Fractional-order Control And Observer Design For Time-Delay Systems

Time-delay systems are widely existed in practical engineering problems, the presence of time delay makes the analysis and synthesis of systems become more complex and difficult, in addition, the presence of time delay is the source of leading to system instability and making performance of the system worse. In recent years, the study of time delay systems which has attracted people's great attention has a very important theoretical and practical significance. Fractional calculus which is a promotion of traditional integer-order calculus theory is a both old and young field in the field of mathematical research. So far, the research on the fractional calculus theory has made a great number of achievements, which lay a new theoretical foundation for its applications to various subjects. This paper studies the time delay systems on the basis of fractional calculus theory.The main research contents of this paper are as follows:Firstly, For time-delay system with parametric uncertainty, the fractional calculus is introduced to the traditional integer-order integral compensator of sliding mode, which is applied to the control of the time-delay system with parametric uncertainty, so that it can track the given ideal model effectively. Simulation results prove the superiority of fractional-order integral compensation of sliding mode, time-delay system with parametric uncertainty controlled by fractional-order integral compensator of sliding mode can well track the given ideal system. Compare to integer-order integral compensator of sliding mode, fractional-order integral compensator of sliding mode has a better control effect in no matter the steady error time, the steady error size or the chattering.Then, this paper designs a fractional-order observer for time-delay systems, by simulation in MATLAB software, it can be found that both integer-order observer and fractional-order observer can track the system very well. However, the track performance of fractional-order observer achieves is better than that of integer-order. When time delay increases, integer-order observer can not track the system while the fractional-order observer still achieves an effective tracking of the system.Finally, the paper mainly studies on chaotic dynamics of the integer order logistic delay system and fractional order logistic delay system respectively by simulink simulation. simulation results demonstrate that there exist limit circle and chaos not only in integer-order logistic delay system but also in the fractional order logistic delay system with order less than 1.Through bifurcation curves with respect to different parameters, we can find the parametric ranges in the presentation of stability, limit circle (periodic solutions) and chaos of the system. For fractional logistic delay system, in the bifurcation curve with respect to fractional order, when fractional order decreases, it shows a period doubling bifurcation to chaotic motion. To have a further research on the system, phase diagrams with different parameters of fractional-order logistic delay system which are limit circles are given, which demonstrate the complex characteristics of fractional-order logistic delay system. Then, a simply effective feedback control method is introduced to control the fractional order logistic delay system, simulation results demonstrate that by tunning the feedback gain K, chaotic motion can be effectively controlled to a limit circle or an equilibrium point. Finally, the chaotic lag synchronization issue is proposed to synchronize two fractional order logistic delay systems with matched parameters and unmatched parameters. Simulation results demonstrate the effectiveness of the proposed control method and synchronization method. When systems' parameters are unmatched, the dynamic performance can be greatly improved by adjusting the fractional order q.