Uncertain Fractional-order Systems Research Articles

Robust disturbance rejection for a fractional-order system based on equivalent-input-disturbance approach

This paper presents a disturbance rejection method that is based on the equivalent-input-disturbance approach for uncertain fractional-order (FO) systems. An FO observer is used to reconstruct the plant states and estimate the disturbances. A disturbance estimator is incorporated in the construction of the FO control system to actively compensate for the entire disturbance. A robust stability condition for the control system and the parameters of the controller are derived using an indirect Lyapunov method. The presented method effectively rejects disturbances and handles modeling uncertainties without requiring prior knowledge of the disturbance. Comparison simulations on both numerical and practical examples demonstrate the effectiveness of the proposed method.

LMI stability conditions and stabilization of fractional-order systems with poly-topic and two-norm bounded uncertainties for fractional-order α: the 1 < α < 2 case

This article addresses the problem of robust stability and stabilization for linear fractional-order system with poly-topic and two-norm bounded uncertainties, and focuses particularly on the case of a fractional order α such that 1 < α < 2. First, the robust asymptotical stable condition is presented. Second, the design method of the state feedback controller for asymptotically stabilizing such uncertain fractional order systems is derived. In the proposed approach, linear matrix inequalities formalism is used to check and design. Lastly, two simulation examples are given to validate the proposed theoretical results.

Robust stability and stabilization of LTI fractional-order systems with poly-topic and two-norm bounded uncertainties

This paper investigates the problems of robust stability and stabilization of LTI fractional-order systems with poly-topic and two-norm bounded uncertainties. Firstly, some sufficient conditions of the robust asymptotical stable for such fractional-order uncertain systems are derived. Secondly, the robust stabilizing state-feedback controller is designed. All the results are obtained in terms of linear matrix inequalities (LMIs). Lastly, three numerical examples are provided to demonstrate the correctness and effectiveness of the proposed approaches.

Adaptive Robust Tracking Control for Multiple Unknown Fractional-Order Nonlinear Systems

By applying the fractional Lyapunov direct method, we investigate the robust consensus tracking problem for a class of uncertain fractional-order multiagent systems with a leader whose input is unknown and bounded. More specifically, multiple fractional-order systems with heterogeneous unknown nonlinearities and external disturbances are considered in this paper, which include the second-order multiagent systems as its special cases. First, a discontinuous neural network-based (NN-based) distributed robust adaptive algorithm is designed to guarantee the consensus tracking error exponentially converges to zero under a fixed topology. Also the derived results are further extended to the case of switching topology by appropriately choosing multiple Lyapunov functions. Second, a continuous NN-based distributed robust adaptive algorithm is further proposed to eliminate the undesirable chattering phenomenon of the discontinuous controller, where the consensus tacking error is uniformly ultimately bounded and can be reduced as small as desired. It is worth noting that all the proposed NN-based robust adaptive algorithms are independent of any global information and thus are fully distributed. Finally, numerical simulations are provided to validate the correctness of the proposed algorithms.

Adaptive Fuzzy Output Tracking Control of a Class of Uncertain Fractional Order Systems Subject to Unknown Disturbance

In this paper, a novel adaptive fuzzy backstepping control method is designed for a class of incommensurate fractional-order nonlinear systems with unknown nonlinearities and external disturbance, in which the frequency distributed model is employed so that the indirect Lyapunov method can be used to design the controller. In each step of the backstepping, the complicated unknown nonlinear function coming from a fractional differential function is approximated using a fuzzy logic system, and the virtual control law and parameters update law are designed, in which the order of the parameters update law cannot be fixed to the system order and more degree of freedom can be obtained. In the last step, an adaptive fuzzy controller is given, which can ensure the convergence of the tracking error. Compared with the previous methods, the proposed backstepping method is first adopted to solve the tracking problem of incommensurate order systems with unknown nonlinearities and external disturbance, in which the stabilization and tracking can be achieved. Two careful simulation studies are provided to illustrate the effectiveness of this novel scheme.

Robust disturbance rejection for uncertain fractional-order systems

This paper describes a disturbance rejection scheme that adopts equivalent-input-disturbance (EID) approach for uncertain fractional-order (FO) systems. An EID estimator that contains an FO observer is designed to actively compensate for the disturbances and process modeling uncertainties without requiring their prior knowledge. Under the construction of the FO control system, a robust stability condition and the parameters of the controller are derived using a linear matrix inequality based method. Finally, numerical and practical examples are illustrated to demonstrate the validity and superiority of the method.

