Necessary and sufficient stability condition of fractional-order interval linear systems

This paper establishes a necessary and sufficient stability condition of fractional-order interval linear systems. It is supposed that the system matrix A is an interval uncertain matrix and fractional commensurate order belongs to 1≤α<2. Using the existence condition of Hermitian P=P^∗ for a complex Lyapunov inequality, we prove that the fractional-order interval linear system is robust stable if and only if there exists Hermitian matrix P=P^∗ such that a certain type of complex Lyapunov inequality is satisfied for all vertex matrices. The results are directly extended to the robust stability condition of fractional-order interval polynomial systems.     

A note on fractional-order derivatives of periodic functions

In this paper, it is shown that the fractional-order derivatives of a periodic function with a specific period cannot be a periodic function with the same period. The fractional-order derivative considered here can be obtained based on each of the well-known definitions Grunwald-Letnikov definition, Riemann-Liouville definition and Caputo definition. This concluded point confirms the result of a recently published work proving the non-existence of periodic solutions in a class of fractional-order models. Also, based on this point it can be easily proved the absence of periodic responses in a wider class of fractional-order models. Finally, some examples are presented to show the applicability of the paper achievements in the solution analysis of fractional-order systems.     

A computing method on stability intervals of time-delay for fractional-order retarded systems with commensurate time-delays

This paper investigates the stability intervals of time-delays for fractional-order retarded time-delay systems. By the Orlando formula, the existence of the crossing frequencies is brought to verify the stability related to the commensurate time-delay. For each crossing frequency, the corresponding critical time-delays are determined by the generalized eigenvalues of two matrices constructed by the crossing frequency, the commensurate fractional-order and the coefficients of the characteristic function. The root tendency (RT) is defined to provide a method to analyze the number of the unstable roots for a given crossing frequency and critical time-delay. Based on the RT values and the number of the unstable roots for fractional-order systems with no time-delay, a computing method on the stability intervals of time-delay is proposed in this paper. Finally, a numerical example is offered to validate the effectiveness of this method.     

区间分数阶系统的鲁棒稳定性判别准则:0 < α < 1

针对同元阶次在0和1之间的区间分数阶系统,提出了类似Kharitonov定理的鲁棒稳定性判别准则. 研究了区间分数阶系统分母的主分支函数值集不包含原点所需满足的条件.根据除零原理, 给出了区间分数阶系统鲁棒稳定的顶点和棱边条件. 定义了由分母函数系数构成的矩阵,通过检验矩阵是否在负实轴上存在特征值来检验棱边条件. 最后,通过对两个数值算例的分析说明了这种方法的有效性.     

Comments on “Necessary and sufficient stability condition of fractional-order interval linear systems” [Automatica 44 (2008), 2985–2988]

This paper shows that the main thesis of the article “Necessary and sufficient stability condition of fractional-order interval linear systems” by Ahn and Chen (Automatica, vol. 44, pp. 2985–2988) is only a sufficient (and not a necessary) condition for the robust stability of fractional-order linear systems.     

On type number concept in fractional-order systems

In this note, the type number concept is defined for fractional-order systems. Based on this definition, it is shown that fractional-order systems having type numbers more than 1 can not track some classes of reference inputs without any overshoot.     

Robust passivity and feedback passification of a class of uncertain fractional-order linear systems

Theoretical results on robust passivity and feedback passification of a class of uncertain fractional-order (FO) linear systems are presented in the paper. The system under consideration is subject to time-varying norm-bounded parameter uncertainties in both the state and controlled output matrices. Firstly, some suitable notions of passivity and dissipativity for FO systems are proposed, and the relationship between passivity and stability is obtained. Then, a sufficient condition in the form of linear matrix inequality (LMI) for such system to be robustly passive is given. Based on this condition, the design method of state feedback controller is proposed when the states are available. Moreover, by using matrix singular value decomposition and LMI techniques, the existing condition and method of designing a robust observer-based passive controller for such systems are derived. Numerical simulations demonstrate the effectiveness of the theoretical formulation.     

