Stability criteria based on integral quadratic constraints

Stability criteria based on Lyapunov functions, dissipativity and absolute stability have been developed over several decades. However, a new perspective on the theory has recently emerged with the development of new numerical methods. For linear time-invariant systems with uncertainty, Doyle and others developed efficient computational tools based on the notion structured singular value. For nonlinear and time-varying systems, the search for a quadratic Lyapunov function can be written as a convex optimization problem with linear matrix inequality (LMI) constraints. Such problems can be solved with great efficiency using interior point methods. A large variety of results of this kind were previously unified and generalized using the notion integral quadratic constraint. The general computational problem to find multipliers that prove stability was stated as an LMI optimization problem. Furthermore, it was shown that previous technical problems associated with anti-causal multipliers can be avoided using a homotopy argument. This article further extends the theory in two respects. The improvements are instrumental for treatment of several important phenomena, such as time delays, ideal relays and hysteresis. In particular, the framework of exponential stability eliminates the need for external signals in the feedback loop, so that attention can be restricted to the type of signals that appear naturally