Optimal and Adaptive Control for Volterra Difference Equations

Abstract

This thesis seeks to fill a knowledge gap in the area of dynamical systems and control theory concerning so-called Volterra Difference Equations (VDEs). A particular focus lies on controlled VDEs and related concepts, including characterizations of controllability and observability, and optimal and adaptive control designs for VDEs. Two different system classes of Volterra Difference Equations are being con- sidered: systems of convolution and non-convolution type. The concept of the resolvent or fundamental matrix for both types is introduced. The fundamental mat- rix allows derivation of solutions for VDEs, and in this thesis it is utilized to establish novel necessary and sufficient conditions for the controllability and observability of VDEs for both convolution and non-convolution types, and for VDEs of the first and second kind. The results develop respective controllability and Gramian matrices, for which invertibility or rank conditions are established respectively. The control- lability and observability characterization results are complementary. Examples are presented to illustrate the theory. Furthermore, this thesis investigates several approaches to the control designs for Volterra Difference Equations. First optimal control problems for VDEs of convolution type of the first kind are considered. The first results establish a solution for a finite horizon optimal control problem. The solution of this problem can be developed by solving a difference Riccati equation, or alternatively, and a novel contribution of this thesis, by derivation of a system of linear equations as an analogy for the optimal control problem. Solving this system of linear equations allows then for a direct approach to obtain the optimal control input. While this direct method is mathematically very convenient, a drawback is that constraints are not easily integrated in this optimal control approach. For this purpose, alternative approaches based on linear quadratic regulator design and a differential dynamic programming approach are developed. Second feedback and adaptive control designs are developed for VDEs of the convolution type of the first kind. Two adaptive control designs are explored, a simple adaptive controller and a direct adaptive controller. To illustrate the results of this thesis an epidemiological model and respective management of the epidemic is considered. We employ the various control strategies and designs to the respective Volterra Difference Equation and explore the efficacy of the designs with respect to different model parameters.