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.     

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.     

Stability and Stabilization of Two‐Dimensional LTI Switched Systems with Potentially Unstable Focus

This paper investigates stability and stabilization of two‐dimensional switched linear time‐invariant (LTI) systems with potentially unstable focus. For the case that the origin is a single common focus of all subsystems, we first give continuous positive definite functions related only to the elements of subsystems' state matrices. Then, based on the continuous positive definite functions obtained, this paper proposes several sufficient conditions of stability/asymptotic stability/instability of the kind of switched LTI systems. By means of the stability results proposed, global asymptotic stabilizing controls (GASC), global asymptotic stabilizing switching paths (GASSP) and corresponding algorithms are designed for two‐dimensional switched LTI systems with focus. Finally, two illustrative examples and numerical simulations demonstrate the effectiveness of the new stability and stabilization results obtained in this paper.     

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.     

Optimal fractional order PIλDμ controller for stabilization of cart‐inverted pendulum system: Experimental results

The cart‐inverted pendulum is a non‐minimum phase system having right half s‐plane pole and zero in close vicinity to each other. Linear time invariant (LTI) classical controllers cannot achieve satisfactory loop robustness for such systems. Therefore, in the present work the fractional order PI^λD^μ (FOPID) controller is addressed for robust stabilization of the system, since fractional order controller design allows more degrees of freedom compared to its integer order counterparts by virtue of its two parameters λ and μ. The controller parameters are tuned by three evolutionary optimization techniques. In order to select the controller parameters optimally, a novel non‐linear fitness function using integral time square error (ITSE), settling‐time, and rise time is proposed here. The control algorithm is implemented successfully in real‐time. Moreover, stability analysis of the system compensated with a fractional order controller is presented using Riemann surface. Robustness of the physical cart‐inverted pendulum system towards multiplicative gain variations and plant parameter variations is verified. In this regard, it is shown that the fractional order controller provides satisfactory robust performance in both simulation and real‐time system.     

Stability analysis of a class of nonlinear fractional‐order systems under control input saturation

This paper studies the stability of a class of nonlinear fractional‐order (FO) systems under input control saturation. Based on the Gronwall‐Bellman lemma and the sector‐bounded condition, sufficient conditions are provided to stabilize such systems by means of a state‐feedback controller. The performance of the proposed controller is tested with 2 FO chaotic systems, namely, the FO brushless direct‐current motor and the Chen systems. The results illustrate the good performance of the new controller under saturation effects.     

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.

     

On the positive invariance of polyhedral sets in fractional-order linear systems

In this paper, we derive conditions for a given polyhedral set to be a positively invariant set with respect to fractional-order linear time-invariant (FO-LTI) systems with fractional order of 0<α<1$0<\alpha <1$. FO-LTI systems are described using Riemann–Liouville operator with initialization response. Then, the conditions are obtained from Farkas’ lemma and the definition of the Mittag-Leffler function. Furthermore, we apply these conditions to the constrained stabilization.     

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.     

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.     

Stability and Control of Fractional Chaotic Complex Networks with Mixed Interval Uncertainties

This paper investigates the stability and control in fractional complex networks with inner and outer interval uncertainties. Each node is defined as a chaotic system. Stability theorems for fractional order 0 < α < 1 and 1 ≤ α < 2 are derived in the chaotic complex network. Instead of removing the nonlinear part directly, for a class of nonlinear function, we use an interval matrix to deal with this problem. By using the Kronecker product and LMI toolbox, stability conditions are provided in terms of linear matrix inequalities, and feasible feedback controllers are solved. In numerical simulations, two examples (real‐valued complex network and complex‐valued complex network) are provided to demonstrate the robustness and effectiveness of our methods.     

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.     

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.     

Sequential circle criterion approach for robustly stabilizing individual generators with SVC

Absolute stability with the spatially defined linear time‐invariant (LTI) state‐space modelings is scrutinized by means of what we call the sequential Lyapunov approach, which possesses independent significance in stabilization when gain‐scheduling control laws are adopted. Then, this theoretical result is exploiting for stabilization of individual generators via SVC actions. More precisely, by remodeling the perturbed swing equations of synchronous generators in multimachine networks through spatially defined LTI state‐space expressions subjected to uncertainties and power disturbance, which are viewed as sector nonlinearities, we introduce frequency responses for coping with nonlinear power swing dynamics of individual generators. By sequentially relating the frequency responses to the circle criterion (substantially, the KYP theorem or the positive real lemma) claimed for LTI systems subject to sector disturbances, output feedback control laws for static VAR compensators are worked out to stabilize individual generators. The frequency‐domain approach is also useful in steady‐state specification besides stabilization in individual generators. Examples show efficacy of the suggested stabilization and steady‐state specification technique. Copyright © 2014 John Wiley & Sons, Ltd.     

An alternative approach to designing stabilizing compensators for saturating linear time‐invariant plants

We present a new methodology for designing low‐gain linear time‐invariant (LTI) controllers for semi‐global stabilization of an LTI plant with actuator saturation, which is based on the representation of a proper LTI feedback using a pre‐compensator‐plus‐static‐output‐feedback architecture. We also mesh the new design methodology with time‐scale notions to develop lower‐order controllers for some plants. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

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.     

An Approach to Design MIMO FO Controllers for Unstable Nonlinear Plants

This paper develops an approach to control unstable nonlinear multi-inputs multi-output (MIMO) square plants using MIMO fractional order (FO) controllers. The controller design uses the linear time invariant (LTI) state space representation of the nonlinear model of the plant and the diagonal closedloop transfer matrix (TM) function to ensure decoupling between inputs. Each element of the obtained MIMO controller could be either a transfer function (TF) or a gain. A TF is associated in turn with its corresponding FO TF. For example, a D (Derivative) TF is related to a FO TF of the form Dδ, δ = [0, 1]. Two applications were performed to validate the developed approach via experimentation: control of the angular positions of a manipulator, and control of the car and arm positions of a translational manipulator.      

Integral partitioning approach to robust stabilization for uncertain distributed time‐delay systems

In this paper, the problems of robust delay‐dependent stability analysis and stabilization are investigated for distributed delay systems with linear fractional uncertainties. By introducing an integral partitioning technique, a new form of Lyapunov functional is constructed and improved distributed delay‐dependent stability conditions are established in terms of linear matrix inequalities. Based on the criterion, a design algorithm for a state‐feedback controller is proposed. Following similar lines, we extend these results to uncertain distributed delay systems. The results developed in this paper can tolerate larger allowable delay than existing ones in the literature, which is illustrated by several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

采用多Lyapunov函数的混杂系统稳定性研究

针对一类由离散事件监控的连续动态子系统组成的混杂动态系统, 首先分析利用多Lyapunov函数方法已有成果, 指出切换超平面为滑动模时, 利用这种方法不能确保混杂系统的稳定. 基于Filipov理论给出了能活稳定性结果. 对于混杂系统的连续动态子系统为线性时不变情况下, 研究了混杂系统二次镇定条件. 最后给出一个例子来说明本文方法.