a b s t r a c tStability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag-Leffler stability and generalized Mittag-Leffler stability notions. With the definitions of Mittag-Leffler stability and generalized Mittag-Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.Published by Elsevier Ltd
This paper deals with the design of fractional order PI l D m controllers, in which the orders of the integral and derivative parts, l and m, respectively, are fractional. The purpose is to take advantage of the introduction of these two parameters and fulfill additional specifications of design, ensuring a robust performance of the controlled system with respect to gain variations and noise. A method for tuning the PI l D m controller is proposed in this paper to fulfill five different design specifications. Experimental results show that the requirements are totally met for the platform to be controlled. Besides, this paper proposes an auto-tuning method for this kind of controller. Specifications of gain crossover frequency and phase margin are fulfilled, together with the iso-damping property of the time response of the system. Experimental results are given to illustrate the effectiveness of this method. r
Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of noninteger order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.
In this paper we study ℓth-order (ℓ⩾3) consensus algorithms, which generalize the existing first-order and second-order consensus algorithms in the literature. We show necessary and sufficient conditions under which each information variable and its higher-order derivatives converge to common values. We also present the idea of higher-order consensus with a leader and introduce the concept of an ℓth-order model-reference consensus problem, where each information variable and its high-order derivatives not only reach consensus, but also converge to the solution of a prescribed dynamic model. The effectiveness of these algorithms is demonstrated through simulations and a multivehicle cooperative control application, which mimics a flocking behavior in birds.
Organic sodium-ion batteries (OSIBs) have numerous promising advantages for foreseeable large-scale applications, particularly including the convenience of performance optimization through molecular design. However, the reported organic cathodes still suffer from limited capacity, low cyclability, and poor rate performance. The tailoring of the p-conjugated system reported here can enhance the p-p intermolecular interactions, leading to insolubility, long-range layer-by-layer p-p stacking, fast-charge transport, and extraordinary stability and ionic conductivity (10 À9 cm 2 s À1 ). Consequently, the obtained cathodes delivered high electrochemical performance with high capacity ($290 mAh g À1 ), superior fast-chargedischarge ability ($160 and 100 mAh g À1 at 10 and 50 A g À1 , respectively), and ultra-long cycle life (capacity as high as 97 mAh g À1 after 10,000 cycles at 50 A g À1 ).
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