Design of distributed PID-type dynamic matrix controller for fractional-order systems

With the continuous requirements for product quality and safety operation in industrial production, it is difficult to describe the complex large-scale processes with integer-order differential equations. However, the fractional differential equations may precisely represent the intrinsic characteristics of such systems. In this paper, a distributed PID-type dynamic matrix control method based on fractional-order systems is proposed. First, the high-order approximate model of integer order is obtained by utilising the Oustaloup method. Then, the step response model vectors of the plant is obtained on the basis of the high-order model, and the online optimisation for multivariable processes is transformed into the optimisation of each small-scale subsystem that is regarded as a sub-plant controlled in the distributed framework. Furthermore, the PID operator is introduced into the performance index of each subsystem and the fractional-order PID-type dynamic matrix controller is designed based on Nash optimisation strategy. The information exchange among the subsystems is realised through the distributed control structure so as to complete the optimisation task of the whole large-scale system. Finally, the control performance of the designed controller in this paper is verified by an example.     

Non-fragile observer-based robust control for a class of fractional-order nonlinear systems

The paper is concerned with the problem of the observer-based robust control for a class of fractional-order nonlinear systems. First, by introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for robust asymptotic stability of the closed-loop system via state observer-based control is presented, which is less conservative than some existing ones in recent literature. Next, using matrix’s singular value decomposition (SVD) and linear matrix inequality (LMI) techniques, the existence condition and method of designing a robust non-fragile observer-based controller are derived. Finally, the validity of the proposed methods are demonstrated by two numerical examples.     

Notes on integral performance indices in fractional-order control systems

Integral performance indices as quantitative measures of the performance of a system are commonly used to evaluate the performance of designed control systems. In this paper, it is pointed out that due to existence of non-exponential modes in the step response of a fractional-order control system having zero steady state error, integral performance indices of such a system may be infinite. According to this point, some simple conditions are derived to guarantee the finiteness of different integral performance indices in a class of fractional-order control systems. Finally, some numerical examples are presented to show the applicability of the analytical achievements of the paper.     

Fractional-order adaptive backstepping control of a class of uncertain systems with external disturbances

Fractional control schemes are powerful tools for fulfilling robust tracking performance of different systems. This paper is the pioneering one in developing a fractional-order adaptive backstepping controller (FOABC) for a general class of integer-order and fractional-order (FO) systems. Model uncertainties and external disturbances can perturb system response and the controller is designed such that it can suppress the performance degradation caused by these factors. Moreover, rigorous mathematical analyses are carried out based on fractional Lyapunov theorems to ensure stability of the controlled systems. To justify the claims, worked-out examples including integer-order and FO systems are simulated. Good tracking performance of the proposed controller as well as robustness against uncertainties and insensitivity to external disturbances make it a good candidate for a broad range of systems. The results of implementing the proposed controller on different systems are compared with some newly proposed control approaches which highlight the outperformance of the FOABC.     

分数阶混沌系统的动态仿真方法研究

分数阶混沌系统是非线性科学的研究热点.由于目前研究分数阶混沌的理论和硬件实验分析方法比较烦琐,提出了分数阶混沌系统的动态仿真方法.通过利用分数阶微分算子及其复域表示方式设计分数阶微分算子仿真模块,根据分数阶系统方程构建分数阶混沌系统仿真模型,可动态地观察系统变量的演化规律,并利用仿真过程的输出数据分析分数阶混沌系统的动力学特性.研究结果表明分数阶混沌系统的动态仿真方法足一种有效的分析方法.     

Fractional cyber-neural systems — A brief survey

Neurotechnology has made great strides in the last 20 years. However, we still have a long way to go to commercialize many of these technologies as we lack a unified framework to study cyber-neural systems (CNS) that bring the hardware, software, and the neural system together. Dynamical systems play a key role in developing these technologies as they capture different aspects of the brain and provide insight into their function. Converging evidence suggests that fractional-order dynamical systems are advantageous in modeling neural systems because of their compact representation and accuracy in capturing the long-range memory exhibited in neural behavior. In this brief survey, we provide an overview of fractional CNS that entails fractional-order systems in the context of CNS. In particular, we introduce basic definitions required for the analysis and synthesis of fractional CNS, encompassing system identification, state estimation, and closed-loop control. Additionally, we provide an illustration of some applications in the context of CNS and draw some possible future research directions. Advancements in these three areas will be critical in developing the next generation of CNS, which will, ultimately, improve people’s quality of life.     

