Research On Stability And Feedback Control Of Fractional-order Systems

Fractional-order system refers to the dynamical system described by di?erential equation whose di?erentiation order is arbitrary real number or even complex number.Whereas fractional-order control system refers either to the one whose controlled system is a fractional-order system, or to the one whose controller is the fractional-order controller. This dissertation focused on the fractional-order systems, in which the stability and feedback control of fractional-order singular systems, fractional-order T-S fuzzy systems, same order fractional-order nonlinear systems and multi-order fractional-order nonlinear systems were investigated. The following outlined the research:(1) The problem of stability and feedback control for a class of fractional-order singular uncertain systems was investigated. Firstly, based on the stability theorem of fractional-order singular system, it was the su?cient conditions for ensuring robust asymptotical stability of fractional-order singular uncertain systems with fractional-orderα belonging to 0 < α < 1 that was presented. Secondly, by applying the matrix’s singular value decomposition and Linear Matrix Inequality(LMI) technics, the problem of feedback controller for fractional-order singular uncertain systems was discussed; the suitable state feedback and output feedback controllers that guaranteed the asymptotical stability of closed-loop control systems were designed. Finally, the validity of conclusions and e?ectiveness of the proposed design method were veri?ed by three numerical simulation examples.(2) The problems of stability and feedback control for two kinds of fractional-order Takagi-Sugeno(T-S) fuzzy uncertain systems were explored. Firstly, based on the stability theorem of fractional-order linear system, it was the su?cient conditions for ensuring robust asymptotical stability of fractional-order T-S fuzzy uncertain systems with fractional-order α belonging to 0 < α < 1 and 1 ≤ α < 2 that was presented, respectively. Secondly, by applying the LMI technics, the problem of feedback controller for T-S fuzzy uncertain systems was discussed; the suitable fuzzy output feedback controllers were designed, which were used to guarantee the asymptotical stability of closed-loop control systems for all admissible uncertainties. Finally, the validity of the conclusions and the e?ectiveness of the proposed design method were illustrated with two numerical simulation examples, respectively.(3) The problem of stability and feedback control for a class of same order fractionalorder nonlinear uncertain systems was researched. Firstly, based on the fractional order Lyapunov stability theorem, the su?cient condition was examined, which was used to ensure robust asymptotical stability of the same order fractional-order nonlinear uncertain systems with fractional-order α belonging to 0 < α < 1. Secondly, by using the LMI technics, the problem of feedback controller for the same order fractional-order nonlinear uncertain systems are discussed; the suitable state feedback controller was devised, which was taken to guarantee the asymptotical stability of closed-loop control systems. Finally,the validity of the conclusions and the e?ectiveness of the proposed design method were examined by numerical simulation of same order fractional-order chaotic Liu system.(4) The problem of feedback control for a class of multi-order fractional-order nonlinear systems were studied. Firstly, based on the stability theorem of multi-order fractionalorder nonlinear systems, the su?cient conditions were presented, of which the asymptotical stability at equilibrium points for multi-order fractional-order nonlinear controlled systems with fractional-order α belonged to 0 < α < 1. Secondly, by applying the eigenvalue for Jacobian matrix of controlled system at equilibrium point, the appropriate feedback controller is programmed, which was used to guarantee the asymptotical stability of closed-loop control systems at equilibrium points. Finally, the validity of the conclusions and the e?ectiveness of the proposed design method were proven by numerical simulation of multi-order fractional-order non-chaotic Lotka-Volterra predator-prey system and chaotic Chen system, respectively.