Tracking Control of Linear Systems

Appendix J

Proof of Theorem 142

Proof. [148]. (i) The inverse of left Laplace transform of Y⁻(s) leads to the inverse of left Laplace transform of (7.3) (Subsection 7.1.1):

$\begin{array}{l}Y\left(t;Y_0^{v-1};I\right)={ℒ}^{-1}\left\{Y^{-}\left(s\right)\right\}=\frac{1}{2\pi j}{\displaystyle {\int }_{c-j\infty }^{c+j\infty }{Y}^{-}\left(s\right)e^{st}=}\\ =\frac{1}{2\pi j}{\displaystyle {\int }_{c-j\infty }^{c+j\infty }\left\{F_{IO}\left(s\right){\left({\left(I-\left(s\right)\right)}^T⋮{\left(I_{0-}^{\mu -1}\right)}^T⋮{\left(Y_{0^{-}}^{v-1}\right)}^T\right)}^T\right\}{e}^{st}ds,}\end{array}$

where $j=\sqrt{-1}$. Let

$\Xi \left(t\right)=\left[{\Xi }_I\left(t\right)⋮{\Xi }_{io}\left(t\right)⋮{\Xi }_{yo}\left(t\right)\right]={ℒ}^{-1}\left\{F_{IO}\left(s\right)\right\},$ J.1

$\begin{array}{l}\left[{\Xi }_I\left(t\right)⋮{\Xi }_{io}\left(t\right)⋮{\Xi }_{yo}\left(t\right)\right]={ℒ}^{-1}\left\{\left[G_{IO}\left(s\right)⋮G_{IO}{io}_{}\left(s\right)⋮G_{IOyo}\left(s\right)\right]\right\},\\ ℒ\left\{\left[{\Xi }_I\left(t\right)⋮\Xi io\left(t\right)⋮{\Xi }_{yo}\left(t\right)\right]\right\}=\left[G_{IO}\left(s\right)⋮G_{IOio}\left(s\right)⋮G_{IOyo}\left(s\right)\right],\end{array}$ J.2

which transforms the preceding result as follows:

$$\begin{array}{rl}\mathbf{Y}(t;{\mathbf{Y}}_0^{\nu -1};\mathbf{I})=& \frac{1}{2\pi j}{\int }_{c-j\mathrm{\infty }}^{c+j\mathrm{\infty }}\left\{\begin{array}{c}[G_{IO}(s)⋮G_{\text{IOio}}(s)⋮G_{\text{IOyo}}(s)]\\ \bullet {\left[{({\mathbf{I}}^{-}(s))}^T⋮{\left({\mathbf{I}}_{0-}^{\mu -1}\right)}^T⋮{\left({\mathbf{Y}}_{0-}^{\nu -1}\right)}^T\right]}^T\end{array}\right\}{e}^{st}ds=\\ =\frac{1}{2\pi j}{\int }_{c-j\mathrm{\infty }}^{c+j\mathrm{\infty }}\left\{\left[\begin{array}{c}{G}_{IO}(s){\mathbf{I}}^{-}(s)+\\ +G_{IOio}(s){\mathbf{I}}_{0-}^{\mu -1}+\\ +G_{IOyo}(s){\mathbf{Y}}_{0-}^{\nu -1}\end{array}\right]\right\}{e}^{st}ds=\end{array}$$

$=\frac{1}{2\pi j}{\displaystyle {\int }_{c-j\infty }^{c+j\infty }{G}_{IO}\left(s\right)}{I}^{-}\left(s\right)e^{st}ds+$

$+\frac{1}{2\pi j}{\displaystyle {\int }_{c-j\infty }^{c+j\infty }[G_{IOio}\left(s\right){\text{I}}_{0^{-}}^{\mu -1}+G_I}$