Controller and Observer Designs for Partial Differential-Algebraic Equations

Abstract

Partial differential-algebraic equations (PDAEs) arise in numerous situations, including the coupling between differential-algebraic equations (DAEs) and partial differential equations (PDEs). They also emerge from the coupling of partial differential equations where one of the equations is in equilibrium, as seen in parabolic-elliptic systems. Stabilizing PDAEs and achieving certain performance necessitate sophisticated controller designs. Although there are well-developed controllers 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. The well-known results for PDEs are generalized to this class of PDAEs. Here, the existence of a unique minimizing optimal control is established. In addition, a projection is used to derive a system of differential Riccati-like equation coupled with an algebraic equation, yielding the solution of the optimization problem in a feedback form. These equations, and hence the optimal control, can be calculated without constructing the projected PDAE.

Lastly, the thesis examines the linear-quadratic (LQ) control problem for linear DAEs of arbitrary index over a finite horizon. Without index reduction or a 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.