Abstract
The Hamilton–Jacobi–Issacs (HJI) inequality is the most basic relation in nonlinear H∞ design, to which no effective analytical solution is currently available. The sum of squares (SOS) method can numerically solve nonlinear problems that are not easy to solve analytically, but it still cannot solve HJI inequalities directly. In this paper, an HJI inequality suitable for SOS is firstly derived to solve the problem of nonconvex optimization. Then, the problems of SOS in nonlinear H∞ design are analyzed in detail. Finally, a two‐step iterative design method for solving nonlinear H∞ control is presented. The first step is to design an adjustable nonlinear state feedback of the gain array of the system using SOS. The second step is to solve the L2 gain of the system; the optimization problem is solved by a graphical analytical method. In the iterative design, a diagonally dominant design idea is proposed to reduce the numerical error of SOS. The nonlinear H∞ control design of a polynomial system for large satellite attitude maneuvers is taken as our example. Simulation results show that the SOS method is comparable to the LMI method used for linear systems, and it is expected to find a broad range of applications in the analysis and design of nonlinear systems.
Highlights
Space vehicles, underwater vehicles, and mechanical arms are often required to make large-angle and fast maneuvers during use
L2 gain control is known as nonlinear H∞ control [1, 2]. is is because when we go to the time domain, the H∞ norm, de ned in terms of the transfer function of a linear system, it would be a L2-induced norm and is called L2 gain in a nonlinear system
Recent years have seen the emergence of several sum of squares (SOS) methods for nonlinear H∞ control [14,15,16], but problems exist with the application of these methods. e existing SOS methods need some formulas in solving problems, so they will be reviewed later in Section 4, based on which we propose a new method for solving nonlinear H∞ control
Summary
Underwater vehicles, and mechanical arms are often required to make large-angle and fast maneuvers during use. If the polynomial of the corresponding system can be arranged in SOS form, it must be nonnegative It is a newly developed method, SOS has shown several unique advantages in some important applications, such as estimation of regions of attraction in nonlinear systems [6, 7], satellite attitude control under large maneuvers [8, 9], aircraft attitude control [10], nonlinear model predictive control [11], and stability analysis of time-delay systems [12, 13]. Vafamand et al [19] propose an approach that uniquely considers the stability of input saturated polynomial systems together with the nonlinear control law Another advantage of this approach over the recently guaranteed cost controller design methods is that it can solve the proposed conditions without relying on any iterative techniques. Vafamand and Khorshidi [20] propose a new polynomial observer and controller based on SOS. e proposed approach employs the polynomial representation and numerical SOS convex optimization technique to design a novel polynomial synchronizer for hyper (chaotic) systems. e focus of our paper is how to use the SOS method to define a nonlinear H∞ control law and design a nonlinear H∞ control for large satellite attitude maneuvers as an example to illustrate the use of this method
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