Systems & Control Letters 46 (2002) 99–110www.elsevier.com/locate/sysconle

Balanced performance preserving controller reduction

G. Wanga ; ∗, V. Sreerama, W.Q. Liub

aDepartment of Electrical and Electronic Engineering, University of Western Australia, Nedlands, WA 6907, AustraliabSchool of Computing, Curtin University of Technology, WA 6102, Australia

Received 3 November 2000; received in revised form 29 November 2001

Abstract

Two methods for controller reduction based on the closed-loop system performance-preserving criterion are proposed.They are the closed-loop block balanced and the closed-loop structurally balanced truncating methods. Reduced-ordercontrollers are obtained directly by solving the closed-loop system reduction problem with performance-preserving weights.c© 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

In the linear control system design, one of the ways to obtain low-order controllers is to design thehigh-order controller 5rst and then reduce the order of controller by controller order reduction [2]. Controllerorder reduction has been extensively studied for more than a decade. It is well known that preferable controllerreduction methods are those in which the information about the closed-loop system has been utilized inreduction process.Many controller reduction approaches have been developed based on stabilizing the closed-loop system with

the reduced-order controller or reducing the closed-loop system H∞ performance error between the full-ordercontroller and the reduced-order controller [2,19,16,10]. Numerical examples show that the approaches basedon reducing the closed-loop system H∞ performance error are better than those based on the closed-loopsystem stability criterion. Therefore, most of the existing controller reduction methods were developed withan e;ort to reduce the closed-loop system H∞ performance error. The closed-loop block balanced method[16] and the closed-loop structurally balancing method [19] were developed from the famous model reductionmethod—balanced truncation.The more recent development in controller reduction is the introduction of the performance preserving

criterion by Goddard and Glover [6–8]. Further extension to H2 controller reduction and relative controllerapproximation appeared in [4,3]. Here performance preservation means the H∞ norm bound of the closed-loopsystem with the reduced order controller is not greater than that with the full order controller. In [14,15] anotherperformance preserving controller reduction method was proposed. Instead of using the additive perturbationof the controller proposed by Goddard and Glover, the method in [14,15] is based on the additive perturbationof the closed-loop transfer function.

∗ Corresponding author. Tel.: +61-8-9380-3069; fax: +61-8-9380-1065.E-mail addresses: wanquan@cs.curtin.edu.au (G. Wang), sreeram@ee.uwa.edu.au (V. Sreeram).

0167-6911/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0167 -6911(02)00122 -6

100 G. Wang et al. / Systems & Control Letters 46 (2002) 99–110

P

K

wz

uy

Fig. 1. Closed-loop system.

In this paper we propose two new controller reduction methods: the closed-loop block balanced and theclosed-loop structurally balanced truncating methods. Actually, the new methods are two practical applicationsof the theory presented in [14,15]. Reduced-order controllers are obtained directly by reducing the closed-loopsystem with performance preserving weights. In the next Section, we formulate the problem and then presentthe main results from [14,15]. In Section 3, we present the analysis of the technique presented in [14,15]. InSections 4 and 5, we present the block balanced and the structurally balanced truncating algorithms. Finally,a numerical example is given in Section 6 to illustrate the new method and to compare it with the techniquesof [8].

2. Preliminaries

Consider the closed-loop system shown in Fig. 1, with external input w, controlled output z, control inputu, and measured output y.In the above 5gure,

P =

[P11 P12

P21 P22

]:

Using LFTs, transfer function from w to z is given by

G =Fl(P; K) = P11 + P12K(I − P22K)−1P21: (1)

In practice, it is desirable to design a controller K that both stabilizes P, and makes ‖Fl(P; K)‖∞ less thana given criterion � (�¿ 0).

De�nition 2.1. A controller K is said to be (P; �)-admissible if K stabilizes P and ‖Fl(P; K)‖∞ ¡�.

