The Multi-step Transformation Control and Exponentially Stability for Linear Discrete-time Systems with Additional Control Delay

Background: Few results about the controller of linear discrete-time system with control delay are reported. Methods: By means of the multi-step transformation with memory, the linear discrete-time systems with additional control delay can be transformed to the equivalent linear discrete-time systems without control delay, and the dimension of the transformed system is not increased. By designing the optimal controller of the finite horizon optimal controller of linear discrete time systems without time delay, the controller of linear discrete time systems with additional control delay can be obtained. At the same time, by designing the optimal controller of the infinite horizon optimal controller of linear discrete time systems without time delay, the controller of linear discrete time systems with additional control delay can be obtained as well. Results: The corresponding finite horizon optimal controller has proved to render the closed-loop system exponentially stable. And the corresponding infinite horizon optimal controller has proved to render the closed-loop system exponentially stable when the open-loop system is either controllable or stabilizable. Finally, two examples are used to verify the theoretical results of this paper. Conclusion: The controller design and the exponentially stability of discrete-time linear system with additional state delay will be investigated in the future.