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.     

一种线性分数阶系统稳定性的频域判别准则

在分析了分数阶系统稳定性与传递函数分母相角增量的关系的基础上, 提出了一种线性分数阶系统的频域稳定性判别定理.定义了关于分数阶系统分母各项系数的两个函数,通过分析这两个函数正实数解的大小关系以及解的数目与分母最高阶数的关系,给出了分数阶系统稳定所需满足的条件.将用于在频域上对整数阶系统稳定性判别的Hermite-Biehler定理推广到对分数阶系统稳定性的判定.最后,通过对两个数值算例的分析,说明了提出的稳定性判别准则的正确性.     

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.     

Stabilization of Uncertain Multi‐Order Fractional Systems Based on the Extended State Observer

The extended state observer (ESO) based controller has been used successfully with integer‐order systems involving large uncertainties. In this paper, the robust control of uncertain multi‐order fractional‐order (FO) systems based on ESO is investigated. First, we transform the multi‐order FO system into an equivalent system in the form of a same‐order state‐space equation. Then, the ESO for the new system is established for estimating both the state and the total disturbance. Sufficient conditions for bounded‐input and bounded‐output stability are derived, and the asymptotic stability of the closed loop system is analyzed, based on whether the states are available or not. Finally, numerical simulations are presented to demonstrate the validity and feasibility of the proposed methodology.     

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.     

A proof for non existence of periodic solutions in time invariant fractional order systems

The aim of this note is to highlight one of the basic differences between fractional order and integer order systems. It is analytically shown that a time invariant fractional order system contrary to its integer order counterpart cannot generate exactly periodic signals. As a result, a limit cycle cannot be expected in the solution of these systems. Our investigation is based on Caputo’s definition of the fractional order derivative and includes both the commensurate or incommensurate fractional order systems.     

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。     

On the existence of periodic solutions in time-invariant fractional order systems

Periodic solutions and their existence are one of the most important subjects in dynamical systems. Fractional order systems like integer ones are no exception to this rule. Tavazoei and Haeri (2009) have shown that a time-invariant fractional order system does not have any periodic solution. In this article, this claim has been investigated and it is shown that although in any finite interval of time the solutions do not show any periodic behavior, when the steady state responses of fractional order systems are considered, periodic orbits can be detected.     

The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form

The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities (LMIs). Two numerical examples illustrate the effectiveness of the developed method.      

Robust Control of Positive Fractional‐Order Interconnected Systems with Heterogeneous Delays

The problem of robust decentralized control of positive fractional‐order interconnected systems with heterogeneous time‐varying delays is studied in this paper. Necessary and sufficient conditions are first derived for internal positiveness of the system. By exploiting the monotonicity induced from positivity of the system, robust stability conditions subject to uncertain system parameters are derived. The derived stability conditions are then utilized to address the controller synthesis problem. The design conditions for obtaining controller gains of stabilizing decentralized controllers are formulated using linear programming, which can be effectively solved by various convex optimization algorithms. Finally, the effectiveness of the obtained results is validated by two numerical examples.     

分数阶混沌系统耦合同步及混沌键控通信设计

分数阶混沌系统同步在混沌通信领域有着重要的应用价值。文中研究分数阶Chen混沌系统的单向耦合同步的问题,基于分数阶混沌系统的Lyapunov稳定性理论,设计分数阶Chen混沌系统单变量线性耦合同步控制器,实现分数阶Chen混沌系统的耦合同步。基于上述分数阶Chen混沌同步系统,设计混沌键控通信系统,分析通信系统的误码率等系统性能。研究表明,分数阶混沌通信系统比整数阶具有更高的保密性,分数阶混沌键控通信系统与整数阶混沌键控通信系统抗噪性能几乎一样。     

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.     

Constrained Swarm Stabilization of Fractional Order Linear Time Invariant Swarm Systems

This paper deals with asymptotic swarm stabilization of fractional order linear time invariant swarm systems in the presence of two constraints: the input saturation constraint and the restriction on distance of the agents from final destination which should be less than a desired value. A feedback control law is proposed for asymptotic swarm stabilization of fractional order swarm systems which guarantees satisfying the above-mentioned constraints. Numerical simulation results are given to confirm the efficiency of the proposed control method.      