In this paper we assume that the plant P has been scaled with

‖Fl(P; K)‖∞ ¡�= 1

or K is (P; 1)-admissible.In controller order reduction, let Kr be the reduced-order controller. Then the closed-loop transfer function

with the controller Kr is given by

Gr =Fl(P; Kr) = P11 + P12Kr(I − P22Kr)−1P21: (2)

Normal controller reduction methods [2,19,3,16] are based on reducing the error ‖G − Gr‖∞. Obviously thisis not a good criterion if we are going to design a (P; 1)-admissible controller; a better criterion is making

G. Wang et al. / Systems & Control Letters 46 (2002) 99–110 101

‖Gr‖∞ ¡ 1 if ‖G‖∞ ¡ 1. In other words, we want Kr to be (P; 1)-admissible if K is (P; 1)-admissible. AsuKcient condition [14,15] for achieving this objective is stated in the following theorem.

Theorem 2.1. Assume ‖G‖∞ = �¡ 1; W1 and W2 are stable; invertible weighting functions with minimumphase and also satisfy[

I −W2W∼2 G

G∼ I −W∼1 W1

]¿ 0 (3)

and

‖W−12 (G − Gr)W−1

1 ‖∞ ¡ 1: (4)

Then

‖Gr‖∞ ¡ 1:

If the weights W1 and W2 chosen are large in some sense, then there is more room for Gr in ‖W−12 (G −

Gr)W−11 ‖∞ reduction. In other words, it makes the reduction from G to Gr satisfying (4) relatively easy. In

[14,15], two formulae for choosing the weights W1 and W2 were derived. They are:

W∼1 W1 = I −

√G∼G; (5)

W2W∼2 = I −

√GG∼ (6)

and

W∼1 W1 = I − �−1G∼G; (7)

W2 = (1− �)1=2I: (8)

When W∼1 W1 = I −�−1G∼G, we can obtain an exact solution to W1 by spectral factorization. When W∼

1 W1 =I − √

G∼G, we need to compute a rational approximation for√

G∼G 5rst which is not easy. While theexact solution is not possible, a solution which is arbitrarily close to

√G∼G can be obtained using rational

approximation algorithms [11].Note that once W1 and W2 are obtained, 5nding Gr from ‖W−1

2 (G−Gr)W−11 ‖∞ reduction is straightforward

using frequency weighted model reduction techniques [1,9,5,13,12]. However, merely 5nding Gr is not enough;our objective is to 5nd Kr corresponding to this Gr .Two di;erent methods can be employed here for 5nding the low-order controller Kr . First, solve ‖W−1

2 (G−Gr)W−1

1 ‖∞ reduction directly to obtain Kr . Methods to solve ‖W−12 (G − Gr)W−1

1 ‖∞ reduction directly arediscussed in Sections 4 and 5.Second, approximate ‖W−1

2 (G−Gr)W−11 ‖∞ reduction by ‖W−1

2 V2(K −Kr)V1W−11 ‖∞ reduction [15] where

V1 = (I − P22K)−1P21;V2 = P12(I + K(I − P22K)−1P22)

and then use frequency weighted model reduction methods to obtain Kr .

3. Analysis of the performance preserving techniques

In this section we compare two types of performance preserving controller reduction techniques: (i) con-troller reduction based on minimizing ‖W−1

2 (G−Gr)W−11 ‖∞ and (ii) controller reduction based on minimizing

‖G − Gr‖∞.

102 G. Wang et al. / Systems & Control Letters 46 (2002) 99–110

The suKcient condition for ‖Gr‖∞ ¡ 1 when using ‖W−12 (G − Gr)W−1

1 ‖∞ reduction is given inTheorem 1.Although ‖G − Gr‖∞ reduction is not classi5ed as performance preserving, it is straightforward to derive

the suKcient condition for ‖Gr‖∞ ¡ 1 when using ‖G − Gr‖∞ reduction as shown below.Since

‖Gr‖∞6 ‖G‖∞ + ‖G − Gr‖∞ = � + ‖G − Gr‖∞then

� + ‖G − Gr‖∞ ¡ 1

or

‖G − Gr‖∞ ¡ 1− �

is the suKcient condition for ‖Gr‖∞ ¡ 1.We rewrite the above inequality as

‖(1− �)1=2I(G − Gr)(1− �)1=2I‖∞ ¡ 1:

