Sliding control of discretized continuous systems via the Euler operator

The idea of “sliding mode control” (SMC) as a robust control method is utilized to control systems whose dynamics can be described by x(t)=(A+ΔA)x(t)+(B+ΔB)u(t)+d(t) where ΔA, ΔB and d(t) characterize unknown plant parameters and unexpected disturbances respectively. The design is described in the discrete form using the Euler operator, which approaches the Laplace operator as the sampling interval approaches zero. Uncertainty transformation from continuous to discrete time domain, which has not been adequately addressed in the control literature, is included throughout the analysis. Selection of the sliding surface, s=0, is not arbitrary but is dependent on the nominal parameters of the system and on the chosen sampling time. The control is constructed based on the estimation of system perturbations using time delay control. This methodology, towards SMC design, offers a robust control action which is continuous with possibly lower gains than the conventional “bang-bang” input, thus avoiding undesired chattering. Such issues are addressed and the asymptotic stability of the controlled system is analyzed. Finally the proposed controller is studied through simulations and experiment on a direct drive NSK robot arm. The results indicate that the controller is robust to parameter variations and unexpected disturbances