The problem of optimal sampled-data vibration control for nonlinear systems with time delays and uncertainties is considered. With the purpose of simplifying the nonlinear optimal vibration control (NOVC) design, the original time-delay sampled-data system is converted into a discrete-time nondelayed system first, as well as the nonlinear and uncertain terms are treated as external excitations. Therefore, the design procedure for NOVC law is reduced and the successive approximation approach is sequentially developed in it. The obtained NOVC law is derived from a Riccati equation, a Stein equation, and sequences of adjoint vector difference equations. It is combined with a feedforward term, the nonlinearity and uncertainty compensator terms, and some control memory terms, which compensate for the effects produced by the disturbance, the nonlinearity and uncertainties, and the time delays. Moreover, the existence and uniqueness of NOVC law are proved and the stability of the closed-loop system is analyzed. In order to make the controller physically realizable, an observer is constructed and the corresponding dynamical control law is given. Furthermore, by this means, the NOVC law for a sampled-data quarter-car suspension model with actuator and sensor delays is designed. The results of numerical simulations illustrate that the NOVC gives satisfactory conclusions in effectiveness of suspension performance responses and feasibility of the proposed design approach.
The Gamma Stein equation and noncentral de Jong theorems
