## Analysis of feedback control systems by the averaging method.(English. Russian original)Zbl 0742.93031

Autom. Remote Control 51, No. 11, 1487-1494 (1990); translation from Avtom. Telemekh. 1990, No. 11, 37-44 (1990).
The considered class of feedback control systems is described by the system of ordinary differential equations $$\varepsilon dy/dt=Ay+F(t/\varepsilon, u(x)-Gy)$$, where $$x$$ is the state vector, $$u$$ the control and $$Gy$$ the actual control signal. $$F\in \mathbb{R}^ k$$ is assumed periodic in $$t$$ and can contain discontinuous functions. This is typical for relay systems and systems with pulse-width modulations. In order to have a controller with small llag, the parameter $$\varepsilon$$ must be small. The control motions are described by $$dx/d\theta=\varepsilon f(x)+\varepsilon HGy$$, $$\theta=t/\varepsilon$$ (fast variable), $$f$$ continuous, $$H$$ is a matrix. The main object is the extension of the averaging principle, developed by Volosov, to systems with discontinuous nonlinear functions as well as to the case of nonlinear functions of locally functional type. Defining an averaged function $$\bar f(\bar x)$$ and assigning the averaged ODE $$d\bar x/dt=\bar f(\bar x)$$, then in the main result, an upper bound for the differences $$| x(t)-\bar x(t)|$$, $$| y(t)-Y_ *(t,x(t))|$$ is derived. Here, $$Y_ *$$ is the (unique) fixed point of an ODE of the form above.

### MSC:

 93C15 Control/observation systems governed by ordinary differential equations 93B52 Feedback control 34C29 Averaging method for ordinary differential equations