The phenomenon of the second order steady modes in the relay control systems with time delay

Abstract

The higher order relay control systems with delay (HORCSD) in the input are considered. For the second order relay control systems with delay (SORCSD) the notations of the frequency, oscillatory solutions and the second order steady modes are introduced. It is shown that the frequency of any oscillatory solution of SORCSD after a finite time interval becomes a constant. The sufficient conditions are found under which there exists a solution of SORCSD with any nonnegative integer constant frequency which is called the steady mode. This means that any oscillatory solution coincides with some steady mode. It is shown that for any nonnegative frequency there exist the periodic steady modes of the autonomous SORCSD. It is justified that under some conditions any periodic solutions of the SORCSD with zero limit frequency are orbitally asymptotically stable. The theorem about existence and stability of the zero frequency steady modes of the singularly perturbed HORCSD is proved. It is shown that there exist the orbitally asymptotically stable zero frequency steady modes for the relay control systems of any order at list for the small value of parameter. This means that there exists some analogy in the behavior of the second order sliding modes systems and the properties of the steady modes in SORCSD. From the other hand the sliding modes with order of sliding more than second are unstable but there exist the stable zero frequency periodic steady modes in relay control systems with delay.