Stability Theory
Wednesday, 11 October, 2023
Stability theory, specifically stability in the sense of Lyapunov, comprises one of the fundamental components of modern dynamical systems theory and control theory. It is reasonable to equate stability with safety in many engineering systems. It is also reasonable to equate strong forms of stability with efficiency and indeed it is arguable that much of classical control theory is based upon this premise. Unfortunately much of stability theory, and in particular that part which makes reference to eigenvalues or poles, applies predominantly, if not exclusively, to linear, time-invariant systems. At the same time the growth in the use of switched systems has been unabated for well over a decade.
The best stability theory for nonlinear and/or time-varying systems employs quadratic Lyapunov functions. The Kalman-Yakubovich-Popov (KYP) lemma essentially identifies and summarizes the core algebraic ideas in this area and the circle criteria comprise the primary outcomes from the perspective of stability theory. However, the circle criteria are, in general, rather conservative in their estimations of the region of stability. Accordingly there is much ongoing study of the problem and in this regard the use of non-quadratic Lyapunov functions is widespread.
It is fair to say that much of the work on such Lyapunov functions has been directed towards the development of efficient numerical algorithms for exploring whether Lyapunov functions of this kind exist or not for a given system. The golden age of stability theory (essentially the 1960s) was not so concerned with this issue, but rather sought necessary and sufficient algebraic conditions for the existence of Lyapunov functions of special forms for given systems. The current project looks back to that earlier goal and seeks similar algebraic conditions which are sufficient (and ideally necessary) for the existence of Lyapunov functions of other forms. In effect we seek to achieve for non-quadratic Lyapunov functions what the KYP lemma achieves for quadratic functions.