Design of a class of nonlinear controllers via state dependent Riccati equations

In this brief, infinite-horizon nonlinear regulation of second-order systems using the State Dependent Riccati Equation (SDRE) method is considered. By a convenient parametrization of the A(x) matrix, the state-dependent algebraic Riccati equation is solved analytically. As a result, the closed-loop system equations are obtained in analytical form. Global stability analysis is performed by a combination of Lyapunov analysis and LaSalle's Principle. Accordingly, a relatively straightforward condition for global asymptotic stability of the closed-loop system is derived. This is one of the first global results available for this class of systems controlled by SDRE methods. The stability results are demonstrated on an experimental magnetic levitation setup and are found to provide a great deal of flexibility in the control system design.

I. Introduction

After several decades of enlightening research, numerous techniques currently exist for the synthesis of control laws for nonlinear systems. These techniques include Jacobian linearization, gain scheduling, feedback linearization, sliding mode control, recursive backstepping, and adaptive control [6]. However, with many of the techniques, the actual control law is determined via some analysis method. This is particularly true for many Lyapunov-based techniques where the stability of the closed loop system is determined via a Lyapunov analysis. With these types of methods, the resulting control may not have a very intuitive design orientation for practical implementation. For example, if a Lyapunov-based control happens to result in a design that causes control saturation, the designer oftentimes will have to go through the analysis again to determine a more favorable control input. Currently, there does not seem to be a good approach to making relatively straightforward control design choices between allowable state error and control input effort for nonlinear systems.