Dennis S. Bernstein is currently Professor of Aerospace Engineering at the University of Michigan in Ann Arbor. From 1992-94 he was a member of the Technical Staff of Lincoln Laboratory in Lexington, MA, and from 1984-91 he was employed by Harris Corporation Government Aerospace Systems Division in Melbourne, FL. At Harris Corporation he was a member of the Structural Controls Group, where he developed control algorithms for vibration control of deployable space structures. At the University of Michigan, Professor Bernsteinís interests include optimal, nonlinear, and adaptive control as well as system identification. To motivate and demonstrate new techniques for control engineering, he founded the Noise, Vibration, and Motion Control Laboratory, where he has developed experiments in acoustic noise suppression, vibration control, control of rotating imbalance, and attitude control. He has published more than 140 journal papers and has 1 patent. He has an intense interest in control education, control history, and the control profession, which is reflected by numerous pedagogical papers published in the Control Systems Magazine including "A Student's Guide to Classical Control" (August 1997), "A Student's Guide to Research" (February 1999), "On Bridging the Theory/Practice Gap" (December 1999), "The Frequency Domain" (April 2000), and "Peer Review" (June 2000). Prof. Bernstein has been a visiting faculty member at the University of Leeds and the University of Glasgow.


From Latency to Potency: Lyapunovís Second Method and the Past, Present, and Future of Control Theory

Lyapunovís Second Method (L2M) 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 birdís 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 Lyapunovís 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.