Tracking Control of Linear Systems

Appendix C

From ISO system to IO system

The ISO system (2.29), (2.30) (Section 2.2),

$\frac{dX\left(t\right)}{dt}=AX\left(t\right)+BI\left(t\right),t\in 𝔗,$ (C.1)

$Y\left(t\right)=CX\left(t\right)+DI\left(t\right),t\in 𝔗,$ (C.2)

can be transformed into the IO system as follows. Laplace transform of (C.1), (C.2) under all zero initial conditions reads

$\begin{array}{c}(sI-A)X(s)=BI(s)\Rightarrow \\ X(s)={(sI-A)}^{-1}BI(s)\\ Y(s)=\left[C{\left(sI-A\right)}^{-1}B+D\right]I(s)=\\ Y(s)=\left[\begin{array}{c}\frac{Cadj (sI-A) B}{\det (sI-A)}+D\end{array}\right]I(s)\\ \Rightarrow \\ \left[\det \left(sI-A\right)\right]Y(s)=\left[Cadj\left(sI-A\right)B+D\det \left(sI-A\right)\right]I(s).\end{array}$

For the sake of simplicity let

$f(s)=\det (sI-A)={\displaystyle \sum _{i=0}^{i=n}{c}_i}{s}^i,c_n=1,$

$Cadj(sI-A)B+D\det (sI-A)={\displaystyle \sum _{i=0}^{i=n}{K}_i}{s}^i,K_i\in {\Re }^{N\times M}.$

Altogether,

$\begin{array}{c}\left({\displaystyle \sum _{i=0}^{i=n}{c}_i}{s}^i\right)Y(s)=\left({\displaystyle \sum _{i=0}^{i=n}{K}_i}{s}^i\right)I(s)\Rightarrow \\ {\displaystyle \sum _{i=0}^{i=n}{c}_i}\left[s^{i}Y(s)\right]={\displaystyle \sum _{i=0}^{i=n}{K}_i}\left[s^{i}I(s)\right].\end{array}$

Inverse Laplace transform of the last equation for all zero initial conditions reads

${\displaystyle \sum _{i=0}^{i=n}{c}_i}{Y}^{(i)}(t)={\displaystyle \sum _{i=0}^{}}$