This is equivalent to W−12 =W−1

1 = (I − �I)−1=2 in ‖W−12 (G − Gr)W−1

1 ‖∞6 1.Let us now suppose there are three pairs of weightings:

The 5rst pair is WF2 =WF1 = (1− �)1=2I .The second pair is WS2 = (I − �I)1=2 and W∼

S1WS1 = I − �−1G∼G.The third pair is WT2W∼

T2 = I −√GG∼ and W∼

T1WT1 = I −√G∼G.

It is not diKcult to prove that

WF2W∼F2 6WS2W∼

S2 ;

W∼F1WF16W∼

S1WS1

and

WF2W∼F26WT2W∼

T2;

W∼F1WF16W∼

T1WT1:

This means if there is a Gr satisfying

‖W−1F2 (G − Gr)W−1

F1 ‖∞6 1

then it must satisfy

‖W−1S2 (G − Gr)W−1

S1 ‖∞6 1;

‖W−1T2 (G − Gr)W−1

T1 ‖∞6 1

but if there is a Gr satisfying

‖W−1S2 (G − Gr)W−1

S1 ‖∞6 1

or

‖W−1T2 (G − Gr)W−1

T1 ‖∞6 1

it is not necessary that

‖W−1F2 (G − Gr)W−1

F1 ‖∞6 1:

G. Wang et al. / Systems & Control Letters 46 (2002) 99–110 103

Therefore, theoretically, there is more room for Gr in ‖W−1S2 (G − Gr)W−1

S1 ‖∞ or ‖W−1T2 (G − Gr)W−1

T1 ‖∞reduction than in ‖W−1

F2 (G − Gr)W−1F1 ‖∞ reduction. Hence, we have a better chance of 5nding low order

controllers using ‖W−12 (G − Gr)W−1

1 ‖∞ reduction than using ‖G − Gr‖∞ reduction.Since there is no optimization method for minimizing ‖W−1

2 (G − Gr)W−11 ‖∞, we present two satisfac-

tory approaches which can be used for 5nding the reduced order controller Kr based on making the norm,‖W−1

2 (G − Gr)W−11 ‖∞, small.

4. Block balanced truncating (BBT) algorithm

In this section, we present a method for deriving the reduced order controller Kr which is (P; 1) admissibledirectly from ‖W−1

2 (G − Gr)W−11 ‖∞ reduction.

Let

P =

[P11 P12

P21 P22

]=

A B1 B2

C1 D11 D12

C1 D21 D22

; (9)

K =

Ak Bk

Ck Dk

; (10)

where the order of K is n. Then

G =Fl(P; K); (11)

=

A+ B2LDkC2 B2LCk B1 + B2LDkD21

BkFC2 Ak + BkFD22Ck BkFD21

C1 + D12DkFC2 D12LCk D11 + D12DkFD21

; (12)

where L= (I − DkD22)−1 and F = (I − D22Dk)−1.In the following discussion, suppose D11 + D12DkFD21 = 0 without loss of generality.

Let W−11 =

[Ai Bi

Ci Di

]and W−1

2 =

[Ao Bo

Co Do

]. Then

W−12 GW−1

1 =

[ MA MB

MC MD

];

where

MA=

Ai 0 0 0

0 Ao Bo(C1 + D12DkFC2) BoD12LCk

(B1 + B2LDkD21)Ci 0 A+ B2LDkC2 B2LCk

BkFD21DiCi 0 BkFC2 Ak + BkFD22Ck

;

104 G. Wang et al. / Systems & Control Letters 46 (2002) 99–110

MB=

Bi

0

(B1 + B2LDkD21)Di

BkFD21Di

;

MC =[0 Co Do(C1 + D12DkFC2) DoD12LCk

];

MD= 0:

