Robust Control with Complexity Constraint: A
Nevanlinna-Pick Interpolation Approach
Abstract:
This thesis consists of five papers that present new techniques and
algorithms to design robust controllers of low complexity. The robust
control problems that we are interested in are the ones reducible to
the Nevanlinna-Pick interpolation problem. The basic tool for
controller design is a recent theory of Nevanlinna-Pick interpolation
with degree
constraint, which provides a complete parameterization of all
interpolants of degree less than the number of interpolation points
and a family of convex optimization problems to determine each of
these interpolants.
In Paper A, a numerically robust algorithm is developed to solve
these optimization problems.
The algorithm is based on a homotopy continuation
method with predictor-corrector steps.
In Paper B, a new technique for shaping of closed-loop frequency
responses is presented. It is based on Nevanlinna-Pick interpolation
theory with degree constraint. It turns out that spectral zeros of
a certain function related to the closed-loop transfer function, as
well as additional interpolation constraints, are
useful as design parameters for improving performance, while keeping
low controller degree. Tuning strategies of these design
parameters are provided and an upper bound of the controller degree is
derived.
In Paper C, a controller design method for a robust
regulation problem with robust stability is proposed for the scalar
case. It is shown that such a
regulation problem can be formulated as a boundary Nevanlinna-Pick
interpolation problem. A degree restriction is imposed
on the interpolants, which leads to controllers of low degrees.
In Paper D, the algorithm developed in Paper A is extended to
interpolation problems including derivative
constraints. This extension is important since control problems often
give rise to derivative constraints.
In Paper E, a multivariable extension of the theory and the algorithm
for scalar Nevanlinna-Pick interpolation with degree constraint
is presented. A matrix version of the generalized entropy is
introduced to obtain the complete parameterization of a set of
interpolants with a bounded McMillan degree. Spectral
zeros are again the
characterizing parameters and will be used as design parameters in robust
control applications. The homotopy continuation method for
computing each interpolant is extended to this multivariable setting.
Keywords:
Robust control, Nevanlinna-Pick interpolation,
Complexity constraint, Continuation method, Model matching, H-infinity
control, Closed-loop shaping, Convex optimization, Newton's method,
Robust regulation, Performance limitations.
Mathematics Subject Classification (2000):
93D15, 93A30, 30E05, 34A12, 93D21, 93D09, 42A15, 42A70, 90C25, 94A17.