Computational Complexity in Robust Control Theory

                   Cedric Langbort
   Center for the Mathematics of Information
       California Institute of Technology

Robust control theory is concerned with assessing or ensuring (via the
design of a controller) that a dynamical system performs as desired,
in spite of the presence of uncertainties.  These uncertainties
typically come from the engineer's inability to and/or conscious
decision not to use a fully accurate model of the system.  For
example, if the exact value of some of the model's parameters is
difficult to measure, one may choose to treat them as uncertain and
try to design a controller guaranteeing proper functioning for all the
possible values at once. This problem, as it turns out, can be
arbitrarily hard (NP-hard to be precise), depending on what we know
about the uncertainties.  In fact, a large number of robust analysis
or control questions are known to be equally intractable.

These two lectures will constitute an introduction to these
tractability issues. After introducing some models of uncertain
dynamical systems and describing what constitutes "satisfactory
performance" (stability, input/ouput gain) and how to evaluate it, we
will prove that what may be the most natural robust analysis question
-- testing the existence of a stable matrix in a matrix cube -- is
NP-hard.  Then, as time permits, we will discuss some of the advances
of the past decade towards 1) identifying particular analysis and
design problems that can be solved efficiently (LPV control design,
particular "mu" structures) and 2) providing relaxations for
untractable ones (S-procedure, randomized methods) and evaluating the
ensuing conservatism (Nesterov's matrix cube relaxation theorem). In
many cases, tractability results from the possibility of reducing the
original control problem to a convex program (typically LP or SDP).