Robust $H_{\infty }$ Observer-Based Control of Fractional-Order Systems With Gain Parametrization

This paper investigates the robust $H_{\infty }$ observer-based control (OBC) for linear time-invariant disturbed uncertain fractional-order systems (DU-FOS). First, the existence conditions for robust $H_{\infty }$ OBC are given. Then, based on the $H_{\infty }$ -norm analysis using the generalized Kalman–Yakubovich–Popov lemma for FOS, and following the fractional derivative order $\alpha$ , new sufficient linear matrix inequalities (LMIs) conditions are obtained to ensure the stability of the estimation errors and the stabilization of the DU-FOS simultaneously. All observer matrices gains and control laws can be computed by solving a unique LMI condition in one step. Numerical simulation is given to illustrate the validity of the proposed method.

Adaptive control of nonlinear fractional-order systems using T–S fuzzy method

Owing to the superior capability of fractional differential equations in modeling and characterizing accurate dynamical properties of many high technology real world systems, the design and control of fractional-order systems have captured lots of attention in recent decades. In this paper, an adaptive intelligent fuzzy approach to controlling and stabilization of nonlinear non-autonomous fractional-order systems is proposed. Since dynamic equations of applied fractional-order systems usually contain various parameters and nonlinear terms, the Takagi–Sugeno (T–S) fuzzy models with if-then rules are adopted to describe the system dynamics. Also, as the nonlinear system parameters are assumed to be unknown, adaptive laws are derived to estimate such fluctuations. Simple adaptive linear-like control rules are developed based on the T–S fuzzy control theory. The stability of the resulting closed loop system is guaranteed by Lyapunov’s stability theory. Two illustrative numerical examples are presented to emphasize the correct performance and applicability of the proposed adaptive fuzzy control methodology. It is worth to notice that the proposed controller works well for stabilization of a wide class of either autonomous nonlinear uncertain fractional-order systems or non-autonomous complex systems with unknown parameters.

Robust stability and stabilization of uncertain fractional order systems subject to input saturation

This paper studies the stability and the stabilization for a class of uncertain fractional order (FO) systems subject to input saturation. The Lipschitz condition and the Gronwall–Bellman lemma are adopted and sufficient conditions are derived to stabilize systems by designing a state feedback controller. Numerical examples demonstrate the effectiveness of the proposed method.

Synchronization of uncertain fractional-order hyperchaotic systems by using a new self-evolving non-singleton type-2 fuzzy neural network and its application to secure communication

This paper presents a robust control method for synchronization of the uncertain fractional-order hyperchaotic systems by using a new self-evolving non-singleton type-2 fuzzy neural network (SE-NT2FNN). The proposed SE-NT2FNNs are used for estimating the unknown functions in the dynamic of system. The effects of approximation error and external disturbance are eliminated by linear matrix inequality control scheme. The proposed SE-NT2FNN has one rule initially, the new rules and membership functions (MFs) are added based on the proposed simple algorithm and unnecessary rules and MFs are deleted. The proposed synchronization scheme is applied in a secure communication scheme. To show the effectiveness of the proposed method, three simulation examples are given. The results are compared with other methods, and it showed that the proposed control scheme results in the better performance than other methods.

Optimal Sliding Mode Robust Control for Fractional-Order Systems with Application to Permanent Magnet Synchronous Motor Tracking Control

This paper presents an optimal sliding mode output tracking control scheme for a class of fractional-order uncertain systems. Firstly, an augmented fractional-order system, composed of the original system and the external system, is constructed to transform the optimal output tracking issue into the design problem of linear quadratic regulator. The optimal tracking control problem for the nominal augmented fractional-order system is then studied. Secondly, the fractional-integral sliding mode controller is introduced to robustify the augmented fractional-order system, which satisfy the matching conditions. As a result, the original system output can track the external system output trajectory effectively even the uncertainties exist. Finally, the developed design techniques are applied to the tracking control of fractional-order permanent magnet synchronous motor. The simulation results demonstrate the validity of this approach.

Design of fuzzy output feedback stabilization for uncertain fractional-order systems

This paper focuses on the fuzzy output feedback stabilization for uncertain fractional-order systems (FOS) in the case of the fractional-order satisfying 0<α<1 and 1≤α<2. Firstly, the uncertain FOS is described by the so-called fractional-order Takagi–Sugeno (T–S) fuzzy model, and the fuzzy output feedback controller is correspondingly designed. Secondly, based on the slack variable method and FOS stability criteria, sufficient conditions in the form of linear matrix inequality (LMI) are derived for the robust asymptotical stability of the closed-loop control systems with fractional-order 0<α<1. Furthermore, the results are extended to stabilize the FOS with fractional-order 1≤α<2. Finally, two numerical simulation examples are given to show the effectiveness of the proposed method.

Adaptive Fuzzy Control for Nonlinear Fractional-Order Uncertain Systems with Unknown Uncertainties and External Disturbance

In this paper, the problem of robust control of nonlinear fractional-order systems in the presence of uncertainties and external disturbance is investigated. Fuzzy logic systems are used for estimating the unknown nonlinear functions. Based on the fractional Lyapunov direct method and some proposed Lemmas, an adaptive fuzzy controller is designed. The proposed method can guarantee all the signals in the closed-loop systems remain bounded and the tracking errors converge to an arbitrary small region of the origin. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results.

