Control design for discrete‐time bilinear systems using the scalarized Schur complement

Abstract

Summary In this paper, controller design for discrete‐time bilinear systems is investigated by using sum of squares programming methods and quadratic Lyapunov functions. The class of rational polynomial controllers is considered, and necessary conditions on the degree of controller polynomials for quadratic stability are derived. Next, a scalarized version of the Schur complement is proposed. For controller design, the Lyapunov difference inequality is converted to a sum of squares problem, and an optimization problem is proposed to design a controller, which maximizes the region of quadratic stability of the bilinear system. Input constraints can also be accounted for. Copyright © 2017 John Wiley & Sons, Ltd.

Three‐stage least squares‐based iterative estimation algorithms for bilinear state‐space systems based on the bilinear state estimator

“…• The 3S-LSI algorithm saves 38.5% ≈ 6 5.12568×10 6 computational cost compared with the LSI algorithm in terms of Table 3.…”

“…4,5 As a special class of nonlinear systems, the bilinear systems exist widely in industry, and the control and identification of the bilinear systems have attracted great attention in the control community. [6][7][8] Their relatively simple mathematical structures make the bilinear models a bridge between linear and nonlinear models, providing an approach to deal with complex nonlinear control problems. 9,10 Furthermore, bilinear systems arise in many engineering fields (eg, nuclear fission, heat transfer, automobile braking systems) and nonengineering fields.…”

“…Building mathematical models of systems is basic for the system analysis and system control 1‐3 and the mathematical models have important applications in engineering systems 4,5 . As a special class of nonlinear systems, the bilinear systems exist widely in industry, and the control and identification of the bilinear systems have attracted great attention in the control community 6‐8 . Their relatively simple mathematical structures make the bilinear models a bridge between linear and nonlinear models, providing an approach to deal with complex nonlinear control problems 9,10 .…”

Global feedback stabilization of discrete‐time bilinear systems

“…Tognetti et al [20] designed a bilinear dynamic output feedback controller ensuring local exponential stability of the origin. Vatani et al [21] conceived a feedback controller for discrete time bilinear systems by using sum of squares programming methods and quadratic Lyapunov functions. Khlebnikov [15] considered the problem of stabilization of discrete bilinear control systems using the linear matrix inequality technique and quadratic Lyapunov functions.…”

A piecewise affine control Lyapunov function for robust control

“…In the case of linear optimal control, suitable quadratic cost functions represent common control Lyapunov candidates, see for instance [1], [8], [10]. Also, polynomial control Lyapunov functions are used for linear/bilinear systems with nonlinear control laws in [37], [38]. Moreover, model predictive control (MPC) usually employs infinite horizon quadratic control Lyapunov functions, as shown in [9], [22], [24].…”