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.