For convenience, let

MA11 =

Ai 0 0

0 Ao Bo(C1 + D12DkFC2)

(B1 + B2LDkD21)Ci 0 A+ B2LDkC2

;

MA12 =

0

BoD12L

B2L

;

MA21 =[FD21DiCi 0 FC2

];

MB1 =

Bi

0

(B1 + B2LDkD21)Di

;

MC1 =[0 Co Do(C1 + D12DkFC2)

]:

Then

MA=

[MA11 MA12Ck

Bk MA21 Ak + BkFD22Ck

]; (13)

MB =

[MB1

BkFD21Di

]; (14)

MC =[MC1 DoD12LCk

]: (15)

Let MP and MQ be the controllability and observability Gramians of W−12 GW−1

1 or ( MA; MB; MC; MD) and be partitionedcompatibly with MA in (13) as

MP =

[ MP1 MP12

MP′12

MP2

];

MQ =

[ MQ1MQ12

MQ′12

MQ2

]:

Then there exists a nonsingular matrix T such that

T MP2T ′ = (T−1)′ MQ2T−1 = diag(�1; : : : ; �n) = �;

where �i¿ �i+1¿ 0.

G. Wang et al. / Systems & Control Letters 46 (2002) 99–110 105

Transform and partition K as

K =

[TAkT−1 TBk

CkT−1 Dk

]=

Ar Ak12 Br

Ak21 Ak22 Bk2

Cr Ck2 Dk

and Kr is obtained as

Kr =

[Ar Br

Cr Dk

]:

Note that by substituting

K =

[Ak Bk

Ck Dk

]=

Ar Ak12 Br

Ak21 Ak22 Bk2

Cr Ck2 Dk

for Ak , Bk , Ck in (13)–(15), we get

MA=

MA11 MA12Cr MA12Ck2

Br MA21 Ar + BrFD22Cr Ak12 + BrFD22Ck2

MA21 Ak21 + Bk2FD22Cr Ak22 + Bk2FD22Ck2

;

MB=

MB1

BrFD21Di

Bk2FD21Di

;

MC =[MC1 DoD12LCr DoD12LCk2

];

where

MA11 MA12Cr MB1

Br MA21 Ar + BrFD22Cr BrFD21Di

MC1 DoD12LCr MD

=W−1

2 GrW−11 :

Therefore, in this algorithm, truncating K to Kr is equivalent to truncating W−12 GW−1

1 to W−12 GrW−1

1 .There are two advantages of this algorithm. Firstly, the computational procedure is very simple and the

result always exists. Secondly, K need not be separated into stable part K1 and an anti-stable part K2 as in thefrequency weighted model reduction procedure. Therefore, the information contained in both the anti-stablepart K2 and stable part K1 is used in the reduction procedure. The main drawback of this algorithm is thatthe stability of Gr and the performance of Gr cannot be predicted, but the e;ectiveness of the method can beexplained as follows:From the state-space realization of W−1

2 GW−11 , we know that transforming K with T is equivalent to

transforming W−12 GW−1

1 with[

I 00 T

]. This implies controllability and observability Gramians of W−1

2 GW−11

become[

MP1 MP12T′

T MP′12 �

]and

[MQ1

MQ12T−1

(T−1)′ MQ′12 �

], respectively, which means W−1

2 GW−11 is block balanced. We also

know W−12 GrW−1

1 is actually obtained from truncating W−12 GW−1

1 which is block balanced. From the experi-ence of frequency weighted balanced truncation we can conclude that this method will work well in reducingthe following error: ‖W−1

2 (G − Gr)W−11 ‖∞.

