Boundary control of parabolic PDE using adaptive dynamic programming

In this dissertation, novel adaptive/approximate dynamic programming (ADP) based state and output feedback control methods are presented for distributed parameter systems (DPS) which are expressed as uncertain parabolic partial differential equations (PDEs) in one and two dimensional domains. In the first step, the output feedback control design using an early lumping method is introduced after model reduction. Subsequently controllers were developed in four stages; Unlike current approaches in the literature, state and output feedback approaches were designed without utilizing model reduction for uncertain linear, coupled nonlinear and two-dimensional parabolic PDEs, respectively. In all of these techniques, the infinite horizon cost function was considered and controller design was obtained in a forward-in-time and online manner without solving the algebraic Riccati equation (ARE) or using value and policy iterations techniques. Providing the stability analysis in the original infinite dimensional domain was a major challenge. Using Lyapunov criterion, the ultimate boundedness (UB) result was demonstrated for the regulation of closed-loop system using all the techniques developed herein. Moreover, due to distributed and large scale nature of state space, pure state feedback control design for DPS has proven to be practically obsolete. Therefore, output feedback design using limited point sensors in the domain or at boundaries are introduced. In the final two papers, the developed state feedback ADP control method was extended to regulate multi-dimensional and more complicated nonlinear parabolic PDE dynamics --Abstract, page iv

Boundary Control of Two Dimensional Burgers PDE Using Approximate Dynamic Programming

An approximate dynamic programming (ADP) based near optimal boundary control of distributed parameter systems (DPS) governed by uncertain two dimensional (2D) Burgers equation under Neumann boundary condition is introduced. First, Hamilton-Jacobi-Bellman (HJB) equation is formulated without any model reduction. Next, optimal boundary control policy is derived in terms of value functional which is obtained as the solution to the HJB equation. Subsequently, a novel identifier is developed to estimate the unknown nonlinearity in the partial differential equation (PDE) dynamics. The suboptimal control policy is obtained by forward-in-time approximation of the value functional using a neural network (NN) based online approximator and the identified dynamics. Adaptive weight tuning laws are proposed for online learning of the value functional and identifier. Local ultimate boundedness (UB) of the closed-loop system is verified by using Lyapunov theory