Tracking Control of Linear Systems

Appendix G

Proof of Theorem 91

Proof. We refer to Theorem 90 and its Equations (3.33),

$\begin{array}{l}\left[\begin{array}{cc}-B_P& \left(s{I}_n-A_P\right)\\ H_P& C_P\end{array}\right]\left[\begin{array}{c}U*\left(s\right)\\ X_P^{\text{*}}\left(s\right)\end{array}\right]=\left[\begin{array}{ccc}{L}_P& I_n& O_n,N\\ -D_P& O_n& I_N\end{array}\right]\left[\begin{array}{c}D\left(s\right)\\ X_{P0}^{*}\\ Y_d\left(s\right)\end{array}\right],\\ {\text{X}}_{P0}^{*}\in {\Re }^n.\end{array}$ G.1

The system matrix

$\left[\begin{array}{cc}-B_P& \left(s{I}_n-A_P\right)\\ H_P& C_P\end{array}\right]\in {ℭ}^{\left(n+N\right)\times \left(n+r\right)}.$

Case 1: N > r. There is not a solution $\left[{\text{U}}^{*}\left(s\right),X_P^{*}\left(s\right)\right]$ to (G.1). For details see Appendix E.

Case 2: N = r. For the existence of the unique solution to (G.1) it is necessary and sufficient that

$\exists s\in ℭ\Rightarrow \det \left[\begin{array}{cc}-B_P& \left(s{I}_n-A_p\right)\\ H_P& C_P\end{array}\right]\ne 0.$

The solution is determined by

$\begin{array}{l}\left[\begin{array}{l}U*\left(s\right)\\ X_P^{*}\left(s\right)\end{array}\right]={\left[\begin{array}{cc}-B_P& \left(s{I}_n-A_P\right)\\ H_P& C_P\end{array}\right]}^{-1}\left[\begin{array}{ccc}{L}_P& I_n& On,N\\ -D_P& O_n& I_N\end{array}\right]\left[\begin{array}{l}D\left(s\right)\\ X_{P0}^{*}\\ Y_d\left(s\right)\end{array}\right],\\ X_{P0}^{*}\in {\Re }^n.\end{array}$ G.2

Case 3: N < r. For the existence of the solution to (G.1) it is necessary and sufficient ...