System order reduction in robust stabilization problems

Robust stabilization via state-feedback is considered. Using a fixed quadratic Lyapunov function approach (quadratic stabilization) we investigate the possibility of reducing the quadratic stabilization problem of a given uncertain system to a similar problem for an uncertain subsystem with a fewer number of states, this subsystem is the so-called regular subsystem associated with the original system. It is shown that when some of the control input channels of the given uncertain system are 'free of uncertainty', this reduction is possible. We show that a given uncertain system is quadratically stabilizable via linear state-feedback if and only if the same holds for its regular subsystem. When the regular subsystem is quadratically stabilizable via linear state-feedback, a simple formula for a controller that quadratically stabilizes the original system is given. We also present an example of an uncertain system that is not quadratically stabilizable even though its regular subsystem can be quadratically stabilized via nonlinear state-feedback. Thus, the above equivalence between the original system and its regular subsystem from the point of view of quadratic stabilizability breaks down if nonlinear controllers are considered.