Control systems

Flower; Parr

1.2.1 Introduction

Examples of the conscious application of feedback control ideas have appeared in technology since very early times: certainly the float-regulator schemes of ancient Greece were notable examples of such ideas. Much later came the automatic direction-setting of windmills, the Watt governor, its derivatives, and so forth. The first third of the 1900s witnessed applications in areas such as automatic ship steering and process control in the chemical industry. Some of these later applications attracted considerable analytical effort aimed at attempting to account for the seemingly capricous dynamic behaviour that was sometimes found in practice.

However, it was not until during, and immediately after, World War II that the fundamentals of the above somewhat disjointed control studies were subsumed into a coherent body of knowledge which became recognised as a new engineering discipline. The great thrust in achieving this had its main antecedents in work done in the engineering electronics industry in the 1930s. Great theoretical strides were made and the concept of feedback was, for the first time, recognised as being all pervasive. The practical and theoretical developments emanating from this activity, constitute the classical approach to control which are explored in some detail in this chapter.

Since the late 1940s, tremendous efforts have been made to expand the boundaries of control engineering theory. For example, ideas from classical mechanics and the calculus of variations have been adapted and extended from a control-theoretic viewpoint. This work is based largely on the state-space description of systems (this description is briefly described in Section 1.2.11). However, it must be admitted that the practical uses and advantages of many of these developments have yet to be demonstrated. Most control system design work is still based on the classical work mentioned previously. Moreover, nowadays these applications rely, very heavily, on the use of computer techniques; indeed, computers are commonly used as elements in control loops.

Techniques from the ‘classical’ period of control engineering development is easily understood, wide-ranging in application and, perhaps most importantly, capable of coping with deficiencies in detailed knowledge about the system to be controlled.

These techniques are easily adapted for use in the computer-aided design of control systems, and have proved themselves capable of extension into the difficult area of multi-variable system control; however, this latter topic is beyond the scope of this chapter. So with the above comments in mind, a conventional basic approach to control theory is presented, with a short discussion of the state-space approach and a more extensive forage into sampled-data systems. These latter systems have become important owing to the incorporation of digital computers, particularly microcomputers, into the control loop. Fortunately, an elementary theory for sampled data can be established which nicely parallels the development of basic continuous control theory.

The topics covered in this introduction, and extensions of them, have stood practitioners in good stead for several decades now, and can be confidently expected to go on delivering good service for some decades to come.

1.2.2 Laplace transforms and the transfer function

In most engineering analysis it is usual to produce mathematical models (of varying precision) to predict the behaviour of physical systems. Often such models are manifested by a differential equation description. This appears to fit in with the causal behaviour of idealised components, e.g. Newton’s law relating the second derivative of displacement to the applied force. It is possible to model such behaviour in other ways (for example, using integral equations), although these are much less familiar to most engineers. All real systems are non-linear; however, it is fortuitous that most systems behave approximately like linear ones, with the implication that superposition holds true to some extent. We further restrict the coverage here in that we shall be concerned particularly with systems whose component values are not functions of time—at least over the time-scale of interest to us.

In mathematical terms this latter point implies that the resulting differential equations are not only linear, but also have constant coefficients, e.g. many systems behave approximately according to the equation

2xdt2+2ζωndxdt+ωn2x=ωn2ft

where x is the dependent variable (displacement, voltage, etc.), f(t) is a forcing function (force, voltage source, etc.), and n2 and ζ are constants the values of which depend on the size and interconnections of the individual physical components making up the system (spring-stiffness constant, inductance values, etc.).

Equations having the form of Equation (1.2.1) are called ‘linear constant coefficient ordinary differential equations’ (LCCDE) and may, of course, be of any order. There are several techniques available for solving such equations but the one of particular interest here is the method based on the Laplace transformation. This is treated in detail elsewhere, but it is useful to outline the specific properties of particular interest here.

1.2.2.1 Laplace transformation

Given a function f(t), then its Laplace transformation F(s) is defined as

ft=Fs=∫0∞ftexp−stdt

where, in general, s is a complex variable and of such a magnitude that the above integral converges to a definite functional value.

A list of Laplace transformation pairs is given in Table 1.2.1.