Almost disturbance decoupling with stability by dynamic output feedback: A sufficient condition (Q1105526)

This note presents a sufficient condition for the problem of almost disturbance decoupling with internal stability where one is only allowed to use the output for achieving the design goal. We will need the linear space \(D_ n:=\{x(\cdot):[0,\infty)\to {\mathbb{R}}^ n| x(\cdot)\) is piecewise continuous\} and for \(p\in [1,\infty)\) the normed space \(L^ p:= \{x(\cdot):[0,\infty)\to {\mathbb{R}}^ n| x(\cdot)\) is measurable, \(\| x(\cdot)\|_ p<\infty \}\), as well as \(D^ p:=L^ p\cap D_ n\). Consider the disturbed system (1) \(\dot x=Ax+Bu+Gv\), \(y=Cx\), \(z=Dx\) with \({\mathbb{R}}\)-matrices, where B, G and C, D have maximal column and row rank, respectively. Applying dynamic output feedback \(\dot w=Nw+My\), \(u=- Fw-Ly\) gives the closed-loop system \[ \left[ \begin{matrix} \dot x\\ \dot w \end{matrix} \right] = \left[ \begin{matrix} A-BLC & -BF \\ MC & N \end{matrix} \right] \left[ \begin{matrix} x \\ w \end{matrix} \right] + \left[ \begin{matrix} G \\ 0 \end{matrix} \right] v,\quad z = [D^ 0]\left[ \begin{matrix} x\\ w\end{matrix} \right]. \] The disturbance in (1)is almost decoupled from the output if given any \(\epsilon >0\) there are matrices \(N(\epsilon)\), \(M(\epsilon)\), \(F(\epsilon)\), \(L(\epsilon)\) for (1) implying the following properties for the closed-loop system (without output \(z(\cdot,\epsilon)):\) (a) The closed-loop system matrix is stable, (b) For every function \(v(\cdot)\in D^ p\) and initial condition 0 one has \(\| z(\cdot \epsilon)\|_ q\leq \epsilon \| v(\cdot)\|_ p\) where \(1\leq p\leq q\leq \infty.\) Let \({\mathbb{C}}^- := \{s\in {\mathbb{C}}| Re(s)<0\}\) and \({\mathbb{C}}^+ := {\mathbb{C}}/{\mathbb{C}}^-\). \({\mathcal V}^+\) is the largest subspace of Ker(D) such that there is an F with \((A-BF){\mathcal V}^-(D,A,B)\subset {\mathcal V}^-(D,A,B),\sigma((A-BF)| {\mathcal V}^-(D,A,B)) \subset {\mathbb{C}}^- .\) By duality we define \({\mathcal S}_+(C,A,G) := {\mathcal V}^+(G^ T,A^ T,C^ T)^{\perp}\). The set of transmission zeros of \((D,A,B)\) is denoted by \(\sigma(D,A,B)\). The main result of this paper is the following theorem: Theorem. For an (A,B)-stabilizable and (C,A)-detectable system (1) almost disturbance decoupling by dynamic output feedback with stability can be realized if the following conditions are satisfied: (a) \(Im(G)\subset {\mathcal S}_+(D,A,B),\) (b) \({\mathcal V}^+(C,A,G)\subset Ker(D),\) (c) \(\sigma(D,A,B)\cap \sigma(C,A,G)\cap {\mathbb{C}}^+=\phi\).