Essentials of Nonlinear Control Theory
The book is concerned with understanding the structure of nonlinear dynamic systems within a control engineering context After a discussion of theoretical foundations, the development moves to specific techniques (describing function method, phase plane portrait, linearisation methods). The treatment then becomes oriented to qualitative analysis and maintains this emphasis to the end of the book. The broad aim is to develop methods that will allow the topology of system behaviour to be visualised. The main tools are Lyapunov methods, extending to include recent work on system decomposition. A bibliography lists both earlier seminal and recent literature to allow the reader to follow up particular aspects.
Inspec keywords: nonlinear dynamical systems; describing functions; limit cycles; linearisation techniques; nonlinear control systems; gradient methods
Other keywords: nonlinear secondorder system linearisation; system decomposition; dynamic systems; gradient systems; Lienard's equation; phase plane portrait; Limit cycles; relaxation oscillations; nonlinearity; describing function method; envelope methods
Subjects: Nonlinear control systems; Control system analysis and synthesis methods; Stability in control theory
 Book DOI: 10.1049/PBSP006E
 Chapter DOI: 10.1049/PBSP006E
 ISBN: 9780906048962
 eISBN: 9781849194464
 Page count: 104
 Format: PDF

Front Matter
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1 Initial orientation
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One group of approaches involves replacement of the nonlinear system by (in some sense) a linear approximation. Analysis of the linear system then gives information about the original nonlinear system. The describing function method and Lyapunov's first method are in this group. A second approach attempts to find useful information about a system without solving the system equations. Lyapunov's second or direct method has such an approach to determine approximate regions of stability. A third approach uses what I have called 'envelope techniques'. Here, the nonlinearity is enveloped graphically in a linear segment and a 'worstcase' stability analysis is then possible. Numerical solution of nonlinear equations is, of course, always possible but is of limited usefulness in obtaining an overall insight into system behaviour. A useful graphical technique for the display of nonlinear system behaviour is the phaseplane. Despite the fact that the display is virtually restricted to secondorder systems, it allows most of the important concepts of nonlinear analysis to be illustrated.
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2 Dynamic Systems  definitions and notation
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The book is concerned with the study of nonlinear dynamic systems. This chapter sets out some definitions and notation for this subject.
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3 The describing function method
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The describing function method can be considered as an extension to nonlinear systems of the usual Nyquist stability criterion. The origins of the method can be traced back to nonlinear mechanics (Krylov and Bogoliuboff, 1932) but the first workers to apply the approach specifically to control systems were Tustin (1947) and Kochenburger (1950).
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4 The phase plane portrait
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Since frequency response techniques and root locus diagrams are not applicable to a nonlinear process, there is an important need for a graphical tool to allow nonlinear behaviour to be displayed. This need is filled by the phase plane diagram. The method is applicable to secondorder processes without input, although effects equivalent to step or ramp inputs can be obtained by choice of initial conditions.
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5 Linearisation
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The replacement of a nonlinear function by a linear approximation is known as linearisation. The motivation for linearisation is to allow analysis of a nonlinear problem by linear techniques. The results of such analysis need to be interpreted with care to ensure that the linearising approximation does not cause unacceptably large errors.
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6 Determination of the qualitative behaviour of a nonlinear secondorder system by linearisation (Lyapunov's first method)
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Whereas a linear system has only one type of behaviour everywhere in the phase plane, a nonlinear system may exhibit qualitatively different types of behaviour in different regions. Let Σ be a nonlinear system having k qualitatively different types of behaviour in the phase plane. Suppose the system Σ to be linearised in each of the k regions, to yield linear approximations Σι, ..., Σk. It is a reasonable proposition that a knowledge of the behaviour of the system Σ in each of the k regions, can be obtained from a knowledge of the behaviour of each of the k linear approximations. The extent to which the proposition is valid was determined in a doctoral thesis in 1892 by A. M. Lyapunov. (Available in translation as Lyapunov (1966)). The approach outlined above and described below is sometimes called Lyapunov's first method.
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7 Lyapunov's second or direct method
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When stability is guaranteed by the second method of Lyapunov, the system response may still be unsatisfactory in that it is highly oscillatory and/or takes an inordinately long time to settle down. For instance, a numerical iterative loop whose convergence was guaranteed by the second method, converged so slowly as to be quite useless in practice. This aspect is particularly important when the Lyapunov method is used as a design tool. The designer must ensure that a satisfactory rate of convergence is obtained.
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8 Envelope methods  the Popov and circle criteria for graphical analysis of a single feedback loop containing a nonlinear element
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This chapter is concerned with stability analysis of the loop. The two methods to be described enclose the nonlinearity in a linear envelope. The linear envelope rather than the particular nonlinearity is then used in the subsequent analysis. Such an approach naturally leads to sufficient but not necessary stability conditions.
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9 Limit cycles and relaxation oscillations
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Oscillations were first studied as part of nonlinear mechanics and a long standing and comprehensive literature exists. In particular, there are fundamental theories concerning the existence of stable periodic solutions. Some knowledge of this mathematical background is helpful in interpreting the results of nonlinear system analyses. Conversely, if it is required to construct a mathematical model of a stable periodic system, such as a biological oscillator, it will be essential to postulate a mechanism that has the necessary properties.
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10 Liénard's equation
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Certain nonlinear equations have been extensively studied and the solutions and stability properties are well understood. One such equation is Lidnard's equation. It is studied here because it is able to represent a wide class of simple but important nonlinear physical problems.
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11 Gradient systems and system decomposition
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Gradient systems have some particularly simple properties. They cannot exhibit oscillatory behaviour and their stability can be investigated using a natural Lyapunov function that can always be found by integration. An arbitrary dynamic system can be represented as the sum of a gradient system and a conservative system. An analysis of the system in its decomposed form is shown to lead to a Lyapunovlike algorithm. The algorithm can be applied to yield directly a graphical indication of the qualitative behaviour of nonlinear secondorder systems.
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Back Matter
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