Stabilization of linear systems with distributed input delay using reduction transformation

Abstract

We propose a new stabilization method for linear systems with distributed input delay via reduction transformation and Riccati equation approach. In the presented stabilization scheme, the gain matrix of controller is constructed by the well-known linear control technique for delay-free systems. The transformation kernel matrix can be determined by solving the non-symmetric matrix Riccati equation backward with the boundary condition. When point delay systems are considered, it will be shown that the proposed control law degenerates to the standard one for input delay systems.delay systems, delay-free systems, distributed delay, reduction transformation, riccati equation Citation:Choon K A. Stabilization of linear systems with distributed input delay using reduction transformation. Chinese Sci Bull, 2011Bull, , 56: 1413Bull, −1416Bull, , doi: 10.1007 In industrial processes, time delays often occur in the transmission of information or material between different parts of a system. The presence of time delays causes serious deterioration on the stability and performance of the system and considerable researches have been done on the control problems of time delay systems [1,2]. Time delay systems are a special case of infinite dimensional systems which have an infinite number of poles. This feature makes the control of time delay systems a difficult task to handle. The easiest way is to reduce time delay systems to delayfree systems and then apply the well developed control techniques for the finite dimensional systems. The Smith predictor may be one of the most well known delay compensation methods utilizing this concept. Although a number of improved versions of the Smith predictor have been proposed by many researchers, the Smith predictor methods suffer from some inherent shortcomings. First, the Smith predictor cannot be applied to unstable systems [3]. The second drawback is that the initial state of the system is ignored in the Smith predictor method. The degradation of the performance is an inevitable result, when it is applied to the systems with nonzero initial conditions. The third shortcoming is the lack of robustness. The performance of the Smith predictor method is sensitive to the accuracy of the model of the system and time delay. It also suffers from sluggish responses to disturbances. To overcome the above mentioned difficulties, a number of improved versions of the Smith predictor have been proposed by many researchers [4]. The reduction transformation method is a different approach from the state-space setting to overcome these problems. The explicit reduction transformation method of input delay systems was suggested by Kwon and Pearson [5]. In [5], Kwon and Pearson employed the receding horizon method to input delay systems, where an unexpected good stabilization method (reduction transformation) was founded. The reduction transformation method was extended to the output-feedback case in [6], where a delay-free observer was introduced. This observer can directly estimate the linear funct...

Spectrum assignment and stabilization of linear differential equations with delay by static output feedback with delay

“…This approach is a development of the second Lyapunov method and allows one to obtain sufficient conditions for asymptotic and exponential stabilization of delayed systems. On the base of the Lyapunov-Krasovsky approach, various methods were obtained in [3][4][5][6]: in [3], a new technique is introduced based on the barycentric representation of a distributed delay system; in [4], a new stabilization method is proposed for linear systems with distributed input delay via reduction transformation and Riccati equation approach; in [5], conditions for stabilization are obtained by using the full-block S-procedure and a convex-hull relaxation in terms of a LMI; in [6] the problem of optimal stabilization is studied. Other approaches to solving the problem of stabilization of systems with distributed delay are presented in [7,8].…”