New sufficient conditions for observer-based control of fractional-order uncertain systems

New simple linear matrix inequalities are proposed to ensure the stability of a class of uncertain fractional-order linear systems by means of a fractional-order deterministic observer. It is shown that the conditions of existence of an observer-based feedback can be split into a set of linear matrix inequalities that are numerically tractable. The presented results show that it is possible to decouple the conditions containing the bilinear variables into separate conditions without imposing equality constraints or considering an iterative search of the controller and the observer gains. Simulations results are given to approve the efficiency and the straightforwardness of the proposed design.

Multiconsensus of fractional-order uncertain multi-agent systems

In this study, the multiconsensus problem of fractional-order uncertain multi-agent systems is investigated. Output feedback controller is designed for fractional-order uncertain systems when fractional-order α satisfies 0<α<2. By introducing appropriate transformation, the multiconsensus problem is converted into the stability problem of fractional-order systems. Based on the stability theory of fractional-order systems, some sufficient conditions are presented to ensure that the states of all the agents can reach more than one consistent values asymptotically, and the feedback matrix of the proposed protocol is also determined according to linear matrix inequalities. Simulation examples are given to demonstrate the effectiveness of the theoretical analysis.

Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

SummaryThis paper focuses on the graphical tuning method of fractional order proportional integral derivative (FOPID) controllers for fractional order uncertain system achieving robust ‐stability. Firstly, general result is presented to check the robust ‐stability of the linear fractional order interval polynomial. Then some alternative algorithms and results are proposed to reduce the computational effort of the general result. Secondly, a general graphical tuning method together with some computational efficient algorithms are proposed to determine the complete set of FOPID controllers that provides ‐stability for interval fractional order plant. These methods will combine the results for fractional order parametric robust control with the method of FOPID ‐stabilization for a fixed plant. At last, two important extensions will be given to the proposed graphical tuning methods: determine the ‐stabilizing region for fractional order systems with two kinds of more general and complex uncertainty structures: multi‐linear interval uncertainty and mixed‐type uncertainties. Numerical examples are followed to illustrate the effectiveness of the method. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Intelligent Chaos Synchronization of Fractional Order Systems via Mean-Based Slide Mode Controller

In this paper, a new mean-based adaptive fuzzy neural network sliding mode control is developed to perform the chaos synchronization among the master–slave fractional order uncertain systems. The mean-based expansion is adopted to replace the traditional Taylor expansion to transform a nonlinear function into a partially linear form for the linearization of nonlinear systems. In comparison with the traditional Taylor method, the proposed mean-based method can estimate the first-order derivative term on the identifier model, which will somehow alleviate the computational burden. Based on the learning algorithms, the adaptive laws and control laws can be tuned on-line to synchronize the master–slave fractional order uncertain systems. Furthermore, the stability of the closed-loop system can not only be assured but the synchronization deviation of external perturbation can also be alleviated. Finally, simulation examples are illustrated to demonstrate the feasibility and the synchronization performance of this new approach.

Robust analysis and synthesis for a class of fractional order systems with coupling uncertainties

A class of uncertain fractional order systems is concerned where intervals of system matrices parameters are affected by the fractional order. Moreover, the order is also uncertain and belongs to an interval containing the integer one. This kind of system models are reasonable results of fractional order system identifications. In this note, we derive the robust stability analysis and synthesis of such uncertain fractional order systems. Two numerical examples are presented to illustrate the effectiveness and potential of the developed theoretical results.

Robust Nonfragile Controllers Design for Fractional Order Large-Scale Uncertain Systems with a Commensurate Order1&lt;α&lt;2

The paper concerns the problem of stabilization of large-scale fractional order uncertain systems with a commensurate order1&lt;α&lt;2under controller gain uncertainties. The uncertainties are of norm-bounded type. Based on the stability criterion of fractional order system, sufficient conditions on the decentralized stabilization of fractional order large-scale uncertain systems in both cases of additive and multiplicative gain perturbations are established by using the complex Lyapunov inequality. Moreover, the decentralized nonfragile controllers are designed. Finally, some numerical examples are given to validate the proposed method.

Robust second-order PD type iterative learning control for a class of uncertain fractional order singular systems

This paper presents a robust second-order feedback type iterative learning control (ILC) for a class of uncertain fractional-order singular systems. Sufficient conditions for the robust convergence of the proposed type of learning control algorithm, with respect to the bounded external disturbance and uncertainity, have been established and specified for time domain. Finally, simulation examples are presented to illustrate the performance of the proposed fractional order ILC.