Gramian matrices and balanced model of ... - Semantic Scholar
Applied Mathematics and Computation 148 (2004) 341–350 www.elsevier.com/locate/amc
Gramian matrices and balanced model of generalized systems q Rafael Bru, Carmen Coll *, Nestor Thome Departamento de Matem atica Aplicada, Universidad Polit ecnica de Valencia, 46071 Valencia, Spain
Abstract In this work, the generalization of reachability and observability Gramian matrices of control generalized systems have been studied. The properties of these matrices to be solutions of Lyapunov equations have been analyzed. In addition, an algorithm to obtain balanced generalized systems is given. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Linear discrete-time generalized systems; Reachability; Observability; Lyapunov equation; Gramian matrix; Balanced systems
1. Introduction The reachability and observability properties have special attention in control theory. Many classical problems of linear control systems have been studied using structural properties, for instance the balanced realization problem was introduced by Moore in [1]. Ober and McFarlane used Gramian matrices of standard systems by means of the corresponding reachability and observability matrices of the system in [2]. Gramian matrices have gained attention in the control theory because these matrices contain interesting information on the input–output behaviour of the system, but really there are no many papers dealing with this topic in the last decade. It is well known that
q
Supported by DGI grant BFM2001-2783. Corresponding author. E-mail addresses: [email protected] (R. Bru), [email protected] (C. Coll), njthome@mat. upv.es (N. Thome). *
0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00848-2
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R. Bru et al. / Appl. Math. Comput. 148 (2004) 341–350
Gramian matrices can be used for checking if a system is reachable and/or observable. In addition, these matrices play an important role in the study of the stability of a system using the respective Lyapunov equations. Recently, different methods studying model reduction of standard system using Gramian matrices have appeared [3,4]. In [5], Bender gave a definition of reachability and observability Gramian 1 matrices using the Laurent expansion of the matrix ðzE AÞ . A sufficient condition on the orthogonality of some blocks of matrices B and C allows to state that Gramian matrices are the solution of the Lyapunov equations EWr ET AWr AT ¼ BBT and ET Wo E AT Wo A ¼ C T C. In [6], Gramian matrices of a generalized system ðE; A; B; CÞ have been introduced as the sum of the causal part Wrc ¼
1 X
/k BBT /Tk ;
Woc ¼
k¼0
1 X
/Tk C T C/k
k¼0
and the noncasual part Wrnc ¼
1 X
/k BBT /Tk ;
Wonc ¼
k¼q
1 X
/Tk C T C/k ;
k¼q
where q is the nilpotence index and /k are the Laurent parameters of the serie expansion of the matrix ðzE AÞ1 . An extension of Gramian matrices to generalized linear discrete-time systems is discussed in this work. We use the reachability and observability Gramian matrices of generalized systems defined as a natural extension of the standard case [7]. We show that these matrices are solutions of the corresponding Lyapunov equations. Further, we use an algorithm to obtain balanced models of generalized systems. First we provide some useful background which is used in the paper. Let us consider a generalized system given by Exðk þ 1Þ ¼ AxðkÞ þ BuðkÞ; yðkÞ ¼ CxðkÞ;
ð1Þ
where A; E 2 Rnn , B 2 Rnm , C 2 Rpn , k 2 Z. If rankðEÞ < n the system (1) is called singular system and if rankðEÞ ¼ n, then we can consider the system e xðkÞ þ B e uðkÞ; xðk þ 1Þ ¼ A yðkÞ ¼ CxðkÞ;
ð2Þ
e ¼ E1 A and B e ¼ E1 B. A system with the structure like that in the where A system (2) is called a standard system. We denote by ðE; A; B; CÞ the singular system and by ðA; B; CÞ the standard system. It is well known (see [8]), that if there exists an escalar k 2 C such that detðkE AÞ 6¼ 0, (i.e. the system (1)