106 G. Wang et al. / Systems & Control Letters 46 (2002) 99–110

5. Structurally balanced truncating algorithm

In this section, we present another practical method for obtaining the low-order controller Kr which is (P; 1)admissible, based on the structurally balanced truncating algorithm [19].Suppose we already have ( MA; MB; MC; MD) from (13)–(15).Then the next step depends on the existence of MP and MQ such that

MP =

[MP1 0

0 MP2

]= diag( MP1; MP2)¿ 0; (16)

MQ =

[MQ1 0;

0 MQ2

]= diag( MQ1; MQ2)¿ 0 (17)

and

MA MP + MP MA′+ MB MB

′6 0; (18)

MA′ MQ + MQ MA+ MC

′ MC6 0: (19)

Here MP and MQ are not the same as in the last section and their existence is not obvious. There are no simplesuKcient conditions for the solvability of inequalities (18) and (19), but there is a simple necessary condition.Substituting Eqs. (13)–(17) into inequalities (18) and (19) and simplifying, we have

MA11 MP1 + MP1 MA′11 + MB1 MB

′16 0; MA

′11

MQ1 + MQ1MA11 + MC

′1MC16 0;

(Ak + BkFD22Ck) MP2 + MP2(Ak + BkFD22Ck)′ + (BkFD21Di)(BkFD21Di)′6 0;

(Ak + BkFD22Ck)′ MQ2 + MQ2(Ak + BkFD22Ck) + (DoD12LCk)′(DoD12LCk)6 0:

These equations imply that MA11 and Ak + BkFD22Ck must be stable. Hence, this algorithm cannot continue ifeither MA11 or Ak + BkFD22Ck is unstable.The next step of the algorithm involves simultaneous diagonalization of MP2 and MQ2

T MP2T ′ = (T−1)′ MQ2T−1 = diag(�1; : : : ; �n) = �2;

where �i¿ �i+1¿ 0.Transform and partition K as

K =

[TAkT−1 TBk

CkT−1 Dk

]=

Ar Ak12 Br

Ak21 Ak22 Bk2

Cr Ck2 Dk

:

Then Kr is obtained as

Kr =

[Ar Br

Cr Dk

]:

The above algorithm has an important advantage as stated in the following theorem.

G. Wang et al. / Systems & Control Letters 46 (2002) 99–110 107

Theorem 5.1. Let Gr be the closed-loop system with the reduced order controller Kr which is obtained fromthe above algorithm. Then Gr is stable and

‖W−12 (G − Gr)W−1

1 ‖∞6 2n∑

i=r+1

�i: (20)

To prove the theorem we need the following lemma which was proved in [10].

Lemma 5.1 (Zhou [19]). Let G be stable with a state space realization

G =

[A B

C D

]=

A1 A2 B1;

A3 A4 B2:

C1 C2 D

and suppose there exists a diagonal matrix � = diag(�1; �2)¿ 0 satisfying

A� + �A′ + BB′6 0;A′� + �A+ C′C6 0:

De3ne

G =

[A1 B1

C1 D

]:

Then G is stable and ‖G − G‖∞6 2tr(�2).

The symbols used in the above lemma are independent of the symbols used in the rest of the paper.

Proof of Theorem 4.1. Let T1 be such that

T1 MP1T ′1 = (T−1

1 )′ MQ1T−11 = diag(�1; : : : ; �r) = �1¿ 0

and de5ne MT = diag(T1; T ) and � = diag(�1; �2). Then transform W−12 GW−1

1 as[MT MA MT

−1 MT MB

MC MT−1 MD

],

[A B

C MD

]:

The new realization of W−12 GW−1

1 satis5es

A� + �A′+ BB

′6 0;

A′� + �A+ C

′C6 0:

We already know that truncating K to Kr equals truncating W−12 GW−1

1 to W−12 GrW−1

1 . Applying Lemma 3.1

to

[A B

C MD

]; we know that W−1

2 GrW−11 is stable and

‖W−12 (G − Gr)W−1

1 ‖∞6 2n∑

i=r+1

�i:

Since W1 and W2 are stable; Gr =W2W−12 GrW−1

1 W1 is also stable.

Remark 5.1. It is easy to see that MP and MQ are not unique if they exist. Considering the error bounds givenin Theorem 3.1; it is desirable to minimize

∑ni=r+1 !1=2i ( MP; MQ) when we choose MP and MQ; but such MP and MQ

108 G. Wang et al. / Systems & Control Letters 46 (2002) 99–110

are diKcult to 5nd since the involved optimization is not convex. A practical way to choose MP and MQ is tominimize the tr( MP2) subject to the constraint MA MP + MP MA

′+ MB MB

′6 0 and to minimize the tr( MQ2) subject to the

constraint MA′ MQ + MQ MA+ MC

′ MC6 0. These optimizations can be done using the LMI-LAB.

