Deadbeat Control for Multivariable Discrete Time Systems with Time Varying Delays

Abstract

In this chapter a novel approach for the deadbeat control of multivariable discrete time systems is proposed. Deadbeat control is a well known technique that has been implemented during the last decades in SISO and MIMO discrete time systems due to the ripple free characteristics and the designer selection of the output response. Deadbeat control consist in establishing the minimum number of steps in which the desired output response must be reached, this objective is achieved by placing the appropriate number of closed loop poles at the origin and cancelling the transmission zeros of the system. On the other side, constant time delays in the state or the input of the system is a phenomena found in many continuous and discrete time systems, produced by delays in the communication channels or other kind of sources, yielding unwanted effects on the systems like performance deterioration, or instability on the system. Even when the analysis and design of appropriate controllers with constant time delays in the state or the input has been studied by several researchers applying several control techniques such as state and output feedback, in this chapter the development of a deadbeat control for discrete time systems with constant delays is explained as a preamble of the main topic of this chapter related to the deadbeat control of discrete time systems with time varying delays. This first approach is derived by implementing a state feedback controller, and in opposition of the implementation of traditional techniques such as optimal control where a stable gain is obtained by solving the required Riccati equations, the deadbeat controller is obtained by selecting the appropriate gain matrix solving the necessary LMI’s placing the required number of poles at the origin and eliminating the finite transmission zeros of the system in order to obtain the required deadbeat characteristics in which the desired system response is reached in minimun time steps. After this overview, deadbeat controllers are designed considering the time varying delays, following a similar approach such as the constant time delay counterpart. In order to obtain an appropriate deadbeat controller, a state feedback controller gain is obtained by solving the required LMI’s, placing the required poles in order to obtain the desired response cancelling the finite transmission zeros. The theoretical background is tested by several illustrative examples and finally the discussion and conclusions of this work are shown in the end of this chapter.