On robust stability of LTI fractional-order delay systems of retarded and neutral type

This paper deals with the analysis of robust BIBO-stability of LTI fractional order delay systems in the presence of real parametric uncertainties. Two large classes of these systems, namely retarded and neutral types, are considered. Two theorems are given to check the robust BIBO-stability of these two families of fractional order systems. One of these theorems provides necessary and sufficient conditions for the case of retarded type and another one presents only sufficient conditions for the case of neutral type. Furthermore, upper and lower bounds (cutoff frequencies) are provided for drawing the value sets. To illustrate the results, two numerical examples are presented.     

一类分数阶复杂网络混沌系统的投影同步

根据分数阶系统的相关理论研究了一类分数阶复杂网络混沌系统的投影同步问题,给出了分数阶复杂网络以及分数阶时滞复杂网络系统实现投影同步的充分性条件,仿真结果表明了方法的正确性.     

Finite‐time stability of fractional order impulsive switched systems

This paper considers the finite‐time stability of fractional order impulsive switched systems. First, by using the fractional order Lyapunov function, Mittag–Leffler function, and Gronwall–Bellman lemma, two sufficient conditions are given to verify the finite‐time stability of fractional order nonlinear systems. Then, the concept of finite‐time stability is extended to fractional order impulsive switched systems. A sufficient condition is given to verify the finite‐time stability of fractional order impulsive switched systems by combining the method of average dwell time with fractional order Lyapunov function. Finally, two numerical examples are provided to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

具有多个非线性项的一般Lurie间接系统的绝对稳定性

为了研究具有多个非线性项的一般Lurie间接系统的绝对稳定性问题,介绍了关于变 元σ绝对稳定的概念.通过对关于变元σ绝对稳定性的研究,得到了此类系统绝对稳定的充要 条件,并给出了一些简洁实用的充分条件.     

Robust H∞ control for discrete-time polytopic uncertain systems with linear fractional vertices

The robust H∞ control problem for discrete-time uncertain systems is investigated in this paper. The uncertain systems are modelled as a polytopic type with linear fractional uncertainty in the vertices. A new linear matrix inequality (LMI) characterization of the H∞ performance for discrete systems is given by introducing a matrix slack variable which decouples the matrix of a Lyapunov function candidate and the parametric matrices of the system. This feature enables one to derive sufficient conditions for discrete uncertain systems by using parameter-dependent Lyapunov functions with less conservativeness. Based on the result, H∞ performance analysis and controller design are carried out. A numerical example is included to demonstrate the effectiveness of the proposed results.     

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     

Positivity and stability of mixed fractional‐order systems with unbounded delays: Necessary and sufficient conditions

This article provides a comprehensive study on quantitative properties of linear mixed fractional‐order systems with multiple time‐varying delays. The delays can be bounded or unbounded. We first obtain a result on existence and uniqueness of solutions to these systems. Then, we prove a necessary and sufficient condition for their positivity. Finally, we provide a necessary and sufficient criterion to characterize asymptotic stability of positive linear mixed fractional‐order systems with multiple time‐varying delays.     

Stabilization of Fractional‐Order Linear Systems with State and Input Delay

This paper describes a variable structure control for fractional‐order systems with delay in both the input and state variables. The proposed method includes a fractional‐order state predictor to eliminate the input delay. The resulting state‐delay system is controlled through a sliding mode approach where the controller uses a sliding surface defined by fractional order integral. Then, the proposed control law ensures that the state trajectories reach the sliding surface in finite time. Based on recent results of Lyapunov stability theory for fractional‐order systems, the stability of the closed loop is studied. Finally, an illustrative example is given to show the interest of the proposed approach.     

Necessary and Sufficient Conditions for Consensus of Delayed Fractional‐order Systems

In this paper, we study the consensus problem of fractional‐order systems with input delays. Using Laplace transform method, the stability of the fractional‐order systems is first discussed in the frequency domain. Based on the generalized Nyquist stability criterion, a necessary and sufficient condition is further derived to ensure the consensus of fractional‐order systems with identical input delays over directed interaction topology. Furthermore, when the interaction topology is undirected, the consensus condition of fractional‐order systems with heterogeneous input delays is explicitly given. Finally, some illustrative examples are presented to show the effectiveness and advantages of the theoretical results.     

一类单输入线性切换系统的可镇定性研究

考虑一类单输入线性切换系统的可镇定性问题,利用变结构控制对系统进行降维．通过对系统滑动模态的研究,得出了系统一致可镇定的充分条件和系统存在容许镇定策略的充分条件．给出了具体的容许镇定策略集合,并对二阶切换系统给出了详细的容许镇定策略．仿真实例验证了所提出结论的有效性．     

二次型耗散线性离散系统的鲁棒性分析与控制

考虑一类不确定离散多变量系统的鲁棒二次型耗散性分析和控制,其中各不确定参数矩阵具有线性分式形式.首先对确定系统建立二次型耗散性与正实性之间的等价关系,由此导出线性系统二次型耗散的充分必要条件;然后证明不确定系统的鲁棒耗散性分析和控制可转化为确定系统的耗散性分析和设计,给出了这类不确定系统鲁棒耗散的充分必要条件以及鲁棒耗散控制问题的线性矩阵不等式解法.所得结果可将H∞控制与正实控制统一起来,提供一种较为灵活、保守性较小的系统设计方法.仿真例子说明了所提方法的有效性.     

