Control Theory (2nd Edition)
This fully revised and updated edition of Control Theory: a guided tour addresses recent developments in the field. It discusses how the rise of Hoo and similar approaches has allowed a combination of practicality, rigour and user interaction to be brought to bear on complex control problems and has helped bridge the gap between control theory and practice. The book also examines the effects of the rise of artificial intelligence techniques and the increasing availability of comprehensive software packages.
Inspec keywords: control nonlinearities; distributed control; state estimation; optimisation; feedback; artificial intelligence; learning systems; discrete systems; genetic algorithms; control system synthesis; linearisation techniques; fuzzy logic; multivariable systems; frequency response; Laplace transforms; robust control; neural nets
Other keywords: artificial intelligence technique; mathematical modelling; control design; discrete time control; control concept; Laplace transform; frequency response method; state estimation; neural network; Kalman filter; multivariable linear process; intelligent system; fuzzy logic; distributed system; learning system; genetic algorithm; robust control design; automatic feedback control loop
Subjects: Discrete control systems; Formal logic; Multivariable control systems; Artificial intelligence (theory); Neurocontrol; Simulation, modelling and identification; Optimisation techniques; Control system analysis and synthesis methods; Stability in control theory; Integral transforms; Nonlinear control systems
Since control theory deals with structural properties, it requires system representations that have been stripped of all detail, until the main property that remains is that of connectedness. (The masterly map of the London Underground system is an everyday example of how useful a representation can be when it has been stripped of all properties except that of connectedness.) Connectedness is a concept from topology. Topology, the discipline that studies the underlying structure of mathematics, offers fascinating reading to aspiring systems theorists. Recommended reading is given in the Bibliography. Clearly, a system is a very general concept; control theory is most interested in certain classes of system and to make progress we delineate the classes. First it is interested in dynamic systems these are systems whose behaviour over a time period is of interest. Thus if a system were concerned with population aspects, a similar dynamic system would be concerned with population growth. Secondly, it is most interested in and most powerful when dealing with linear systems. The upper part of the figure shows a system's response to some arbitrary stimulus. The lower part shows how, in the presence of linearity, the response to a scaled-up version of the stimulus is simply a scaled-up version of the previous response, with proportionality being preserved. Finally, it is interested in feedback systems -these are systems where information flows in one or more loops, so that part of the information entering an element may be information that previously left that element.
In the previous chapter we saw that prerequisites for control design were broadly: 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. This model is assumed able to rapidly calculate the expected behaviour of the system when subjected to any particular action.
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 pre-determined at the design stage. The controller may be driven by time alone or it may be driven in a more complex way by a combination of signals. In feedback control, the controller is error driven. That is, the controller receives a continuous measurement of the difference between required behaviour and actual behaviour and its output is some function of this error.
Many useful techniques depend on the Laplace transform. The Laplace transform of a function f(t) is denoted sometimes by X {/(?)} and sometimes by F(s). The inverse Laplace transform of F(s) is denoted sometimes by £~l{F(s)} and sometimes by f ( t ).
Frequency response methods have a physical explanation that is readily understand able 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 servo-mechanism, process control and aerospace applications, and they have been rather resistant to applications outside these areas.
Whereas control theory is a fairly coherent well-defined body of concepts and knowledge, supported by techniques, the activity of mathematical modelling is ill-defined and its practitioners are scattered amongst many disciplines. Thus, in science, models are often used to explain phenomena as, for instance, the Bohr model of the atom or the wave theory of electromagnetic propagation. Such models are essentially visualisations of mechanisms. Far removed from this are those models, usually implicit and sometimes fictitious, by which politicians claim to predict future rates of employment or inflation. We can propose that the science models contain and this is their fundamental characteristic-a representation of Whereas control theory is a fairly coherent well-defined body of concepts and knowl edge, supported by techniques, the activity of mathematical modelling is ill-defined and its practitioners are scattered amongst many disciplines. Thus, in science, models are often used to explain phenomena as, for instance, the Bohr model of the atom or the wave theory of electromagnetic propagation. Such models are essentially visualisations of mechanisms. Far removed from this are those models, usually implicit and sometimes fictitious, by which politicians claim to predict future rates of employment or inflation. We can propose that the science models contain and this is their fundamental characteristic-a representation of physical variables. The second group may be, in the extreme, no more than extrapolations of past trends. Constructing a model in the first category is primarily a matter of bringing together, combining and refining concepts to produce an object called a model (usually it will consist of a set of equations).physical variables. The second group may be, in the extreme, no more than extrapolations of past trends. Constructing a model in the first category is primarily a matter of bringing together, combining and refining concepts to produce an object called a model (usually it will consist of a set of equations).
