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.

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**Friday, November 16, 2001**

**3:30 - 5:00 p.m.**