SUMMARYThis article presents a Lyapunov-Krasovskii formulation of scaled small gain problem for systems described by coupled differential-difference equations. This problem includes H ∞ problem with blockdiagonal uncertainty as a special case. A discretization may be applied to reduce the conditions into linear matrix inequalities. As an application, the stability problem of systems with time-varying delays is transformed into the scaled small gain problem through a process of either one-term approximation or two-term approximation. The cases of time-varying delays with and without derivative upper-bound are compared. Finally, it is shown that similar conditions can also be obtained by a direct Lyapunov-Krasovskii functional method for coupled differential-functional equations. Numerical examples are presented to illustrate the effectiveness of the method in tackling systems with time-varying delays.

SUMMARYThis paper proposes a sliding mode observer-controller design method for uncertain Markovian jump systems with time delays and uncertain switching probabilities. Both the structures of a sliding mode observer and a sliding mode controller are given. By the mode-dependent Lyapunov functional approach, a sufficient condition for the stochastic stability of the closed-loop system is given, which can be converted into a convex optimization problem. The reachability of the sliding surfaces in both the estimation error space and the state estimate space can be ensured by the presented control scheme. Finally, the effectiveness of the proposed design method is demonstrated by a simulation example.

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