Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996

FP02 5 : l O

The a-Reciprocal System for Model Reduction of Ic-Power Bilinear Systems'

Baohui Xie and Vassilis L. Syrmos Department of Electrical Engineering

University of Hawaii at hlanoa Honolulu, HI 96822 USA

Abstract

In this paper the a-reciprocal system for the Ic-power bilinear system is defined. We introduce the a - reciprocal bilinear system and we show that the bi- linear generalized singular perturbation approximation (GSPA) is related to the direct truncation of its a - reciprocal bilinear system. Model reduction algorithms based on the analysis of the a-reciprocal bilinear sys- tem are given. Numerical example is reported for the proposed model reduction methods.

1 Introduction

Bilinear systems are an important subclass of nonlin- ear systems. Bilinear processes are common in various areas and frequently arise naturally in nuclear fission, heat transfer models [4, 71. A bilinear system is one which is linear in state, linear in control, but not linear jointly. This model has been found to be very useful in describing many nonlinear phenomena. A subclass of bilinear systems, which is called Ic-power (or degree-k [l]) homogeneous system, have the property that ap- plication of the input au( t ) , where a is a scalar, yields the output cr'y(t), where y(b) is the response to u( t ) . Homogeneous systems arise in various areas of engi- neering, e.g., modeling of hydraulic drives for machine tools, robots, etc. [ 3 ] .

Recently, more and more attention has been given to the model reduction of the bilinear systems [3, 81. The complexity of the bilinear state-space algorithms gen- erally increases drastically with the dimension of the state space. The solution of the state equations for a bilinear system is more complicated than that for a lin- ear system of equal order. Therefore, the motivation for performing model reduction is even stronger with bilinear control systems.

The generalized singular perturbational approximation

'This research was supported by the National Science Foun- dation under grant NCR-9210408 and by the Advanced Research Projects Agency under contract MDA-972-93-1-0032 and MDA- 972-95-3-0016.

(GSPA) method is one of the most popular methods for obtaining reduced-order representations of linear time- invariant systems. The main advantages of this tech- nique is due to the application of the balanced GSPA (BGSPA) method, which has the same H,-norm error- bound as the balanced truncation in LTI case. Never- theless, the frequency characteristic of the error of the BGSPA can vary according to the requirements of the designer [a ] . The BGSPA model reduction method is based on searching through all possible a for different GSPA over a balanced realization so that the H,-norm of error is minimum. Therefore, by using the BGSPA one can achieve less or equal H,-norm of the error than by using the balanced truncation or the balanced SPA (BSPA) [lo]. In this p p e r , we extend the BGSPA to the bilinear case.

The a-reciprocal system [2], introduced to emphasize the properties of the GSPA, is used to show that the GSPA can be converted to a direct truncation of the a-reciprocal system. Consequently, many properties of the direct truncation can be extended to the GSPA of the systems. In the LTI case, the truncated system by BGSPA retains the stability and the same error-bound as those systems obtained by the balanced truncation.

In this paper, we define the a-reciprocal system for the t-power bilinear system and we show that the bilinear GSPA can be converted to a direct truncation of its a-reciprocal system. Stability analysis is given for the bilinear GSPA of t-power bilinear systems. Simulation result shows that the H,-norm of the error of the bilin- ear BGSPA is less or equal to the one in the balanced truncation or BSPA. An algorithm is also given for the bilinear approximation with matched DC-gain.

The paper is organized as follows: Section 2 and 3 sum- marize the background materials needed; t.he GSPA of the bilinear systems is presented in Section 3 ; In Sec- tion 4, the a-reciprocal system is given and discussed. The algorithms for the best frequency domain perfor- mance, i.e, least H,-norm of the error, and matched DC-gain model reduction are also given in this section. In Section 5 , simulation results are given for such algo- rithms. Section 6 concludes this pa.per.

