Tracking Control of Linear Systems

Appendix H

Proof of Lemma 102 (Basic Lemma)

Proof. [148]. Let the conditions of Lemma 102 hold. From (6.3) and (6.4) we determine the original z(t). Let the ij-th element of M(s) be m_ij(s), the i-th element of z(t) be z_i(t), the i-th element of Z(s) be Si(s), and the j-th element of W(s) be w_j(s), so that

$z_i\left(t\right)={ℒ}^{-1}\left\{{\varsigma }_i\left(s\right)\right\}={ℒ}^{-1}\left\{{\displaystyle \sum }_j{m}_{ij}\left(s\right)w_j\left(s\right)\right\}.$ H.1

Since M(.) is a real rational proper matrix function of s, Z(.) and W(.) are real rational proper vector functions of s; then, the same holds for their entries, which can be presented in the factorized forms,

$m_{ij}\left(s\right)=\frac{{\displaystyle \prod _{k=1}^{{\mu }_{ij}}\left(s-s_{mk}^{oij}\right)}}{{\displaystyle \prod _{k=1}^{v_{ij}}\left(s-s_{mk}^{*ij}\right)}},{\mu }_{ij}\le v_{ij},w_j\left(s\right)=\frac{{\displaystyle \prod _{k=1}^{{\upsilon }_j}\left(s-s_{mk}^{oj}\right)}}{{\displaystyle \prod _{k=1}^{w_j}\left(s-s_{mk}^{*j}\right)}},{\upsilon }_j\le w_j.$

These equations set (H.1) into the following form

$zi\left(t\right)={ℒ}^{-1}\left\{{\varsigma }_i\left(s\right)\right\}={ℒ}^{-1}\{{\displaystyle \sum _j\frac{{\displaystyle \prod _{k=1}^{{\mu }_{ij}}\left(s-s_{mk}^{oij}\right)}}{}}$