Robust control of uncertain systems in the absence of matching conditions: Scalar input

Robust control is derived for general linear uncertain systems with scalar control. Matching conditions are not assumed to hold. The solution to this general class is made possible by transforming the original system to canonical controllable form. A constant switching surface Fx = 0 is constructed by designing the closed-loop characteristics as a function of the uncertainties so that the uncertainties in the positive definite solution of Lyapunov´s equation cancel the uncertainties in the canonical controllable transformation. The derived control law u(x) has a linear negative feedback term -Kx and a nonlinear term ??u(x) that switches on the switching surface Fx = 0. The nonlinear component ??u(x) of the robust control law is shown to depend on the product of the canonical controllable transformation and the difference between the closed-loop and open-loop characteristics. Examples are presented.