R. Bru et al. / Appl. Math. Comput. 148 (2004) 341–350
343
satisfies the regularity condition), then the system (1) is equivalent to the forward–backward canonical form given by Exðk þ 1Þ ¼ AxðkÞ þ BuðkÞ;
ð3Þ
yðkÞ ¼ CxðkÞ; with n1 þ n2 ¼ n, and E ¼ diagðIn1 ; N Þ;
A ¼ diagðA1 ; In2 Þ;
B1 B¼ ; B2
C ¼ ½ C1
C2
and N is nilpotent. We can split the system (3) into two subsystems, the forward subsystem given by x1 ðk þ 1Þ ¼ A1 x1 ðkÞ þ B1 uðkÞ; y1 ðkÞ ¼ C1 x1 ðkÞ
ð4Þ
and the backward subsystem given by Nx2 ðk þ 1Þ ¼ x2 ðkÞ þ B2 uðkÞ; y2 ðkÞ ¼ C2 x2 ðkÞ:
ð5Þ
In the following we suppose that singular systems satisfy the regularity condition. Note that, if rankðEÞ ¼ n, then detðkI E1 AÞ 6¼ 0 and so detðkE AÞ 6¼ 0, for some k 2 C. That is, a standard system can be interpreted as a generalized system with only the forward part. Then, we focus our attention on singular systems. Definition 1. Consider the singular system ðE; A; B; CÞ. i(i) A state x 2 Rn is reachable if there exists a time k 2 Z and a control sequence uðjÞ, j ¼ 0; 1; . . . ; k 1 transferring the state of the system from the origin at time 0 to x at time k. The system ðE; A; B; CÞ is reachable if every x 2 Rn is reachable. (ii) The system ðE; A; B; CÞ is observable if the initial condition xð0Þ may be uniquely determined by control and output sequences, uðjÞ and yðjÞ, j ¼ 0; 1; . . . ; k 1, k 2 Z. The reachability matrices of subsystems (4) and (5) are given by Xr ðA1 ; B1 Þ ¼ ½B1 A1 B1 A21 B1 and by Xr ðN ; B2 Þ ¼ ½ N 2 B2 NB2 B2 , respectively. The observability matrices of subsystem (4) and (5) are given by 2 3 2 3 .. C1 6 C1 A1 7 6 . 7 27 6 7 6 2 Xo ðA1 ; C1 Þ ¼ 6 C1 A 7 and Xo ðN ; C2 Þ ¼ 6 C2 N 7; respectively: 15 4 4 C2 N 5 .. . C2
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It is known that the system (3) is reachable if and only if rank½Xr ðA1 ; B1 Þ ¼ n1 and rank½Xr ðN ; B2 Þ ¼ n2 and the system (3) is observable if and only if rank½Xo ðA1 ; C1 Þ ¼ n1 and rank½Xo ðN ; C2 Þ ¼ n2 . A reachable and observable system is minimal. In the following the spectral radius of a square matrix M will be denoted by qð M Þ. The set Sm;p denote the systems ðE; A; B; CÞ satisfying the following n properties: (i) E; A 2 Rnn , B 2 Rnm and C 2 Rpn , (ii) ðE; A; B; CÞ is a minimal system and (iii) ðE; A; B; CÞ is asymptotically stable. Now we define the Gramian matrices associated to generalized systems. m;p Definition 2. Let ðE; A; B; CÞ 2 Sn and consider its associated forward– backward system E; A; B; C . The reachability Gramian matrix of the system ðE; A; B; CÞ is given by
Wr ðE; A; BÞ ¼ diagðWr ðA1 ; B1 Þ; Wr ðN ; B2 ÞÞ; where Wr ðA1 ; B1 Þ ¼ Xr ðA1 ; B1 ÞXTr ðA1 ; B1 Þ and Wr ðN ; B2 Þ ¼ Xr ðN ; B2 ÞXTr ðN ; B2 Þ. The observability Gramian matrix of the system ðE; A; B; CÞ is given by Wo ðE; A; CÞ ¼ diagðWo ðA1 ; C1 Þ; Wo ðN ; C2 ÞÞ; where Wo ðA1 ; C1 Þ ¼ XTo ðA1 ; C1 ÞXo ðA1 ; C1 Þ and Wo ðN ; C2 Þ ¼ XTo ðN ; C2 ÞXo ðN ; C2 Þ. Note that, the uncoupled structure of Gramian matrices in the above definition permits to construct parallel algorithms for computing them, taking advantage respect to the other definitions. 2. Reachability and observability Gramian matrices In this section we study the reachability Gramian matrix for singular systems. We start with the following lemma. Lemma 1. Let ðE; A; B; CÞ be a singular system and consider its associated for ward–backward system E; A; B; C . If qðA1 Þ < 1 then (a) Wr ðA1 ; B1 Þ is the only nonnegative definite solution of the Lyapunov equation P A1 PAT1 ¼ B1 BT1 : (b) )Wr ðN ; B2 Þ is the only nonnegative definite solution of the Lyapunov equation Q NQN T ¼ B2 BT2 : Proof. Using the stability characterization of standard systems, it is easy to prove part (a). Now, we show condition (b). Since N is a nilpotent matrix, then
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345
