Positivity And Stability Of Fractional-order Systems

In this thesis,the control of fractional-order positive systems with time-delay and T-S fuzzy fractional-order systems is mainly studied.The specific works are as follows:Chapter 1 The research background and the research status of two classes of fractionalorder systems are introduced.Chapter 2 The positivity and the asymptotic stability of fractional-order linear timedelay systems(FOLTDSs)which are composed of N(N ? 2)subsystems are investigated.Firstly,a sufficient and necessary condition is given to ensure the positivity of FOLTDSs.The solutions of the studied systems are obtained by using the Laplace transform method,and the positivity of FOLTDSs is completely determined by the series of matrices and independent of the magnitude of time-delays.Secondly,a theorem is given to prove the asymptotic stability of positive FOLTDSs.By considering the monotonicity and asymptotic properties of systems with constant time-delay,it is further shown that the asymptotic stability of positive FOLTDSs is independent of the time-delay.Next,a state-feedback controller is designed such that the state variables of the systems are nonnegative and asymptotically convergent.When the order of the FOLTDSs is greater than one,by utilizing a property of Caputo derivative,a sufficient condition for the positivity of FOLTDS is presented.Finally,simulation examples are presented to verify the validity of the theoretical analysis.Chapter 3 By adopting the Takagi-Sugeno(T-S)fuzzy method,the adaptive control of fractional-order time-delay systems involving unknown parameters and input saturations is considered.T-S fuzzy systems with “IF-THEN” rules are adopted to describe the fractionalorder nonlinear systems.The influence of input saturation is handled by designing an auxiliary system.By using a norm conversion,the system's time-delay term is converted into a nondelayed form.Adaptive updating rules are devised to evaluate the uncertainty of the model,and a new control scheme is proposed.The proposed controller can not only ensure the boundedness of the variables involved,but also make state variables converge to a sufficiently small region of the origin,in the presence of uncertainties,saturation functions,and system parameters.Finally,simulation studies are given to verify the correctness of the designed method.