Absolute stability of dynamic system containing non-linear functions of several state variables

Abstract

Recently the convenient criterion of Popov and its extensions have been developed to examine the absolute stability of a dynamic system with a special class of non-linearities. Each non-linear element of this system is a function of a linear combination of the state variables and possibly a function of the independent variable, time. In this paper, a system of non-linear differential equations of a general nature is considered. The equations, in general, contain multiple non-linearities where each non-linear element is a function of several linear combinations of the state variables, and of the independent variable, time. The equations in addition have forcing terms. These equations can be written in the form: x ̇ =A x+B f(δ,t) δ=C x+ r(t) where x, σ, and r are n-vectors; A, B, and C are n × n constant matrices; and f is n-vector whose transpose is denoted by f′(δ,t)=[f 1(δ 1,δ 2,…,δ n,t) , f 2(δ 1,δ 2,…,δ n,t),…,f n(δ 1,…,δ n,t] . A special case of such a non-linear system is the system which contains a “Product Type” non-linearity. Sufficient conditions for absolute stability of these systems are derived. These conditions are in the frequency domain and are easy to verify. Two examples are given to show the practical application of the results. The results are applicable to the corresponding class of integral equations as well as differential equations. The proof of the theorem is developed along the lines of the proof of Popov's theorem and the lemmas are basically extensions of the lemmas developed for proof of Popov's theorem.