Chapter 4 An introduction to multivariable Systems In this chapter a number of tools for the analysis of multivariable systems will be given. In particular, this will include poles and zeros of multivariable sy stems the generalized Ny quist Theorem for st ability analysis of multivariable sy stems, and frequency ses for multi- variable sy stems using singular values. 4.1 Poles and zeros of Multivariable s ystems For a single input single output(SISO) system with the transfer function G(s (4.1) the poles are defined as the values of the complex values s for which G(s)=oo, and the zero as the values of s for which G(s)=0 For a multiple input multiple output(MIMO) system with the transfer matria Gn,1(s Gn,2(s) the poles can be defined in analogy with the definition for SISo system. I.e., as the poles of every scalar transfer function Gll(s). Gn,n,s) in G(s). This definition is reasonable, as at ole. Hence, at least one of the entries in the transfer matrix equals Zeros for multivariable systems, however, can not in a reasonable way be defined simply as the values for which one of the entries of the transfer matrix equals zero, so the multivariable zeros are not directly related to the zeros of the indiviual transfer functions G11(s).Nun,(s) Instead, the zeros of a multivariable sy stem are defined as the values of the complex variable s for which G(s) loses rank. The rank of a matrix A is defined as the number of linearly independent columns of A and denoted by
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Thus, the definition of zeros for multivariable systems becomes Definition 4.1(Zeros for Multivariable Syst ems) The zeros of a transfer matriz G(s are defined as those values of the compler variable s for which rank G(s) is less than its morima valule Zeros defined in this way are called transmission zeros. The reason is the following transfer function G(s)loses rank for 8= %o, it can be shown, see Section 4.3.3, that the exist s an input vector uo#0, such that =0 Hence, the transmission of cert ain input signals is blocked for 8= Zo 4.1.1 Smith-McMillan form of a Transfer matrix Poles and transmission zeros of a transfer matrix G(s can be found, e.g. by transforming G(s)to its Smith-McMillan form. It can be shown that any propertransfer matrix G (s)can be written in its Smith-MCMillan form G(8)=U1(s)M(s)U2(s (4.5) U1(8)(1(8) 0}U2(s) (4.6) where U1(s)and U2(s)are unimodular matrices U(s)is said to be a unimodular matrix,if and only if its determinant det U (s is in dependent of s, i. e. if det U (s) is const ant. M(s) is a pseudo-diagonal matrix, and is called the Smith-McMillan form of G(s). G(s)and M(s) said to be similar, denoted as G(s) M(s). The polyn Ixi(s),i(s) have to be common factors or-equivalently -no common root s). Finally, xi(s),i(s) have to possess the following divisibility pr x(8)ki+1(s) +1(8)(8) The not ati s that the polynomial xi+1(s) is a factor of the poly nomial xi(s)(with no remainder ). Next, the following pole and zero polynomials are defined p(8)=1(s)φ2(s)…q(s x(8)=X1(8)X2(s)…Xr(s) (4.9) Now, it can be shown, that the poles and transmission zeros of the transfer function G(s)can e founds as the roots of p(s) and z(s), respectively. The degree of the pole polynomial p(s) called the McMillan degree of G (s) It can be shown that the Smith-McMillan form M(s of a transfer matrix G(s) can be deter mined by a series of elementary row and column operations on G(). let d(s) be the smallest I A transf (s)is said to be proper if all its entries satisfy Gi;(s)1-Ci< oo for rictly proper if Gil(s)|→0fors→∞
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common denominator for the entries Gij (s)of G(s). Then, G(s) can be written as N1(s) 1n(s) G(s)=dsN(s)=ds where N(s)is a poly nomial matrix (not a transfer matrix). There are three element ary types of row or column operations to perform on N(s) Interchange of two rows or columns Multiplication of a row or a column with a const ant Addition of one row or column multiplied with a polynomial to another A common property of these operations is, that they do not change the rank of the matrix ( s). Each of these elementary operations can be represented as a pre- or post multiplication of N(s by a suit able matrix L(s) called an elementary matric. It can be shown, that all element ary matrices are unimodular. Now, N(s can be rewritten as a sequen ce of row and column operations (8)=L1(s)S(s)L2(8) Ln(s)diag{∈1(s),E2(8),……,6r(s),0,0,…,0}L2(s) (4.12) Here, S(s is a pseudo-diagonal polynomial matrix. S(s) is called the Smith form of N(s The polynomials Ei(s) are monic and have the following divisibility properties 1 (4.13) Hence, the Smith form of a polynomial matrix is equivalent to the Smith-McMillan form of a transfer matrix. Now, the point is that the polynomials Ei(s) can be determined from the determinant divisors (4.14) Di()=greatest common divisor for all i x i sub determinants of N(s) (4.15) where every of the greatest common divisors are normalized to a monic polynomial. It can be shown, see [Mac89, Pages 40-43 that the polynomials Ei() are given by (4.16) Hence, the Smith-McMillan form of G(s) is given by M()=(s) 17) In summary, this est ablishes the following procedure for determining the Smith-McMillan form of a transfer matrix G(s)
