Tracking Control of Linear Systems

Appendix K

Proof of Theorem 145

Proof. What follows is from [148]. (i) The inverse of left Laplace transform of (7.16) (Subsection 7.1.2) gives the following:

$x\left(t\right)={ℒ}^{-1}\left\{{\text{X}}^{-}\left(s\right)\right\}={ℒ}^{-1}\left\{F_{ISOIS}\left(s\right){\left[I^{{-}^T}\left(s\right)⋮{\text{x}}_{0^{-}}^T\right]}^T\right\}=$

$=\frac{1}{2\pi j}{\displaystyle {\int }_{c-j\infty }^{c+j\infty }\left\{F_{ISOIS}\left(s\right){\left[{\text{I}}^{{-}^T}\left(s\right)⋮{\text{x}}_{0-}^T\right]}^T\right\}{e}^{st}ds=}$

$=\frac{1}{2\pi j}{\displaystyle {\int }_{c-j\infty }^{c+j\infty }\left\{{\displaystyle {\int }_{0^{-}}^{\infty }{ℒ}^{-1}\left\{F_{ISOIS}\left(s\right)\right\}e{-}^{s\tau }d\tau }\right\}\left[I^{{-}^T}\left(s\right)⋮{\text{x}}_{0^{-}}^T\right]e^{st}ds=}$

$={\displaystyle {\int }_{0^{-}}^{\infty }{ℒ}^{-1}\left\{F_{ISOIS}\left(s\right)\right\}\left[\frac{1}{2\pi j}{\displaystyle {\int }_{c-j\infty }^{c+j\infty }\left[I^{{-}^T}\left(s\right)⋮x_{0^{-}}^T\right]e^{s\left(t-\tau \right)}ds}\right]d\tau =}$

$={\displaystyle {\int }_{0^{-}}^{\infty }{ℒ}^{-1}\left\{F_{ISOIS}\left(s\right)\right\}\left[i^T\left(t-\tau \right)⋮\delta \left(t-\tau \right){\text{x}}_{0-}^T\right]}d\tau .$

This and (i) of Definition 144 (Subsection 7.6.2) imply

${\Phi }_{ISOIS}\left(t\right)={ℒ}^{-1}\left\{F_{ISOIS}\left(s\right)\right\},$

that is, the equation (7.114).

(ii) Left Laplace transform of (7.114) is the equation (7.115) ■