Location
M3 4206 and MS Teams (please email amgrad@uwaterloo.ca for the meeting link)
Candidate
Ala' Alalabi | Applied Mathematics, University of Waterloo
Title
Controller and Observer Design for Partial Differential-Algebraic Equations
Abstract
Partial differential-algebraic equations (PDAEs) emerge in a variety of contexts, notably in systems that involve coupling of partial differential equations (PDEs) with differential-algebraic equations (DAEs). They also arise in scenarios where partial differential equations are coupled, with one of the equations being in equilibrium. Stabilizing PDAEs and achieving certain performance necessitate sophisticated controller designs. Although there are well-developed controller designs for each of PDEs and DAEs, research into controllers for PDAEs remains limited. Discretizing PDAEs to DAEs, or reducing PDAE systems to PDEs when feasible, often results in undesirable outcomes or a loss of the physical meaning of the algebraic constraints. Consequently, this thesis concentrates on the direct design of controllers based on PDAEs, using two control techniques: linear-quadratic and boundary control.
The thesis first addresses the stabilization of coupled parabolic-elliptic systems, an important class of PDAEs with wide applications in fields such as biology, incompressible fluid dynamics, and electrochemical processes. Even when the parabolic equation is exponentially stable on its own, the coupling between the two equations can cause instability in the overall system. A backstepping approach is used to derive a boundary control input to stabilize the system, resulting in an explicit expression for the control law in a state-feedback form. Since the system state is not always available, exponentially convergent observers are designed to estimate the system state using boundary measurements. The observation error system is shown to be exponentially stable, again by employing a backstepping method. This leads to the design of observer gains in closed form. By integrating these observers with state feedback boundary control, the thesis also tackles the output-feedback problem.
Next, the thesis considers finite-time linear-quadratic control of PDAEs that are radial with index 0; this corresponds to a nilpotency degree of 1. Here, the uniqueness of an optimal control minimizing solution is established, and a projection method is employed to derive a system of a differential Riccati-like equation coupled with an algebraic equation. This method not only extends known results from PDEs to PDAEs but also enables the determination of optimal controller without the need to construct a projected PDAE system.
Lastly, the thesis examines the linear-quadratic (LQ) control problem for linear DAEs of arbitrary index over a finite horizon. Without index reduction or behavioral approach, it is shown that a certain projection can lead to the derivation of a differential Riccati equation, from which the optimal control is obtained. Numerical simulations are presented to illustrate the theoretical findings for each objective of the thesis.