From Latency to Potency: Lyapunov’s Second Method and the Past, Present, and Future of Control Theory
University of Michigan
Lyapunov's Second Method is one of the cornerstones of systems and control theory. Although virtually every CDC attendee is familiar with the basic theory and its application to control, this talk will provide fresh and novel perspectives on this amazingly simple but powerful and still-developing idea.
After giving a historical account of Lyapunov the mathematician, one of my first key points, which may come as a surprise to control scientists, is that L2M as a basis for stability analysis has had little impact on science and technology as a whole. Next, I will trace the impact of L2M on the development of optimal, robust and adaptive control theory. A bird’s eye view of this development reveals that the impact of L2M on control theory has been circuitous and tortuous, with significant lags in the application of ideas. These lags eventually give way to astounding successes when L2M is used as the basis for controller synthesis.
Next, I will highlight recent advances that have expanded the breadth and depth of Lyapunov's original ideas. These advances include finite-time convergence, partial stability, and semistability. Finite-time convergence, which occurs in the (discontinuous) bang-bang control of the double integrator, is of practical interest when the feedback control is required to be continuous. Homogeneity theory with negative relative degree provides a crucial tool for analysis and synthesis. Next, partial stability is relevant in applications where not all states converge. This is the case in adaptive control where the output error is required to vanish, but the gains need not converge. Next, semistability is concerned with convergence to a nonunique Lyapunov stable equilibrium determined by initial conditions. This property arises in systems with conservation laws, such as thermodynamics and chemical reaction kinetics, where the limiting temperatures and concentrations converge to values that depend on the initial heat distribution and concentrations. Semistability is of interest in these applications since asymptotic stability is impossible due to the continuum of nonisolated equilibria.
Finally, I will discuss open problems in control theory and control technology whose solution depends on the continued development of L2M and its associated concepts.
Friday, November 16, 2001
3:30 - 5:00 p.m.