### Doctoral Thesis Defense, Optimization and Systems Theory

Friday, September 20, 2002, 10.00, Kollegiesalen, Administration building,
Valhallavägen 79, KTH

**Ryozo Nagamune**

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Robust control with complexity constraint: A Nevanlinna-Pick
interpolation approach

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Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan
framlägges till offentlig granskning för avläggande av teknologie doktorsexamen
fredagen den 20:e september 2002 kl 10.00 i Kollegiesalen,
Administrationsbyggnaden, Kungliga Tekniska Högskolan,
Valhallavägen 79.
*
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.

Calendar of seminars

*Last update: September 5, 2002 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*