Computation of Structural Decomposition for Linear Singular System

Computation of Structural Decomposition for Linear Singular Systems

MINGHUA HE and BEN M. CHEN Department of Electrical and Computer Engineering The National University of Singapore 10 Kent Ridge Crescents, Singapore 117576 REPUBLIC OF SINGAPORE

Abstract: We present in this paper computation algorithms for the structural decomposition of general linear multivariable singular systems. Such kind of decomposition has a distinct feature of capturing and displaying all the structural properties, such as the finite and infinite zero structures, invertibility structures and redundant dynamics of the given system. The computation will make it a powerful and convenient tool in solving control problems for singular systems.

Keywords: Computation; MATLAB programs; singular systems; structure decomposition; structure properties.

1. Introduction paper more self-contained, we briefly describe the structural decomposition theorem in Section 2. And Consider the following linear singular system in Section 3, the main MATLAB computation programs are presented while several numerical Ex˙  Ax  Bu n p m  :  x   , y  ,u   , examples will be included in Section 4. Finally, a y  Cx  Du  conclusion will be drawn in Section 5. (1) where rank(E)  n . It is also alternatively called a descriptor system, an implicit system or a generalized linear system in literature. Linear 2. Structural Decomposition Theorem singular system has attracted researchers for more and Its Properties than three decades since many real systems, such as electrical power systems, can be naturally modeled We first summarize the structural decomposition of as singular systems. Among the issues discussed for general multivariable singular systems in compact linear singular system [7], system structure and matrix form. And its essential properties will also equivalence is an essential topic. Campbell [1] be given in brief in this section. presented an effective structural decomposition method and got the corresponding equivalent Theorem 2.1 Consider the general multivariable system, while Verghese et al. [11] defined a strong linear singular system  in (1). Then, there exist system equivalence using a trivial augmentation nonsingular state and output transformations nn pp and deflation technique. On the other hand, s  and o  , and a nonsingular structural invariants also received much attention in nn literature. Further, Misra et al. [10] and Liu et al. transformation e  , as well as an mm input

[9] have presented their algorithms to compute the transformation i (s) , whose inverse has all its invariant structural indices of singular systems. elements being some polynomials of s (i.e., its More recently, He and Chen [5] and He et al [6] inverse contains various differentiation operators), have developed a structural decomposition method which together give a structural decomposition of for single-input single-output and multivariable  and display explicitly its structural properties. linear singular systems respectively. Such a structural decomposition can not only give the This structural decomposition can be described in invariant structural indices but also explicitly the following equation form, display the structural properties, such as the finite and infinite zero dynamics, invertibility structures  xz  and redundant dynamics of the given systems. And    xe  it is expected to be a powerful tool in solving  y0  u0   x      ~  a  ~ ~ system and control problems as its counterpart in x  s x  s , y  o y   yb  , u  i (s)u  uc  ,  xb      nonsingular linear system [3].   yd ud x      c    This paper focuses on the computation algorithms  xd  of the structural decomposition. First, to make this (2) and I m 0 0 ~ ~ 0 ~ D   1D (s)  D  D (s)   0 0 0  D (s), o i v k   k  0 0 0  x   y   u   d1   d1   d1  where x y u  d2   d2   d2   xd  , yd  , ud  ,  ⋮   ⋮   ⋮  B0  0 0 B0a B0b B0c B0d  ,        x   y  u  C0  0 0 C0a C0b C0c C0d  ,  dmd   dmd   dmd  (6) and xz  0, x  B u  B u  B u , e e0 0 ec c ed d ~ ~ ~ ~ ~ ~ ~ ~ x˙a  Aaa xa  B0a y0  Lad yd  Lab yb , Ak x  Bk (s) u  0, Ck x  Dk (s) u  0. (3) x˙ b  Abb xb  B0b y0  Lbd yd , yb  Cb xb , (7) x˙c  Acc xc  B0c y0  Lcd yd  Lcb yb  BcM ca xa  Bcuc y0  C0a xa  C0b xb  C0c xc  C0d xd  u0 The equation form of this theorem and detail proof can be found in He, Chen and Lin [6]. Here, we and for each i 1,2,⋯, m , briefly introduce the essential properties of this d structural decomposition. x˙  A x  L y  L y  B [u  di qi di i0 0 id d qi di md (4) Property 2.1 The given system  in (1) is M x  M x  M x  M x ], ia a ib b ic c  ij dj stabilizable if and only if A , B  is j1 con con stabilizable, and it is detectable if and only if ydi  Cqi xi , yd  Cd xd . Aobs , Bobs  is detectable, where

The structural decomposition can also be expressed Aaa LabCb  B0a Lad  in the following compact form. Acon :   , Bcon :   ,  0 Abb  B0b Lbd  J 0 0 0 0 0  (8) nz and E 0 0 0 0 0   ez   Aaa 0   C0a C0c  ~ Eaz 0 I n 0 0 0  1 a Aobs : , Bcon : . E  e Es  Ev   ,     E 0 0 I 0 0 Bc M ca Acc  M da M dc   bz nb 

