Stabilizability of constrained uncertain linear systems via smooth control Lyapunov R-functions

The stabilization problem of constrained uncertain linear systems is addressed via the class of control Lyapunov R-functions that are obtained reformulating the classic geometric intersection operator in terms of R-functions. The feasibility test of the proposed smooth control Lyapunov functions can be casted into (bi)linear matrix inequalities conditions. Like polyhedral Lyapunov functions, the maximal estimate of the controlled invariant state space set is achieved. The advantage of the proposed approach is that the inner sublevel sets are smooth and can be made everywhere differentiable. This smoothing technique is very general and it can be used to smooth both polyhedral and truncated ellipsoidal control Lyapunov functions to improve the control performances, as shown in some benchmark examples.