qðN Þ < 1. So the solution of the Lyapunov equation Q NQN T ¼ B2 BT2 is Xr ðN ; B2 ÞXTr ðN ; B2 Þ. Hence Wr ðN ; B2 Þ is the required nonnegative definite solution. Using the above lemma, the matrix Wr ðN ; B2 Þ satisfies NWr ðN ; B2 ÞN T Wr ðN ; B2 Þ ¼ B2 BT2 : To obtain the solution of the Lyapunov equation T
T
T
EP E AP A ¼ BB ; where E, A and B are given in (4) and (5) and P ¼ ½Pij i;j¼1;2 , we must solve the following equations T T P11 A1 P11 A1 ¼ B1 BT1 ; T
P12 N A1 P12 ¼
ð6Þ
B1 BT2 ;
ð7Þ
NP21 P21 AT1 ¼ B2 BT1 ;
ð8Þ
NP22 N T P22 ¼ B2 BT2 :
ð9Þ
When qðA1 Þ < 1, the Eq. (6) has only one solution but Eqs. (7) and (8) can even not have solution. A sufficient condition to guarantee the solution of Eq. (7) is that B1 BT2 ¼ O. We use this condition in the following result. Theorem 2. Let ðE; A; B; CÞ be a singular system and consider its associated forward–backward system E; A; B; C . If qðA1 Þ < 1 and B1 BT2 ¼ O, then the reachability Gramian matrix Wr ðE; A; BÞ satisfies the Lyapunov equation T
T
T
EP E AP A ¼ BB : Proof. We must prove that the matrix Wr ðE; A; BÞ satisfies the Lyapunov equation. In fact, T
T
EWr ðE; A; BÞE AWr ðE; A; BÞA A1 O In1 O Wr ðA1 ; B1 Þ O In1 O ¼ O In2 O N O Wr ðN ; B2 Þ O NT T O Wr ðA1 ; B1 Þ A1 O O Wr ðN ; B2 Þ O In2 Wr ðA1 ; B1 Þ A1 Wr ðA1 ; B1 ÞAT1 O ¼ : O NWr ðN ; B2 ÞN T Wr ðN ; B2 Þ Using part (a) of the above lemma, Wr ðA1 ; B1 Þ A1 Wr ðA1 ; B1 ÞAT1 ¼ B1 BT1 and also NWr ðN ; B2 ÞN T Wr ðN ; B2 Þ ¼ B2 BT2 . Further,
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B1 T BB ¼ B1 B2 T
BT2
B1 BT1 ¼ B2 BT1
B1 BT2 B2 BT2
and since B1 BT2 ¼ O we obtain the result.
Now, we give the corresponding results of the observability Gramian matrix for singular systems, for which proofs are similar to the reachability case. Lemma 3.Let ðE; A; B; CÞ and consider its associated forward–backward system E; A; B; C . If qðA1 Þ < 1 then: (a) Wo ðA1 ; C1 Þ is the only nonnegative solution of the Lyapunov equation P AT1 PA1 ¼ C1T C1 ; (b) Wo ðN ; C2 Þ is the only nonnegative solution of the Lyapunov equation Q N T QN ¼ C2T C2 : Theorem 4. Let ðE; A; B; CÞ be a singular system and consider its associated forward–backward system E; A; B; C . If qðA1 Þ < 1 and C1T C2 ¼ O, then observability Gramian Wo ðE; A; CÞ satisfies the Lyapunov equation T
T
T
E QE A QA ¼ C C:
3. Balanced model The need of balancing the gains between inputs and states just like that between states and outputs permits to develop algorithms for obtaining balanced standard system, that is, linear standard systems whose reachability and observability Gramian matrices are diagonal and coincide. This kind of system is used to obtain models of smaller size (and then cheaper) than the original system by different techniques. For instance, in [9] singular perturbation approximation and direct truncation are used to obtain reduced model of balanced standard system. Different algorithms have been developed to obtain balanced standard realizations, for instance using Markov parameters in [10] and using singular value decomposition of the Hankel matrix in [11]. The purpose of this section is to construct a balanced singular system associated to a generalized system. First, we give the following definition. Definition 3. Let ðE; A; B; CÞ 2 Sm;p n . The system ðE; A; B; CÞ is called balanced if
R. Bru et al. / Appl. Math. Comput. 148 (2004) 341–350
347
