LEARNING OBJECTIVES FOR THIS CHAPTER

1. 7–1 To recognize the A-, T- and D-type elements of electrical systems.

2. 7–2 To develop the mathematical equations that model the dynamic behavior of RLC electrical circuits.

3. 7–3 Develop the equations that describe the dynamic behavior of RLC circuits with time-varying capacitors and inductors.

4. 7–4 Analyze simple operational-amplifier circuits.

INTRODUCTION

The A-type, T-type, and D-type elements used in modeling electrical systems, which correspond to the mass, spring, and damper elements discussed in Chap. 2, are the capacitor, inductor, and resistor elements.

A capacitor, the electrical A-type element, stores energy in the electric field induced in an insulating medium between a closely spaced pair of conducting elements, usually plates of metal, when opposite charges are applied to the plates. Capacitance is a measure of the ability of a capacitor to accept charge and hence its ability to store energy. It occurs naturally between the conductors of a coaxial cable, between closely spaced parallel cables, and in closely packed coils of wire. In these cases the capacitance is distributed, along with resistance and inductance, along the line, and the analysis of such situations is beyond the scope of this text. However, in some cases the resistance and inductance are negligible, making it possible to use a lumped-capacitance model. More frequently, specially designed off-the-shelf capacitors are used that have negligible inductance and series resistance.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 16–1 To develop open-loop and closed-loop transfer functions in the z domain for simple digital control systems.

2. 16–2 To evaluate stability and transient performance of linear discrete-time systems.

3. 16–3 To assess steady-state performance of discrete-time systems.

4. 16–4 To implement a discrete-time equivalent of a PID controller.

INTRODUCTION

Unprecedented advances in electronics have revolutionized control technology in recent years. Digital controllers, built around microcomputer chips as stand-alone units or implemented with ubiquitous personal computers, have dominated modern industrial process control applications. The computational power and operational speed of digital controllers allow for performance of much more sophisticated algorithms than were possible with analog controllers. Even for relatively simple control tasks, digital controllers are superior to analog controllers by virtue of their improved flexibility, greater reliability, and, more and more often, lower cost.

The main objective of this chapter is to introduce the very basic concepts of analysis of digital control systems. Only linear, stationary models are considered. In Section 16.2, a pulse transfer function block diagram of a single-loop digital control system is presented. Section 16.3 deals with transient characteristics determined by the locations of roots of the system characteristic equation; methods for determining stability are also briefly discussed. In Section 16.4, steady-state performance characteristics of digital control systems are reviewed. Section 16.5 provides introductory material on digital control algorithms. A digital version of the PID controller is given special attention because of its popularity in industrial process control applications.

OVERVIEW OF SIMULINK

This tutorial is intended to acquaint the reader with the rudiments of Simulink, the MATLAB add-on that allows engineers and researchers to rapidly develop and run computer simulations of dynamic systems by using a block-diagram-oriented graphical environment. As with the MATLAB environment itself, the Simulink package is a very rich and versatile tool that is constantly developing. The best we can hope to accomplish with this tutorial is to inspire interest and start you on the path to developing a strong competence in the area of computer simulation of dynamic systems.

LAUNCHING THE SIMULINK LIBRARY BROWSER

Figure A4.1 shows the default configuration when MATLAB launches in a Windows environment. If Simulink is installed as part of the package, the multicolored icon will appear in the tool bar. The first step in building models is launching the Simulink Library Browser by clicking on that icon.

Figure A4.2 shows a typical library browser window. Your installation may look different because the installation of different MATLAB and Simulink add-ons such as the Signal Processing, Artificial Neural Networks, or Fuzzy Logic toolboxes.

Starting a New Model

Start modeling in Simulink by clicking on the standard “new document” icon on the Library Browser toolbar. That opens a new, blank window on your desktop similar to the one shown in Fig. A4.3.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 6–1 To know the common elements of computer simulations.

2. 6–2 To be able to build block diagrams of systems based on the model equations.

3. 6–3 To become familiar with the basic block set of Simulink.

4. 6–4 To understand different approaches to simulation utilizing different features of Simulink.

5. 6–5 To be able to simulate both linear and nonlinear systems.

