Control Theory: A guided tour (3rd Edition)
Using clear tutorial examples, this fully updated new edition concentrates on explaining and illustrating the concepts that are at the heart of control theory. It seeks to develop a robust understanding of the underlying principles around which the control subject is built. This simple framework is studded with references to more detailed treatments and also has interludes that are intended to inform and entertain. The book is intended as a companion on the journey through control theory, and although the early chapters concentrate on fundamental ideas such as feedback and stability, later chapters deal with more advanced topics such as state variables, optimisation, estimation, Kalman filtering and robust control.
Inspec keywords: frequency response; discrete time systems; feedback; optimisation; multivariable systems; control theory; Laplace transforms; state estimation; stability; digital control; nonlinear control systems; mathematical analysis
Other keywords: control design; digital control system; robust control design; nonlinear systems; stability; control theory; multivariable linear systems; discrete time control systems; state estimation; mathematical modelling; Laplace transform; frequency response methods; optimisation; state space approach
Subjects: Integral transforms; General and management topics; Simulation, modelling and identification; Mathematical analysis; Optimisation techniques; Control engineering computing; Control theory
 Book DOI: 10.1049/PBCE072E
 Chapter DOI: 10.1049/PBCE072E
 ISBN: 9781849192279
 eISBN: 9781849192286
 Page count: 472
 Format: PDF

Front Matter
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1 Control concepts: a nonmathematical introduction
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In this chapter the prerequisites for control design is discussed: a defined objective, a set of available actions and a model that could be interrogated to establish which of the available actions would best move the system towards meeting the objective. Now we add more structure to the concepts to put forward a possible design methodology. In this methodology, central use is made of a system model.

2 Control design ideas: a nonmathematical treatment
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The model assumed here is able to rapidly calculate the expected behaviour of the system when subjected to any particular action. Automatic feedback control overcomes both the problems of possible unmeasurability of disturbances and difficulty of obtaining a sufficiently accurate model by being error driven.

3 Synthesis of automatic feedback control loops: a more quantitative view
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In automatic control, a device called a controller issues commands that are physically connected to a process with the intention to influence the behaviour of the process in a particular way. The commands that will be issued by the controller in a particular set of circumstances are completely determined by the designer of the controller. Thus, automatic control can be seen to be completely predetermined at the design stage.

4 How the Laplace transform greatly simplifies system representation and manipulation
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Many useful techniques depend on the Laplace transform. The Laplace transform of a function f(t) is denoted sometimes by L{f(t)} and sometimes by F(s). The inverse Laplace transform of F(s) is denoted sometimes by L^{1}{F(s)} and sometimes by f(t).

5 Frequency response methods
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Frequency response methods have a physical explanation that is readily understandable without any mathematics. In addition, the methods are design oriented, link easily between practical results and differential equation methods, and have been proven to work well in many practical design situations. The 'home territory' for frequency response methods has traditionally been in servomechanism, process control and aerospace applications, and they have been rather resistant to applications outside these areas.

6 Mathematical modelling
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Some of the approaches of mathematical modelling in various control systems are presented. The modelling approach is done for distributed systems as well as other physical complex systems.

7 Nonlinear systems
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There is no unifying feature present in nonlinear systems except the absence of linearity. For a nonlinear system, it is generally meaningless to speak of global behaviour. Approaches to the analysis of nonlinear systems include: Lyapunov's second or direct method, Lyapunov's first method, describing function method, sector bound methods. The most powerful tools for analysis and design of control systems operate only on linear models. It is therefore, potentially, very attractive when undertaking the design of a controller for a nonlinear system to replace the nonlinear system model by a linear approximation. Thus, linearisation amounts to a local approximation of differentiable functions by derivatives and is only valid for small perturbations.

8 Limits to performance
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The following topics are dealt with: closed loop system; stability; control system performance; and equipment limitation.

