ON SOME EXTENSION AND APPLICATION OF POPOV CONDITION FOR STABILITY

Abstract

The thesis is divided into eight chapters. The first chapter contains (a) general introduction to the thesis, (b) statements of the problems, (c ) outline of the scope of the present work. A critical review of the results available elsewhere in the literature and their limitations are discussed in the second chapter. In the next five chapters the results of the present investigations are presented. The contents of these chapters are arranged in the following order: Chapter III deals with the application of the multivariable Popov condition to a class of single variable system. More specifically it has been applied to find the sufficient condition for stability of a Lur'e type system with two nonlinearities. It has been shown that if the second nonlinearity is considered as a rate feedback channel then the nonlinearity can be replaced by its sector gain for deriving the stability condition. This verifies Aizerman conjecture for the nonlinearity in the rate channel. In Chapter IV Popov type stability condition is obtained for Lur'e type system with hysteretic nonlinearity. It is shown that the nonlinearity can be decomposed into two single valued characteristics one of which appears as a rate feedback. Aizerman linearisation as -itestablished in Chapter III is then utilised for t'his 'system and stability condition is derived. Illustrative examples are worked out. In Chapter V stability condition for discrete data system with hysteretic nonlinearity has been obtained, applying the above decomposition technique. In Chapter VI Popov type stability condition is applied for finding the amplitude and frequency of osci ation for system having limit cycle oscillation. This is also extended to the double valued case. It may be mentioned here that this application is based on the presumption that the Popov condition is both necessary and sufficient for the specific system under consider ation. This in a way provides a test for the necessity of the condition, for the system in question. Two examples are discussed. Results obtained by other methods are also presented. In Chapter VII the problem of estimating bounds for the speed of transient response of Lur'e type system, is dealt with. If the system is found to be asymptotically stable through a Liapunov function V(x^ of the form V(x) = XTBX +B/Yf (y)dy, then the ratio Max t-V(x)/v(xV] o x and Min L-V(x)/V(x)] give correspondingly the estimatx ion of largest and smallest time constants for the tran sient response of the system. The extremal values of

the time constant are dependent on the eigen values of the linear subsystem matrix as well as on the constant 6 and k the sector gain of the nonlinear ity. The main features of this work are summarized in the concluding chapter.