Wr ðE; A; CÞ ¼ Wo ðE; A; CÞ ¼ diagðr1 ; r2 ; . . . ; rn Þ; where r1 ; r2 ; . . . ; rn are called the Hankel singular values of the system. A first step to obtain a balanced system associated to the system ðE; A; B; CÞ 2 Sm;p is to compute the reachability and observability Gramian n matrices of this system. Thus, consider a minimal and stable system ðE; A; B; CÞ given by (4) and (5), with B1 BT2 ¼ O and C1T C2 ¼ O: An algorithm to compute the reachability (observability) Gramian matrix is as follows: To take X0 ¼ O and Y0 ¼ O. The matrix Wr ðA1 ; B1 Þ is obtained using the iterative squeme Xiþ1 ¼ A1 Xi AT1 þ B1 BT1 , and the matrix Wr ðN ; B2 Þ is obtained using the iterative squeme Yiþ1 ¼ NYi N T þ B2 BT2 ; and making iterations until convergence. Assign Wr ðA1 ; B1 Þ ¼ Xi , Wr ðN ; B2 Þ ¼ Yi and construct the matrix O Wr ðA1 ; B1 Þ Wr ðE; A; BÞ ¼ : O Wr ðN ; B2 Þ By a similar way the corresponding observability Gramian matrices can be obtained, only changing the matrices involving in the iterative squemes. Remember that O Wo ðA1 ; C1 Þ Wo ðE; A; CÞ ¼ : O Wo ðN ; C2 Þ Now, we give the algorithm to obtain the balanced realization. Step 1: To obtain the singular value decomposition of Wr ðE; BÞ using the block structure of this matrix. Then there exists an orthogonal matrix U ¼ diagðU1 ; U2 Þ such that Wr ðE; BÞ ¼ diagðU1 ; U2 ÞR2r diagðU1T ; U2T Þ. T Step 2: To construct matrix M ¼ ðU Rr Þ Wo ðE; A; CÞU Rr . Then M ¼ diagðM1 ; M2 Þ: Step 3: To obtain the singular value decomposition of M, that is M ¼ diagðV1 ; V2 ÞR2o diagðV1T ; V2T Þ: Step 4: To construct matrix T ¼ diagðT1 ; T2 Þ 1
T T T ¼ R2o diagðV1T ; V2T ÞR1 r diagðU1 ; U2 Þ:
Step 5: To construct matrices Eb ¼ T ET 1 ;
Ab ¼ T AT 1 ;
Bb ¼ T B
C b ¼ CT 1 :
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R. Bru et al. / Appl. Math. Comput. 148 (2004) 341–350
Note that, all the above matrices are diagonal matrices and the system T satisfies that Bb1 BTb2 ¼ O and Cb1 Cb2 ¼ O. Then, the obtained balanced model keeps the uncoupled structure between the forward and backward subsystems and Gramian matrices of the balanced model satisfy the Lyapunov equations T
T
T
Eb Wr ðEb ; Ab ; Bb ÞEb Ab Wr ðEb ; Ab ; Bb ÞAb ¼ Bb Bb ; T
T
T
Eb Wo ðEb ; Ab ; C b ÞEb Ab Wo ðEb ; Ab ; C b ÞAb ¼ C b C b : Note that the Gramian matrices satisfy the following relationships Wr ðEb ; Ab ; Bb Þ ¼ TWr ðE; A; BÞT T and Wo ðEb ; Ab ; C b Þ ¼ T T Wo ðE; A; CÞT 1 : Next an example illustrates the above algorithm. Example 1. Consider the system ðE; A; B; CÞ given by 2
1 60 6 60 E¼6 60 6 40 0 2
1 62 6 61 B¼6 60 6 40 1
0 1 0 0 0 0
0 0 1 0 0 0
3 1 2 7 7 1 7 7; 0 7 7 0 5 1
0 0 0 0 0 0
0 0 0 1 0 0
3 0 07 7 07 7; 07 7 15 0
2
0:5 6 0 6 6 0 A¼6 6 0 6 4 0 0
1 1 C¼ 0 0
1 0
0 1
1 0:3 0 0 0 0
0 0 0:8 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
3 0 07 7 07 7; 07 7 05 1
0 0 : 1 1
Using the algorithm, the matrix T is given by 2
0:3915 0:4612 6 0:5478 0:7040 6 6 0:2613 0:6634 T ¼6 6 0 0 6 4 0 0 0 0
0:3696 0:7831 1:0880 0 0 0
0 0 0 0:5177 0:3051 0:4183
0 0 0 0:3379 0:8785 0:2226
3 0 7 0 7 7 0 7 0:9730 7 7 0:4674 5 0:1453
R. Bru et al. / Appl. Math. Comput. 148 (2004) 341–350
349
and the balanced realization is given by 3 2 1 0 0 0 0 0 7 6 0 0 0 7 60 1 0 7 6 7 60 0 1 0 0 0 7 6 Eb ¼ 6 7; 6 0 0 0 0:1158 0:5656 0:2693 7 7 6 7 6 0:6265 0:2431 5 4 0 0 0 0:5656 2
0
0
0
0:0065
6 6 0:3412 6 6 0:0240 6 Ab ¼ 6 6 0 6 6 0 4 2
0 1:6834
6 6 0:0771 6 6 0:0226 6 Bb ¼ 6 6 0:9730 6 6 4 0:4674 " Cb ¼
0:1453
0:2693 0:3412
0:0240 0
0
0
3
0:6303
0:2381
0
0
0:2381
0:9632
0
0
0
0
1
0
0
0
0
1
7 07 7 07 7 7; 07 7 7 05
0
0
0
1
0 3 1:6834 7 0:0771 7 7 0:0226 7 7 7; 0:9739 7 7 7 0:4674 5 0:1453
2:3807 0:1090 0
0:2431 0:5107
0:0319
0
0
0
0
1:3760
0:6611
0:2054
0
# :
Finally, the reachability and observability Gramian matrices are Wr ðEb ; Ab ; Bb Þ ¼ Wo ðEb ; Ab ; Cb Þ ¼ diagð5:8129; 1:2397; 1:0336; 2:6579; 2:1667; 0:4912Þ:
References [1] B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Autom. Control AC-26 (1) (1981) 17–32. [2] R. Ober, D. McFarlane, Balanced canonical forms minimal systems a normalized coprime factor approach, Linear Algebra Appl. 122/123/124 (1989) 23–64. [3] E.J. Ang, V. Sreeram, W.Q. Liu, Identification/reduction to balanced realization via the extended impulse response Gramian, IEEE Trans. Autom. Control 40 (12) (1995) 2153–2158. [4] M. Diab, V. Sreeram, W.Q. Liu, Frequency weighted identification and model reduction via extended impulse response Gramian, Int. J. Control 70 (1) (1998) 103–122.
