Uniform ultimate boundedness of control systems with matched and mismatched uncertainties by Lyapunov-like method

The recently proposed control method using a Lyapunov-like function can give global asymptotic stability to a system with mismatched uncertainties if the uncertainties are bounded by a known function and the uncontrolled system is locally and asymptotically stable. In this paper, we modify the method so that it can be applied to a system not satisfying the latter condition without deteriorating qualitative performance. The assured stability in this case is uniform ultimate boundedness which is as useful as global asymptotic stability in the sense that it is global and the bound can be taken arbitrarily small. By the proposed control law we can deal with both matched and mismatched uncertain systems. The above facts conclude that Lyapunov-like control method is superior to any other Lyapunov direct methods in its applicability to uncertain systems.