9 Some practical aspects of control design, implementation and justification
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This paper discusses control system design for industrial process control. In a typical industrial process, the online available measurements will usually be rather peripheral to the variables that are needed for control or that relate to 'customer satisfaction' (in the widest sense). For example, it is all too easy to measure the (unimportant) temperature of the flames inside a heating furnace but all too difficult to measure or even estimate, as a function of time, the internal temperature contours of a massive object that is being heated. Important (to the customer) product variables, (say) the clarity of plate, glass or the texture and taste of a food product, are typically difficult to define and even more difficult to measure online.

10 Discrete time and digital control systems
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This chapter discusses the discrete time and digital control system. Advantages of digital control systems over their analogue equivalents, the sampling on which they depend does introduce inevitable performance degradation. In summary, this chapter discusses the delays introduced into a feedback loop by sampling are necessarily destabilising, and very rapid sampling of noisy continuous signals can amplify the noise content, particularly if differentiation of the signals is envisaged.

11 Multivariable linear systems and the state space approach
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This chapter concentrates on establishing the mainstream structure of the state space approach. This chapter also establishes statevariable techniques for the representation and analysis of both continuous time and discrete time systems with an analogous development for the two cases. Canonical forms are introduced for the structural insight that they create and it is also indicated how canonical forms may be useful in control system design.

12 Links between state space and classical viewpoints
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The purpose of this chapter is to increase intuitive understanding of state space representations by linking to situations that are already familiar in a classical transfer function or frequency response method.

13 Optimisation
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The term optimisation is often used very loosely in general speech, but in control theory it has a precise meaning: the action of finding the best possible solution as defined by an unambiguous criterion. Timeoptimal control and linear quadratic optimisation are also discussed in this chapter.

14 State estimation: observers and the Kalman filter and prediction
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Most of this chapter will be devoted to methods for determining the current value of a not directly measurable state vector from measurements of the outputs y and the inputs u of some linear system. A more realistic problem situation, in which models are known only approximately and measurements are subject to noise, goes by the name state estimation and is the main topic that follows.

15 An introduction to robust control design using H _{∞} and related methods
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This paper discusses H_{∞} approach by specifically taking into account modelling uncertainty, and doing so in a worstcase sense, allow complex control design problems to be solved in a theoretically rigorous way while guaranteeing robustness of the implemented solutions over a prespecified range of model incorrectness or (equivalently) of process variability. Here, we review the linear spaces that underlie much of the modern operatorbased control theory with particular emphasis on the theory underlying H_{∞} approach. Some of the H_{∞} control design methodology is then introduced in very simple terms to establish the basic principles.

16 A miscellany of control techniques
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This chapter describes a selection of what are sometimes referred to as artificial intelligence (AI) techniques. Neural networks, fuzzy logic and genetic algorithms; control switching, gain scheduling, adaptive and learning techniques; intelligent systems, agentbased and cooperative systems are discussed.

17 Review: the development of the control systems discipline and the mathematical roots of control systems theory
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During the period of early industrial development, control was not identified as anything significant since the main preoccupations were with wider basic issues. For instance, the main problems in the early coal industry were with explosions, roof falls, carbon monoxide poisoning and dustborne diseases. Once these problems had been largely solved, control systems technology came into play, for instance, in the design of remotely operated coal cutters. Presentday coal mine managers are now preoccupied with logistics, reliability, information and maintenance. The evolutionary patternmechanisation/automation and control/organisation and logisticscan be discerned in almost every industry. Thus, automatic control was scarcely needed until mechanisation had produced the devices and processes that needed to be controlled, and, in fact, it was the requirements of telephony that drove Nyquist (1932), Black (1934), Bode (1945) and coworkers to develop their frequency response and feedback techniques that were to have such wide applicability much later. Control theory rests on a very substantial mathematical foundation that gives rigour to mathematical representations of dynamical systems and underpins all the approaches to systems analysis and multivariable controller design.

18 Resources, references and further reading
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General remarks on the control literature and on the following references and recommended further reading are discussed in this chapter. Mainstream control literature and mainstream control books regarding control theory and control systems are also discussed.

Appendix A: Case histories
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Back Matter
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