Pole/variant zero placement by periodic output-feedback for multivariable systems (Q580276)

The plant considered is described by \(x_{i+1}=Ax_ i+Bu_ i\), \(y_ i=Cx_ i\), where \(u_ i\), \(y_ i\in R^ m\). \(\{\) C,A,B\(\}\) is assumed to be reachable and observable. Let p be the controllability index. Periodic output-feedback is defined by \(u_{ip+j}=v_{ip+j}+F_ jy_{ip}\). Then, a closed-loop system \(x_{k+1}=\bar Ax+R\bar v_ k\) is obtained which has a sampling interval equal to p times the original sampling period, where R is the controllability matrix, and \(\bar A=A^ p+RFC\), where \(F=[F^ T_{p-1}...F_ 0^ T]^ T.\) Transmission zeros are invariant under the output feedback, but poles may be assigned arbitrarily. In the first, it is shown that the i-th row of the closed-loop transfer matrix \(C(I_ nz-\bar A)^{-1}\bar B\), where \(\bar B=A^{p-1}B\), is independent of the i-th column of F. The desired closed-loop transfer matrix \(G_ d(z)\) must have desired distinct poles \(\{z_ i\}\) (not equal to transmission zeros), variant zeros and the same transmission zeros as the plant. Sufficient conditions for the pole assignment to be achieved are \[ 1)\quad rank\left[ \begin{matrix} z_ iI_ n- A^ p\\ -C\end{matrix} \quad \begin{matrix} \bar B\\ 0\end{matrix} \right]=n+m,\quad i=1,...,n. \] 2) there exists a modal vector \(e_ i\) such that \(e_ i^ TG_ d^{-1}(z_ i)=0\), \(i=1,...,n\). 3) det \(V\neq 0\), where \(V^ T=[v_ 1\) \(v_ 2...v_ n]\) and \[ [v_ i^ T\quad w_ i^ T]\left[ \begin{matrix} z_ iI_ n-A^ p\\ -C\end{matrix} \quad \begin{matrix} \bar B\\ 0\end{matrix} \right]=[0\quad e^ T_ i],\quad i=1,...,n. \] An algorithm to find \(F_ 0,F_ 1,...,F_{p-1}\) is presented. The reviewer is not sure that the proposed method is more efficient than conventional estimated state variable feedback or frequency domain pole assignment method.