Asymptotic optimal control of uncertain nonlinear Euler–Lagrange systems

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 systems is challenging. Previous efforts have utilized feedback linearization methods which assume exact model knowledge, or have developed neural network (NN) approximations of the HJB value function. The current effort builds on our previous efforts to illustrate how a NN can be combined with a recent robust feedback method to asymptotically minimize a given quadratic performance index as the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically track a desired time-varying trajectory despite general uncertainty in the dynamics. A Lyapunov analysis is provided to examine the stability of the developed optimal controller.

Adaptive Inverse Optimal Neuromuscular Electrical Stimulation

“…While this work makes a contribution as the first analysis to explore an optimal controller for NMES, given a nonlinear uncertain muscle model, the development is limited by the restriction to use a derived cost functional. Future efforts will explore the use of implicit learning methods to solve for a value function that minimizes the cost of a given desired functional versus a derived cost, as in our preliminary work for generic nonlinear systems in [44].…”

Adaptive Neural PD Control With Semiglobal Asymptotic Stabilization Guarantee

“…Remark 4: There already exist several continuous control strategies to guarantee global or semiglobal stability of linearly parameterized NN approximation-based adaptive control systems, including the integrated RISE feedback and NN feedforward control [18]- [21], the adaptive NN feedforward control [28], and the variable-gain PD control [30], [31]. Among these approaches, the variable-gain PD control is relatively simple but effective since it does not complicate the control structure.…”

“…To make the NN approximation in Section V-A applicable, x ∈ D x should be ensured such that x e ∈ xe . Choose a Lyapunov function candidate V s (e n ) = e 2 n /2 (21) for the system given by (4). Then, the following result is provided to solve this issue.…”

“…By virtue of a second-order sliding-mode control technique, adaptive approximation-based continuous control strategies using robust integral of the sign of the error (RISE) feedback and NN feedforward were developed to achieve semiglobal asymptotic tracking for various types of uncertain nonlinear systems in [18]- [21], where RISE feedback is applied to compensate for transient-state uncertainties, and NN feedforward is utilized to cancel out steady-state plant dynamics. However, the increase of the plant order during control design in these approaches results in several drawbacks as follows:…”

An Effective Approach Control Scheme for the Tethered Space Robot System

“…[28] Wang et al introduced an adaptive dynamic programming algorithm for the optimal control of the non-affine nonlinear discrete-time systems. [29] Keith Dupree used the implicit learning capabilities to learn the dynamics asymptotically and studied the optimal control of uncertain nonlinear Euler-Lagrange systems. [30] Wu et al presented a min-max optimal control of linear systems with uncertainty and terminal state constraints, which is solved through solving a sequence of semi-definite programming problems.…”