Robust Admissibility and Stabilization of Uncertain Singular Fractional-Order Linear Time-Invariant Systems

This paper addresses the robust admissibility problem in singular fractional-order continuous time systems. It is based on new admissibility conditions of singular fractional-order systems expressed in a set of strict linear matrix inequalities(LMIs). Then, a static output feedback controller is designed for the uncertain closed-loop system to be admissible. Numerical examples are given to illustrate the proposed methods.

Distributed Adaptive Cooperative Tracking of Uncertain Nonlinear Fractional-order Multi-agent Systems

In this paper, the leader-following tracking problem of fractional-order multi-agent systems is addressed. The dynamics of each agent may be heterogeneous and has unknown nonlinearities. By assumptions that the interaction topology is undirected and connected and the unknown nonlinear uncertain dynamics can be parameterized by a neural network, an adaptive learning law is proposed to deal with unknown nonlinear dynamics, based on which a kind of cooperative tracking protocols are constructed. The feedback gain matrix is obtained to solve an algebraic Riccati equation. To construct the fully distributed cooperative tracking protocols, the adaptive law is also adopted to adjust the coupling weight. With the developed control laws,we can prove that all signals in the closed-loop systems are guaranteed to be uniformly ultimately bounded. Finally, a simple simulation example is provided to illustrate the established result.

Reinforcement learning based adaptive control for uncertain mechanical systems with asymptotic tracking

This paper mainly focuses on the development of a learning-based controller for a class of uncertain mechanical systems modeled by the Euler-Lagrange formulation.The considered system can depict the behavior of a large class of engineering systems,such as vehicular systems,robot manipulators and satellites.All these systems are often characterized by highly nonlinear characteristics,heavy modeling uncertainties and unknown perturbations,therefore,accurate-model-based nonlinear control approaches become unavailable.Motivated by the challenge,a reinforcement learning(RL)adaptive control methodology based on the actor-critic framework is investigated to compensate the uncertain mechanical dynamics.The approximation inaccuracies caused by RL and the exogenous unknown disturbances are circumvented via a continuous robust integral of the sign of the error(RISE)control approach.Different from a classical RISE control law,a tanh(·)function is utilized instead of a sign(·)function to acquire a more smooth control signal.The developed controller requires very little prior knowledge of the dynamic model,is robust to unknown dynamics and exogenous disturbances,and can achieve asymptotic output tracking.Eventually,co-simulations through ADAMS and MATLAB/Simulink on a three degrees-of-freedom(3-DOF)manipulator and experiments on a real-time electromechanical servo system are performed to verify the performance of the proposed approach.

Chaotic synchronization for a class of fractional-order chaotic systems

In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.

No-chattering sliding mode control in a class of fractional-order chaotic systems

A no-chattering sliding mode control strategy for a class of fractional-order chaotic systems is proposed in this paper. First, the sliding mode control law is derived to stabilize the states of the commensurate fractional-order chaotic system and the non-commensurate fractional-order chaotic system, respectively. The designed control scheme guarantees the asymptotical stability of an uncertain fractional-order chaotic system. Simulation results are given for several fractional-order chaotic examples to illustrate the effectiveness of the proposed scheme.

Function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems

This paper investigates the function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems using the stability theory of fractional-order systems. The function projective synchronization between three-dimensional （3D） integer-order Lorenz chaotic system and 3D fractional-order Chen chaotic system are presented to demonstrate the effectiveness of the proposed scheme.

Parameter identification of the fractional-order systems based on a modified PSO algorithm

In order to better identify the parameters of the fractional-order system,a modified particle swarm optimization(MPSO)algorithm based on an improved Tent mapping is proposed.The MPSO algorithm is validated with eight classical test functions,and compared with the POS algorithm with adaptive time varying accelerators(ACPSO),the genetic algorithm(GA),a d the improved PSO algorithm with passive congregation(IPSO).Based on the systems with known model structures a d unknown model structures,the proposed algorithm is adopted to identify two typical fractional-order models.The results of parameter identification show that the application of average value of position information is beneficial to making f 11 use of the information exchange among individuals and speeds up the global searching speed.By introducing the uniformity and ergodicity of Tent mapping,the MPSO avoids the extreme v^ue of position information,so as not to fall into the local optimal value.In brief the MPSOalgorithm is an effective a d useful method with a fast convergence rate and high accuracy.

