In this paper, a robust fractional-order PID (FOPID) controller design method for fractional-order delay systems is proposed based on positive stability region (PSR) analysis. Firstly, the PSR is presented to improve the existing stability region (SR) in D-decomposition method. Then, the optimal fractional orders and of FOPID controller are achieved at the biggest three-dimensional PSR, which means the best robustness. Given the optimal and , the other FOPID controller parameters k p , k i , k d can be solved under the control specifications, including gain crossover frequency, phase margin, and an extended flat phase constraint. In addition, the steps of the proposed robust FOPID controller design process are listed at length, and an example is given to illustrate the corresponding steps. At last, the control performances of the obtained robust FOPID controller are compared with some other controllers (PID and FOPI).The simulation results illustrate the superior robustness as well as the transient performance of the proposed control algorithm. KEYWORDSdelay system, FOPID control, fractional-order, positive stability region (PSR), robust analysis INTRODUCTIONFractional calculus is an extension of traditional calculus. The integral and differential orders in fractional calculus are not limited to integers, they can be any arbitrary real number or even complex number. 1 The concepts and mathematical analysis methods of fractional calculus are more complex than traditional calculus. Strictly speaking, most of the physical phenomena and processes can be described more precisely by fractional-order models. 2,3 Early fractional calculus related research works concentrated in the mathematics fields. With the rapid development of science and technology, the fractional calculus related engineering studies have also shown their great potential, for example, bioengineering, 4 signal process, 5 control engineering, 6-11 and applied physics. 12 Among these fractional calculus related fields, the development of fractional-order control technologies is particularly fast. Fractional-order control algorithm has more tuning knobs compared with conventional control algorithm. Therefore, it provides more probabilities in improving system robustness, transient control performance, and disturbance suppression ability. One of the most remarkable fractional-order control methodologies is fractional-order PID (FOPID or PI D ) control, which was proposed by Podlubny. 13 The PID controller has been overwhelmingly applied and was reported to be applied in more than 90% industrial control loops. 14 The FOPID controller is an extension of traditional PID controller with extra two parameters, ie, integral order and differential order. Hence, extra flexibility in meeting different Int J Robust Nonlinear Control. 2019;29:5195-5212.wileyonlinelibrary.com/journal/rnc

Linear matrix inequality (LMI), as a class of stability conditions for non-autonomous fractionalorder (NAFO) systems, is proposed in this paper. Based on the fractional-order Lyapunov direct method, the Mittag-Leffler stability of fractional-order systems without time delay is analyzed under LMI conditions first. For fractional-order systems with multiple time delays, the LMI conditions of Lyapunov asymptotical stability are studied by using the fractional-order comparison principle. Besides, the stability of linear NAFO systems is analyzed by using the LMI approach. What is more, the LMI-based stability method of nonlinear NAFO systems with multiple time delays is presented. Especially, for nonlinear fractional-order systems, a sufficient condition of the existence and uniqueness of the equilibrium point is given in the LMI form. In addition, two examples are provided to verify the effectiveness of the obtained theoretical results. INDEX TERMS Linear matrix inequality, non-autonomous, fractional-order, multiple time delays, Lyapunov asymptotical stability.

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This paper proposes a novel fractional-order delayed Ross–Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.

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