The problem of output regulation of SISO nonlinear time-varying control systems is discussed in this chapter. The control system is designed to provide robust zero steady-state error of the reference input realization. Moreover, the controlled output transients should have a desired behavior. These transients should not depend on the external disturbances and varying parameters of the plant model. The model of the desired output behavior in the form of a desired differential equation is considered, with parameter selection based on the required output step response parameters (overshoot, settling time). Then an insensitivity condition of the output transient behavior with respect to the external disturbances and varying parameters of the system is introduced. The main particularity of the discussed control law lies in the use of the highest derivative in the feedback loop. The closed-loop system properties are analyzed on basis of the two-time-scale technique and, as a result, slow and fast motion subsystems are analyzed separately.

4.1 Controller Design for Plant Model of the 1st Order

4.1.1 Control Problem

In this section we consider a nonlinear system of the form

(4.1)

where t denotes time, t ? [0, ?), y = x is the measurable output of the system (4.1), x ? ? ¹, u is the control, u ? ? _u ? ? ¹, w is the vector of unknown bounded external disturbances or varying parameters, w ? ? _w ? ? ^l, and ? w( t)