Robust sliding mode control for fractional-order chaotic economical system with parameter uncertainty and external disturbance

This paper presents a modified sliding mode control for fractional-order chaotic economical systems with parameter uncertainty and external disturbance. By constructing the suitable sliding mode surface with fractional-order integral, the effective sliding mode controller is designed to realize the asymptotical stability of fractional-order chaotic economical systems. Comparing with the existing results, the main results in this paper are more practical and rigorous. Simulation results show the effectiveness and feasibility of the proposed sliding mode control method.

Impulsive synchronisation of a class of fractional-order hyperchaotic systems

In this paper, an impulsive synchronisation scheme for a class of fractional-order hyperchaotic systems is proposed. The sufficient conditions of a class of integral-order hyperchaotic systems＇ impulsive synchronisation are illustrated. Furthermore, we apply the sufficient conditions to a class of fractional-order hyperchaotic systems and well achieve impulsive synchronisation of these fractional-order hyperchaotic systems, thereby extending the applicable scope of impulsive synchronisation. Numerical simulations further demonstrate the feasibility and effectiveness of the proposed scheme.

A novel study on the impulsive synchronization of fractional-order chaotic systems

The stability of impulsive fractional-order systems is discussed. A new synchronization criterion of fractional-order chaotic systems is proposed based on the stability theory of impulsive fractional-order systems. The synchronization criterion is suitable for the case of the order 0 〈 q ≤ 1. It is more general than those of the known results. Simulation results are given to show the effectiveness of the proposed synchronization criterion.

Fractional-order systems without equilibria:The first example of hyperchaos and its application to synchronization

A challenging topic in nonlinear dynamics concerns the study of fractional-order systems without equilibrium points.In particular, no paper has been published to date regarding the presence of hyperchaos in these systems. This paper aims to bridge the gap by introducing a new example of fractional-order hyperchaotic system without equilibrium points. The conducted analysis shows that hyperchaos exists in the proposed system when its order is as low as 3.84. Moreover, an interesting application of hyperchaotic synchronization to the considered fractional-order system is provided.

Generalized Synchronization Between Different Fractional-Order Chaotic Systems

In this paper,using scalar feedback controller and stability theory of fractional-order systems,a gener-alized synchronization method for different fractional-order chaotic systems is established.Simulation results show theeffectiveness of the theoretical results.

Analysis of the effect on control systems of order variation for fractional-orderPI~λD~μcontrollers

This paper is concerned with fractional-order PI~λD~μcontrollers. The definitions and properties of fractional calculus are introduced. The mathematical descriptions of a fractional-order controller and fractional-order control systems are outlined. The effects on control systems of order variation for fractional-order PI~λD~μ controllers are investigated by qualitative analysis and simulation. The conclusions and simulation examples are given. The results show the fractional-order PI~λD~μ controller is not sensitive to variation of its order.

Robust Fractional-Order Proportional-Integral Observer for Synchronization of Chaotic Fractional-Order Systems

In this paper, we propose a robust fractional-order proportional-integral(FOPI) observer for the synchronization of nonlinear fractional-order chaotic systems. The convergence of the observer is proved, and sufficient conditions are derived in terms of linear matrix inequalities(LMIs) approach by using an indirect Lyapunov method. The proposed FOPI observer is robust against Lipschitz additive nonlinear uncertainty. It is also compared to the fractional-order proportional(FOP) observer and its performance is illustrated through simulations done on the fractional-order chaotic Lorenz system.

Stable Model Order Reduction Method for Fractional-Order Systems Based on Unsymmetric Lanczos Algorithm

This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.

