PERIODIC MOTION AND TESTS FOR ABSOLUTE STABILITY OF NONLINEAR NONSTATIONARY SYSTEMS

In [1-3] it is established that periodic motions occur for a boundary value of the parameter k which defines a sector of change of nonlinearities in the problem of absolute stability in second and third order systems. The use of the limit system together with the above result leads to effective necessary and sufficient conditions for absolute stability. The present paper contains a further generalization of this result for systems of arbitrary order. It is established that there are periodic motions close to the boundary value of the parameter k in systems with an invariant cone. The presence of such motions demonstrates one more property of absolutely stable nonlinear systems analogous to the properties of stationary linear asymptotically stable systems. It is shown that, for any control system of the type under consideration, there is a corresponding system with an invariant cone and equivalent to the original system in the sense of absolute stability.