An SOS-Based Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems

This paper deals with a sum-of-squares (SOS)-based control Lyapunov function (CLF) design for polynomial fuzzy control of nonlinear systems. The design starts with exactly replacing (smooth) nonlinear systems dynamics with polynomial fuzzy models, which are known as universal approximators. Next, global stabilization conditions represented in terms of SOS are provided in the framework of the CLF design, i.e., a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying Sontag\u27s control law, once a CLF for a given nonlinear system is constructed. Furthermore, semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions. Both global and semiglobal stabilization problems are formulated as SOS optimization problems, which reduce to numerical feasibility problems. Five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and SOS approaches

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Nonlinear analysis and control of a reaction wheel pendulum: Lyapunov-based approach

This paper presents a nonlinear analysis, control, and comparison of controllers based on the dynamical model of the reaction wheel pendulum (RWP) in a tutorial style. Classical methodologies such as proportional integral derivative (PID) control and state variables feedback control are explored. Lyapunov's method is proposed to analyze the stability of the proposed nonlinear controllers, and it is also used to design control laws guaranteeing globally asymptotically stability conditions in closed-loop. A swing up strategy is also included to bring the pendulum bar to the desired operating zone at the vertical upper position from an arbitrary initial location. Simulation results show that it is possible to obtain the same dynamical behavior of the RWP system adjusting the control gains adequately. All simulations were conducted via MATLAB Ordinary Differential Equation packages. © 2019 Karabuk Universit