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Procedure 4.1(Smith-McMillan form of a transfer mat rix) 1. Let G(s)be a proper transfer matrit. Find the smallest common denominator d(s) for all entries in G(s)and rewrite G(s)as G(s N(8) (4.18) 2. Determine the Smith form ofN(s) N(8)~S(8)=dag{∈1(8),E2(s),……,6r(s),0,0,…,0} (4.19) where Ei(s)is determined from the determinant divisors D(s) Di(s) ;(s)=D1 8. Then, the Smith-McMillan form of G(s)is given by (8)~M(8) (4.21) The following example has been taken from [ Mac89, Ex. 2.2 Example 4.1(The Smith-McMillan form of a transfer mat rix) Let G(s be given by 2= 22 2 2s-4 The smalles common denominator for Gi(s)is d(s)=82+38+2 and G(s)can be written as 1 G(8)=2+38+282+8-4282-8-8 423) (8-2)(s+2)(2s-4)(s+2) (4.24) The Smith form of the polynomial matrit N(s is given b (8)~S(s)=diag{∈1(8),E2(s)} where Ei( s)is determined from the determinant divisors D(s) e()=D-1(s) i=1,2
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D A Figure 4.1: State space rep res entation of a system. Since C adj(sI-A)B is a poly nomial matrix, obviously all poles of G(s)(i.e. the values of s for which G(s)=oo) have to be roots of the polynomial det(sI-A). The roots of det(sI-A) coincide with the eigenvalues of A. Hence, all poles of G(s) have to be eigenvalues of A. Th opposite needs not be the case al ways, since roots of det(sI - A)might be cancelled in(4.38) and consequently they will not appear as poles of G(s). This is the case, when the realiz ation (A, B, C, D)is uncontrollable, unob servable, or both. On the other hand, if the realiz (A, B, C, D)is both controllable and observable, the roots of det(sI- A)equals the poles of G(s and the pole polynomial p(s) will be given by ps=det(sI-A) 441) This means, that the dimension of A can not be smaller than the McMillan degree of G(s) Hence, a st ate sp ace realization which is both controllable and observable is called a minimal realization. These results can be summarized as the following theorem ThId li maldlaaaw K aaamranlavpacta Let G(s)be a tra fer matric with a minimal realization(A, B, C, D)and let p(s be the Smith-McMillan pole polynomial of G(s. Then dim a= degp(s (4.42) Hence, the McMillan degree of G(s equals the dimension cf a minimal realization. Moreot the eigenvalues f A equal the poles af G(s) Note, that if(A, B, C, D) is a non-minimal realization, then the poles of G(s) constitute a proper sub set of the eigenvalues of A aea Fdh rrtahrvelce dbaspranAauact The transformation of a state space description to a transfer function description is unique given by(4.38 ). In contrast, there are several ways in which a transfer function can be transformed into a st ate space description. A straightforward approach would be to derive separate st ate space descriptions for each column in G(s), i.e. for each input, and then collect these separate state space descriptions into an overall state space model. Let Gp) be the ith ch that G(s)=(Gn(s), Gt(s).,Ghs) O=tt C)ntr)
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32 of Write each column as T(8)+ 444) where di(s)is the common denominator polynomial of G:(s) d(s)=s4+a4-1+…+4 Note, that di(s) is monic. ni (s is a vector of polynomials, each having a degree strictly less than ki. The jth entry of n; (s) can be written as the polynom n1()=n,一1+-2+…+m is a vector consisting entirely of constants a state sp ace description in controllable canonical form of the column Gi(s) is then given by the realization(Ai, Bi, Ci,&)where n B 448) Finally, a realiz ation(A, B, C, D)of G(s) can be found as B=diag D=[51,2,…,na This realization is controllable, but not necessarily observable. If a minimal is required, there exist algorithms to remove the unob servable modes, see e.g. [Mac89, Section 8.3.5 The MATLAB function tfm2ss m from MATLABs Robust Control Toolbox pro duces a similar st ate sp ace realization and minreal m from MATLABTM's Control Toolbox can extract a minimal realiz ation from a non-minimal one Remark In computer aided design, and especially in MATLABTM, it is easier in general to work with st ate space descriptions, since it is difficult to represent transfer matrices, as this requires three dimensional structures. Robust control. howe sponse analysis of a number of transfer matrices, such as the sensitivity function S(s)and the complementary sensitivity function T(s). It is no problem, however, to compute the resp onse based on a st ate sp ace description of the system. Hence, a multivariable often be represented in st ate space representation, although the analy sis is performed in th frequency domain. This mixture of time and frequency domain is actually quite typical for
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aldar and the liccevdshrllte acabsiyIds ali dci+ eCL n=de-nI a A4 (4.68) As described in Section 4.1.2, the st ability of the closed loop system is determined by the of the characteristic poly nomial ocL+). To reach a Ny quist-similar'expression, cl-n) be expressed in terms of the open loop transfer matrix m n)K-n) To that end, the eAv PNfeeFte matrix F-n)is introduced as F (4.69) Moreover, the following lemma, which is presented without a proof, will be instrument al LImma 4. 1(Schur's formula for block partitioned ditirminants) Ie4l Cvle s1- 4anP ye SL 44dReMlo n PP zreP, 4e Nees aFIf4d P HPyeeSecceMlo de-P=de-Pll de-P22 a P21 Pi P12), SctalaMie-Pl1 b0 (4.71) d de-p=de-P22 de-Pu a Pi2 Pr p2), Sctalevie-P22b0 472) Now, write the determinant of the return difference matrix Fn)as de-F+)=de+crIa lBr+ DI With the following variable substitutions P Ar P1 =a cr P21= bl P22=I+Dr (475) the right hand side of(4.73) corresponds to the last term of the right hand side of (4.71) Thus, multiplying by de-Pil= de nI a al gives r a de nI a al de+ Dr+cEnl a Albr=de acr I+ Dr 476) r a nia ar Br AD cr I+Dr Once again, applying Schur's formula, it can be shown that r a de- a a BrDr--de-ae-h-oraBrI +Dr
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