State space modeling of differential-algebraic systems using singularly perturbed sliding manifolds

Abstract

This thesis introduces a new approach for modeling and control of algebraically constrained dynamic systems. The formulation of dynamic systems in terms of differential equations ·and algebraic constraints provides a systematic framework that is well suited for object oriented modeling of thermo-fluid systems. In this approach, differential equations are used to describe the evolution of subsystem states and algebraic equations are used to define the interconnections between the subsystems (boundary conditions). Algebraic constraints also commonly occur as a result of modeling simplifications such as steady state approximation of fast dynamics and rigid body assumptions that result in kinematic constraints. Important examples of algebraically constrained dynamic systems include multi-body problems, chemical processes, and two phase thermo-fluid systems. Differential-algebraic equation (DAE) systems often referred to as descriptor, implicit, or singular systems present a number of difficult problems in simulation and control. One of the key difficulties is that DAEs are not expressed in an explicit state space form required by many simulation and control design methods. This is particularly true in control of nonlinear DAE systems for which there are few known results. Existing control methods for nonlinear DAEs have so far relied on deriving state space models for limited classes of problems. A new approach for state space modeling of DAEs is developed by formulating an equivalent nonlinear control problem. The zero dynamics of the control system represent the dynamics of the original DAE. This new connection between DAE model representation and nonlinear control is used to obtain state space representations for a general class of differential-algebraic systems. By relating nonlinear control concepts to DAE structural properties a sliding manifold is constructed that asymptotically satisfies the constraint equations. Sliding control techniques are combined with elements of singular perturbation theory to develop an efficient state space model with properties necessary for controller synthesis. This leads to the singularly perturbed sliding manifold (SPSM) approach for state space realization. The new approach is demonstrated by formulating a state space model of vapor compression cycles. This allows verification of the method and provides more insight into the problems associated with modeling differential algebraic systems.