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.

     

Adaptive Mittag‐Leffler Stabilization of a Class of Fractional Order Uncertain Nonlinear Systems

This paper deals with the stabilization of a class of commensurate fractional order uncertain nonlinear systems. The fractional order system concerned is of the strict‐feedback form with uncertain nonlinearity. An adaptive control scheme combined with fractional order update laws is proposed by extending classical backstepping control to fractional order backstepping scheme. The asymptotic stability of the closed‐loop system is guaranteed under the construction of fractional Lyapunov functions in the sense of generalized Mittag‐Leffler stability. The fractional order nonlinear system investigated can be stabilized asymptotically globally in presence of arbitrary uncertainty. Finally illustrative examples and numerical simulations are performed to verify the effectiveness of the proposed control scheme.     

复杂系统的分数阶内模控制器设计

针对高阶复杂系统提出一种分数阶内模控制器设计方法。利用微粒群算法(PSO)进行模型化简,基于内模控制(IMC)原理设计分数阶控制器,该控制器仅有一个可调参数,并根据鲁棒性能指标给出控制器参数整定的解析表达式。仿真结果表明,该方法可以使系统同时具有良好的目标值跟踪特性、扰动抑制特性以及克服参数变化的鲁棒性。     

Stabilization of Non‐Linear Fractional‐Order Uncertain Systems

In this paper, we present a stabilization method on the non‐linear fractional‐order uncertain systems. Firstly, a sufficient condition for the robust asymptotic stabilization of the non‐linear fractional‐order uncertain system is presented based on direct Lyapunov approach. Secondly, utilising the matrix's singular value decomposition (SVD) method, the systematic robust stabilization design algorithm is then proposed. Finally, two numerical examples are provided to illustrate the efficiency and advantage of the proposed algorithm.     

An Adaptive Neural Sliding Mode Control with ESO for Uncertain Nonlinear Systems

An adaptive neural sliding mode control with ESO for uncertain nonlinear systems is proposed to improve the stability of the control system. Any control system inevitably exists uncertain disturbances and nonlinearities which severely affect the control performance and stability. Neural network can be utilized to approximate the uncertain nonlinearities. Nevertheless, it produces approximate errors, which will become more difficult to deal with as the order of the system increases. Moreover, these errors and uncertain disturbances will result in a consequence that the control system can be unable to converge quickly, and has to deal with a lot of calculations. Therefore, in order to perfect the performance and stability of the control system, this paper combines sliding mode control and ESO, and designs an adaptive neural control method. The simulation results illustrate that the improved system has superior tracking performance and anti-interference ability.

     

Fractional‐Order Dynamic Output Feedback Sliding Mode Control Design for Robust Stabilization of Uncertain Fractional‐Order Nonlinear Systems

A robust fractional‐order dynamic output feedback sliding mode control (DOF‐SMC) technique is introduced in this paper for uncertain fractional‐order nonlinear systems. The control law consists of two parts: a linear part and a nonlinear part. The former is generated by the fractional‐order dynamics of the controller and the latter is related to the switching control component. The proposed DOF‐SMC ensures the asymptotical stability of the fractional‐order closed‐loop system whilst it is guaranteed that the system states hit the switching manifold in finite time. Finally, numerical simulation results are presented to illustrate the effectiveness of the proposed method.     

Proportional Plus Derivative State Feedback Control of Takagi-Sugeno Fuzzy Singular Fractional Order Systems

This paper investigates the fuzzy normalization and stabilization issues of a class of singular fractional order nonlinear systems with order 0 < α < 1 based on a singular Takagi-Sugeno fuzzy model. First, we present the admissibility theorem of Takagi-Sugeno fuzzy singular fractional order systems. Next, benefited by that the fuzzy model and the state feedback controllers do not share the same membership functions, a proportional plus derivative state feedback controller is designed, which guarantees the closed-loop system normalized and admissible. Finally, a numerical simulation example is given to illustrate the effectiveness of the proposed method.

     

Control and Synchronization Of A Class Of Uncertain Fractional Order Chaotic Systems Via Adaptive Backstepping Control

This paper presents the stabilization and synchronization problem of a class of fractional order chaotic systems with unknown parameters. A systematic step by step approach is explained to derive control results using an adaptive backstepping strategy. The analytically obtained control structure, derived by blending a systematic backstepping procedure with Mittag‐Leffler stability results, helps in obtaining the stability of a strict feedback‐like class of uncertain fractional order chaotic systems. The results are further extended to achieve synchronization of these systems in master–slave configuration. Thereafter, the methodology has been applied to two example systems, that is, chaotic Chua's circuit and Genesio‐Tesi system, which belong to addressed class, in order to show the application of results. Numerical simulation given at the end confirms the efficacy of the scheme presented here.     

基于最大灵敏度的加热系统分数阶内模控制

针对加热系统热传导过程模型不精确和系统参数不确定性问题, 提出一种新的基于最大灵敏度的分数阶内模控制方案. 采用分数阶模型描述加热系统可以提高精度, 而内模控制能够很好地处理系统参数不确定性问题. 利用最大灵敏度整定分数阶控制器参数, 并以此获得强鲁棒性控制系统. 数值结果验证了所提出的分数阶内模控制方案的有效性, 具有比整数阶内模控制方案更好的控制性能.

     

基于自适应模糊滑模控制的分数阶混沌系统的投影同步

基于模糊控制理论和滑模控制理论以及自适应控制理论,研究了一类含有外部扰动的不确定分数阶混沌系统的混合投影同步问题.提出了一种自适应模糊滑模控制的分数阶混沌系统投影同步方法.模糊逻辑系统用来逼近未知的非线性函数和外部扰动,并且对逼近误差采用了自适应控制,同时构造了一种具有较强鲁棒性的分数阶积分滑模面.应用分数阶Barbalat引理设计了自适应模糊滑模控制器和参数自适应律.最后数值仿真结果验证了所提控制方法的有效性.     

