Steady modes and sliding modes in the relay control systems with time delay

The relay control systems with time delay are considered. We show that the time delay does not allow to realize an ideal sliding mode, but implies 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. Such steady modes have all properties of sliding modes. The problem of stability of steady modes in three main cases autonomous, quasiautonomous and periodic is considered. It is shown that in autonomous and quasiautonomous cases only steady modes with zero frequency are stable. At the same time this modes are non-asymptotically stable. The structural stability of steady modes in the periodic case is investigated. The resonances are singled out. The existence of the slow stable periodic solution of the multidimensional singularly perturbed relay system with time delay is proved. At last, we design a direct adaptive control of relay type with time delay that extinguishes parasite auto-oscillations in this model