Steady modes in the relay control systems with time delay

The multidimensional relay control systems with time delay are considered. We show that the time delay does not allow to realize an ideal sliding mode, but indicates oscillations, whose stability is determined by one discrete parameter-oscillation frequency. Any motion of scalar discontinuous delay system turns into a steady mode-a motion with a constant frequency. The existence of the periodic steady modes autonomous multidimensional relay delay control systems for any even frequency is proved. The sufficient conditions for stability of the zero frequency steady modes are discussed. The theorem about existence and stability of the zero frequency steady modes of the singularly perturbed multidimensional relay control systems with time delay is proved. The algorithm of asymptotic representation of periodic solutions using boundary layer method is suggested