Arkadii Kh. Gelig

Mathematical Description of Pulse-Modulated Systems

We begin with the mathematical description of pulse-modulated systems. Basically, we shall follow the classification given in [TP73]. The principal element of an impulsive system is a pulse modulator. In mathematical terms, it is described by a nonlinear operator that maps an input function σ(t) to an output function f (t) (both functions are defined for \(t \geqslant 0\) and have real scalar values). The specific form of this operator and the restrictions imposed the type of modulation and on the mathematical model accepted. The most general property of a pulse modulator is that it produces an increasing sequence of time moments t0 = 0 < t1 < t2 <… called sampling moments. The time interval [tn, tn+1) is called the nth sampling interval. For some types of modulation it is supposed that t n+1 -t n = T = const. (the value T is called a sampling period), whereas for the other types the value t n+1 -t n depends on an input function.

Auto-Oscillations in Pulse Modulated Systems

Two classes of auto-oscillating systems can be considered [Vor81]. The first involves systems for which an auto-oscillation is the main operating mode. For such systems the term “auto-oscillation” designates a stable periodic solution [AWC65]. Hence to investigate such a system we need to solve two problems: to find conditions for the existence of a periodic solution and conditions for its stability. The second class includes systems for which auto-oscillations are undesirable and opposed to stability. For such systems it is inconvenient to consider auto-oscillation as a periodic mode because an unstable solution does not need to be periodic. For these systems we shall employ the notion of auto-oscillation due to V.A. Yakubovich [GLY72]. A system will be called auto-oscillative if all of its trajectories, save equilibria, with initial values in a certain region of the state space have the following properties: they are bounded for t ≥ 0, they do not tend to equilibria, as t→ +∞, and a system output σ(t) changes its sign infinitely many times as t→+∞. Neither the existence of a periodic solution nor its stability are supposed. By developing ideas proposed in the papers [Ge183a, Ge184, Ge185, Yak73a, Yak76, AKLR76], we shall examine pulse-modulated systems which are auto-oscillative in the sense of V.A. Yakubovich.

Pulse-Width Modulated Systems of Phase Synchronization

To clarify the setting of the synchronization problem given in the next section, let us begin with an electromechanical example [Ge182a]. Consider the system shown in Fig10.1 It consists of two flywheels, denoted W 1 and W 2. The flywheel W 1 is a master which rotates with a constant angular velocity. The other flywheel W 2 is a slave and its rotational speed is to be adjusted to that of W 1. The external moment (or external load) N is applied to the shaft of W 2 . The velocities of rotation are measured in the following way. Each of flywheels has a number of holes distributed uniformly along its outer edge. Consider the flywheel W 1. When it rotates, the holes intersect a beam of light produced by the light source L 1. When the beam goes through a hole, it falls on the photoelectric cell Φ1. The photoelectric cell switches on the control device CD which produces a control torque applied to the shaft of the slave flywheel W 2 . The same scheme is applied to the flywheel W 2 , but the photoelectric cell Φ2 is used to switch out the control signal. Let us give a mathematical description of this system. Open image in new window FIGURE 10.1

Periodic Oscillations. Method of Harmonic Balance and Its Justification

The method of harmonic balance is one of the most popular approximate methods of investigating periodic modes in nonlinear systems (see e.g. [PP60, GK83]). The idea of this method was first proposed in [KB37]. Its different versions were developed independently by many authors (see e.g. [Go147, Koc50]). Applications of the harmonic balance to impulsive systems can be found in [PJ65, Vid68, KC70, JI75, Ant76]. Before we start a study of impulsive systems by this method, we shall use a simple system (not of an impulsive nature) shown in Figure 8.1 to illustrate its main ideas. Open image in new window FIGURE 8.1

Stability of Equilibria. Averaging Method

The averaging method is the principal one for stability studies in this book. It is extensively applied in chapters 3, 4, 9 and 10. In this section we describe the history and the main idea of the method. Note that not all the results obtained with the help of the averaging method are covered by this monograph. We can mention studies on the stability of stochastic pulse-modulated systems [GEC94, GE95a, GE95b], on the stability of asynchronous systems with combined modulation [AG93] (where the system investigated previously in [OK87] is considered), on the stabilization of a PWM system by a harmonic external action [GC93b], and on robust stability [Ge196].

Oscillations of PFM Systems.Fixed-Point Method

Practical use of the method of finding forced periodic modes, described in the previous chapter, has two difficulties. First, for a pulse modulation of the second kind, the solvability of the equations of periods is insufficient for the existence of a periodic mode. Some additional research is required. Secondly, if a period of an external action S2 is N times greater than the sampling period T, then, instead of one equation of periods, we obtain a system of N transcendental equations, whose solution becomes substantially more complicated with the increase of N.

Stability of Equilibria. Miscellaneous Methods

In this chapter we describe a number of methods used for investigating the stability of pulse modulated systems. They include the method of integral quadratic bounds proposed by V.A. Yakubovich [Yak68], the method of positive kernels of integral operators, first proposed by A. Halanay [Hal64] and developed for pulse-modulated systems by A.Kh. Gelig [Ge182a], the method of direct integral estimates given by H.O. Gulcur and A.U. Meyer [GM73], and the two versions of applying the direct Lyapunov method proposed by V.M. Kuntsevich and Yu.N. Chekhovoi [KC71b, KC70] and by the authors of this book [GC97]. The averaging method, which is the principal method for stability investigations of this monograph, will be thoroughly described in chapters 3 (stability of equilibria) and 4 (stability of processes). We have limited ourselves to the study of stability in the large, so the problem of stability in the small was not considered at all.

Stability of Processes. Averaging Method

In the preceding sections we studied the stability of the zero equilibrium x = 0 of the system whose CLP is given by (3.1). However, the technique developed there enables us to investigate the stability of an arbitrary solution (a process) of the system

Stabilization of New Classes of Uncertain Systems

\frac{{dx}}{{dt}} = Ax + bf + q\left( t \right),\sigma = {c^*}x + \psi \left( t \right),f = M\sigma