Tracking Control of Linear Systems

Appendix F

Proof of Theorem 72

Proof. [148]. Necessity. Let [I*(.), X*(.)] be a nominal functional (input and state) vector pair for the ISO plant (2.29), (2.30) relative to its desired response Y_d (.). The plant is in its desired regime relative to Yd(.). Definition 71 shows that [I(.), X(.)] = X*(.)] implies Y(.) = Y_d(.). This and the ISO model (2.29), (2.30) yield the following equations:

$\begin{array}{l}\frac{dX*\left(t\right)}{dt}=AX*\left(t\right)+BI*\left(t\right),\forall t\in {𝔗}_0,\\ Y\left(t\right)=Y_d\left(t\right)=CX*\left(t\right)+DI*\left(t\right),\forall t\in {𝔗}_0,\end{array}$ F.1

which can be easily set in the form of the equations (3.17), (3.18). Application of Laplace transform together with its properties sets the equations (3.17), (3.18) into (3.19).

Sufficiency. We accept that all the conditions of the theorem are valid. We chose [I(.), X(.)] = [I*(.), X*(.)]. ...