## 3Study of one-frequency self-oscillation in nonlinear harmonically linearized MIMO systems

### 3.1 INTRODUCTION

The *harmonic linearization* method, also called the *harmonic balance* or *describing function* method (Atherton 1975; Cook 1994; Gibson 1963; Taylor 1999), is one of the most powerful approximate methods used in the nonlinear control systems design and probably the most widespread in the engineering community. Its principal merit is that it allows, in quite simple mathematical form, the main parameters of the investigated system to be related immediately to those general performance indices, which are specified for the design. The mathematical foundations of the harmonic linearization method were laid out in the works of Krilov and Bogolubov (1934, 1937). Later on, an essential contribution to the development of the method was made by Goldfarb (1947), E. Popov (1962, 1973), Kochenburger (1950), Tustin (1947) and many others.

Let us recall briefly the main ideas of the harmonic linearization method for the simplest case of the investigation of *self-oscillation* (or *limit cycle*) in a nonlinear SISO system with a single nonlinear element (Figure 3.1). Suppose that the input signal is zero, that is φ(*t*) = 0, and the nonlinear characteristic *F*(*x*) is odd, symmetrical and memoryless. Suppose also that, in the system, we have steady-state symmetrical self-oscillation with some constant frequency Φ, and linear part *W*(*s*) is a low pass filter, that is it sufficiently attenuates all higher ...