6. 6–6 To be able to simulate configuration-dependent systems.

7. 6–7 To be able to conduct parametric studies of systems through scripting multiple runs of the simulations.

INTRODUCTION

System simulation is one of the most widely used tools in modern society. From weather forecasting to economic analysis, from robotics to computer animation, simulation is becoming a commonplace tool for analysts and designers of all types, not just engineers. Yet, as widespread as these applications are, the fundamental basis of system simulations is common. A computer simulation is a numerical solution of a set of differential equations that are intended to model the way in which a particular system evolves in time.

In Chap. 4, techniques for the analytical solution of differential equations were presented. These methods represent a powerful set of tools for the study of linear dynamic systems. It was shown that by simply inspecting the coefficients of a first- or second-order characteristic equation, the entire characteristic of the step response could be predicted. At this point, it would be fair to ask the question: Why are computer-based methods for finding the responses of dynamic systems needed?

Learning objectives for this chapter

1. 3–1 To derive input–output models of linear mechanical systems.

2. 3–2 To distinguish between input–output and state-space forms of mathematical models of engineering systems.

3. 3–3 To discuss the concept of a system state and the related concepts of state variable, state vector, and state space.

4. 3–4 To derive state-space models of linear and nonlinear mechanical systems.

5. 3–5 To recognize advantages and disadvantages of input–output models and state models and transform input–output models to state-space models and vice versa.

INTRODUCTION

In almost all areas of engineering, and certainly in all those areas for which new processes or devices are being developed, considerable efforts are directed toward acquiring information on various aspects of system performance. This process is generally referred to as a system analysis. Traditionally, system analysis was carried out by investigation of the performance of an existing physical object subjected to selected test input signals. Although there is no doubt that such an experimental approach provides extremely valuable and most reliable information about system characteristics, experimenting with an actual full-scale system is not always feasible and is very often practically impossible, especially in the early stages of the system analysis. There are several reasons for this. First, an actual system must be available for testing or a new test object must be constructed, which may involve high cost in terms of time and money. Second, the extent to which the parameters of an existing engineering system can be varied in order to observe their effects on system performance is usually very limited.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 10–1 To integrate knowledge of mechanical, electrical, thermal, and fluid systems to model physical systems made up of more than one type of element.

2. 10–2 To model coupling transducers as ideal and nonideal devices.

3. 10–3 To use state or input–output models to analyze performance of translational–rotational mechanical, electromechanical, and fluid mechanical systems.

INTRODUCTION

In previous chapters, various types of systems were discussed, each within its own discipline: mechanical, electrical, thermal, and fluid. However, many engineering systems consist of combinations of these elementary single-discipline system elements: electromechanical, fluid mechanical, and so on. To combine single-discipline systems, it is necessary to use coupling devices that convert one kind of energy or signal to another: mechanical to electrical, fluid to mechanical, and so on.

The general term “transducer” will be used here for ideal coupling devices. In cases in which significant amounts of energy or power are involved, these coupling devices will be referred to as energy-converting transducers; when the amount of energy being transferred is minimal, they will be referred to as signal-converting transducers.

Selected nonideal energy convertors, which are modeled graphically, are also discussed in terms of typical characteristic curves that have been derived from performance tests.

ENERGY-CONVERTING TRANSDUCERS AND DEVICES

The energy-converting transducers introduced here are ideal in that they are lossless models that contain no energy-storage or energy-dissipation elements. When energy storage or energy dissipation is present, these effects are modeled with lumped ideal elements connected at the terminals of the ideal transducer.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 1–1 To work comfortably with the engineering concept of a “system” and its interaction with the environment through inputs and outputs.

2. 1–2 To distinguish among various types of mathematical models used to represent and predict the behavior of systems.

3. 1–3 To recognize through (T-type) variables and across (A-type) variables when examining energy transfer within a system.

4. 1–4 To recognize analogs between corresponding energy-storage and energy-dissipation elements in different types of dynamic systems.

5. 1–5 To understand the key role of energy-storage processes in system dynamics.