6. Examples

In this section, we consider a four-disk control system studied by Enns and was quoted in [18] as anexample to show the e;ectiveness of Goddard and Glover’s performance preserving controller reduction. Herewe use this example to illustrate the BBT method presented in the paper and we also compare the resultswith the results of Goddard and Glover’s performance preserving controller reduction [18]. In the BBT-methodcalculations we have used second pair of weights de5ned in Section 3.Consider the system given by (9) where

A=

−0:161 −6:004 −0:58215 −9:9835 −0:40727 −3:982 0 0

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

;

B1 =

1 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

; B2 =

1

0

0

0

0

0

0

0

;

C1 =

[0 0 0 0 0:00055 0:011 0:00132 0:018

0 0 0 0 0 0 0 0

];

C2 =[0 0 0:0064432 0:0023196 0:071252 1:0002 0:10455 0:99551

];

D11 =

[0 0

0 0

]D12 =

[0

1

]D21 =

[0 1

]D22 = 0:

In [18], an eighth order suboptimal controller K is designed such that the H∞ norm of the closed-loop systemFl(P; K) is ¡ 1:2. The controller order is reduced using several methods and the results are listed in Table1. Since we only compare performance preserving balanced controller reduction methods, in Table 1 we onlyquote the results from balanced reduction. The following abbreviations are used in Table 1.

G. Wang et al. / Systems & Control Letters 46 (2002) 99–110 109

Table 1H∞ norms of the closed-loop system (Fl(P; Kr))

Order of Kr 7 6 5 4 3 2 1

UW U 1.321 U U U U USW 1.327 1.199 2.27 1.47 23.5 U UPWA 1.196 1.196 1.199 1.197 U 4.99 UPWRCF 1.2 1.196 1.207 1.195 2.98 1.674 UPWLCF 1.197 1.196 U 1.197 U U UBBT 1.198 1.196 1.204 1.197 3.906 1.954 U

UW Unweighted reduction: ‖K−Kr‖∞. The order of K is reduced directly by balanced truncation withoutany weightings.

SW Stability weighted reduction: ‖Wa(K−Kr)‖∞ where Wa=(I−P22K)−1P22. The order of K is reducedby the frequency weighted balanced truncation method.

PWA Performance weighted additive reduction [18]:PWRCF Performance weighted right coprime factor reduction [18]:PWLCF Performance weighted left coprime factor reduction [18]: These are three variations of Goddard and

Glover’s performance preserving controller reduction method, see [18] for details.U Unstable closed-loop system.

When we applied our proposed block balanced truncating method to this example, we also obtained a eighthorder suboptimal controller K and the H∞ norm of the closed-loop system Fl(P; K) is 1.196. In Table 1,note that some results when using PWA, PWRCF and PWLCF methods are slightly better than the results ofthe proposed BBT method. However, these methods use optimization technique proposed in [17], where asour technique does not require optimization.

7. Conclusion

In this paper, two new techniques for computing performance preserving reduced-order controllers areproposed. The methods are the block balanced and the structurally balanced truncating algorithms. In terms ofcomputational complexity, the block balanced truncating algorithm is easier to implement than the structurallybalanced truncating algorithm. The disadvantage of the structurally balanced method is it requires solvingoptimization problems and the solution may not exist. However, the block balanced truncating algorithm issimple, easy to compute and the solution always exists. Furthermore, unstable controllers need not be separatedinto stable and unstable subsystems which means that the information in the unstable part is also used in thereduction process. Numerical example shows that the block balanced truncating algorithm compares well withother well-known controller reduction methods [18].

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