A notion of solutions and equilibrium points for non smooth systems

Steady state operation of engineering systems is generally around a stable equilibrium point. The concept of an equilibrium point for a dynamic system is well-established as a constant solution in the time domain. In other words, the system is at steady state at an equilibrium point when all time derivatives are equal to zero. In numerous text books, the standard mathematical definition of an equilibrium point x* for the dynamical system x˙=f(x) is that it satisfies the equation f(x*)=0. But it has been shown that this definition may be inadequate from an engineering point-of-view. The aim of the paper is to formally propose a rigorous definition for solutions and equilibrium points, which is also intuitively appealing. Here the concept of an equilibrium point is developed mathematically as a constant solution for the dynamic system, by first precisely defining the notion of an order of a solution. The basic motivation for the proposed definition of an order of a solution comes from the “continuity” concept of solutions in the state space. Order 1 equilibria correspond to the traditional definition of equilibrium points. But such points need not be equilibria in the real system. It is shown that the origin for the classical example x˙=x2/3 is physically not an equilibrium point, but behaves like a regular point. Other phenomena such as the presence of impasse points and unbounded higher order time derivatives can also be associated with conventional equilibrium points which hence are weak in scope. Several examples are presented in the paper to show the versatility of the new definition. The implications of the definition for special cases such as when the function f is smooth are established and are shown to be consistent with the existing notions. It is hoped that the proposed definitions will allow a rigorous extension of the results for equilibria of smooth systems to non smooth systems, but no such attempt is made in this paper