Feedback synthesis by the robust criterion using Riccati equations
type: Item
instance of: scholarly article
title: Feedback synthesis by the robust criterion using Riccati equations
zbMATH Open document ID: 0933.93038
author: L. S. Chechurin
author: Anatoli A. Pervozvanski
published in: Automation and Remote Control
publication date: 2000-04-04T00:00:00Z
review text: 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. Inst…
Mathematics Subject Classification ID: 93B51
Mathematics Subject Classification ID: 93D09
Mathematics Subject Classification ID: 93B50
zbMATH DE Number: 1263158
zbMATH Keywords: robustness optimization
zbMATH Keywords: optimization of the margin of stability
zbMATH Keywords: transfer function
zbMATH Keywords: Nyquist stability criterion
zbMATH Keywords: sensitivity function
zbMATH Keywords: \(H_\infty\) norm
zbMATH Keywords: matrix-Riccati equations
reviewed by: Vadim Komkov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1263158