Averaging method in stability theory of impulse systems
In the paper there is considered a functional differential equation dx  - =Ax + bо, о = Му, у = (с, x), dt where A is a constant Hurwitzian m × m-matrix, b and c are constant m-dimensional columns, and о is an output signal of impulse element, realizing the pulse-width modulation of the first kind. If by the averaging method for an output signal of impulse element, an equivalent continuous system is constructed, then this system may be globally asymptotically stable for any inclination of static characteristic of modulator. At the same time, for arbitrary large impulse frequency, the considered impulse system can have a periodic solution for a certain inclination of static characteristic. This contradiction disappears if the averaging is applied to the system, obtained as a result of discretization, but not the original continuous system. In the paper the considered system is first reduced to the discrete one and then the averaging method is applied. By constructing quadratic Lyapunov function and applying the discrete frequency theorem, sufficient conditions of global asymptotical stability, which impose restrictions on a discrete transfer function and impulse frequency, are obtained.