Ecz 0 0 0 I n 0  (9)  c  E 0 0 0 0 I  dz nd  Property 2.2 The invariant zeros of the given

~ 1 ~ system  are the eigenvalues of Aaa . The normal A  e As  Av  B0C0  Ak I 0 0 0 0 0 rank of  is equal to m0  md . Here md is the  nz    0 I 0 0 0 0 dimension of ud .  ne   0 0 Aaa LabC b 0 LadCd  ~     B0C0  Ak , 0 0 0 A 0 L C m  bb bd d  Property 2.3 The given system  has 0 infinite

 0 0 Bc M ca LcbC b Acc LcdCd  zero of order 0. And its infinite zero structure (of   order greater than 0) is given by  0 0 Bd M da Bd M db Bd M dc Add  S  ()  q , q , ⋯, q   0 0 0   1 2 md , B B B  (10)  0e de ce  ~ 1 ~ B0a 0 0  ~ that is, for each i  1, 2,⋯, md ,  has an infinite B  e Bi (s)  Bv  Bk (s)     Bk (s), B 0 0  0b  zero of order qi , respectively. B0c 0 Bc    B0d Bd 0  Property 2.4 The given system  is right

invertible if and only if xb and hence yb are non- (5) existent, is left invertible if and only if xc and

hence uc are non-existent, and is invertible if and C0z 0 C0a C0b C0c C0d  ~ ~ ~ C   1C  C  C  C 0 0 0 0 C   C , only if both x and x are non-existent. o s v k  dz d  k b c C bz 0 0 C b 0 0  The properties show that our structural decomposition can explicitly display the structure properties of the given singular system, and hence it is expected to be a powerful tool in solving The function transform a matrix pair A, B into its singular system and control problems. control canonical form as follows,

1 Ac Acc  1 Bc  T AT    , T B    , 3. MATLAB Computation Programs  0 Ac   0  for the Structural Decomposition (12) where Ac , Bc  is completely controllable while As mentioned before, a detailed constructive A ,0 is totally uncontrollable. decomposition algorithm will not be given here due c to the limit pages. And in this section, we will give brief descriptions of the main functions for the bdc_cf.m computation. The computation programs introduced are all in MATLAB codes. This function decomposes a complete controllable pair Ac , Bc  into a special block controllability SD.m canonical form [2], in which every submatrix block corresponds to a distinct input channel. The This is the main function, that is, structural decomposition process can be described as follows, decomposition function for general linear singular systems. The function transforms the given singular J1 0 ⋯ 0  system (E, A, B,C, D) into its structural   ~ ~ ~ ~ ~ 1 0 J 2 ⋯ 0 decomposition form (E, A, B,C, D) , which can R A R    , c  0 0 ⋱ ⋮ explicitly display all the structural properties, such   as the finite and infinite zero structures, invertibility  0 0 0 J k  structures and even redundant dynamics of the B B ⋯ B B  given system. 1 12 1k 1l  0 B ⋯ B B  R 1B   2 2k 2l  sys_hat.m c  0 0 ⋱ ⋮ ⋮    This function separates two kinds of redundant  0 0 0 Bk Bkl  states from the original system. One kind of (13) redundant states xz are static and identical zero all where J i ,i  1, 2,⋯, k are Jordan blocks with the time, whereas the other redundant states xe are linear combination of appropriate order of system zero eigenvalue and input's derivatives. Such states are associated with 0  the so called impulse modes, which are introduced 0      by the derivatives of the system input. Bi  , Bij  . ⋮ ⋮     pre_decom.m 1 0 (14) This one is to perform a fast-slow decomposition (see e.g., [4] for details) for the given singular kronecker.m system. With two constant transform matrices P and Q , it transforms the given singular system into The function transform the given system's system two subsystems, one is nonsingular and the other is matrix P (s) to its Kronecker Canonical Form singular. The decomposition can be characterized with two constant transform matrices M and N , as the following transformations, I 0 A 0   n1  1 P~ (s)  M P (s) N PEQ    , PAQ    ,   0 N 0 I A  sE B    n2     M   N  C D B1  T T PB  , CQ  C1 C2 ,  diag sI  J f I  sJ  L ⋯ L L ⋯ L    1 p 1 q B2  (15 (11) ) where N is a nilpotent matrix. ~ Here every block of the diagonal entries in P (s) ctr_cf.m is associated the distinct structure index. 0 1  0.3333  0.7071 0 SCB.m   0 0 0.6667 0 1 Ge  0 1 0 0 0 , This is the function of structural decomposition for   linear nonsingular system. The function was 1 0 0 0 0 developed by Lin and Chen [8], and it decomposes 0 0  0.6667  0.7071 0 a given linear system A, B,C, D and explicitly (17) displays its structural properties. The function SD.m is its natural extension to linear singular 0 0  0.3333  0.7071 0   systems. 0 1 0 0 0 Gs  0 0  0.6667  0.7071 0 ,   r_jordan.m 1 0 0 0 0 0 0 0.6667 0 1 This function transforms a real matrix H to its (18) Jordan canonical form. 1 Go  1, Gi (s)  2 , (19) The functions introduced here are only some main s and procedures, and there are still many other functions  0 0 0 0 0 needed in our computation. But due to the limit of   page, we can not introduce every function here.  1 0 0 0 0 However, this omission will not affect our Ev    6 0 1 0 0 ,Cv  1 0 0 0 1,   illustration of computation process in the following 4.2426 0 0 1 0 section.  5 0 0 0 1 (20) 1 0 0 0 0   0 

