Abstract
We present a sum of squares (SOS) method for the synthesis of nonlinear polynomial control systems. As an emerging numerical solution method in recent years, SOS targets polynomials as the research object. It guarantees that the polynomial we solve for is always nonnegative. In this paper, we give a generalized S-procedure to solve the SOS problem. As an illustration of how the SOS method can be used, the region of attraction (ROA) in a nonlinear polynomial system is analyzed in detail. The method of determining decision variables is given in the SOS problem. We discuss the determination and solution of set-containment constraints and the conservatism problem in solving the SOS problem. SOS provides a convenient numerical method to solve nonlinear problems that are not easy to solve analytically.
Highlights
Since the sum of squares (SOS) problem can be solved by SOSTOOLS, the main problem in the analysis and design of nonlinear systems is how to transform the unsolved problem into the optimization requirements and feasibility constraints shown in (8) and (9)
In this paper, we present a generalized S-procedure for SOS
Using the region of attraction (ROA) analysis of nonlinear polynomial systems, we illustrate the characteristics of SOS as well as how it works in each process, including the determination of decision variables and the solution of set containment problems
Summary
This growing interest is mainly because the SOS technology provides convex relaxation for many difficult problems, such as global constraints and Boolean optimization [17]–[19], and thereby serves as an effective solution to nonlinear system analysis and control design In essence, this toolbox is designed for the control discipline, because it starts from the construction of the Lyapunov function by using programs in analyzing the stability of nonlinear systems. SOS is used to solve nonnegative problems; the design of control system usually requires positive definiteness, while nonnegative and positive definiteness are quite different in numerical calculation It may involve a need, for example, for the Lyapunov function and its derivative, which means some inequality constraints related to each other, namely, VOLUME 8, 2020 set containment. This paper focuses on how to transform the design and analysis problems of nonlinear control systems into an SOS problem, and how to use S-procedure [32] to handle set containment problems in SOS design
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