Tracking Control of Linear Systems

Appendix L

Proof of Theorem 149

Proof. What follows is from [148]. (i) The inverse of left Laplace transform of (7.47) (Subsection 7.2.2) gives the following:

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

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

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

$={\displaystyle {\int }_{0^{-}}^{\infty }{ℒ}^{-1}\left\{F_{ISO}\left(s\right)\right\}\left[\frac{1}{2\text{π}j}\leftrightarrow {\displaystyle {\int }_{c-j\infty }^{c+j\infty }\left[{\text{I}}^{\text{-}T}\left(s\right)⋮{\text{x}}_{0^{-}}^T\right]e^{s\left(t-\text{τ}\right)}ds}\right]d\text{τ}}=$

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

This and (i) of Definition 146 (Subsection 7.6.2) imply

${\text{Φ}}_{\text{ISO}}(t)={ℒ}^{-1}\left\{F_{\text{ISO}}(s)\right\},$

that is, the equation (7.118).

(ii) Left Laplace transform of (7.118) is the equation (7.119) ■