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[5] D. Bender, Lyapunov-like equations and reachability/observability Gramians for descriptor systems, IEEE Trans. Autom. Control AC-32 (4) (1987) 343–348. [6] L. Zhang, J. Lam, Q. Zhang, Lyapunov and Riccati equations of discrete-time descriptor systems, IEEE Trans. Autom. Control. 44 (11) (1999) 2134–2139. [7] D.F. Delchamps, State-Space and Input–Output Linear Systems, Springer-Verlag, New York, 1988. [8] L. Dai, Singular Control Systems, Springer-Verlag, NewYork, 1989. [9] Y. Liu, B.D.O. Anderson, Singular perturbation approximation of balanced systems, Int. J. Control 50 (4) (1989) 1379–1405. [10] J. Sveinsson, F. Fairman, Minimal balanced realization of transfer function matrices using Markov parameters, IEEE Trans. Autom. Control AC-30 (10) (1985) 1014–1016. [11] C. Yang, F. Yeh, A simple algorithm on minimal balanced realization for transfer function matrices, IEEE Trans. Autom. Control 34 (8) (1989) 879–882.
Gramian matrices and balanced model of generalized systems q Rafael Bru, Carmen Coll *, Nestor Thome Departamento de Matem atica Aplicada, Universidad Polit ecnica de Valencia, 46071 Valencia, Spain
Abstract In this work, the generalization of reachability and observability Gramian matrices of control generalized systems have been studied. The properties of these matrices to be solutions of Lyapunov equations have been analyzed. In addition, an algorithm to obtain balanced generalized systems is given. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Linear discrete-time generalized systems; Reachability; Observability; Lyapunov equation; Gramian matrix; Balanced systems
1. Introduction The reachability and observability properties have special attention in control theory. Many classical problems of linear control systems have been studied using structural properties, for instance the balanced realization problem was introduced by Moore in [1]. Ober and McFarlane used Gramian matrices of standard systems by means of the corresponding reachability and observability matrices of the system in [2]. Gramian matrices have gained attention in the control theory because these matrices contain interesting information on the input–output behaviour of the system, but really there are no many papers dealing with this topic in the last decade. It is well known that
q
Supported by DGI grant BFM2001-2783. Corresponding author. E-mail addresses: [email protected] (R. Bru), [email protected] (C. Coll), njthome@mat. upv.es (N. Thome). *
0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00848-2
342
R. Bru et al. / Appl. Math. Comput. 148 (2004) 341–350
Gramian matrices can be used for checking if a system is reachable and/or observable. In addition, these matrices play an important role in the study of the stability of a system using the respective Lyapunov equations. Recently, different methods studying model reduction of standard system using Gramian matrices have appeared [3,4]. In [5], Bender gave a definition of reachability and observability Gramian 1 matrices using the Laurent expansion of the matrix ðzE AÞ . A sufficient condition on the orthogonality of some blocks of matrices B and C allows to state that Gramian matrices are the solution of the Lyapunov equations EWr ET AWr AT ¼ BBT and ET Wo E AT Wo A ¼ C T C. In [6], Gramian matrices of a generalized system ðE; A; B; CÞ have been introduced as the sum of the causal part Wrc ¼
1 X
/k BBT /Tk ;
Woc ¼
k¼0
1 X
/Tk C T C/k
k¼0
and the noncasual part Wrnc ¼
1 X
/k BBT /Tk ;
Wonc ¼
k¼q
1 X
/Tk C T C/k ;
k¼q
where q is the nilpotence index and /k are the Laurent parameters of the serie expansion of the matrix ðzE AÞ1 . An extension of Gramian matrices to generalized linear discrete-time systems is discussed in this work. We use the reachability and observability Gramian matrices of generalized systems defined as a natural extension of the standard case [7]. We show that these matrices are solutions of the corresponding Lyapunov equations. Further, we use an algorithm to obtain balanced models of generalized systems. First we provide some useful background which is used in the paper. Let us consider a generalized system given by Exðk þ 1Þ ¼ AxðkÞ þ BuðkÞ; yðkÞ ¼ CxðkÞ;