0-7803-3590-2/96 $5.00 0 1996 IEEE 4294

2 Preliminaries and Background

A bilinear system can be represented by the following state variable equations:

m

;(t) = ~ z ( t ) + N~z(~)u~(~) + ~ u ( t ) , ( a . l a ) i = l

Y(t) = C 4 ) , (2.lb)

where z E R", U E IRm, ui is the i-th component of ~ ( t ) , A, Nj E IRnxn, B E RnX" and C E RPxn.

We denote the infinite-horizon norm of G to be

It is shown in [l] that the bilinear state equation (2.1) can be described by a Volterra system representation in which the k-power subsystem can be written as

hk(U1, . f . , U k ) U ( U 1 ) U ( U k ) d U k * * * d u i ,

where the kernels are given by hk(ul,...,uk) = C e A ~ I N , ~ A ( ~ ~ - C ~ ) . . . N,eA"Jk-''k-l)B.

(2.3) Clearly, the regular kernels for a bilinear system equa- tion are of a particularly simple form. And taking the Laplace transform of (2.3) shows that the regular trans- fer function also has a simple form, namely

Hk(S1, * . ., sk) = C(sk1- A)-'NI(sk-1I - A)-", . . . N,(s l I - A)-'B.

The regular transfer function for a bilinear state equa- tion is strictly proper rational function if the nu- merator polynomial degree is strictly less than the denominator polynomial degree for each s j , j = 1, . . . k. Furthermore, the denominator polynomial of Hk(s1 , . . . , Sk) can be expressed as a product of single- variable polynomials, so that Hk(s1, . . . , sk) can be written in the form

A rational function of this form is called a recognizable function. The stability of a bilinear sys- tem is defined as follows [l]:

Definition 2.1 Suppose that a k-power homogeneous system is described b y a strictly proper, recognizable regular transfer function wherein all the roots of the de- nominator polynomials have negative real parts. Then i f ( A , N i , B, C ) is a minimal bilinear realization of the system, all the eigenvalues of A will have negative real parts and the system is bounded-input, bounded-output stable.

3 k-Power System and Its Generalized Singular Perturbational Approximation

Let the bilinear system be a k-power and minimal, then it can be transformed into a system with a special struc- ture as in the following representation [3].

r o 1

where N i j can be represented as the product of Eij and Fij-1. Suce canonical structure simplifies the bilinear problem significantly.

Theorem 3.1 Let (2.la) and (2.lb) be a minimal bi- linear realization of a k-power. Then there exists a min- imal bilinear realization of the canonical structure.

The regular transfer function for the canonical struc- tured k-power bilinear system is

H t ( 3 1 , . . * , s , ) = 0 , i # k Hk(sl,*.*,Sk) = C k ( S k 1 -

( ~ k - 1 1 - Ak-1)-1N,(k--2) ... N , , ( s ~ I - Ai ) - 'B i .

Without loss of generality, we consider the canonical structure of the k-power bilinear systems.

Definition 3.1 For a k-power BIB0 system, every A,, j = 1,2,. . , k, is asymptotically stable.

The generalized algebraic Lyapunov equation for the canonical structured bilinear system is give in [3]. The controllability and observability grammians P and Q of the k-power system satisfy the following Lyapunov equations:

A1P1 + PIAT + BIB: = 0 , AjPj + P,AT + AzQk + Q k A k + CTCk = 0, ATQJ + Q j A, +

Nt,-xP,-lN<-l = 0 ,

N<Q,+I N:, = 0 ,

for j = 2 , 3 , . . . k,

for j = k - 1, k - 2, . . .1,

where P = diag[P1, P2,. . . , Pk], Q = diag[Qi, Q 2 , - - , Q k ] .

These equations are standard Lyapunov equations. Balance and truncation of such bilinear system then becomes standard and the algorithm is given in 131.