Most closed loop systems become unstable as gains are increased in attempts to achieve high performance. It is therefore correct to regard stability considerations as forming a rather general upper limit to control system performance. Also, as will be discussed in this chapter, achievable rates of change are always constrained in practice by equipment limitations.
In view of the evident efficiency of feedback controllers in controlling unknown phenomena, is it not feasible to attempt control of all processes by some very simple standard strategy? The simplest possible controller involves just multiplication of the error by a scalar C; the overall transfer function is CG(s)/(l + CG(s)) and if C is very high, then the overall transfer function is approximately CG(s) 1 CG(s) i.e. provided that C ^> 1 near-perfect control can be obtained.
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 non-linear system to replace the non-linear system model by a linear approximation.
By a multivariable process we mean a process with several inputs and several outputs. In general, every input is connected to every output through some dynamic coupling. We can pretend that the ith output yi is connected to the jth input ui through a transfer function gij(s). In practice, multivariable closed loop systems can rarely be diagonalised for all frequencies by choice of controller. However, they can be made diagonally dominant; that is, the diagonal terms can be made large compared with the off-diagonal terms.
A system that can change its state only at discrete points in time is called a discrete time system. Amongst the many examples of discrete time systems in everyday life could be mentioned the rates of exchange for foreign currencies charged by retail banks. Typ ically, these rates may be updated once every working day and stay constant otherwise. Computers are the discrete time systems that interest us here; in particular, com puters that perform the same calculation repeatedly. Such computers are used as controllers within closed loop systems. It turns out, perhaps surprisingly, that the discrete time effects of a computer, when used as a controller, are sufficiently pro found to require a whole new batch of design techniques.
Many powerful feedback control strategies require the use of state feedback. However, in many important practical cases the state is not avail able to be fed back (it is said to be inaccessible). In such cases, a state estimator may be used to reconstruct the state from a measured output.
Any system whose input-output characteristic does not satisfy the conditions for linear function is classified as a nonlinear system. Thus, there is no unifying feature present in nonlinear systems except the absence of linearity. Nonlinear systems sometimes may not be capable of analytic description; they may sometimes be discontinuous or they may contain well understood smooth mathematical functions.
Optimisation is concerned with finding the best possible solution, formally referred to as the optimal solution, to a particular problem. 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 (or cost function). Optimisation has, to some extent deservedly, acquired a reputation for being out of touch with reality. This is because the analytic techniques for optimisation are highly involved and in order to make headway many workers have resorted to drastic modification of the original problem to allow application of some particular optimisation technique; i.e. simplistic assumptions about the problem have, unsurprisingly, produced simplistic solutions. Currently, more healthy attitudes are beginning to prevail. For instance, it is becoming accepted that, for large complex problems, it may be better to encode optimality criteria in more vague but more realistic terms than parallel human evaluation criteria, than to force unwilling problems into an ill-fitting straitjacket to allow rigorous optimisation. With these reservations having been made, it is possible to turn to the ideas and techniques of optimisation theory and practice.
Because of exposure to school physics and what in the UK is called applied mathematics, we are conditioned to accept without question that, for instance, an object, missile or projectile, flying through space, can be truthfully represented by a single point located at the object's centre of mass. This practice, while allowing neat examination questions, leads us into a false sense of simplistic security. For instance, as soon as a projectile is made to spin about its axis of travel (a common practice), we may be unprepared for the escalation of complexity of the problem that this simple addition to the problem causes.
In this chapter, we review the linear spaces that underlie much of modern operator-based control theory with particular emphasis on the theory underlying H∞ approaches. Some of the H∞ control design methodology is then introduced in very simple terms to establish the basic ideas. The chapter ends with an introduction to a deeply satisfying and visualisable approach: the v gap metric method, which is firmly embedded within the H∞ family but which is both powerful and general as well as intuitive.
Neural networks are sets of interconnected artificial neurons that, very simplistically, imitate some of the logical functioning of the brain. Fuzzy logic emulates the reliable but approximate reasoning of humans, who, it is said, distinguish only six or seven different levels of any variable during decision making. Genetic algorithms and genetic programming are powerful evolutionary search methods that can search for structures as well as numerical parameters. Learning systems aim to emulate the human learning-by-experience mechanism so that a system can potentially learn to perform a task with increasing efficiency over time using an iterative algorithm. Intelligent machines and machine intelligence offer future prospects for creating systems with ever increasing autonomy and reasoning ability.
During the period of early industrial development, control was not identified as any thing 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 dust-borne diseases. Once those problems had been largely solved, control systems technology came into play, for instance in the design of remotely operated coal cutters. Present day coal mine managers are now preoccupied with logistics, reliability, information and maintenance. The evolution ary pattern - mechanisation/automation and control/organisation and logistics can be discerned in almost every industry.
This chapter provides a list of references and further reading.
