Feedback synthesis by the robust criterion using Riccati equations
The authors consider the optimization of the margin of stability for a linear system's feedback by studying the transfer function. A presentation of the present state of research in this topic is given by \textit{H. Kwakernaak} [IEEE Trans. Autom. Control AC-30, 994-1004 (1985; Zbl 0576.93022)]. Suppose that the transfer function of the system is given by fractional matrix representation: \(H(s)= M^{-1}(s) N(s)\), where \(M\), \(N\) are polynomial matrices. \(M\) and \(N\) are known only approximately and may be written as: \(M(s)= M_0(s)+\Delta_M(s)\), and \(N(s)= N_0(s)+\Delta_N(s)\), where \(\Delta_M(s)\), \(\Delta_N(s)\) denote the maximum errors from exact values \(M_0(s)\), \(N_0(s)\). We have \(H_0(s)= M^{-1}_0(s) N_0(s)\). The transfer function \(G= (G_0(s)+ \Delta_G)\) is modified by the feedback loop resulting in a familiar representation \(H= [I- GF]^{-1}G= [I- G_0F- \Delta_GF]^{-1}(G+\Delta_G)\). Denoting the transfer function of the feedback loop by \(K\), the authors use a stability criterion derived by \textit{D. C. McFarlane} and \textit{K. Glover} [Robust controller using normalized coprime factor plant descriptions. Lect. Notes Control Inf. Sci. 138. Berlin, Springer-Verlag (1990; Zbl 0688.93044)], which is a consequence of the Nyquist stability criterion, implying that the sensitivity function \(S_0(s)\) obeys the equality: \(S_0(s)= [I- H_0(s)K(s)]^{-1}\). A class of optimization problems is equivalent to minimization of the quantity: \(1-\left(\sup_\omega\text{ norm }\left[\begin{smallmatrix} S_0M^{-1}_0 & U(j\omega)\\ KS_0 M^{-1}_0 & V(j\omega)\end{smallmatrix}\right]\right)\). Here \(U(s)/V(s)\) denotes the output which is also fed back into the feedback loop. Thus the main problem of optimization consists of finding \(K(s)\) such that the \(H_\infty\) norm of the matrix given above inside the square brackets is maximized. The authors discuss numerical difficulties in realizing this optimum, due to singularities present in this formalism. Instead they bound the \(U\) and \(V\) functions and produce a pair of quadratic matrix equations back in the state space, which include a scalar parameter \(\gamma\), following the formalism of \textit{J. C. Doyle}, \textit{K. Glover}, \textit{P. P. Khargonegar} and \textit{B. Francis} [IEEE Trans. Autom. Control 34, No. 8, 831-847 (1989; Zbl 0698.93031)]. A program for numerical solution of these matrix-Riccati equations ends this paper.
scientific article; zbMATH DE number 1263158
scientific article