Structured uncertainty analysis of robust stability for multidimensional array systems

This paper considers one of the fundamental issues in design and analysis of sampled multidimensional systems - that of uncertainty modeling and robust stability analysis. Methods of structured uncertainty analysis (μ-analysis) are extended toward systems with dynamical and noncausal spatial coordinates. The stability is understood in a broad sense and includes decay (localization) of system response along the noncausal spatial coordinates. Robustness of dynamical stability and spatial localization of response and boundary effects are addressed in a unified way. The main technical condition enabling the technical results of the paper is that the feedback loop including a multidimensional plant and controller does not have a feedthrough in the dynamical (time) coordinate sense. As an example, this paper applies the multidimensional structured uncertainty analysis to closed-loop control of a cross-directional paper machine process. The paper formulates multidimensional models of the process, its controller, and a structured uncertainty. The uncertainty corresponds to a combination of errors in the actuator mapping, the cross-directional response gain, and the response width.