An LMI-based iterative algorithm for state and output feedback stabilization of discrete-time Lur’e systems

Abstract

This paper is concerned with the problem of static output-feedback stabilization of discrete-time Lur'e systems. The control law feedbacks both the output and the nonlinearity. By using a quadratic Lyapunov function, new design conditions are provided in terms of new sufficient design linear matrix inequalities where the control gains appear affinely. Using some relaxations, the search for the stabilizing control gains is performed through an iterative algorithm. The approach can be considered as more general than the existing ones thanks to the fact that the gains are treated as decision variables in the optimization problem. Therefore, the approach can handle state or output feedback indistinctly, and can include magnitude or structural constraints (such as decentralization) on the gains. Numerical examples illustrate that the proposed method can provide less conservative results when compared with other techniques from the literature.

Control design of uncertain discrete‐time Lur'e systems with sector and slope bounded nonlinearities

“…The treatment of a more general class of Lur'e systems, where only the sector bound condition (2) is assumed for the nonlinearity, can be easily obtained from the conditions of Theorem 1 by fixing $$$ {R}_1={R}_2=0 $$$. In this case, the resulting condition is an alternative to Reference 19, Theorem 1 (a previous conference version of this article), with the advantage of dealing with a feedforward matrix $$$ {D}_{\phi } $$$ in the model.…”

“…With respect to the implementation of the conditions of Algorithm 1, Theorem 5 is used whenever possible. In this case, when investigating sector bounded nonlinearity through Algorithm 1, the gains provided by BPOV20 19 are used. For slope bounded nonlinearity, the gains provided by Algorithm 1 for the sector bounded case (always better in terms of larger values of $$$ \Omega $$$ than the ones from BPOV20) are employed.…”

“…The first one takes into account only a sector limitation (the system certified as stable in open‐loop for $$$ \Omega \le 0.62 $$$ using the quadratic Lyapunov function from Reference 29). Algorithm 1 is compared with BPOV20, 19 LJD15, 17 and KB17 18 (some methods are extended to cope with polytopic uncertainty, basically defining the LMIs at the vertices of the polytope). The second case considers both sector and slope limitations with $$$ \Omega =\Lambda $$$ (the system is certified as stable in open‐loop using the method from Reference 13 for $$$ \Omega =\Lambda \le 3.85 $$$).…”

“…Note that Algorithm 1 is the only method (among all strategies considered) capable to deal with output‐feedback when both sector and slope ($$$ \Omega =\Lambda $$$) limitations are imposed. The iterative procedure of BPOV20 19 and Algorithm 1 are tested with $$$ i{t}_{\mathrm{max}}=15 $$$. The results considering full state‐ or output‐feedback gains are presented in Table 1 and, clearly, the values of $$$ \mu $$$ and $$$ \Omega $$$ provided by Algorithm 1 are the least conservative ones.…”

“…Numerical examples are presented to illustrate the effectiveness of the approach. The article can be viewed as a generalization of the results in Reference 19, where the stabilization of a Lur'e system with sector bounded nonlinearities has been studied with quadratic Lyapunov functions.…”