Uncertain Fractional-order Systems Research Articles

Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation

Abstract In this article, we address the delay-dependent robust stability of uncertain fractional order neutral-type (FONT) systems with distributed delays, nonlinear perturbations, and input saturation. With the aid of the Lyapunov–Krasovskii functional, criteria on asymptotic robust stability of FONT, expressed in terms of linear matrix inequalities, are constructed to compute the state-feedback controller gains. The controller gains are determined subject to maximizing the domain of attraction via the cone complementarity linearization algorithm. The theoretical results are validated using numerical simulations.

Adaptive fault compensation control for nonlinear uncertain fractional-order systems: Static and dynamic event generator approaches

The tracking problem of the fractional-order nonlinear systems is assessed by extending new event-triggered control designs. The considered dynamics are accompanied by the uncertain strict-feedback form, unknown actuator faults and unknown disturbances. By using the neural networks and the fault compensation method, two adaptive fault compensation event-triggered schemes are designed. Unlike the available control designs, two static and dynamic event-triggered strategies are proposed for the nonlinear fractional-order systems, in a sense that the minimum/average time-interval between two successive events can be prolonged in the dynamic event-triggered approach. Besides, it is proven that the Zeno phenomenon is strictly avoided. Finally, the simulation results prove the effectiveness of the presented control methods.

An ADRC-based backstepping control design for a class of fractional-order systems

A combined fractional tracking controller based on backstepping and active disturbance rejection control (ADRC) is proposed in this work to control uncertain fractional-order systems (FOSs). The proposed method is constructed in two parts: first, all internal and external uncertainties are estimated by a fractional-order extended state observer (FO-ESO). Second, fractional backstepping method is used to determine the control law and lead the system to the desired control objectives. The control system stability is verified by fractional Lyapunov-like lemma and two numerical simulations confirm the efficiency of the proposed combined ADRC-based backstepping controller.

Robust fractional-order fixed-structure controller design for uncertain non-commensurate fractional plants using fractional Kharitonov theorem

This article deals with the problem of robust fractional-order fixed-structure controller design for commensurate and non-commensurate fractional-order interval systems using fractional Kharitonov theorem. The contribution of this study is to develop a simple control methodology to stabilize the fractional-order Kharitonov-defined vortex polynomials. Using the idea of robust stability testing function and extending it to the systems under study, the straightforward graphical and systematic procedures are proposed to investigate the robust stability of the system by searching for a non-conservative fractional-order Kharitonov region in the controller parameters plane. This region can establish all the fractional-order controllers that make the uncertain fractional-order systems stable. The relation between the fractional-order Kharitonov region and the parameters of the stabilizing controller is also found. Finally, comparison results with three relevant works are given to illustrate the feasibility of the proposed method.

T-S Fuzzy Control of Uncertain Fractional-Order Systems with Time Delay

In this article, the adaptive control of uncertain fractional-order time-delay systems (FOTDSs) with external disturbances is discussed. A Takagi-Sugenu (T-S) fuzzy model with if-then rules is adopted to characterize the dynamic equation of the FOTDS. Besides, a fuzzy adaptive method is proposed to stabilize the model. By utilizing the Lyapunov functions, a robust controller is constructed to stabilize the FOTDS. Due to the uncertainty of system parameters, some fractional-order adaptation laws are designed to update these parameters. At the same time, some if-then rules with linear structure based on the fuzzy T-S adaption concept are established. The designed method not only guarantees that the state of closed-loop system asymptotically converges to origin but also keeps the signal in the FOTDS bounded. Finally, the applicability of the control method is proved by simulation examples.

Robust stability of uncertain fractional order systems of neutral type with distributed delays and control input saturation

Time delay occurs naturally due to the limited bandwidth of any real-world system. However, this problem can deteriorate the system performance and can even result in system instability. Input saturation is also an essential issue due to the energy constraint in real actuators that makes the control design procedure more difficult. This article concerns with the stability of uncertain fractional order (FO) delay systems of neutral type including structured uncertainties, distributed delays and actuator saturation. A Lyapunov–Krasovskii functional allows the formulation of the conditions to insure the asymptotic robust stability of such systems via the linear matrix inequalities (LMI) and to compute the gain of a state feedback controller. In addition, by using the cone complementarity linearization method, we obtain the controller gains that extend the domain of attraction. Several simulations validate the theoretical analysis.