0 1 0 0 0  1 4. Some Numerical Examples     Av  0 0 1 0  3 , Bv   0 ,     0 0 0 0 1.4142 0 To illustrate the computation of our structural         decomposition algorithm, we present in this section 0 0 0.6667 0 2   1  two numerical examples, one is of single-input and single-out linear singular system while the other is Dv  0, nz  1, ne  1, na  2, nb  0, nc  0, nd  1. of multi-input and multi-output case. And finally this decomposition result can be verified by the following operation. The statement Let us first look at the following single-input and Pv ,M , N  KroneckerE, A,B,C,D returns single-output system, 1 s 0 0 0 0 0  1 0 0 0 0 1 0 0 0 0  0  s 0 0 0 0  0 0 0 1 1 0 1 0 0 0        0 0 1  s 0 0  E  0 0 0 1 0, A  0 0 1 0 0 Pv    .     0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0        0 0 0 0 1  s 0 0 1 1 0 0 0 0 0 1    0 0 0 0 0 1  1 (21) 0   B  1 , C  2 0  2 1 1, D  0, P   From v , the Kronecker Canonical Form of the 0 given system, we can see clearly that the structure 0 indices are the same as our computation results. (16) Now we look at the following multi-input multi- the statement output linear singular system,

Ge ,Gs ,Gi ,Go ,Ev , Av ,Bv ,Cv ,Dv ,nz ,ne ,na ,nb ,nc ,nd   SDE, A,B,C,D, returns 1 0 0 0 0 0 0  0 0 0 0 0 0 0   0 0 1 0 1 0 1 0 0 0 0 0 0 0     0 0 0 0 0 0 0  0 0 0 0 0 0 0     E  , E  0 0 0 0 1 0 0 , A  I v 0 0 0 1 0 0 0   7 0 0 0 0 1 0 0 0 1 0 0 1 1 1     0 0 0 0 0 1 0 0 0 1 0 1 0 1       0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0   1 1 0 0 1 0 0 0 0 0  1 1 1     0 0 1 0 0 0 0     1 0 0 A  0 0 0  1 0 0 1.4142 ,   0 1 0 v   B   0 0 1 , D    , 0 0 0 1.5215 0.3333 1.8648 0.7172  0 0 0   1 0 1 0 0 0  0.7071 0 0  1      1 2 1 0 0 0 0.7071  0.3873 0.8333  0.3333 1 0 1  0 0 0    1 0 0 0 0 0 1  0  0.2 0.8944 C  .   0 1 1 0 2 1  2  1 0 0      0 0 0 (22) Bv   0 0 0  , Dv    , 0 0 0  0 0 1.0541   Then its structural decomposition form is in the  1 0 0    results of the following statement,  0 1 0  0 0 0 0 0 1 0 G ,G ,G ,G ,E , A ,B ,C ,D ,n ,n ,n ,n ,n ,n  Cv   . e s i o v v v v v z e a b c d 0 0 0 0 0 0 1  SDE, A,B,C,D. And the results is (25)

0 1 1 0.7746 0 0 0  And the corresponding structure indices are   0 0 1  0.2582  0.3162 0 0.6 1 1 0 0 0 0 0  nz  1, ne  2, na  1, nb  0, nc  1, nd  2.   (26) Ge  0 1 0  0.5164 0.3162 1 0.4 , 0 0 0 0.2582 0.3162 0 0.4   Again, this result can be verified by the following 1 0 2  0.2582  0.3162 0 0.6   computation, 1 0 0 0.2582 0.3162 0 0.4 0 0 0 0.7746 0 0 0  P ,M , N  KroneckerE, A,B,C,D   v , 0 1 0 0 0 0 0  1 0 0  0.2582  0.3162 0 0.6 and its computation result is:   Gs  0 0 1  0.2582  0.3162 0 0.6 0 0 0  0.5164 0.3162 1 0.4   1 1 0 0.2582 0.3162 0 0.4 1 s 0 0 0 0 0 0 0 0 0       0  s 1 0 0 0 0 0 0 0 1 0 0  0.5164 0.3162 1 0.4    0 0 0 1  s 0 0 0 0 0 (23)    0 0 0 0 1 0 0 0 0 0 2    s 1  s  s 1 Pv  0 0 0 0 0 1  s 0 0 0 ,   -1   0 0 0 0 0 0 1 0 0 0 Gi (s)    s  2 1 1  ,      0 0 0 0 0 0 0 1 0 0 0.8944s  0.4472 0.8944 1.3416   0 0 0 0 0 0 0 0 1 0 1 0       Go   .  0 0 0 0 0 0 0 0 0 1 0 1 (24)

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