SYSTEMS AND SYSTEM MODELS

The word “system” has become very popular in recent years. It is used not only in engineering but also in science, economics, sociology, and even in politics. In spite of its common use (or perhaps because of it), the exact meaning of the term is not always fully understood. A system is defined as a combination of components that act together to perform a certain objective. A little more philosophically, a system can be understood as a conceptually isolated part of the universe that is of interest to us. Other parts of the universe that interact with the system comprise the system environment, or neighboring systems.

All existing systems change with time, and when the rates of change are significant, the systems are referred to as dynamic systems. A car riding over a road can be considered as a dynamic system (especially on a crooked or bumpy road). The limits of the conceptual isolation determining a system are entirely arbitrary.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 15–1 To use the finite-difference approximation of a derivative to develop an approximate discrete-time model corresponding to a continuous input–output model.

2. 15–2 To derive discrete-time state models for linear dynamic systems.

3. 15–3 To develop block diagrams of a digital control system including sampling and holding devices.

4. 15–4 To use the z transform to develop pulse transfer functions of discrete-time systems.

INTRODUCTION

In almost all existing engineering systems, the system variables (input, output, state) are continuous functions of time. The first 14 chapters of this book deal with this category of systems, classified in Chap. 1 as continuous dynamic systems. The last two chapters are devoted to discrete-time systems in which, according to the definition given in Chap. 1, the system variables are defined only at distinct instants of time. It may seem that there are not many such systems, and, indeed, very few examples of intrinsically discrete engineering systems come to mind. There are, however, many systems involving continuous subsystems that are classified as discrete because of the discrete-time elements used to monitor and control the continuous processes. Any system in which a continuous process is measured and/or controlled by a digital computer is considered discrete. Although some variables in such systems are continuous functions of time, they are known only at distinct instants of time determined by the computer sampling frequency, and therefore they are treated as discrete-time variables.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 9–1 To recognize the A-, T-, and D-type elements of fluid systems.

2. 9–2 To use laws of continuity and compatibility to develop lumped-parameter models of hydraulic fluid systems.

3. 9–3 To model lumped-parameter compressible (pneumatic) fluid systems by use of ideal gas assumptions.

INTRODUCTION

Corresponding to the mass, spring, and damper elements discussed in Chap. 2, the A-type, T-type, and D-type elements used in modeling fluid systems are the fluid capacitor, the fluid inertor, and the fluid resistor elements.

Capacitance occurs as a result of elasticity or compliance in the fluid or in the walls of the container. Although liquids such as water and oil are often considered to be incompressible in many fluid flow situations, these hydraulic fluids are sometimes compressible enough to produce fluid capacitance. In other cases, the walls of the chambers or passages containing the fluid have enough compliance, when the fluid pressure changes rapidly, to produce fluid capacitance. In long lines and passages, fluid capacitance is distributed along the line, together with inertance and resistance, and the analysis of such situations is beyond the scope of this text. If both the resistance and inertance are negligible, however, it is possible to use a lumped-capacitance model of the line.

When fluid capacitance is wanted for energy-storage purposes, specially designed off-the-shelf capacitors with minimal resistance and inertance, called hydraulic accumulators, are used. Storage tanks and reservoirs also serve as fluid capacitors.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 4–1 To use analytical solution methods for ODEs to predict the response of first- and second-order systems to nonzero initial conditions and typical input signals.

2. 4–2 To estimate key parameters (i.e., time constant, natural frequency, and damping ratio) in system responses.

3. 4–3 To use solution methods for ODEs to derive the relationship between the complex roots of underdamped second-order systems and the natural frequency and damping ratio.

4. 4–4 To use the concept of “dominant poles” to estimate the response of higher-order systems when one or two poles dominate the system's dynamic behavior.

INTRODUCTION

In Chap. 3, the state representation of system dynamics was introduced and the derivation of state equations was shown to be a relatively simple and straightforward process. Moreover, the state models take the form of sets of first-order differential equations that can be readily solved by use of one of many available computer programs. Having these unquestionable advantages of state-variable models in mind, one might wonder whether devoting an entire chapter to the methods for solving the old-fashioned input–output model equations is justified. Despite all its limitations, the classical input–output approach still plays an important role in analysis of dynamic systems because many of the systems to be analyzed are neither very complex nor nonlinear. Such systems can be adequately described by low-order linear differential equations. Also, even in those cases in which a low-order linear model is too crude to produce an accurate solution and a computer-based method is necessary, an analytical solution of an approximate linearized input–output equation can be used to verify the computer solution.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 5–1 To explain the fundamental principle of numerical integration as a finite sum of approximate areas.

