Uncertainty structures in adaptive and robust stabilization

In this paper we develop and synthesize results concerning the stabilization of parametrically uncertain linear systems via state feedback. The main purpose of this paper is to bring together the previously disjoint areas of parametric robust control and adaptive control, and to examine and compare the uncertainty structures that can be accommodated by various approaches of adaptive stabilization and robust stabilization. Among adaptive approaches we consider adaptive design based on certainty equivalence approach and the ones based on Lyapunov design, in turn include the case of parameter-independent Lyapunov function and the case of parameter-dependent Lyapunov function. Among robust approaches we consider quadratic stabilization as well as nonquadratic stabilization. A hierarchy of uncertainty structures is revealed. As an example of the results established in this paper, we show that the broadest uncertainty structure stabilizable by adaptive control based on parameter-independent Lyapunov function is the antisymmetric stepwise form which is strictly larger than the so-called pure feedback form