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Next, we introduce some notations used throughout the paper. For convenience, partition the canonical structured bilinear system as follows,

where Ni, can be represented as the product of Ea, and Fij-,. The dimension of E:,, E;,, F( and Fi", are conformly with the partitions of Aj , Ni, , Bj and Cj. Therefore, by reordering the elements in A, we have

1

L where, m, n = 1 , 2 . Similarly, B, B1, B2, C, C1, e', l"i, and NY, m, n = 1 , 2 , can be defined conformly.

Define the generalized singular perturbational approx- imation (GSPA) of the k-power, canonical structured bilinear system as follows-

A 3 ( 0 3 ) = A:' + Ai2(a,I - A , 22 ) -1 A, 21 , BI(U~) = B : + A i 2 ( b i J - A 1 22 ) -1 B1, 2

Ck(ak) = c: + c,"(d - A ? ) - ' A ~ , E,,(nJ) = E:, + A:~(C,I - A ; ~ ) - ~ E , ~ , Pt,(a,) = F:, + F , 2 , W - A , 22 1 -1A21 3 I

Nt,-l (a,, a,-1) = Et, ( U J ) F + , ( U p l ) ,

and A ( o l , m , . . . , uk) = diag{Al ( m ) , . . . , A k ( o k ) } ,

C(U1,U2,'..,ffk) = [ 0 ". 0 C k ( U k ) 3 , B ( 6 1 , 6 2 1 ' . ' t b k ) = [ Bl(o1) 0 ." 0 I T , N*(a1,T2,. . . , a)

1 0

= I

In general, Dj(uj) for j = l , . . . l e contains the nondominant part in the neighborhood of the fre- quency uj . Then the system S(A, Nil B , C) can be approximated by the reduced-order represen- tation ~ ( A ( u 1 , . . . , uk), nTi(u1, .. .,arc), B(u1, . . ., u k ) , C(u1, . . . , u k ) ) . We call this result the GSPA of bi- linear system at frequencies (u1 , . . . , u k ) .

Observe that when uj -+ -00, j = 1,. . ., k , (3.5) is the system obtained by the direct truncation.

4 The u-Reciprocal Systems for the k-Power Systems

Similar to the properties of the LTI SPA and the LTI balanced truncation [9], the reduced order model from the bilinear balanced truncation is a fast singular perturbational approximation of the balanced system, while the reduced order model from bilinear SPA is a slow singular perturbational approximation of the bal- anced system. The bilinear balanced truncation and the bilinear BSPA are two special case of the bilinear GSPA. To understand the scheme of the bilinear GSPA, we introduce the u-reciprocal bilinear system.

A frequency domain property of the u-reciprocal map- ping is given by the following proposition.

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Notice that fit-')(&, S2 . . . , &) are the function of Po, , j = I , . . . , k, which are the nondominant part of the regular transfer function to be truncated. Negli- gence of such portion will not cause eminant effect to the main characteristic of the regular transfer function.

Proposition 4.2 reveals the connection between the a- reciprocal and the generalized singular perturbational approximation. It is shown that the reduced model by using the GSPA is exactly the same system as the one obtained by performing a direct truncation of the a-reciprocal of the full-order system. With such a rela- tionship, the GSPA can be always viewed as the direct truncation of its a-reciprocal system. This property gives an efficient way to explain the characteristic of the GSPA method, such as stability, etc.

- .

Proposition 4.2 The generalized singular perturba- tional approzimation of the k-power bilinear system is the a-reciprocal of the bilinear system obtained b y per- forming a direct truncation of the a-reciprocal of the full-order system.

Proposition 4.2 is useful in all cases in which we want to extend the properties of the reduced-order systems obtained by the direct truncation to systems obtained by using the GSPA. We will see later in Theorem 4.1, that Proposition 4.2 plays the key roll in proving the stability of the GSPA of bilinear systems.

Theorem 4.1 Suppose that k-power bilinear system is minimal. Apply GSPA with A on such system. If A is selected such that S ( A - A, N i , B , C ) is stable and internally balanced, then the reduced-order system S(A(A) -A, Ni(A), B(A), C'(A)) is also stable and in- ternally balanced.

