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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→∞. $ ' * $ + ",, % ""! ) + . % % && ' "/! "-! % % && 0 ! ! ' "1! ! % $ ' "2! "3! && % 4