Observer-based method for synchronization of uncertain fractional order chaotic systems by the use of a general type-2 fuzzy system

Synchronization of the fractional order chaotic systems is extensively studied in recent years due to its potential applications in many branches of science and engineering. The main problems in this field are that the dynamics of the system in hand are often uncertain and are perturbed by external disturbances. Also the unknown nonlinear functions in the system dynamics are generally complicated and in many practical applications we have measurement errors and unavailable states. In this paper, a novel robust and asymptotically stable controller is proposed to synchronize uncertain fractional order chaotic systems. Its design is based on linear matrix inequality (LMI) technique. Furthermore, an observer is presented to estimate the unavailable states. A general type-2 fuzzy system (GT2FS) based on α-plane representation with Gaussian secondary membership functions (MF) and type-2 non-singleton fuzzification is proposed to approximate the unknown complex nonlinear functions in the dynamics of system. The input uncertainties associated with the observer error and the malfunctioning of the input devices are modeled by interval type-2 fuzzy MFs instead of crisp numbers. To decrease the computational cost of the GT2FS, a simple type-reduction method is proposed. The antecedent parameters of GT2FS are tuned based on a modified form of social spider optimization (SSO) algorithm. The simulation examples show that the proposed control scheme gives high performance in the presence of unknown functions, external disturbances and unavailable states. The performance of GT2FS with different α-levels and different fuzzification methods are compared with type-1 and interval type-2 fuzzy systems in several examples.     

Bit-level color image encryption algorithm based on coarse-grained logistic map and fractional chaos

A novel color image encryption algorithm based on coarse-grained fractional chaotic system signals is proposed in this paper. First, color images are divided into three channels, which are encrypted based on the corresponding three states of the chaotic system. Second, the chaotic systems are defined as fractional chaotic, in which the fractional order enlarges the parameter space. Third, the fractional chaotic signals are handled with unfixed coarse-grained methods instead of being utilized directly. In addition, the original image and the chaotic signals are divided into bit signals from the pixel values, and the high and low bits are encrypted, respectively. To demonstrate the effectiveness and robustness of the proposed color image encryption algorithm, its properties, including the key space, information entropy, correlation analysis, key sensitivity, and resistance to differential attacks, are provided using a numerical simulation.

     

Uncertain impulsive differential systems of fractional order: almost periodic solutions

In this paper, we investigate the existence and stability of almost periodic solutions of impulsive fractional-order differential systems with uncertain parameters. The impulses are realised at fixed moments of time. For the first time, we determine the impact of the uncertainties on the qualitative behaviour of such systems. The main criteria for the existence of almost periodic solutions are proved by employing the fractional Lyapunov method. The global perfect robust uniform-asymptotic stability of such solutions is also considered. We apply our results to uncertain impulsive neural network systems of fractional order.     

Time delay margin for islanded microgrid with parameter uncertainty using fractional order proportional-integral-derivative controller

In this article, an islanded microgrid (MG) consisting of the diesel generator (DEG), the photovoltaic panel (PV), the wind turbine generator (WTG), the battery energy storage system (BESS), and the control unit is considered. In the islanded MGs, the control signals are exchanged on the open communication network, which results in the time delays in the input and the output of the islanded MG central controller (MGCC). The time delay has a destructive effect on the islanded MG stability. Thus, finding the maximum allowable time delay bound (MADB) is a significant issue. Since it is shown that the fractional order systems have larger stability region and more robustness rather than the corresponding integer order systems, in this research, we propose the fractional order proportional-integral-derivative (FOPID) controller as the MGCC to achieve a larger MADB value. As another innovation, in this article, a method is presented in which the MADB of the islanded MG system is determined considering the parametric uncertainties related to the damping coefficient (D) and the inertia constant (H). It is shown that the percentage improvement in the MADB of the uncertain MG system with the designed FOPID controller over the integer order proportional-integral-derivative (IOPID) controller is 9.64%. The accuracy of the proposed method is verified by simulation results in Matlab.     

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.     

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.     

Passivity and passification for stochastic systems with Markovian switching and generally uncertain transition rates

The paper deals with the problems of passivity and passification for stochastic systems with Markovian switching and generally uncertain transition rates. The considered systems are more general, which cover uncertain transition rates and partly known transition rates as two special cases. By employing the multiple Lyapunov function and some free-weighting matrices, a state feedback controller is constructed such that the resulted closed-loop system is stochastically passive. Some sufficient conditions for the solution to the problem are derived in the form of linear matrix inequalities (LMIs). Finally, a numerical example is given to demonstrate the validity of the main results.

     

Robust D-Stability Test of LTI General Fractional Order Control Systems

This work deals with the robust D-stability test of linear time-invariant(LTI) general fractional order control systems in a closed loop where the system and/or the controller may be of fractional order. The concept of general implies that the characteristic equation of the LTI closed loop control system may be of both commensurate and non-commensurate orders, both the coefficients and the orders of the characteristic equation may be nonlinear functions of uncertain parameters, and the coefficients may be complex numbers. Some new specific areas for the roots of the characteristic equation are found so that they reduce the computational burden of testing the robust D-stability. Based on the value set of the characteristic equation, a necessary and sufficient condition for testing the robust D-stability of these systems is derived. Moreover, in the case that the coefficients are linear functions of the uncertain parameters and the orders do not have any uncertainties, the condition is adjusted for further computational burden reduction. Various numerical examples are given to illustrate the merits of the achieved theorems.