Robustness of iterative learning control for a class offractional-order linear continuous-time switched systems in the sense of L^p norm

For a class of fractional-order linear continuous-time switched systems specified by an arbitrary switching sequence,the performance of PDα-type fractional-order iterative learning control(FOILC)is discussed in the sense of L^p norm.When the systems are disturbed by bounded external noises,robustness of the PDα-type algorithm is firstly analyzed in the iteration domain by taking advantage of the generalized Young inequality of convolution integral.Then,convergence of the algorithm is discussed for the systems without any external noise.The results demonstrate that,under some given conditions,both convergence and robustness can be guaranteed during the entire time interval.Simulations support the correctness of the theory.

Synchronization between fractional-order chaotic systems and integer orders chaotic systems (fractional-order chaotic systems)

Based on the idea of tracking control and stability theory of fractional-order systems, a controller is designed to synchronize the fractional-order chaotic system with chaotic systems of integer orders, and synchronize the different fractional-order chaotic systems. The proposed synchronization approach in this paper shows that the synchronization between fractional-order chaotic systems and chaotic systems of integer orders can be achieved, and the synchronization between different fractional-order chaotic systems can also be realized. Numerical experiments show that the present method works very well.

Controllability, Observability and Stability for a Class of Fractional-Order Linear Time-Invariant Control Systems

The definitions of controllability, observability and stability were presented for fractional-order linear systems. Using the Cayley-Hamilton theorem and Mittag-Leffler function in two parameters, the sufficient and necessary conditions of controllability and observability for such systems were derived. In terms of Lyapunov’s stability theory, using the theorems of Mittage-Leffler function in two parameters this paper directly derived the sufficient and necessary condition of stability for such systems. The results obtained are useful for the analysis and synthesis of fractional-order linear control systems.

Robust Stability Analysis of Smith Predictor Based Interval Fractional-Order Control Systems:A Case Study in Level Control Process

The robust stability study of the classic Smith predictor-based control system for uncertain fractional-order plants with interval time delays and interval coefficients is the emphasis of this work.Interval uncertainties are a type of parametric uncertainties that cannot be avoided when modeling real-world plants.Also,in the considered Smith predictor control structure it is supposed that the controller is a fractional-order proportional integral derivative(FOPID)controller.To the best of the authors'knowledge,no method has been developed until now to analyze the robust stability of a Smith predictor based fractional-order control system in the presence of the simultaneous uncertainties in gain,time-constants,and time delay.The three primary contributions of this study are as follows:ⅰ)a set of necessary and sufficient conditions is constructed using a graphical method to examine the robust stability of a Smith predictor-based fractionalorder control system—the proposed method explicitly determines whether or not the FOPID controller can robustly stabilize the Smith predictor-based fractional-order control system;ⅱ)an auxiliary function as a robust stability testing function is presented to reduce the computational complexity of the robust stability analysis;andⅲ)two auxiliary functions are proposed to achieve the control requirements on the disturbance rejection and the noise reduction.Finally,four numerical examples and an experimental verification are presented in this study to demonstrate the efficacy and significance of the suggested technique.

Optimal Tuning of FOPID-Like Fuzzy Controller for High-Performance Fractional-Order Systems

This paper addresses improvements in fractional order(FO)system performance.Although the classical proportional-integral-derivative(PID)-like fuzzy controller can provide adequate results for both transient and steady-state responses in both linear and nonlinear systems,the FOPID fuzzy controller has been proven to provide better results.This high performance was obtained thanks to the combinative benefits of FO and fuzzy-logic techniques.This paper describes how the optimal gains and FO parameters of the FOPID controller were obtained by the use of a modern optimizer,social spider optimization,in order to improve the response of fractional dynamical systems.This group of systems had usually produced multimodal error surfaces/functions that occasionally had many variant local minima.The integral time of absolute error(ITAE)used in this study was the error function.The results showed that the strategy adopted produced superior performance regarding the lowest ITAE value.It reached a value of 88.22 while the best value obtained in previous work was 98.87.A further comparison between the current work and previous studies concerning transient-analysis factors of the model’s response showed that the strategy proposed was the only one that was able to produce fast rise time,low-percentage overshoot,and very small steady-state error.However,the other strategies were good for one factor,but not for the others.