Many engineering applications are modeled by systems of coupled differential and algebraic equations (DAEs) that cannot be directly reduced to ordinary differential equations (ODEs). This research note presents, in a unified framework, recent results on the stabilization, output tracking and disturbance elimination for a large class of nonlinear DAE systems. More specifically, the book focuses on continuous-time systems of differential and algebraic equations in semi-explicit form (the most common form for the applications in mind).
The chapter “DAEs: Background and Concepts” provides a discussion of results on the analysis and numerical simulation of DAE systems and the control of linear DAE systems. Emphasis is placed on illustrating the fundamental differences between DAE and ODE systems, their implications in simulation and control, and the available approaches to address these issues. The concepts of index (a measure of the “singularity” of the DAE system) and regularity (the existence of a control-invariant state space), that play an important role in the subsequent chapters, are also introduced and discussed.
The chapter “Chemical Process Applications” describes examples of generic classes of chemical processes with fast rates of mass transfer, heat transfer, reaction or gaseous flow, that are naturally modeled by high-index DAE systems under the (quasi-steady-state) assumptions of phase, thermal or reaction equilibrium, and negligible pressure drop. The occurrence of high-index DAE models in networks (interconnections) of chemical processes is also documented.
The chapter “Feedback Control of Regular DAE Systems” addresses the derivation of state-space realizations of regular high-index DAE systems, and the synthesis of state feedback controllers for stabilization and tracking on the basis of the state-space realizations.
The chapter “Feedback Control of Nonregular DAE Systems” deals with nonregular DAE systems, for which a state-space realization does not exist independently of the controller design. The controller synthesis for such systems is addressed through a regularizing feedback modification and the derivation of state-space realizations of the feedback regularized system.
The chapter “Control of DAE Systems with Disturbance Inputs” focuses on high-index DAE systems with disturbance inputs, and addresses the feedforward/state feedback control of systems that possess a well-defined state-space realization either independently of the disturbances, or, if they are measured, after a feedforward/feedback regularization. The chapter “DAEs and Singularly Perturbed Systems” addresses the connections between the high-index DAE systems considered in this work and a class of singularly perturbed systems in nonstandard form. The results provide a rigorous justification for the quasi-steady-state approximations and the resulting high-index DAE models of the chemical processes.
Finally, the chapter “Simulation Studies of Chemical Process Applications” includes several simulation case studies on representative chemical processes, that illustrate the application of the control methods and the advantages of using the (quasi-steady-state) high-index DAE models as a basis for controller design.
The book is intended primarily as a reference for control and chemical engineers and applied mathematicians interested in the analysis and control of dynamical systems.