Tracking Control of Linear Systems

Appendix D

Proof of Theorem 64

Proof. What follows is from [148].

Necessity. Let a vector function I*(.) be nominal for the IO plant (2.1), i.e., for (2.9), relative to its desired response Y_d(.). Definition 45 (Subsection 3.3.1) holds. It and (2.1), i.e., (2.9), imply

${\displaystyle \sum _{k=0}^{k=\nu }{A}_k}{Y}_d^{(k)}(t)={\displaystyle \sum _{k=0}^{k=\mu }{B}_k}{I}^{*(k)}(t),\forall t\in {\mathfrak{T}}_0,$

$A^{(\nu )}{Y}_d^{\nu }(t)=B^{(\mu )}{I}^{*\mu }(t),\forall t\in {\mathfrak{T}}_0.$

These equations are (3.9) and (3.10) in another forms, respectively. Their Laplace transforms solved in I*(s) are given in (3.11), (3.12), respectively. Since they are solvable in I*^(k)(.), it follows that the conditions 1) and 2) hold.

Sufficiency. Let the conditions 1) and 2) be valid. The input vector function I*(.) to the IO plant (2.1), i.e., (2.9),

$A^{(\nu )}{Y}^{\nu }(t)=B^{(\mu )}{I}^{*\mu }(t),\forall t\in {\mathfrak{T}}_0,$

satisies (3.9), hence ...