H∞ Model Reduction for Positive Fractional Order Systems

This paper focuses on the H_∞ model reduction problem of positive fractional order systems. For a stable positive fractional order system, we aim to construct a positive reduced‐order fractional system such that the associated error system is stable with a prescribed H_∞ performance. Then, based on the bounded real lemma for fractional order systems, a sufficient condition is given to characterize the model reduction problem with a prescribed H_∞‐norm error bound in terms of a linear matrix inequality (LMI). Furthermore, by introducing a new flexible real matrix variable, the desired reduced‐order system matrices are decoupled with the complex matrix variable and further parameterized by the new matrix variable. A corresponding iterative LMI algorithm is also proposed. Finally, several illustrative examples are given to show the effectiveness of the proposed algorithms.     

Pseudo-state feedback stabilization of commensurate fractional order systems

This paper addresses the problem of pseudo-state feedback stabilization of commensurate fractional order systems (FOS). In the proposed approach, Linear Matrix Inequalities (LMI) formalism is used to check if the pseudo-state matrix eigenvalues belong to the FOS stability region of the complex plane. A review of LMI stability conditions is first proposed for fractional order 0<ν<1 and 1<ν<2. The paper then focuses particularly on the case 0<ν<1 as the stability region is non-convex and associated LMI condition is not as straightforward to obtain as in the case 1<ν<2. A new LMI stability condition is thus proposed. Based on this condition, a necessary and sufficient LMI method for the design of stabilizing controllers is given. This method paves the way for extension to FOS of various LMI-based results. Among these possible extensions, a first result on robust control of polytopic fractional order systems is given in this paper.     

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．     

On the sliding‐mode control of fractional‐order nonlinear uncertain dynamics

This paper deals with applications of sliding‐mode‐based fractional control techniques to address tracking and stabilization control tasks for some classes of nonlinear uncertain fractional‐order systems. Both single‐input and multi‐input systems are considered. A second‐order sliding‐mode approach is taken, in suitable combination with PI‐based design, in the single‐input case, while the unit‐vector approach is the main tool of reference in the multi‐input case. Sliding manifolds containing fractional derivatives of the state variables are used in the present work. Constructive tuning conditions for the control parameters are derived by Lyapunov analysis, and the convergence properties of the proposed schemes are supported by simulation results. Copyright © 2015 John Wiley & Sons, Ltd.     

分数阶系统线性矩阵不等式稳定判据的改进与鲁棒镇定: 0<α<1的情况

针对确定的分数阶系统,通过发掘其稳定区域与线性矩阵不等式(LMI)之间的联系,提出改进的LMI稳定判据.与已有结果相比,该判据形式简洁,求解变量少,且不含复矩阵变量,能够直接应用于控制器设计而不带来任何保守性,克服了原判据存在的主要缺点.然后基于该判据,进一步研究了当分数阶系统具有多胞型不确定时的鲁棒镇定控制器的设计方法.最后,仿真结果表明该方法的有效性.     

An Exploration on Adaptive Iterative Learning Control for a Class of Commensurate High-order Uncertain Nonlinear Fractional Order Systems

This paper explores the adaptive iterative learning control method in the control of fractional order systems for the first time. An adaptive iterative learning control (AILC) scheme is presented for a class of commensurate high-order uncertain nonlinear fractional order systems in the presence of disturbance. To facilitate the controller design, a sliding mode surface of tracking errors is designed by using sufficient conditions of linear fractional order systems. To relax the assumption of the identical initial condition in iterative learning control (ILC), a new boundary layer function is proposed by employing Mittag-Leffler function. The uncertainty in the system is compensated for by utilizing radial basis function neural network. Fractional order differential type updating laws and difference type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. The hyperbolic tangent function and a convergent series sequence are used to design robust control term for neural network approximation error and bounded disturbance, simultaneously guaranteeing the learning convergence along iteration. The system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapnov-like composite energy function (CEF) containing new integral type Lyapunov function, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.      

Full‐Order and Reduced‐order Observer Design for a Class of Fractional‐order Nonlinear Systems

The paper is concerned with problem of the full‐order and reduced‐order observer design for a class of fractional‐order one‐sided Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full‐order observer error dynamic system is presented. Furthermore, the proposed design method was extended to reduced‐order observer design for fractional‐order nonlinear systems. All the stability conditions are obtained in terms of LMI, which are less conservative than some existing ones. Finally, a numerical example demonstrates the validity of this approach.