follows to balance observability of (A + BG, G’) and controllability of (A - FM, F’).

Yousuff and Skelton proposed in [l] to balance the observability of

(A+BG-FM,G’) and the controllability of

(A+BG-FM,F’).

Hence in [l] a balanced controller algorithm de- letes the least observable and least controllable states directly from the controller S,(n) by balanc- ing the following Liapunov equations:

BCRA: XA;+A,X+F’F’T=O,

ZA +ATZ+G’TG’=O. c c

(74

(7b)

We label this method Balanced Controller Algo- rithm (BCRA). If the optimal controller is unsta- ble then [l] shows that BCRA must be modified to BCRAM:

BCRAM : O=X(A+BG)T

+(A + BG)$+ F’F’T, (84 O=Z(A-FM)

+(A-FM) Z+G’TG’. (8b) Of course, all of these balancing methods yield

different results. As was shown in [l] the order of the optimal controller can be less than the order of system S(n), and the advantage of algorithms (7) (8) is that they always find a minimal realization of the controller of order < n if one exists, whereas the algorithm (6) will not do this.

Another controller reduction scheme which en- joys this desirable minimal property is the so-called ‘energy equivalent controller’ or EEC [7]. This method uses component cost analysis of the opti- mal controller and deletes those states of the con- troller (2) which can be deleted without affecting the optimal mean squared controller output (E,]]u]]i). Hence, these are states which have a

zero component cost, and the reduced controller is ‘cost equivalent’ with respect to the control cost E,]]u!& The essential calculations are these: com- pute X and Sz from

EEC : O=&A+BG)T

+(A + BG)ri+ F’FrT, (94 22 = mT, Y’( p a, [ s2TG’TG’i2], (9b)

where controller states 8, associated with the smal- lest singular values of [QTG’TG’s)] (denoted by the component cost q) are deleted. This component cost method is not a balancing method in the sense of Moore but it will be included in the comparison which follows.

3. The new balancing scheme

The BCRA balances the observability of the states of the controller (2) in the control (u) and the controllability of the controller from z (see eq. (4)). In contrast, the BRRA balances the control and filter Riccati solutions. The following method combines the ideas of both.

Consider an alternative set of expressions for the control Riccati equation (6a):

K=K,+K,, 004

K, (A + BG) + (A + BG)TK, + C’TC’ = 0, (lob)

K,(A+BG)+(A+BG)TK,+G’TG’=O. (10~)

Note that K, represents the observability gram- mian of the controller expressed in the form

~=(A+BG)~++z,

u=Gi

(Z white noise with intensity V). The alternative set of expressions for the filter Riccati equations (6b) is

P=P,+P,, 014

P,(A-FM)T+(A-FM)P,+D’D’T=O, (llb)

P,(A-FM)T+(A-FM)P,+F’F’T=O, (11~)

where P, is the controllability grammian associ- ated with the plant noise w in the estimation error system,

k = (A - FM)2 + D’w - F’u,

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Volume 4, Number 2 SYSTEMS t CONTROL LETTERS April 1984

and Pz is the controllability grammian associated with the sensor noise u (the noisy input to the controller). Instead of balancing the entire Riccati solutions (P and K ), the new method balances only those parts which are directly related to the controller (Pz and K2). This leads to balancing (10~) and (11~). We label this method Balanced Riccati Reduction Algorithm Modified (BRRAM).

The new method therefore attempts to delete those states of the controller which least change the control (u) and which least contribute to the co- variance of the estimation error. Finally, it is noted that like the BRRA [2-41, the BRRAM might not extract the minimal realizations of the controllers when they exist [l].

BRRAM: ~Y,(A+BG)+(A+BG)~K,

•t G”G’ = 0, 024 4. Example

P,(A -FM) +(A -FM)P, The new method BRRAM is shown not to be

+F’FT=O. W) the ‘best’ method under all circumstances, but the

Table 1 Controller reduction comparisons, example (13)

Defining eqs.

(6) (8) (7) (9)

(12)

Method I(%) I,(%) I”(W IzyI+ IZ.K%) ZF(W ZFJW

h = - 1, = 0.01 p

BRRA [2-41 412.0254 412.7969 - 26.9497 439.7466 48.5241 545.1680 BCRAM [l] 1.7735 1.9382 - 91.9336 93.8718 91.8096 1.8167 BCRA [l] 0.0816 0.0818 - 0.0037 0.0854 0.1063 1.5894 EEC [7] 0.0819 0.0820 - 0.0014 0.0835 0.1066 1.5937 BRRAM 0.0817 0.0818 0.0005 0.0823 0.1063 1.5907

x=-l,p=l.O

P-0. 0.0573 0.0605 0.1611 0.2216 0.6609 0.9493 BRRA 0.1131 0.1021 0.8621 0.9642 1.4975 1.9158 BCRAM 0.0660 0.1622 - 6.5084 6.6706 5.5846 1.1254 BCRA 0.0736 0.0752 - 0.0390 0.1142 0.9989 1.4644 EEC 0.0761 0.0770 0.0139 0.0909 1.0249 1.5032 BRRAM 0.0741 0.0760 - 0.0108 0.0862 1.0050 1.4741

