Relationship Between Integer Order Systems and Fractional Order Systems and Its Two Applications

Existence of periodic solutions and stability of fractional order dynamic systems are two important and difficult issues in fractional order systems (FOS) field. In this paper, the relationship between integer order systems (IOS) and fractional order systems is discussed. A new proof method based on the above involved relationship for the non existence of periodic solutions of rational fractional order linear time invariant systems is derived. Rational fractional order linear time invariant autonomous system is proved to be equivalent to an integer order linear time invariant non-autonomous system. It is further proved that stability of a fractional order linear time invariant autonomous system is equivalent to the stability of another corresponding integer order linear time invariant autonomous system. The examples and state figures are given to illustrate the effects of conclusion derived.      

Fractional‐order–dependent global stability analysis and observer‐based synthesis for a class of nonlinear fractional‐order systems

This paper focuses on proposing novel conditions for stability analysis and stabilization of the class of nonlinear fractional‐order systems. First, by considering the class of nonlinear fractional‐order systems as a feedback interconnection system and applying small‐gain theorem, a condition is proposed for L₂‐norm boundedness of the solutions of these systems. Then, by using the Mittag‐Leffler function properties, we show that satisfaction of the proposed condition proves the global asymptotic stability of the class of nonlinear fractional‐order systems with fractional order lying in (0.5, 1) or (1.5, 2). Unlike the Lyapunov‐based methods for stability analysis of fractional‐order systems, the new condition depends on the fractional order of the system. Moreover, it is related to the H_∞‐norm of the linear part of the system and it can be transformed to linear matrix inequalities (LMIs) using fractional‐order bounded‐real lemma. Furthermore, the proposed stability analysis method is extended to the state‐feedback and observer‐based controller design for the class of nonlinear fractional‐order systems based on solving some LMIs. In the observer‐based stabilization problem, we prove that the separation principle holds using our method and one can find the observer gain and pseudostate‐feedback gain in two separate steps. Finally, three numerical examples are provided to demonstrate the advantage of the novel proposed conditions with the previous results.     

Stability and Stabilization of Fractional‐Order Systems with Different Derivative Orders: An LMI Approach

Stability and stabilization analysis of fractional‐order linear time‐invariant (FO‐LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single‐order equivalent system for the given different‐order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional‐order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The proposed stability and stabilization theorems are applicable to FO‐LTI systems with different fractional orders in one or both of 0 <  α  < 1 and 1 ≤  α  < 2 intervals. Finally, some numerical examples are presented to confirm the obtained analytical results.     

分数阶混沌系统的延迟同步

基于分数阶线性系统的稳定性理论，结合反馈控制和主动控制方法，提出了实现分数阶混沌系统的延迟同步的一种新方法．该方案通过设计合适的控制器将分数阶混沌系统的延迟同步问题转化为分数阶线性误差系统在原点的渐近稳定性问题．分数阶Chen系统的数值模拟结果验证了该方案的有效性．     

分数阶线性定常系统的稳定性及其判据

介绍了分数阶微分方程和分数阶系统 ,给出分数阶线性定常系统的传递函数描述和状态空间描述 .给出了分数阶线性定常系统的稳定性条件 ,并结合分数阶状态方程给出定理的证明 .直接从复分析中的辐角原理出发 ,推导出分数阶线性定常系统 2个有效的稳定性判据 :分数阶系统奈奎斯特判据和分数阶系统对数频率判据 .通过实例验证了其有效性     

分数阶超混沌系统的线性广义同步观测器设计

首先利用分数阶的常微分动力系统的稳定性理论,通过判断线性化后平衡点的稳定不变特性、辅助以分岔图分析等数值手段,给出了新近提出的改进型超混沌L讧系统对应分数阶系统产生混沌现象的阶次参数范围;进一步,设计了一类广义线性同步观测器,该观测器的动力学行为能与原系统实现任意的线性关系的广义同步,而经典的完全同步、反相同步以及投影同步可以视为本文提出方法的特例．最后的数值仿真进一步证实了本文提出的观测器设计方案的有效性．     

Finite‐time stability of linear fractional‐order time‐delay systems

In this paper, a finite‐time stability results of linear delay fractional‐order systems is investigated based on the generalized Gronwall inequality and the Caputo fractional derivative. Sufficient conditions are proposed to the finite‐time stability of the system with the fractional order. Numerical results are given and compared with other published data in the literature to demonstrate the validity of the proposed theoretical results.     

