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Optimization and pole assignment in control system design. (English) Zbl 1031.93089

Consider the linear controllable multi-input system \(x'(t)= Ax (t)+ Bu(t)\), where \(x(t)\in \mathbb{R}^n\), \(u(t)\in \mathbb{R}^m\). Given a collection \(\Omega= \{\lambda_1, \dots, \lambda_n\}\) of \(n\) (complex) numbers, symmetric relative to the real axis, the problem is to determine a gain matrix \(K\) such that the spectrum of the closed-loop system matrix \(A+BK\) is equal to \(\Omega\). The general solution of this problem is an \(n(m-1)\)-dimensional variety in \(\mathbb{R}^{m \times n}\). The paper deals with several optimization techniques to find a particular solution \(K\) satisfying additional requirements, e.g. \(\text{cond}(X)= \|X\|\|X^{-1}\|\to\min\), \(\|X\|+ \|X^{-1} \|\to \min\) or \(0.5\|X\|^2 -\ln|\det(X)|\to \min\), where \(X\) is the modal matrix of \(A+BK\) (all these techniques require that the desired poles are pairwise disjoint – a restriction that excludes some important design problems such as e.g. the dead-beat control of discrete-time systems when all poles are set to zero). In particular the convergence of the corresponding minimization procedures is studied.
Reviewer’s remark: When dealing with the so-called “robust” pole assignment, one must distinguish between the influence of three different (and generally independent) factors: the sensitivity of the gain matrix \(K\) to perturbations in the data \(A,B,\Omega\), the size \(\|K\|\) of the gain matrix and the sensitivity of the eigenstructure of \(A+BK\), see [P. Petkov, N. Christov and the reviewer, Comments on “On computational algorithms for pole assignment”, IEEE Trans. Autom. Control 33, 892-893 (1988)].

MSC:

93B55 Pole and zero placement problems
93B40 Computational methods in systems theory (MSC2010)
65K10 Numerical optimization and variational techniques
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