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Robust dissipativity of interval uncertain linear systems. (English) Zbl 1049.93020

The paper describes another interesting control theory application of the powerful “matrix cube theorem” proposed by A. Ben-Tal and A. Nemirovski in their excellent textbook [Lectures on modern convex optimization, SIAM, Philadelphia, PA (2001; Zbl 0986.90032)]. The matrix cube theorem is a central result in robust optimization, a promising new branch of mathematical programming which focuses on solving decision problems in the presence of modelling uncertainty. The matrix cube theorem consists in a conservative sufficient condition for feasibility of a linear matrix inequality (LMI) in the presence of structured interval coefficient uncertainty, an NP-hard problem in general. Most importantly, one can measure the amount of conservatism in the sufficient condition, and this measure is tight within an absolute constant factor (most of the time between 1.57 and 2). This is a striking result since, even though conservative sufficient LMI approximations are well known for most of the difficult (infinite-dimensional or non-convex) optimization problems faced by control theoreticians, very few tight measures of conservatism are available.
Here, the matrix cube theorem is applied to derive various results relevant to dissipativity theory in the presence of structured interval uncertainty. Specific applications of this general theory, developed by Willems, include Lyapunov stability and linear-quadratic control. Convincing numerical examples are described in Section 5: positive-real analysis and design for an RC bridge circuit, and linear-quadratic optimal control for a mechanical mass-spring system. The main limitation of the approach, in the reviewer’s opinion, is the use of a common Lyapunov dissipativity certificate shared by all the uncertain system instances, in the spirit of the so-called quadratic stability framework. One may therefore wonder whether similar results can be obtained for, e.g., parameter-dependent Lyapunov certificates. Another limitation of the approach is the size of the sufficient LMI conditions, which remains polynomial in the problem’s dimensions, but may prove impractically large in view of the current limitations in available semidefinite programming methods for solving LMI problems. This issue is carefully studied by the authors in Section 4.6 and several simplifying approximation and reduction schemes are proposed.

MSC:

93B35 Sensitivity (robustness)
93D09 Robust stability
90C22 Semidefinite programming
65G30 Interval and finite arithmetic
15A39 Linear inequalities of matrices

Citations:

Zbl 0986.90032
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