ð1Þ
where A; E 2 Rnn , B 2 Rnm , C 2 Rpn , k 2 Z. If rankðEÞ < n the system (1) is called singular system and if rankðEÞ ¼ n, then we can consider the system e xðkÞ þ B e uðkÞ; xðk þ 1Þ ¼ A yðkÞ ¼ CxðkÞ;
ð2Þ
e ¼ E1 A and B e ¼ E1 B. A system with the structure like that in the where A system (2) is called a standard system. We denote by ðE; A; B; CÞ the singular system and by ðA; B; CÞ the standard system. It is well known (see [8]), that if there exists an escalar k 2 C such that detðkE AÞ 6¼ 0, (i.e. the system (1)
R. Bru et al. / Appl. Math. Comput. 148 (2004) 341–350
343
satisfies the regularity condition), then the system (1) is equivalent to the forward–backward canonical form given by Exðk þ 1Þ ¼ AxðkÞ þ BuðkÞ;
ð3Þ
yðkÞ ¼ CxðkÞ; with n1 þ n2 ¼ n, and E ¼ diagðIn1 ; N Þ;
A ¼ diagðA1 ; In2 Þ;
B1 B¼ ; B2
C ¼ ½ C1
C2
and N is nilpotent. We can split the system (3) into two subsystems, the forward subsystem given by x1 ðk þ 1Þ ¼ A1 x1 ðkÞ þ B1 uðkÞ; y1 ðkÞ ¼ C1 x1 ðkÞ
ð4Þ
and the backward subsystem given by Nx2 ðk þ 1Þ ¼ x2 ðkÞ þ B2 uðkÞ; y2 ðkÞ ¼ C2 x2 ðkÞ:
ð5Þ
In the following we suppose that singular systems satisfy the regularity condition. Note that, if rankðEÞ ¼ n, then detðkI E1 AÞ 6¼ 0 and so detðkE AÞ 6¼ 0, for some k 2 C. That is, a standard system can be interpreted as a generalized system with only the forward part. Then, we focus our attention on singular systems. Definition 1. Consider the singular system ðE; A; B; CÞ. i(i) A state x 2 Rn is reachable if there exists a time k 2 Z and a control sequence uðjÞ, j ¼ 0; 1; . . . ; k 1 transferring the state of the system from the origin at time 0 to x at time k. The system ðE; A; B; CÞ is reachable if every x 2 Rn is reachable. (ii) The system ðE; A; B; CÞ is observable if the initial condition xð0Þ may be uniquely determined by control and output sequences, uðjÞ and yðjÞ, j ¼ 0; 1; . . . ; k 1, k 2 Z. The reachability matrices of subsystems (4) and (5) are given by Xr ðA1 ; B1 Þ ¼ ½B1 A1 B1 A21 B1 and by Xr ðN ; B2 Þ ¼ ½ N 2 B2 NB2 B2 , respectively. The observability matrices of subsystem (4) and (5) are given by 2 3 2 3 .. C1 6 C1 A1 7 6 . 7 27 6 7 6 2 Xo ðA1 ; C1 Þ ¼ 6 C1 A 7 and Xo ðN ; C2 Þ ¼ 6 C2 N 7; respectively: 15 4 4 C2 N 5 .. . C2
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It is known that the system (3) is reachable if and only if rank½Xr ðA1 ; B1 Þ ¼ n1 and rank½Xr ðN ; B2 Þ ¼ n2 and the system (3) is observable if and only if rank½Xo ðA1 ; C1 Þ ¼ n1 and rank½Xo ðN ; C2 Þ ¼ n2 . A reachable and observable system is minimal. In the following the spectral radius of a square matrix M will be denoted by qð M Þ. The set Sm;p denote the systems ðE; A; B; CÞ satisfying the following n properties: (i) E; A 2 Rnn , B 2 Rnm and C 2 Rpn , (ii) ðE; A; B; CÞ is a minimal system and (iii) ðE; A; B; CÞ is asymptotically stable. Now we define the Gramian matrices associated to generalized systems. m;p Definition 2. Let ðE; A; B; CÞ 2 Sn and consider its associated forward– backward system E; A; B; C . The reachability Gramian matrix of the system ðE; A; B; CÞ is given by
