Truncated Predictor Based Feedback Designs for Linear Systems with Input Delay

Time delay systems refer to control systems whose change of current state depends on the past values of its state and/or input. Input delay in a control system emerges when the transmission of the control signal from the controller to the actuator takes a certain amount of time. This lagging effect in the input can be caused by long-distance transmission of the control signal or time-consuming computation of a control algorithm carried out by the controller.

A fundamental problem in the control of linear systems with input delay is the problem of stabilization. The importance of such a problem is clear from the observation that feedback laws designed without consideration of the delay mostly fail to stabilize given a significantly large delay. To overcome an arbitrarily large delay, a predictor feedback law predicts the future state of the system as the sum of the zero input solution and the zero state solution of the system. Such a prediction of the future state cancels the effect of the input delay and results in a stable closed-loop system free of delay. However, the term corresponding to the zero state solution in the predictor feedback law is distributed because the zero state solution is a convolution between the state transition matrix and the input term. This causes difficulty in the implementation of the predictor feedback law. A truncated predictor feedback law discards the distributed term to avoid such difficulty. A delay independent truncated predictor feedback law further drops the delay-dependent state transition matrix in the truncated predictor feedback law to overcome uncertain or time-varying delay.

This dissertation presents the stabilization of linear systems with input delay via truncated predictor based feedback designs. The first part of the dissertation is dedicated to the design of truncated predictor based feedback laws with a constant feedback parameter. It is shown through examples that such feedback laws cannot stabilize general, possibly exponentially unstable, linear systems with a sufficiently large delay. Admissible delay bounds with stability guarantee are then established. Comprehensive analysis of stabilization is presented for linear systems with time-varying delay under either state feedback or output feedback.

The second part of the dissertation is dedicated to devising truncated predictor based feedback laws for linear systems with all open loop poles at the origin or in the open left-half plane. A time-varying feedback parameter design for the truncated predictor based feedback laws is motivated by improving the closed-loop performance over constant feedback parameter designs and by enabling the feedback laws to accommodate a completely unknown delay. In particular, time-varying feedback parameters in a delay independent truncated predictor feedback law are proposed to result in a smaller overshoot and a higher convergence rate of the closed-loop system. Such a time-varying parameter design still requires an upper bound of the delay to be known. In the absence of any knowledge of the delay, a control scheme is proposed that equips the delay independent truncated predictor feedback law with a delay independent update algorithm for the feedback parameter. The implementation of such a control scheme is simple because only the current state, and no knowledge of the delay, is required. Further work would focus on devising novel control schemes that completely adapt to an unknown input delay in general linear systems with possibly exponentially unstable open loop poles.