Introduction

We first study a very simple class of mechanical systems – those consisting of a single rigid body or of two rigid bodies simply connected. The bodies will be restricted to planar motion in most cases, but they may rotate as well as translate in the plane. We recognize that most practical mechanical systems feature parts that move in three dimensions, but our restrictions are necessary here because the dynamics of massive bodies in three-dimensional motion is, in most cases, beyond the scope of this introductory book on automatic control. However, our use of simple systems has some advantages. We can use the freebody diagram to formulate directly from Newton's and Euler's laws the differential equations that describe the dynamics of the motion without resorting to the more abstract approach of using variational principles, which is usually necessary in the three-dimensional case. It is also possible in this simple setting to illustrate an important principle of mechanics that one must observe when expressing Euler's law for a body undergoing angular acceleration with respect to inertial space. Furthermore, since the motions of bodies in many practical applications are approximately planar, our simple approach in such cases will yield useful results.

The mathematical model we seek for each of the systems studied here is a set of differential equations that describe the physical system and its environment, plus certain auxiliary information that permits us to use these equations to determine the dynamic behavior of our system.

Introduction

In the type of dynamic system analysis that concerns us here, we normally identify the input quantity (or quantities) and the system state variables as our first step. We then write the differential equation (or equations) that describe the relationships between the input variables, the state variables, and their derivatives. The process of establishing these equations requires an understanding of the physical principles that govern the dynamics of our system. We have seen in the first four chapters that the principles of mechanics (the laws of Newton and Euler), those of electromechanics (the laws of Faraday, Ampere, Ohm, and others), and those of fluid mechanics, including aerodynamics, are all basic to the systems of interest here. The equations that result are generally nonlinear, ordinary, differential equations. In our work the differential equations are also restricted to those having constant physical parameters. In much of our work we also concentrate on the study of the dynamics of systems in a restricted regime of operation, usually for motions of the system near an equilibrium state (called a bias point in electronic circuits, or a trim condition in aircraft flight-control systems). With this further restriction on our analysis, the nonlinear differential equations may usually be approximated by linear differential equations having constant coefficients. We have seen several examples of this form of approximation in the first four chapters, and in the ensuing chapters our attention will be focused almost exclusively on linear systems of differential equations.

Background

Dynamics plays the central role in automatic control engineering. The analytical techniques and design principles examined in this book are simply methods of dealing with dynamics problems from the specialized point of view of the automatic feedback control system. This collection of methods and procedures – known as servomechanism theory, basic control theory, and, in recent years, as classical control theory – constitutes the basic subject to be mastered by a beginning control system engineer.

A typical automatic control system consists of several interconnected devices designed to perform a prescribed task. For example, the task may be to move a massive object such as the table of a machine tool in response to a command. The interconnected devices of the system are typically electromechanical actuators, sensors that measure the position and velocity of the controlled object and the currents or voltages at the actuator, and a control computer that processes the sensor signals along with the command. These interconnected dynamic elements work simultaneously, and they also embody a feedback connection. Frequently the engineer must determine the dynamic response of the entire system to a given command when only the physical properties of the individual component elements of the system are known. This formidable task requires quantitative dynamic analysis even in relatively simple systems.

Modern Control Theory and the Digital Computer

Classical control theory is directly applicable to systems which have only single input variables and single response variables.

Introduction

Electric circuits composed of discrete elements in which the voltages and currents are described in continuous time are sometimes called analog circuits, to distinguish them from circuits used in digital signal processors. We now model some analog circuits by writing ordinary differential equations in the state-variable form just as we modeled mechanical systems in Chapter 2. We apply the basic principles of circuit analysis, which are Kirchhoff's voltage and current laws, Ohm's law, Faraday's law, and the law relating the voltage across a capacitor to the current through it. We consider only circuits composed of the following elements: voltage sources, current sources, resistors, capacitors, inductors, and operational amplifiers. Our treatment here is limited to the type of circuits commonly encountered in the signal-processing parts of control systems, as opposed to those in power supplies or in other highpowered devices.

In Section 3.2 we confine our attention to passive circuits. These circuits contain no energy sources (such as operational amplifiers or batteries) except for the voltage or current sources that drive them. We can apply nearly all the basic circuit laws in this simple setting, and we can demonstrate an important principle of control engineering pertaining to interconnected elements, called the loading effect.

In Section 3.3 we model active circuits having operational amplifiers. This extends significantly our capability to design dynamic circuits for control system purposes. We also study simple instrumentation circuits for measuring dynamic variables.

Introduction

We now develop methods for calculating the dynamic response of a linear system that is modeled by its transfer functions. The response depends upon the physical parameters of the system, the input function, and any nonzero initial conditions. We are particularly interested in relating the dynamic features of the response to the physical properties of the system. If the input is a step function, the response may fluctuate temporarily and eventually reach a constant value. The nature of the fluctuations – the length of time during which they persist, and whether they cause the response to overshoot its final value excessively or to oscillate with both positive and negative values – are dynamic features of vital importance. The initial values of the response and its derivatives and the final value of the response are also important characteristics that depend on the physical parameters. In simple systems with simple inputs – for example, a firstor second-order system with a step input – the dynamic features of the response are directly related to simple combinations of the parameter values. But in higher-order systems these important relationships are less obvious because the significant dynamic features of the response depend on complicated combinations of the parameter values. We must now employ a mixture of analysis tools, computer calculations, and approximation techniques to determine which of our system parameters have the most influence on the significant dynamic features of the response.