Inverse optimal adaptive control of a nonlinear Euler-Lagrange system, part I: Full state feedback

Abstract

A sufficient condition to solve an optimal control problem is to solve the Hamilton-Jacobi-Bellman (HJB) equation. However, finding a value function that satisfies the HJB equation for a nonlinear system is challenging. Inverse optimal control is an alternative method to solve the nonlinear optimal control problem by circumventing the need to solve the HJB equation. Inverse optimal adaptive control techniques have been developed that can handle structured (i.e., linear in the parameters (LP)) uncertainty for a particular class of nonlinear systems that do not include Euler-Lagrange systems with an uncertain time-varying inertia matrix. In this paper, an inverse optimal adaptive controller is developed to asymptotically minimize a meaningful performance index while the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically track a desired time-varying trajectory despite LP uncertainty. A Lyapunov analysis is provided to examine the stability of the developed optimal controller, and preliminary simulation results illustrate the performance of the controller.

Inverse optimal adaptive output feedback control of a class of Euler–Lagrange systems: A nonlinear filter based approach

“…This approach is used in various controllers where a control Lyapunov function is defined in conjunction with a controller to ensure stability of the closed-loop system. 6–9 It is then proven that the designed controller is optimal with respect to a meaningful cost function that increases with increasing norms of state and control input vector. 10…”

“…When full-state information is not available, inverse optimal output feedback controllers are preferred. 7,15–18 In Ezal et al, 16 nonlinear systems with input disturbance is considered and an output feedback controller is presented with near-optimal and semi-global inverse optimal result. In Perez and Haimovich, 17 another near-optimal solution is presented.…”

“…In this study, the unknown states were estimated but used in the control input replacing actual unknown states. In Dupree et al, 7 the results in Dupree et al 15 were extended with an output feedback structure using a nonlinear filter. In our recent work, 18 we designed an observer based inverse optimal output feedback controller where semi-global asymptotic stability and inverse optimality were proven.…”

Inverse optimal adaptive output feedback control of Euler-Lagrange systems: A variable structure observer based approach

“…Aside from finding a proper control-Lyapunov function, the inverse optimal approach is easier compared with the forward optimal approach as it does not involve the solution of HJB equation for a highly nonlinear system which may or may not exists. Some applications of inverse optimal controller approach to nonlinear systems can be found in [1]- [5].…”

“…In [8], an inverse optimal adaptive backstepping technique is applied to the design of a pitch control law for a surface-to-air nonlinear missile model with a constant inertia matrix [2]. In [1], and [2], the authors initially developed an inverse optimal adaptive controller for an Euler-Lagrange system with full state feedback, then extended their result to output feedback type using a nonlinear filter based approach.…”