Wr ðE; A; BÞ ¼ diagðWr ðA1 ; B1 Þ; Wr ðN ; B2 ÞÞ; where Wr ðA1 ; B1 Þ ¼ Xr ðA1 ; B1 ÞXTr ðA1 ; B1 Þ and Wr ðN ; B2 Þ ¼ Xr ðN ; B2 ÞXTr ðN ; B2 Þ. The observability Gramian matrix of the system ðE; A; B; CÞ is given by Wo ðE; A; CÞ ¼ diagðWo ðA1 ; C1 Þ; Wo ðN ; C2 ÞÞ; where Wo ðA1 ; C1 Þ ¼ XTo ðA1 ; C1 ÞXo ðA1 ; C1 Þ and Wo ðN ; C2 Þ ¼ XTo ðN ; C2 ÞXo ðN ; C2 Þ. Note that, the uncoupled structure of Gramian matrices in the above definition permits to construct parallel algorithms for computing them, taking advantage respect to the other definitions. 2. Reachability and observability Gramian matrices In this section we study the reachability Gramian matrix for singular systems. We start with the following lemma. Lemma 1. Let ðE; A; B; CÞ be a singular system and consider its associated for ward–backward system E; A; B; C . If qðA1 Þ < 1 then (a) Wr ðA1 ; B1 Þ is the only nonnegative definite solution of the Lyapunov equation P A1 PAT1 ¼ B1 BT1 : (b) )Wr ðN ; B2 Þ is the only nonnegative definite solution of the Lyapunov equation Q NQN T ¼ B2 BT2 : Proof. Using the stability characterization of standard systems, it is easy to prove part (a). Now, we show condition (b). Since N is a nilpotent matrix, then
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qðN Þ < 1. So the solution of the Lyapunov equation Q NQN T ¼ B2 BT2 is Xr ðN ; B2 ÞXTr ðN ; B2 Þ. Hence Wr ðN ; B2 Þ is the required nonnegative definite solution. Using the above lemma, the matrix Wr ðN ; B2 Þ satisfies NWr ðN ; B2 ÞN T Wr ðN ; B2 Þ ¼ B2 BT2 : To obtain the solution of the Lyapunov equation T
T
T
EP E AP A ¼ BB ; where E, A and B are given in (4) and (5) and P ¼ ½Pij i;j¼1;2 , we must solve the following equations T T P11 A1 P11 A1 ¼ B1 BT1 ; T
P12 N A1 P12 ¼
ð6Þ
B1 BT2 ;
ð7Þ
NP21 P21 AT1 ¼ B2 BT1 ;
ð8Þ
NP22 N T P22 ¼ B2 BT2 :
ð9Þ
When qðA1 Þ < 1, the Eq. (6) has only one solution but Eqs. (7) and (8) can even not have solution. A sufficient condition to guarantee the solution of Eq. (7) is that B1 BT2 ¼ O. We use this condition in the following result. Theorem 2. Let ðE; A; B; CÞ be a singular system and consider its associated forward–backward system E; A; B; C . If qðA1 Þ < 1 and B1 BT2 ¼ O, then the reachability Gramian matrix Wr ðE; A; BÞ satisfies the Lyapunov equation T
T
T
EP E AP A ¼ BB : Proof. We must prove that the matrix Wr ðE; A; BÞ satisfies the Lyapunov equation. In fact, T
T
EWr ðE; A; BÞE AWr ðE; A; BÞA A1 O In1 O Wr ðA1 ; B1 Þ O In1 O ¼ O In2 O N O Wr ðN ; B2 Þ O NT T O Wr ðA1 ; B1 Þ A1 O O Wr ðN ; B2 Þ O In2 Wr ðA1 ; B1 Þ A1 Wr ðA1 ; B1 ÞAT1 O ¼ : O NWr ðN ; B2 ÞN T Wr ðN ; B2 Þ Using part (a) of the above lemma, Wr ðA1 ; B1 Þ A1 Wr ðA1 ; B1 ÞAT1 ¼ B1 BT1 and also NWr ðN ; B2 ÞN T Wr ðN ; B2 Þ ¼ B2 BT2 . Further,
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B1 T BB ¼ B1 B2 T
BT2
B1 BT1 ¼ B2 BT1
B1 BT2 B2 BT2
and since B1 BT2 ¼ O we obtain the result.
Now, we give the corresponding results of the observability Gramian matrix for singular systems, for which proofs are similar to the reachability case. Lemma 3.Let ðE; A; B; CÞ and consider its associated forward–backward system E; A; B; C . If qðA1 Þ < 1 then: (a) Wo ðA1 ; C1 Þ is the only nonnegative solution of the Lyapunov equation P AT1 PA1 ¼ C1T C1 ; (b) Wo ðN ; C2 Þ is the only nonnegative solution of the Lyapunov equation Q N T QN ¼ C2T C2 : Theorem 4. Let ðE; A; B; CÞ be a singular system and consider its associated forward–backward system E; A; B; C . If qðA1 Þ < 1 and C1T C2 ¼ O, then observability Gramian Wo ðE; A; CÞ satisfies the Lyapunov equation T
T
T
E QE A QA ¼ C C:
3. Balanced model The need of balancing the gains between inputs and states just like that between states and outputs permits to develop algorithms for obtaining balanced standard system, that is, linear standard systems whose reachability and observability Gramian matrices are diagonal and coincide. This kind of system is used to obtain models of smaller size (and then cheaper) than the original system by different techniques. For instance, in [9] singular perturbation approximation and direct truncation are used to obtain reduced model of balanced standard system. Different algorithms have been developed to obtain balanced standard realizations, for instance using Markov parameters in [10] and using singular value decomposition of the Hankel matrix in [11]. The purpose of this section is to construct a balanced singular system associated to a generalized system. First, we give the following definition. Definition 3. Let ðE; A; B; CÞ 2 Sm;p n . The system ðE; A; B; CÞ is called balanced if