The bilinear GSPA model reduction technique based on the analysis of bilinear a-reciprocal system consists of the following steps:

1. Compute a balanced realization of the system and decide the order of the reduced model.

2. Varying the parameter (61, u2, . . . I uk), perform a search process which minimizes the H , norm of the error between the original system and the approximated one, obtained via generalized sin- gular perturbational approximation.

The key of the proposed approach lies in varying the parameter (a1 , 6 2 , . . ., ak) to search the minimum of the error between the original system and the approxi- mated one. Therefore, the resulting approximation has an H,-norm of the error less or a t least equal to the H,-norm of the error obtained when SPA or direct truncation are used. Furthermore, when requiring the reduced-order system has a matched DC-gain, i.e., the reduced-order system and the full-order one have same steady-state response, the algorithm becomes:

1. Compute a balanced realization of the system and decide the order of the reduced model.

2. Varying the parameter (a1, 172, + . ., a k ) , perform a search process by comparing the step response of the reduced-order system with that of the full- order system so that the DC-gain of the trun- cated system is exactly equal to the DC-gain of the original system.

5 Simulation Results

In this section, we provide one example to obtain 3-th order model approximations of a 11-th order 2-power bilinear system using the algorithms proposed in the

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Section 4. The matrices A, N , B , C of the full-order model are given as follows:

where A1 = diag ([-9.9104 - 3.6534 - 2.4704 - 9.8255 - 7.2266 - 7.53361); A2 = diag([-6.5152 - 0.7269 - 6.3163 - 8.8471 - 2.72711); NI1 =

0.1716 0.0795 0.2324 0.4288 0.4344 0.0289 ,

0.1568 0.0727 0.2123 0.3917 0.3968 0.0264 0.2754 0.1276 0.3729 0.6880 0.6969 0.0464

0.0854 0.0396 0.1157 0.2134 0.2162 0.0144 1 0.0988 0.0458 0.1337 0.2468 0.2499 0.0166 B1 = [ 0.9047 0.5045 0.5163 0.3190 0.9866 0.4940 1' , C2 = [ 0.2661 0.0907 0.9478 0.0737 0.5007 ] .

The Hankel singular values of original system are C = diag([38.29271 2.749266 0.050239 0.001791 0 0

50.48320 9.558843 0.528794 0.002671 01) x

1 At first, we apply a direct truncation to the balanced realization S(&, Nb, Bb, Cb). The last four states in the first power the last four states in the second power are to be removed. The balance and truncation of the system is

L 2.9714 r 0.6790 i

-2.7444 -5.0048

0 -3.3633

0.7260 0

Bi = 0.1659 , C,' = [ 0 0 0.5827 1 . L 0 1

Applying the SPA (u1 = 0, 6 2 = 0) to the balanced realization, the resulting SPA realization is [ -5.9861 -2.6353

A'' = -2.6353 -4.6398 0 -1.4981

0

1.9778 0.4665 0

, C1 = [ 0 0 0.3889 3 . When 'TI = -8.1 and u2 = -7.2, we get the least H,- error GSPA system, which means we get the approx- imation system with the best frequency performance. Such reduced-order system is,

A I 1 = -1.4467 -0.6561 :] , [ -5.632: -1.4467

0 -5.3897

0 : " , 3.8871 0.5479 0

fill =

0.1

* - GSPA wilh lenst hequency domain (~ru

SPA

+ - GSPA rid? matdKtd DC-gain

0 5 10 15 20 25 30 time(=)

Figure 1. The step responses of the full-order and the reduced-order systems

8' = 0.0926 , e' == [ 0 0 0.7876 3 . [ 0.657: 1 When 6 1 = -8.0 and u2 = -7.0, the reduced-order system and the full-order system have same DC-gain. Therefore, the GSPA system with the matched DC- gain is,

[ -5.697: -1.6636 2" = -1.6636 -1.3829

0 -5.5714 0

3.9987 0.6347 0

, 2.' = [ 0 0 0.8058 1 .