2. 5–2 To implement simple numerical integration methods by use of MATLAB or a similar computing platform.

3. 5–3 To articulate the advantages of higher-order approximation methods and adaptive-step-size algorithms as more accurate and more efficient methods for integration.

4. 5–4 To recognize numerically stiff systems and use methods that alleviate the difficulties that they present in computer solutions.

INTRODUCTION

For centuries, engineers and scientists have sought help from calculating machines of all kinds in solving mathematical equations that model dynamic systems. A digital computer, the most recent version of the calculating machine, has come a long way from the more than 5000-year-old Babylonian abacus. Most engineers would probably prefer a computer to an abacus because of its superior computational power, but both devices are capable of performing only those tasks that engineers already know how to perform but either choose not to do, for some reason, or cannot do because of lack of sufficient speed or memory or both. Despite today's fascination with computers, it is important to remember that they can do only what they are programmed to do within their vast yet finite performance limits.

Generally speaking, correctness of computer results depends on two conditions: the correctness of the formulation of the problem and the computational capability of the computer to solve it. It does not seem widely recognized how often at least one of these conditions is not met, leading to worthless computer results.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 8–1 To recognize the A- and D-type elements of thermal systems.

2. 8–2 To identify and model the three fundamental modes of heat transfer.

3. 8–3 To use the energy-balance method to develop models of lumped-parameter thermal systems.

INTRODUCTION

Fundamentals of mathematical methods used today to model thermal systems were developed centuries ago by such great mathematicians and scientists as Laplace, Fourier, Poisson, and Stefan. The analytical solution of the equations describing the basic mechanisms of heat transfer – conduction, convection, and radiation– was always considered to be an extremely challenging mathematical problem. The study of energy transfer in thermal systems continues to be an important topic in engineering because it forms the basis of analysis of energy efficiency for indoor environmental controls, industrial processes, and all forms of energy transformation.

As described in Chap. 1, temperature is an A-type variable, determining the amount of energy stored in a thermal capacitance, the A-type energy-storing element corresponding to a mass in mechanical systems or a capacitor in electrical systems. All matter has thermal capacitance (which is proportional to mass), and energy is stored as internal energy because of its temperature. The T-type variable in thermal systems is heat flow rate Qh; however, as was pointed out in Section 1.2, there is no T-type element in thermal systems that would be capable of storing energy as a result of heat flow rate. The D-type element in thermal systems is the thermal resistor and will be defined as the resistance to heat transfer.

After studying this chapter you will be able to

• define random variables and signals;

• describe a random variable by the cumulative distribution function and by the probability density function;

• compute the expected value, mean, variance, standard deviation, correlation, and covariance of a random variable;

• define a Gaussian random signal;

• define independent and identically distributed (IID) signals;

• describe the concepts of stationarity, wide-sense stationarity, and ergodicity;

• compute the power spectrum and the cross-spectrum;

• relate the input and output spectra of an LTI system;

• describe the stochastic properties of linear least-squares estimates and weighted linear least-squares estimates;

• solve the stochastic linear least-squares problem; and

• describe the concepts of unbiased, minimum-variance, and maximum-likelihood estimates.

Introduction

In Chapter 3 the response of an LTI system to various deterministic signals, such as a step input, was considered. A characteristic of a deterministic signal or sequence is that it can be reproduced exactly. On the other hand, a random signal, or a sequence of random variables, cannot be exactly reproduced. The randomness or unpredictability of the value of a certain variable in a modeling context arises generally from the limitations of the modeler in predicting a measured value by applying the “laws of Nature.” These limitations can be a consequence of the limits of scientific knowledge or of the desire of the modeler to work with models of low complexity. Measurements, in particular, introduce an unpredictable part because of their finite accuracy.