Adaptive Inverse Optimal Control of a Magnetic Levitation System

Abstract

In recent years, control Lyapunov functions (CLFs) and CLF-based control designs have attracted much attention in nonlinear control theory. Particularly, CLF-based inverse optimal controllers are some of the most effective controllers for nonlinear systems [Sontag (1989); Freeman & Kokotovic (1996); Sepulchre et al. (1997); Li & Krstic (1997); Krstic & Li (1998)]. These controllers minimize a meaningful cost function and guarantee the optimality and a stability margin. Moreover, we can obtain the optimal controller without solving the Hamilton-Jacobi equation. An inverse optimal controller with input constraints has also been proposed [Nakamura et al. (2007)]. On the other hand, these controllers assume that the desired state of the controlled system is an equilibrium state. Then, if the controlled system does not satisfy the assumption, we have to use a pre-feedback control design method to the assumption is virtually satisfied. However, a pre-feedback control design causes the luck of robustness. This implies that a stability margin of inverse optimal controllers is lost. Hence the designed controller does not asymptotically stabilize the system if there exists a parameter uncertainty in the system. In this article, we study how to guarantee a stability margin when the pre-feedback controller design is used. We consider a magnetic levitation system as an actual control example and propose an adaptive inverse optimal controller which guarantees a gain margin for the system. The proposed controller consists of a conventional inverse optimal controller and a pre-feedback compensator with an adaptive control mechanism. By introducing adaptive control law based on adaptive control Lyapunov functions (ACLFs), we can successfully guarantee the gain margin for the closed loop system. Furthermore, we apply the proposed method to the actual magnetic levitation system and confirm its effectiveness by experiments. This article is organized as follows. Section 2 introduces some mathematical notation and definitions, and outlines the previous results of CLF-based inverse optimal control design. Section 3 describes the experimental setup of the magnetic levitation system and its mathematical model. In section 4, we design an inverse optimal controller with a prefeedback compensator for the magnetic levitation system. The problem with the designed controller is demonstrated by the experiment in section 5. To deal with the problem, we