Computational methods for infinite-dimensional control systems

Control system analysis and synthesis problems associated with linear time-invariant infinite-dimensional systems have received much attention in recent years. This dissertation focuses on the use of the eigenstructure of certain operators on infinite-dimensional Hilbert spaces to compute answers to control problems. In particular, the results presented here can be applied to systems that do not have eigenvalues and eigenvectors available in closed form, e.g., systems described by partial differential equations (PDEs) with spatially variant parameters, systems with two- or three-dimensional domains with complicated boundary shapes, etc. Three basic problems are solved for classes of systems that can be formulated as bounded spectral systems. First, power, inverse power, and orthogonal iteration methods are formulated for directly calculating eigenvalues and eigenfunctions of classes of spectral operators associated with the systems of interest. Second, bounds are derived on the error incurred by approximating canonical parabolic and hyperbolic systems with finite-dimensional modal models. These bounds require only a finite number of eigenvalues and eigenfunctions, which can be calculated with the proposed power methods. Next, a computable test is formulated for verifying the stability or instability of the feedback connection of a class of spectral systems and either state feedback or a finite-dimensional linear time-invariant controller. This test also requires only a finite number of eigenvalues and eigenfunctions of the spectral system in question. Finally, a collection of M scATLAB functions is presented which implements the the power methods, frequency-domain model and bound calculations, and stability tests developed in the dissertation. The package is designed to be used in conjunction with the $\mu$-Tools toolbox to enable the user to perform modeling, $H\sp\infty$ and D-K iteration controller synthesis, stability analysis, simulation, and animated visualization for systems described by parabolic and hyperbolic PDEs, possibly with spatially variant parameters. The utility of the collection is demonstrated via several examples.