Digital Control A Setting For A Mixture Of Computing Strategies:Numerics, Symbolics And Simulation

The authors report on a course that is a blend of classical and modern digital control, and digital filtering. Exercises in this course reflect a variety of computing resources and strategies and as a result low order system problems are no longer assigned. The authors have incorporated symbolic computing in several appropriate places where it can increase understanding and yield error-free algebraic manipulations. In most exercises, the computing is of the numerical type using algorithms developed in the course. Simulation, employing a graphical-oriented simulation language, lends reality to implementation of control laws and estimation algorithms. Introduction At the University of Wyoming a lower level graduate course is offered to introduce graduate students to discrete-time control and digital filtering. The assumed prerequisite material is that contained in first courses in control and discrete signals and systems. The students are typically from the controls, power systems and signal processing areas with the presence of an occasional mechanical or chemical engineer. The topics are a blend of classical discrete-data control topics, modern digital control topics and digital filter synthesis techniques. A topical outline is given in Table 1. Table 1. Course Outline Review of z-transforms The pulse transfer function Single-loop controller design PID and lead-lag controllers Ragazzini controller design Digital filter synthesis Discrete-time state variable representation State feedback and associated algorithms Prediction, current and reduced order observers Nonzero setpoints versus regulators L-Q optimal control Reciprocal root locus P ge 313.1 The work reported here represents an update of that reported previously based on a second class offering. In the past, essentially the identical course material has been covered, but due to computational limitations the problem assignments have been rather simple, usually of the second-order variety. With the convenience offered by MATLAB for Windows 2 and VisSim, the authors have made the assigned problems more realistic and can concentrate class time on algorithms for the design process rather with detailed classroom calculations. The Symbolic Toolbox of MATLAB allows some rather tedious, error-prone algebraic manipulations to be accomplished without error. The authors and others have reported previously the use of symbolic computation in a digital filter design course . VisSim is a simulation environment that handles mixed continuousand discrete-time elements with a graphical user interface. The learning curve for VisSim is essentially negligible, and students progress rapidly in their ability to simulate complex systems. Simulink 5 is an alternative to VisSim and runs seamlessly with MATLAB. The third order “theme problem” employed for most of the exercises is one that is interesting from both classical and a state space design points-of-view and one not amenable to hand calculations, thus motivating the students to develop an interest in the use of modern software tools. Theme Problem Plant The plant chosen for the student exercises is the thermal environmental chamber illustrated in Fig. 1. The students are given a handout that explains the physical situation and explores the small perturbation analysis for the system. The outer chamber is heated by a flow of steam that is controlled by a valve the dynamics of which may not be neglected. Fig. 1 is labeled with the perturbed temperature variables ( x1(t), x2(t), w(t)) and the perturbed valve gate motion variable (x3(t)). The current used to produce that perturbation is denoted as u(t). The perturbation in the environmental temperature is denoted as w(t). Figure 1. Thermal Chamber System to be Controlled N N S S u(t) b