This paper considers the simultaneous stabilization problem which concerns finding a single controller to simultaneously stabilize a finite collection of discrete-time nonlinear systems with time-varying state delay and norm-bounded uncertainties. For this purpose, a simultaneously stabilizing discrete-time sliding mode controller (DT-SMC) is synthesized. Based on the delay-dependent Lyapunov–Krasovskii functional method, sufficient conditions are derived which guarantee the existence of a common sliding surface in terms of a nonconvex feasibility problem. The cone complementarity linearization approach is exploited to transform the nonconvex conditions into a convex optimization problem. Finally, the robust simultaneously stabilizing DT-SMC is designed in order to draw the closed-loop trajectories of each system onto the common sliding surface in finite time and to maintain it thereafter. The effectiveness of the proposed method is investigated by simulation of a practical example.

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The simultaneous stabilization is addressed for a collection of nonlinear systems with multiple time‐varying delays. The time delays vary not only from one system to the other but even from one state to another within a system. The nonlinear functions may also be different for each system of the collection. A delay‐independent simultaneously stabilizing control law is presented by developing a Control Lyapunov‐Krasovskii Functional (CLKF) method. This controller has a unified structure for all the systems and, despite differences in nonlinearities and delays, leads to the stabilization of all the systems, simultaneously. During the design procedure, first, a systematic approach is provided to construct a CLKF of a particular form for each system. Second, a simultaneously stabilizing control law is defined based on the constructed CLKFs. Finally, a practical example is given to verify the efficiency and applicability of the given method.