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Wr ðE; A; CÞ ¼ Wo ðE; A; CÞ ¼ diagðr1 ; r2 ; . . . ; rn Þ; where r1 ; r2 ; . . . ; rn are called the Hankel singular values of the system. A first step to obtain a balanced system associated to the system ðE; A; B; CÞ 2 Sm;p is to compute the reachability and observability Gramian n matrices of this system. Thus, consider a minimal and stable system ðE; A; B; CÞ given by (4) and (5), with B1 BT2 ¼ O and C1T C2 ¼ O: An algorithm to compute the reachability (observability) Gramian matrix is as follows: To take X0 ¼ O and Y0 ¼ O. The matrix Wr ðA1 ; B1 Þ is obtained using the iterative squeme Xiþ1 ¼ A1 Xi AT1 þ B1 BT1 , and the matrix Wr ðN ; B2 Þ is obtained using the iterative squeme Yiþ1 ¼ NYi N T þ B2 BT2 ; and making iterations until convergence. Assign Wr ðA1 ; B1 Þ ¼ Xi , Wr ðN ; B2 Þ ¼ Yi and construct the matrix O Wr ðA1 ; B1 Þ Wr ðE; A; BÞ ¼ : O Wr ðN ; B2 Þ By a similar way the corresponding observability Gramian matrices can be obtained, only changing the matrices involving in the iterative squemes. Remember that O Wo ðA1 ; C1 Þ Wo ðE; A; CÞ ¼ : O Wo ðN ; C2 Þ Now, we give the algorithm to obtain the balanced realization. Step 1: To obtain the singular value decomposition of Wr ðE; BÞ using the block structure of this matrix. Then there exists an orthogonal matrix U ¼ diagðU1 ; U2 Þ such that Wr ðE; BÞ ¼ diagðU1 ; U2 ÞR2r diagðU1T ; U2T Þ. T Step 2: To construct matrix M ¼ ðU Rr Þ Wo ðE; A; CÞU Rr . Then M ¼ diagðM1 ; M2 Þ: Step 3: To obtain the singular value decomposition of M, that is M ¼ diagðV1 ; V2 ÞR2o diagðV1T ; V2T Þ: Step 4: To construct matrix T ¼ diagðT1 ; T2 Þ 1
T T T ¼ R2o diagðV1T ; V2T ÞR1 r diagðU1 ; U2 Þ:
Step 5: To construct matrices Eb ¼ T ET 1 ;
Ab ¼ T AT 1 ;
Bb ¼ T B
C b ¼ CT 1 :
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Note that, all the above matrices are diagonal matrices and the system T satisfies that Bb1 BTb2 ¼ O and Cb1 Cb2 ¼ O. Then, the obtained balanced model keeps the uncoupled structure between the forward and backward subsystems and Gramian matrices of the balanced model satisfy the Lyapunov equations T
T
T
Eb Wr ðEb ; Ab ; Bb ÞEb Ab Wr ðEb ; Ab ; Bb ÞAb ¼ Bb Bb ; T
T
T
Eb Wo ðEb ; Ab ; C b ÞEb Ab Wo ðEb ; Ab ; C b ÞAb ¼ C b C b : Note that the Gramian matrices satisfy the following relationships Wr ðEb ; Ab ; Bb Þ ¼ TWr ðE; A; BÞT T and Wo ðEb ; Ab ; C b Þ ¼ T T Wo ðE; A; CÞT 1 : Next an example illustrates the above algorithm. Example 1. Consider the system ðE; A; B; CÞ given by 2
1 60 6 60 E¼6 60 6 40 0 2
1 62 6 61 B¼6 60 6 40 1
0 1 0 0 0 0
0 0 1 0 0 0
3 1 2 7 7 1 7 7; 0 7 7 0 5 1
0 0 0 0 0 0
0 0 0 1 0 0
3 0 07 7 07 7; 07 7 15 0
2
0:5 6 0 6 6 0 A¼6 6 0 6 4 0 0
1 1 C¼ 0 0
1 0
0 1
1 0:3 0 0 0 0
0 0 0:8 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
3 0 07 7 07 7; 07 7 05 1
0 0 : 1 1
Using the algorithm, the matrix T is given by 2
0:3915 0:4612 6 0:5478 0:7040 6 6 0:2613 0:6634 T ¼6 6 0 0 6 4 0 0 0 0
0:3696 0:7831 1:0880 0 0 0
0 0 0 0:5177 0:3051 0:4183
0 0 0 0:3379 0:8785 0:2226
3 0 7 0 7 7 0 7 0:9730 7 7 0:4674 5 0:1453
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and the balanced realization is given by 3 2 1 0 0 0 0 0 7 6 0 0 0 7 60 1 0 7 6 7 60 0 1 0 0 0 7 6 Eb ¼ 6 7; 6 0 0 0 0:1158 0:5656 0:2693 7 7 6 7 6 0:6265 0:2431 5 4 0 0 0 0:5656 2
0
0
0
0:0065
6 6 0:3412 6 6 0:0240 6 Ab ¼ 6 6 0 6 6 0 4 2
0 1:6834
6 6 0:0771 6 6 0:0226 6 Bb ¼ 6 6 0:9730 6 6 4 0:4674 " Cb ¼
0:1453
0:2693 0:3412
0:0240 0
0
0
3
0:6303
0:2381
0
0
0:2381
0:9632
0
0
0
0
1
0
0
0
0
1
7 07 7 07 7 7; 07 7 7 05
0
0
0
1
0 3 1:6834 7 0:0771 7 7 0:0226 7 7 7; 0:9739 7 7 7 0:4674 5 0:1453
2:3807 0:1090 0
0:2431 0:5107
0:0319
0
0
0
0
1:3760
0:6611
0:2054
0
# :
Finally, the reachability and observability Gramian matrices are Wr ðEb ; Ab ; Bb Þ ¼ Wo ðEb ; Ab ; Cb Þ ¼ diagð5:8129; 1:2397; 1:0336; 2:6579; 2:1667; 0:4912Þ:
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