A=-l,p=lO

BRRA 0.0653 0.0653 0.5462 0.6115 1.8404 1.4738 BCRAM 0.1056 0.1056 - 1.7941 1.8998 1.3554 0.9965 BCRA 0.0630 0.0630 - 0.1371 0.1993 1.4605 1.1750 EEC 0.0627 0.0633 0.0419 0.1052 1.5746 1.2669 BRRAM 0.0620 0.0620 - 0.0759 0.1379 1.4788 1.1900

x=+1,p=o.o1

BCRAM 8.2945 8.3979 - 91.5728 99.9707 90.6472 6.7093 (all others unstable)

h=+l,p=l.O

BCRAM 6.4633 7.3200 - 46.6960 54.0160 34.5760 5.8236 (all others unstable)

x=+1,p=10

BRRA BCRAM BCRA EEC BRRAM

642.3772 3.5007

35.5893 unstable

10.5552

640.8054 658.7120 1229.5265 623.8083 506.8053 5.1614 - 13.7684 18.9298 8.4912 4.0817

36.0975 30.3037 66.3735 61.1810 74.7581

9.3605 22.9789 32.3393 14.4572 7.4868

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Volume 4, Number 2 SYSTEMS & CONTROL LETTERS April 1984

following example taken from [1] demonstrates its potential.

S(2): i= [o” “*,1x+( ,;)u

(134

(*3b)

v= 1 0 [ 1 0 1’ (134

Y-= E,{ IIYWII; + Il4wJ~ (13e)

R=P, Q=[,; ,;;].

(forX= -l), u= [-1 -1o]a.

(1%)

(*W First order controllers are derived from the

BRRA of [2-41, the BCRA of [l] and its modified version BCRAM, the Energy Equivalent Con- troller (EEC) method [7], and the proposed method, BRRAM. For reference purposes only, we include the optimal reduced order controller labeled ‘P.O.’ in Table 1 for the case (A = - 1, p = l), obtained from a direct parameter optimization (choosing the parameters of the controller of specified order (one) to minimize v).

5. Performance measures and results

The ‘goodness’ of the five algorithms is com- pared using six indices, three of which are:

(14a)

Iy k Yp<*

^v,* ’ (*4b)

where v* is the optimal value of the ‘cost function (13e) and

Y”* = 5* + vu*.

82

v, 5 and VU are the corresponding values calcu- lated when the system is driven by the reduced order controller. Thus (14a, b, c) represent the change from optimal of the performance index, the output contribution and the control contribu- tion. The fourth index considers the absolute change from optimal of the output and control contributions (i.e. ]I,] + ]I,]).

The preceding indices are based on scalars which are prominent in optimal control design. One could argue the relative importance of these scalars as a measure of goodness in reduced order controller design. Presented below are two additional scalars for evaluation of the reduced order controllers. Define

where the R subscript denotes outputs of the sys- tem driven by the reduced order controller. Then

1, 4 II& - %/ll~llF (154 where I] . I] r is the Frobenius norm. I, is a normal- ized measure of the absolute change in the covari- ante of (i;) for the reduced order system. Varia- tions on (Isa) could include

1, A IIYR - ~llF/ll~llF (*5b)

where Y A cov y, Ya 2 cov y,. The results from the example for the control

weighting (p) equal to 0.01, 1.0 and 10 are shown in Table 1 for a stable system (h = - 1) and an unstable system (A = 1).

From this example a number of observations can be made. For the stable system, BCRA, EEC and BRRAM provided consistently low values for all indices shown, while the BCRAM method did well for isolated cases (usually associated with output performance). For the unstable system, only BCRAM maintained closed loop stability after controller reduction in all cases. Note that it is possible to make large changes in both y and u as measured by (14b), (14~) and yet keep I small in (14a). In fact, this is how BCRAM wins the com- petition by criterion I in the (A = - 1, p = 1) case (but see that Iy and I,, are largest for the BCRAM in this case).

No performance index can be singled out as ‘best’ out of those presented (since what is best can only be defined with knowledge of the final

Volume 4, Number 2 SYSTEMS & CONTROL LETTERS April 1984

use of the controller) and the list of error indices above is certainly not a complete list. A logical design question might be to minimize the change in the nature of the output cov(y) while taking advantage of the largest control effort reduction possible (i.e. maximize -~u/IFy). BCRAM best satisfies this criterion in all above cases.

6. Conclusions

The new method of balancing is a modification to the balanced Riccati method [2-41 and deals more directly with the states of the controller. For the example considered this method compared favorably with the other balancing methods. How- ever, in the examples shown only the BCFLAM method of [l] successfully accommodated unstable plants. One conclusion from the numerical com- parison with earlier methods [2-4,1,7] is that many ‘balanced’ controller reductions can be derived, that there are many measures of performance, and

that depending on the measure chosen, a balanced reduction procedure can be ‘good, bad or optimal’.

References

[I] A. Yousuff and R.E. Skelton, Balanced controller reduc- tion, IEEE Trans. Automat. Control 29 (3) (Feb. 1984).

[2] E.A. Jonckheere and L.M. Silverman, A new set of in- variants for linear systems - Application and approxima- tion, Internat. Symp. Theory Networks and Systems, Santa Monica, CA (1981).

[3] E.E. Verriest, Low sensitivity design and optimal order reduction for the LQG-problem, in: Proc. 24th Symp. Cir- cuits and Systems (June 1981) pp. 365-369.

[4] E.I. Verriest, Suboptimal LQG-design and balanced realiza- tions, in: Proc. 20th IEEE, CDC (Dec. 1981) pp. 686-687.

[5] B.C. Moore, Principal component analysis in linear sys- tems; controllability, observability, and model reduction, IEEE Trans. Automat. Control 26 (1981) 17-32.

[6] H. Kwakernaak and R. Sivan, Linear Optimal Control Sys- tems (Wiley, New York, 1972).

[7] A. Yousuff and R.E. Skelton, Controller reduction by com- ponent cost analysis, IEEE Trans. Automat. Control 28 (Oct. 1983).

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