CONTROL AND SYSTEMS THEORY
(FOR THE ENVIRONMENTAL SCIENTIST)Purpose
Though it may seem strange, perhaps even offensive at first sight, environmental systems may be viewed as more or less complicated chemical process engineering systems. Within a given volume, substrate chemical species interact with biological species, which in turn interact with other biological species. Conceptual hydrological models are based on the notion of stores (or volumes, or tanks) of water; and the routing of water and solutes along a river system can be simulated likewise, as a cascade of tanks (or a sequence of reactors). The purpose in drawing this parallel is to allow the environmental scientist access to the methods of systems theory and some elementary notions of process control. And these in turn are generic methods and notions, capable of being deployed for analyzing the behavior of any system not at steady state. In short, this is the introduction to a sequence of classes in Environmental Systems Analysis, composed of the following: System Identification (for the Environmental Scientist) (FORS 8150); Environmental Process Control Laboratory (FORS 8160); and the Environmental Systems Analysis and Control Seminar (FORS 8170).Outline
The class begins by taking a number of contemporary environmental problems and demonstrating that these can be addressed through the conceptual analog of a chemical processing system, the continuously stirred tank reactor (CSTR). While acknowledging that water quality in a river may be described by a set of partial differential equations (the classical advection-dispersion equation), on the one hand, and by a set of simple algebraic equations (the aggregated dead-zone model), on the other hand, models in terms of ordinary differential equations (ODE) will be the focus of the class. This covers therefore: formulation of mass balances; analytical integration of the ODEs for single and double CSTR systems; time-constant, steady-state gain, step and impulse responses; and state-space representation of multivariable systems. Whereas understanding of the analytical solution of these ODEs is important, it is clear that models of even modest size must be solved numerically, for which purpose an introduction to MATLAB® is provided. This requires "vectorized thinking" and an elementary prior familiarity with vector-matrix algebra; tutorials are set using field data for the balance of dissolved oxygen in a river system. Constructing models in the iconic framework of the block diagrams of Control Theory are then addressed using the companion software of SIMULINK®.A large portion of the class is devoted to developing further these ideas of the state-space representation of a system's dynamic behavior, touching upon phase-plane portraits, natural and forced responses, Laplace transforms (s), and culminating in derivation of the state transition matrix. This, in turn, allows the class to move on to discrete-time, algebraic equations for the dynamics of a system — a vital point, since such equations lie at the heart of digital computation, filtering theory, recursive time-series analysis, and the algebraic relationships of input-output, transfer-function models. Throughout, these abstract concepts are referred back to case-study material, principally of models for describing the propagation of water, solutes, and the interactions among solutes and microorganisms within the water environment. At the same time, these case-study problems are used to introduce other basic features of dynamic systems analysis and control, including the notion of stability, and the elementary principles of feedback and feedforward control, optimal control, and hierarchical control.
There is a term project, to be conducted within the MATLAB-SIMULINK® framework, but otherwise open to the student's choice of specific case study for extending his/her experience of dynamic systems and their control.