分数阶奇异系统的Lie对称性与守恒量

对称性与守恒量可以简化动力学问题从而进一步求出力学系统的精确解,这样更加有利于研究动力学行为.分数阶模型相比于整数阶模型,能够描述复杂系统的动力学过程,因此在分数阶模型下研究对称性与守恒量是不可或缺的.首先介绍两个分数阶奇异系统,一个系统包含混合整数和Caputo分数阶导数,另一个系统仅含Caputo分数阶导数.由两个分数阶奇异系统分别给出两个分数阶固有约束,并给出对应的分数阶约束Hamilton方程.然后,基于微分方程在无限小变换下的不变性,给出了分数阶约束Hamilton方程Lie对称性的定义,导出了相应的确定方程,限制方程和附加限制方程.第三,建立并证明了两个分数阶约束Hamilton系统的Lie对称性定理,得到了相应的分数阶约束Hamilton系统的Lie守恒量.在特定条件下,本文所得结果可以退化为整数阶约束Hamilton系统的Lie守恒量.最后通过两个算例来说明此结果的应用.     

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.     

Full-order and Reduced-order Observer Design for One-sided Lipschitz Nonlinear Fractional Order Systems with Unknown Input

This paper studies the problem of designing the unknown input observers (UIOs) for fractional order one-sided Lipchitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using the matrix generalized inverse approach, sufficient conditions for asymptotic stability of the observer error dynamic systems are presented, which guarantee the existence of the full-order and reduced-order UIOs. All the conditions are obtained in terms of linear matrix inequality (LMI). Furthermore, we show that the obtained results can be applied to a fractional order electrical circuit with the unknown input signal. Two examples are given to demonstrate the applicability of the proposed approach.     

A note on the stability of fractional order systems

In this paper, a new approach is suggested to investigate stability in a family of fractional order linear time invariant systems with order between 1 and 2. The proposed method relies on finding a linear ordinary system that possesses the same stability property as the fractional order system. In this way, instead of performing the stability analysis on the fractional order systems, the analysis is converted into the domain of ordinary systems which is well established and well understood. As a useful consequence, we have extended two general tests for robust stability check of ordinary systems to fractional order systems.     

Consensus of Fractional‐Order Uncertain Multi‐Agent Systems Based on Output Feedback

This paper is devoted to the consensus protocol design for a set of agents with fractional‐order uncertainty dynamics where the fractional order α satisfies 0 < α < 2. For multi‐agent systems (MASs) with fixed undirected topology, a distributed static output feedback protocol is proposed with an undetermined system matrix. Based on model transformation and fractional‐order stability theory, sufficient conditions are derived to ensure the consensus of MASs in terms of linear matrix inequalities (LMIs). Finally, a simulation example is employed to validate the effectiveness of the proposed consensus protocol.     

Analytical solution of the linear fractional system of commensurate order

A useful representation of fractional order systems is the state space representation. For the linear fractional systems of commensurate order, the state space representation is defined as for regular integer state space representation with the state vector differentiated to a real order. This paper presents a solution of the linear fractional order systems of commensurate order in the state space. The solution is obtained using a technique based on functions of square matrices and the Cayley-Hamilton theorem. The technique developed for linear systems of integer order is extended to derive analytical solutions of linear fractional systems of commensurate order. The basic ideas and the derived formulations of the technique are presented. Both, homogeneous and inhomogeneous cases with usual input functions are solved. The solution is calculated in the form of a linear combination of suitable fundamental functions. The presented results are illustrated by analyzing some examples to demonstrate the effectiveness of the presented analytical approach.     

Analytic study on linear systems of fractional differential equations

An analytic study on linear systems of fractional differential equations with constant coefficients is presented. We briefly describe the issues of existence, uniqueness and stability of the solutions for two classes of linear fractional differential systems. This paper deals with systems of differential equations of fractional order, where the orders are equal to real number or rational numbers between zero and one. Exact solutions for initial value problems of linear fractional differential systems are analytically derived. Existence and uniqueness results are proved for two classes. The presented results are illustrated by analyzing some examples to demonstrate the effectiveness of the presented analytical approaches.