分数阶控制系统的仿真方法

该文对分数阶控制系统提出了一种数值仿真方法。该方法根据Riemann—Liouville定义的分数阶积分和微分，利用δ函数的过滤性质，得到分数阶积分和分数阶微分的近似计算，推导出了相应的分数阶微分方程的数值求解方法，从而得到了分数阶控制系统的一种仿真算法。利用“短记忆法则”对此仿真方法进行改进，可以减少计算量、提高运行速度。文中给出了一些仿真例子，仿真研究结果表明该方法是有效的，它可适用与不同的分数阶系统。对于分数阶控制系统的仿真研究具有一定的实际意义。     

A numerical algorithm for stability testing of fractional delay systems

This paper presents an effective numerical algorithm for testing the BIBO stability of fractional delay systems described by fractional-order delay-differential equations. It is based on using Cauchy's integral theorem and solving an initial-value problem. The algorithm has a reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of feedback control for both integer- and fractional-order systems having time delays.     

Identification and PID Control for a Class of Delay Fractional-order Systems

In this paper, a new model identification method is developed for a class of delay fractional-order system based on the process step response. Four characteristic functions are defined to characterize the features of the normalized fractionalorder model. Based on the time scaling technology, two identification schemes are proposed for parameters' estimation. The scheme one utilizes three exact points on the step response of the process to calculate model parameters directly. The other scheme employs optimal searching method to adjust the fractional order for the best model identification. The proposed two identification schemes are both applicable to any stable complex process, such as higher-order, under-damped/over-damped, and minimum-phase/nonminimum-phase processes. Furthermore, an optimal PID tuning method is proposed for the delay fractionalorder systems. The requirements on the stability margins and the negative feedback are cast as real part constraints (RPC) and imaginary part constraints (IPC). The constraints are implemented by trigonometric inequalities on the phase variable, and the optimal PID controller is obtained by the minimization of the integral of time absolute error (ITAE) index. Identification and control of a Titanium billet heating process is given for the illustration.      

H∞ Output Feedback Control of Linear Time-invariant Fractional-order Systems over Finite Frequency Range

This paper focuses on the H_∞ output feedback control problem of linear time-invariant fractional-order systems over finite frequency range. Based on the generalized Kalman-Yakubovic-Popov (KYP) Lemma and a key projection lemma, a necessary and sufficient condition is established to ensure the existence of the H_∞ output feedback controller over finite frequency range, a desirable property in control engineering practice. By using the matrix congruence transformation, the feedback control gain matrix is decoupled and further parameterized by a scalar matrix. Two iterative linear matrix inequality algorithms are developed to solve this problem. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.      

Non-existence of finite-time stable equilibria in fractional-order nonlinear systems

We note that in the literature it is often taken for granted that for fractional-order system without delays, whenever the system trajectory reaches the equilibrium, it will stay there. In fact, this is the well-known phenomenon of finite-time stability. However, in this paper, we will prove that for fractional-order nonlinear system described by Caputo’s or Riemann–Liouville’s definition, any equilibrium cannot be finite-time stable as long as the continuous solution corresponding to the initial value problem globally exists. In addition, some examples of stability analysis are revisited and linear Lyapunov function is used to prove the asymptotic stability of positive fractional-order nonlinear systems.     

分数阶奇异系统容许性的充分必要条件

探讨了阶数在（0,1）区间的分数阶奇异系统的容许性条件. 文中首先给出了正则性,无脉冲性和容许性的定义,然后给出了分数阶奇异系统容许性的充分必要条件,最后通过数值仿真的例子来说明我们给出的容许性条件.