The full- and reduced-order models are simulated with step responses. The step responses of the third-order models from the two algorithms given in Section 4 are compared with those of the balance and truncation sys- tem and BSPA system. The step responses for these systems are shown in Figure 1. The frequency analysis of the errors between the full-order and the reduced- order systems are given in Figure 2-5.

Note that balance and truncation gives a reasonable approximation. Response to the third-order model is close to that of the original model. However, with prop- erly selected (u1, az), an improved approximation is obtained as can be seen from the error spectrum. Fur- thermore, when requiring the matched DC-gain for the reduced system, we can always find (al, q) to achieve such property. From Figure 5, we can see that in order to match the DC-gain the H,-error of such GSPA has to be sacrificed to achieve such objective. Another im- portant common property among these reduced-order models is that they are all stable as the full-order model.

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6 Conclusion

( 0

0

30

Figure 2. The frequency response of error system of the balance & truncated system

10

0

30

Figure 3. The frequency response of error system of the GSPA system

Figure 4. The frequency response of error system of the SPA system

0

30

F-(r.MaC) F .W(*Y* 40-40 -L“

Figure 5. The frequency response of error system of the matched DC-gain system

In this paper, a definition of the a-reciprocal bilinear system has been introduced. Some preliminary results concerning the properties of this system, with special emphasis on the connections with GSPA, have been presented. On the basis of these results, the bilin- ear GSPA model-order reduction algorithms have been derived. The GBSPA model-order reduction method can achieve better or at least equal H,-error perfor- mance than balance and truncation or BSPA. Such properties of BGSPA can be easily seen when inves- tigating the a-reciprocal systems of the bilinear s y s tems. Numerically reliable algorithms are incorporated in the BGSPA which make it possible to compute the reduced-order bilinear models.

References [l] W.J.Rugh, “Nonlinear System Theory”, The Johns Hopkins Universaty Press, 1985. [2] G . Muscat0 and G . Nunnari, “On the u- Reciprocal System for Model Order Reduction”, Math- ematical Modeling of Systems, Vol. 1, No. 4, pp. 261- 171, 1995. [3] S.A. Al-Baiyat, M. Bettayeb, “A New Model Re- duction Scheme for IC-Power Bilinear Systems”, Proc. of the 32nd Conference on Decision and Control, San Antonio, Taxas, Dec., 1993. [4] R. Mohler, “Bilinear Control Process”, Mathe- matics in Science and Engineering, Vo1.106, Academic Press, 1973. [5] R. Mohler, W.J. Kolodxiej, “An Overview of Bi- linear System Theory and Applications”, IEEE Trans- actions on systems science and cybernetics, Vol. SMC-

[6] R. Mohler, “Nonlinear Systems, Volume 11: Ap- plications to Bilinear Control”, prentice Hall, Engle- wood Cliffs, New Jersey, 1991. [7] R. W. Brockett, “Applications of lie group theory to nonlinear network problems”, Supplement to IEEE International Sym. on Circuit Theory, April 1974, San Francisco. C . A . Desoer, et al. ppl-11, North Holly- wood, CA : Western Periodicals Co., 1974. [8] C.S. Hsu, U.B.Desai, and C.A.Crawley, “Real- ization and Approximation of Discrete Bilinear Sys- tems”, Applied Digital Control/S. G. Tzafestas(ed.) El- sevier Science Publishers B.V.(North-Holland), pp171- 187, 1985. [9] M. Green and D.J.N. Limebeer, “Linear Robust Control”, Prentice-Hall Inc., 1995. [lo] Y. Liu and B.D.O. Anderson, “Singular Pertur- bation Approximation of Balanced Systems”, Int. J. Control, vol. 50, no. 4, pp1379-1405, 1989.

10, ~ ~ 6 8 3 - 6 8 8 , Oct., 1980.

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