Invariance control for a class of cascade nonlinear systems

We consider the control of partially linear cascade systems using switching control of the states of the linear subsystem. We give sufficient conditions under which feedback of the linear states with switching gains guarantees both exponential stability of the linear subsystem and positive invariance of a prespecified region in state space. We refer to a control scheme incorporating these two objectives as invariance control. Semiglobal asymptotic stabilization follows under some additional conditions. The key idea of our design is to keep a given state space region positively invariant by switching on the boundary of the region. Thus, the transient response of the system can be kept within prescribed bounds which is important in many practical applications. Our approach can also be viewed as an alternative to high gain designs. The results in this note can be extended to nonlinear cascades and even to noncascaded systems.