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This paper considers the problem of inherent robustness analysis for nonlinear discrete–time systems using the concept of a finite–time Lyapunov function. The main contribution is to prove that, for sufficiently continuous dynamics and finite–time Lyapunov functions, inherent global input–to–state stability to general disturbances can be established for nominally stable nonlinear systems. Moreover, under mere continuity of the finite–time Lyapunov function and of the dynamics, inherent input–to–state stability on a compact set is obtained.

This paper proposes a method for interpolating two (or more) polytopic control Lyapunov functions (CLFs) for discrete–time linear systems subject to polytopic constraints, thereby combining different control objectives. The corresponding interpolated CLF is used for synthesis of a stabilizing controller by on–line optimization. Recursive feasibility and computational efficiency of the resulting optimization problem are established under suitable assumptions. The proposed method offers a smooth transient between different polytopic CLFs, while guaranteeing stability and constraints satisfaction. Moreover, it can also be designed such that it automatically yields a desired, optimal control law, within a region around the equilibrium. An illustrative example is presented to demonstrate the effectiveness of the developed synthesis method.