Mixed H2/H∞ control for nonlinear time-varying systems with application to robotic systems

The first goal of the present paper is to discuss some sufficient conditions for the mixed H₂/H_∞ optimal control. with internal stability for nonlinear affine time-varying systems. The control design is based on the derivation of a state-feedback law that ensures the attenuation of any energy-bounded disturbance on the system output, while guaranteeing both the minimum output energy, when a "worst-case" disturbance is applied to the system, and some internal stability properties. The well-known theory of nonzero-sum Nash games is used to solve this problem. It is shown that this control law is obtained from the solutions of two cross-coupled Hamilton-Jacobi equations. The second goal of the paper is to illustrate the effectiveness of this approach for the H₂/H_∞ optimal control of some robotic systems, where explicit solutions to the cross-coupled Hamilton-Jacobi equations can be obtained.