Time Domain Dynamics and Control
In this section we study the time-dependent behavior of some chemical engineering systems, both openloop (without control) and closedloop (with controllers included). Systems are described, by differential equations and solutions are given in terms of time-dependent functions. Thus, our language for this part of the book will be “English.” In thenext part we will learn a little “Russian” so that we can work in the Laplace domain, where the notation is simpler than “English.” In Part Three we will study some “Chinese” because of its ability to easily handle much more complex systems. Most chemical engineering systems are modeled by equations that are quite complex and nonlinear. In the remaining parts of this book only systems described bylinear ordinary differential equations will be considered (linearity is defined in Chapter 2). The reason coverage is limited to linear systems is that practically all the analytical mathematical techniques currently available ,are applicable only to linear equations. Since most chemical engineering systems are nonlinear, studying methods that are limited to linear systems might initially appear tobe a waste of time. However, linear techniques are of great practical importance, particularly for continuous processes, because the nonlinear equations describing most systems can be linearized around some steady-state operating condition. The resulting linear equations adequately describe the dynamic response of the system in some region around the steady-state conditions. The size of theregion over which the linear model is valid varies with the degree of nonlinearity of the process and the magnitude of the disturbances. In many processes the linear model can be successfully used to study dynamics and, more important, to design controllers. Complex systems can usually be broken down into a number of simple elements. We must understand the dynamics of these simple systems before wetackle the more
Time Domain Dynamics and Control
complex ones. We start out looking at some simple uncontrolled processes in Chapter 2. We examine the openloop dynamics or the response of the system with no feedback controllers to a disturbance starting from some initial condition. In Chapters 3 and 4 we look at closedloop systems. Instrumentation hardware, controllertypes and performance, controller tuning, and various types of control system structures are discussed.
Time Domain Dynamics
Studying the dynamics of systems in the time domain involves the direct solution of differential equations. Computer simulations are general in the sense that they can give solutions to very complex nonlinear problems. However, they are also very specific in the sensethat they provide a solution to only the particular numerical case fed into the computer. The classical analytical techniques discussed in this chapter are limited to linear ordinary differential equations. But they yield general analytical solutions that apply for any values of parameters, initial conditions, and forcing functions. We start by briefly classifying and defining types of systems andtypes of disturbances. Then we learn how to linearize nonlinear equations. It is assumed that you have had a course in differential equations, but we review some of the most useful solution techniques for simple ordinary differential equations. The important lesson of this chapter is that the dynamic response of a linear process is a sum of exponentials in time, such as eQr. The sk termsmultiplying time are the roots of the characteristic equation or the eigenvalues of the system. They determine whether the process responds quickly or slowly, whether it is oscillatory, and whether it is stable.
CLASSIFICATION AND DEFINITION
Processes and their dynamics can be classified in several ways: 1. Number of independent variables a. Lumped: if time is the only independent variable;...
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