The Design of a Coprime‐Factorized Predictive Functional Controller for Unstable Fractional Order Systems

In this paper, a new approach, called coprime‐factorized predictive functional control method (CFPFC‐F) is proposed to control unstable fractional order linear time invariant systems. To design the controller, first, a prediction model should be synthesized. For this purpose, coprime‐factorized representation is extended for unstable fractional order systems via a reduced approximated model of unstable fractional order (FO) system. That is, an approximated integer model of fractional order system is derived via the well‐known Oustaloup method. Then, the high order approximated model is reduced to a lower one via a balanced truncation model order reduction method. Next, the equivalent coprime‐factorized model of the unstable fractional‐order plant is employed to predict the output of the system. Then, a predictive functional controller (PFC) is designed to control the unstable plant. Finally, the robust stability of the closed‐loop system is analyzed via small gain theorem. The performance of the proposed control is investigated via simulations for the control of an unstable non‐laminated electromagnetic suspension system as our simulation test system.     

Global Practical Mittag Leffler Stabilization by Output Feedback for a Class Of Nonlinear Fractional‐Order Systems

In this paper, the problem of a global practical Mittag Leffler feedback stabilization for a class of nonlinear fractional order systems by means of observer is described. The linear matrix inequality approach is used to guarantee the practical stability of the proposed feedback fractional order system. An illustrative example is given to show the applicability of the results.     

Fuzzy Adaptive Control of a Fractional Order Chaotic System With Unknown Control Gain Sign Using a Fractional Order Nussbaum Gain

In this paper we propose an improved fuzzy adaptive control strategy, for a class of nonlinear chaotic fractional order (SISO) systems with unknown control gain sign. The online control algorithm uses fuzzy logic sets for the identification of the fractional order chaotic system, whereas the lack of a priori knowledge on the control directions is solved by introducing a fractional order Nussbaum gain. Based on Lyapunov stability theorem, stability analysis is performed for the proposed control method for an acceptable synchronization error level. In this work, the Grünwald-Letnikov method is used for numerical approximation of the fractional order systems. A simulation example is given to illustrate the effectiveness of the proposed control scheme.      

FINITE TIME STABILITY ANALYSIS OF LINEAR AUTONOMOUS FRACTIONAL ORDER SYSTEMS WITH DELAYED STATE

For the first time, in this paper, a stability test procedure is proposed for linear time‐invariant fractional order systems (LTI FOS). Paper extends some basic results from the area of finite time and practical stability to linear, continuous, fractional order time invariant time‐delay systems given in state space form. Sufficient conditions of this kind of stability, for particular class of fractional time‐delay systems is derived.     

Sensitivity analysis of the linear nonconservative systems with fractional damping

In this paper, the influence of small structural perturbation on a linear, nonconservative dynamical system exhibiting fractional bifurcation was investigated. In considering design problems for nonconservative systems, the integral structural characteristics as fundamental frequencies, critical loads for instability, and the sensitivity analysis play an important part. In this paper, the influence of small perturbation on a linear, nonconservative dynamical system exhibiting a flutter type bifurcation was investigated. The hereditary damping is described by means of fractional derivatives. To study the dynamical instability for nonconservative governing equations with fractional damping, the method of auxiliary eigenvalue problem is applied. The stability conditions of generalized Lyapunov type for the system with hereditary damping were derived. A new analytical framework for the coupled optimization of aero-structural, fractionally damped systems is presented. The approach to obtain aerodynamic sensitivities is based on adjoint systems.     

Robust stability analysis of uncertain multiorder fractional systems: Young and Jensen inequalities approach

Robust stability analysis of multiorder fractional linear time‐invariant systems is studied in this paper. In the present study, first, conservative stability boundaries with respect to the eigenvalues of a dynamic matrix for this kind of systems are found by using Young and Jensen inequalities. Then, considering uncertainty on the dynamic matrix, fractional orders, and fractional derivative coefficients, some sufficient conditions are derived for the stability analysis of uncertain multiorder fractional systems. Numerical examples are presented to confirm the obtained analytical results.     

