Abstract
This paper deals with the second-order nonlinear control system of which the restoring force and the damping factor contain sinusoidal functions of the displacement. A control system taken up in this paper is the following oscillator with an automatic phase control.
x2=x1
(1+α)x2+(1/√β+α√βcosx1)x2+sinx1=γ
The parameters α, β, γ are all positive, but their magnitudes are not restricted. Therfore, there may be cases in which the damping factor changes alternatively between positive and negative.
First, a phase plane z=x1+jx2 is transferred into a new w plane by the conformal representation with w=exp(jz)=y1+jy2. The stable and unstable singular points in the w plane are, respectively, (√1-γ2, γ) and (-√1-γ2, γ) Locating on the unit circle.
A limit cycle, if exists, is a closed curve enclosing an origin of the w plane, and is the stable second-kind limit cycle. In order to obtain the region of its existence, the Liapunov's function is taken as
V(y1, y2)=R2exp(-2α√β/1+α·y2/R)>0
where R2=y12+y22. The closed curve V(y1, y2)=C is determined corresponding to dV/dt>0 as follows:
C=exp[-2√β(1+γ)]
The second-kind limit cycle may exist in the region lying between this closed curve and the unit circle R=1. A first-kind limit cycle does not exist.
Next, the region of initial values converging to the stable singular point is obtained constructing the Liapunov's function by the method of Infante.
