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Stability and oscillations of nonlinear pulse-modulated systems. (English) Zbl 0935.93001

Boston: Birkhäuser. xvi, 362 p. (1998).
Pulse-modulated systems can be seen as a special class of functional differential or functional integral equations where continuous dependence of the solutions on initial conditions is lacking and where the solutions themselves may be discontinuous. The book is devoted to these sampled-data systems.
In chapter one, various types of pulse modulation are described. One modulates the amplitude, the width, the phase, the frequency of an output signal of the modulator (a nonlinear operator) or a combination. Several generalizations where the parameters of modulation are computed through more complicated procedures depending on functions or on the solutions to integral equations are described, too. The case of output impulse responses to piecewise continuous inputs is examined. The basic feedback structure where one regulates the pulse modulating system cascaded with a stable linear system, is described next. The system response seen as an integral equation and its implication for more complicated linear parts (delay equations, PDE) as well as in the study of stability in the impulsive output case are presented. It is finally seen how to reduce such systems to discrete ones (also from the integral equation) as well as to relay systems when the frequency is modulated.
In chapter two, several sufficient conditions for global stability of the feedback configuration for some types of frequency modulation are obtained. They are often conservative, based on frequency domain conditions (reflecting the impact of the linear part): the method of integral quadratic bounds of V. Yakubovich, a frequency domain property of positive kernels of integral operators, and some more or less direct methods including the use of quadratic Lyapunov functions. Here the linear part is not necessarily stable and the impact of the nonlinearity may be seen in the choice of the Lyapunov function. Here again the linear part is used to compensate for the effects of jumps which may not be desired.
In the next chapter, sufficient conditions for stability are obtained via the averaging method. If \(f\) (resp. \(\sigma\)) is the output (resp. the input) of the modulator then “an equivalent nonlinearity” \(\phi(\sigma)\) is introduced: \[ \phi[\sigma(t^\sim_n)]= {1\over t_{n+1}- t_n} \int^{t_{n+ 1}}_{t_n} f(t) dt. \] The form of the equivalent nonlinearity is obtained for many types of modulation providing a unified framework to handle them. Then, if one assumes that the equivalent nonlinearity satisfies a sector condition, the authors obtain frequency domain criteria generalizing the Popov stability criterion.
Chapter four extends the point of view of the previous chapter to cases where disturbances enter the dynamics. One desires that the asymptotic behaviour of the introduced quantities does not depend on the initial states. In order to apply the previous theorems, certain conditions related to the equivalent nonlinearity have to be satisfied and a detailed study for each case of modulation is performed.
Chapter five investigates the existence of forced periodic oscillations. It is required to find an initial state such that, for the usual feedback configuration, the input and the output variable of the modulator are T-periodic. A T-periodic reference signal is allowed. One distinguishes cases where the output is bounded or a train of impulses. The condition for periodicity is expressed as an integral equation which is called the equation of periods and sufficient conditions for it to hold for several types of modulation and for a stable linear part (in some cases critical situations are also treated) are provided. The Lemma of Rolle is used but has limitations in some cases where the continuity of the translation operator along trajectories does not hold.
Chapter six and seven give sufficient conditions for the existence of periodic modes using a fixed point theorem for the translation operator along trajectories which, as just pointed out, may be discontinuous. The cases investigated are pulse width modulation and pulse frequency modulation, respectively. In the second case the trajectories themselves may be discontinuous. But this problem is typically repaired by introducing ellipsoids whose invariance is obtained via suitable Lyapunov functions.
In chapter eight, the harmonic balance method (or describing function method), which consists in expanding the signal \(f(t)\) in Fourier series, in assuming that the higher-order harmonics can be neglected and in “pushing” the nonlinearity of the feedback system in the amplitude, is applied to frequency modulated systems.
Chapter nine presents several results on auto-oscillations in pulse modulated systems. V. Yakubovich has introduced this notion to describe systems where all trajectories are bounded, unstable, and the output leaves an interval infinitely many times. This could be seen as a form of “chaos”. Several frequency domain criteria are provided.
In the last chapter, stationary modes of a synchronization system where the state of a slave plant follows the one of a master plant are investigated. The master plant contains a pulse-width modulator device. Several sufficient frequency domain conditions for the existence of the mode are obtained.
An appendix contains many technical results used throughout the book. A rich list of references and a modest index close the book.
This volume leaves the impression that “the right word” about the complicated nonlinear systems under investigation has not yet been discovered (how far are sufficient conditions from necessary conditions? periodic modes exist but where are they?). It develops the ideas of V. Yakubovich and of the St. Peterburg school.
As stated in the preface, engineers are embarrassed by the mathematical sophistication required to treat rigorously pulse modulated systems and mathematicians have to face complicated technical schemes devised by engineers in this domain. Moreover, publications in this field are seldom or available only in the Russian literature. So this book is original and useful for engineers and control scientists in the West and other countries.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93C57 Sampled-data control/observation systems
93C10 Nonlinear systems in control theory
93D15 Stabilization of systems by feedback
93C23 Control/observation systems governed by functional-differential equations
34K13 Periodic solutions to functional-differential equations
93D10 Popov-type stability of feedback systems
34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
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