Robust stability analysis for fractional‐order systems with time delay based on finite spectrum assignment

In this paper, the robust stability of a fractional‐order time‐delay system is analyzed in the frequency domain based on finite spectrum assignment (FSA). The FSA algorithm is essentially an extension of the traditional pole assignment method, which can change the undesirable system characteristic equation into a desirable one. Therefore, the presented analysis scheme can also be used as an alternative time‐delay compensation method. However, it is superior to other time‐delay compensation schemes because it can be applied to open‐loop poorly damped or unstable systems. The FSA algorithm is extended to a fractional‐order version for time‐delay systems at first. Then, the robustness of the proposed algorithm for a fractional‐order delay system is analyzed, and the stability conditions are given. Finally, a simulation example is presented to show the superior robustness and delay compensation performance of the proposed algorithm. Moreover, the robust stability conditions and the time‐delay compensation scheme presented can be applied on both integer‐order and fractional‐order systems.     

Robust analysis and synthesis for a class of fractional order systems with coupling uncertainties

A class of uncertain fractional order systems is concerned where intervals of system matrices parameters are affected by the fractional order. Moreover, the order is also uncertain and belongs to an interval containing the integer one. This kind of system models are reasonable results of fractional order system identifications. In this note, we derive the robust stability analysis and synthesis of such uncertain fractional order systems. Two numerical examples are presented to illustrate the effectiveness and potential of the developed theoretical results.

     

分数阶线性系统的内部和外部稳定性研究

介绍了分数阶线性定常系统的状态方程描述和传递函数描述．运用拉普拉斯变换和留数定理，给出并证明了分数阶线性定常系统的内部和外部稳定性条件，并讨论了其相互关系．以一个粘弹性系统的实例验证了上述方法的正确性．     

直线一级倒立摆分数阶控制器设计及仿真

针对直线一级倒立摆的稳定控制问题,设计了分数阶比例积分(FOPI和FO[PI])控制器。首先,根据Newton力学方法建立了倒立摆系统的数学模型。然后,采用基于向量的增益鲁棒性分数阶控制器参数求解简化算法,设计了分数阶比例积分控制器。最后,在MATLAB环境下进行了分数阶比例积分控制器参数整定方法的有效性验证,并且对倒立摆系统分别采用分数阶比例积分控制器和整数阶PID(IOPID)控制器进行了稳定控制仿真实验,并将得到的摆杆角度响应曲线进行了对比分析。结果表明:分数阶比例积分控制器对系统的稳定控制效果优于IOPID控制器,且在分数阶比例积分控制器中,FO[PI]控制器对系统稳定控制最好,响应时间较快、振荡幅值较小且具有鲁棒性。     

不确定分数阶时滞系统的鲁棒稳定性判定准则

基于退化分析方法提出一种判定准则, 用于分析不确定分数阶时滞系统的稳定性. 介绍一种分数阶积分算子的有理逼近方法, 在此基础上采用整数阶系统逼近分数阶系统, 从而将难以判定的分数阶系统稳定性问题转化为由逼近偏差作为不确定项的整数阶系统稳定性问题进行处理. 利用积分不等式法研究逼近系统稳定性, 得到LMI 形式的稳定性判据. 仿真结果表明, 所提出方法能够有效分析这类系统的稳定性.

     

Stability analysis of distributed-order nonlinear dynamic systems

The problem of asymptotic stability analysis of equilibrium points in nonlinear distributed-order dynamic systems with non-negative weight functions is considered in this paper. The Lyapunov direct method is extended to be used for this stability analysis. To this end, at first, a discretisation scheme with convergence property is introduced for distributed-order dynamic systems. Then, on the basis of this tool, Lyapunov theorems are proved for asymptotic stability analysis of equilibrium points in distributed-order systems. As the order weight function assumed for the distributed-order systems is general enough, the results are applicable to a wide range of nonlinear distributed-order systems such as fractional-order systems with multiple fractional derivatives. To verify the applicability of the obtained results, these results are applied for the stability analysis of a distributed-order diffusion system and control of a fractional-order Lorenz system with multiple fractional derivatives.     

The multistage homotopy analysis method: application to a biochemical reaction model of fractional order

In this paper, a new reliable algorithm called the multistage homotopy analysis method (MHAM) based on an adaptation of the standard homotopy analysis method (HAM) is presented to solve a time-fractional enzyme kinetics. This enzyme–substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The new algorithm is only a simple modification of the HAM, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. Numerical comparisons between the MHAM and the classical fourth-order Runge–Kutta method in the case of integer-order derivatives reveal that the new technique is a promising tool for nonlinear systems of integer and fractional order.