Design Of A Proportional Integral Derivative-Like Continuous Sliding Mode Controller

A describing function-based design for the continuous sliding mode controller in the proportional integral derivative (PID) form (PID-CSMC) is provided. Two sets of the PID-CSMC gains are suggested. The Harmonic Balance is used to predict the amplitude and frequency of the main harmonic of chattering caused by the presence of fast-parasitic dynamics in the closed-loop. Predicted values of amplitude and frequency allow to compute the average power needed to maintain the trajectories of the system into real sliding modes. The methodology for selection of suboptimal PID-CSMC gains consists of amplitude of chattering and average power minimization, taking into account the presence of a critically damped actuator parameterizing the effects of parasitic dynamics. A novel homogeneous Lyapunov function proves that suggested sets of the PID-CSMC gains ensure, in theory, finite-time stability of the perturbed double integrator without parasitic dynamics, and exact compensation of Lipschitz perturbations. On the other hand, the Loeb's criterion allows to ensure that the PID-CSMC with suggested gains generates orbitally asymptotically stable fast-oscillations in the presence of fast-actuators.