Robust stability analysis of incommensurate fractional-order systems with time-varying interval uncertainties

This paper focuses on the stability analysis of the incommensurate fractional-order systems describing by the pseudo-state-space form in the presence of interval uncertainties. By using the generalized small-gain theorem and the properties of M-matrices, a condition is achieved for robust finite-gain L2 stability of these systems. In addition, satisfying this condition ensures the asymptotic stability of uncertain fractional-order systems, based on the Caputo definition. Unlike the methods which are based on the eigenvalue argument checking, the proposed robust stability analysis method can be used for systems with irrational fractional orders and time-varying uncertainties. Compared with the Lyapunov based methods, the conservatism due to the lack of relation between stability condition and fractional-order of the system does not occur in the proposed method. An illustrative example is presented to confirm this method does not lead to conservatism generated by reformulation of the interval uncertainty, which is the basis of the commonly used methods in the literature. Finally, the suggested method is employed for robust stability checking of the Sallen–Key filter including fractional capacitors with irrational orders.

Functional interval observers for nonlinear fractional-order systems with time-varying delays and disturbances

This article considers the estimation problem for a class of uncertain fractional-order systems with time-varying delays and disturbances, in which the time-varying delays are present in the state, while the disturbances are present in the output. A new design method of functional interval observers is proposed. In contrast to interval observers currently available in the literature, which dealt with integer-order systems or fractional-order systems without time delays, the functional interval observers with fractional-order [Formula: see text] in this article can deal with fractional-order time-delay systems and integer-order time-delay systems (when [Formula: see text]). Existence conditions of functional interval observers for non-linear fractional-order systems with time-varying delays are established and they are formulated in terms of linear programmings. Two algorithms are provided for determining the functional interval observers. Illustrative examples of the functional interval observer application are presented.

Observer-based event-triggered control for uncertain fractional-order systems

In this paper, the observer-based event-triggered control problem for a class of uncertain fractional-order systems with order 0 < α < 1 is investigated. Owing to the difficulties of measuring system full-state in practice, a fractional-order observer is firstly designed to construct an event-triggered condition, which eventually decides whether the current signal should be transmitted. Then, by utilizing the observer-based feedback controller, a sufficient condition in the form of linear matrix inequalities (LMIs) is presented to guarantee the asymptotic stability of the augmented closed-loop system. After that, the corresponding design conditions for the observer gain and controller gain are provided. Furthermore, the Zeno behavior of the considered systems is discussed, and a positive lower bound of inter-execution intervals is estimated to solve infinite triggering and sampling issues. Finally, an illustrative example and simulations are provided to illustrate the effectiveness of the proposed approaches.

Designing of a Control Approach for Uncertain Fractional Order Systems with Indirect Adaptive Fuzzy Controller and Frational Order Sliding Mode

Designing of a Control Approach for Uncertain Fractional Order Systems with Indirect Adaptive Fuzzy Controller and Frational Order Sliding Mode

Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation

The robust stability and stabilization of a class of fractional order systems under input saturation is studied using the Lyapunov method. In the problem formulation the Lipchitz and bounded conditions are adopted. Moreover, sufficient conditions, in the form of a Linear Matrix Inequality, for stabilizing the system via a state feedback control, are derived. Two numerical examples show the effectiveness of the proposed technique.

H∞-synthesis and control of uncertain fractional-order systems of commensurate type

New Linear-Matrix-Inequality (LMI) conditions are proposed for ℋ∞ analysis and synthesis of uncertain fractional-order systems where the non-integer order of differentiation belongs to the set ]0 2[. The developed conditions are extended LMI conditions involving additional LMI variables needed for numerical calculation of the feedback gains. The stability conditions are embedded with the necessary H∞ LMI conditions leading to new formulation of the bounded-real-lemma result. The stabilizability conditions with H∞ performance are subsequently derived and tested with static-pseudo-state feedbacks and static-output feedbacks as well.

Graphical PID tuning method for uncertain fractional-order multivariable systems

In this paper, a graphical tuning method for controllers parameters based on the open-loop fractional transfer function (FO-EOTF) method is proposed for fractional-order parameter uncertain multivariable system. The FO-EOTF method is proposed to transform the parameter uncertain fractional-order multivariable system into a set of independent parameter uncertain fractional-order univariate systems and determine the parameters regions of the univariate systems. The gain phase margin tester is used to further guarantee the robust performance of the controlled system. Finally, simulation result from the numerical simulation is presented to demonstrate the effectiveness of this method.

Adaptive Stabilization for a Class of Fractional-Order Systems with Nonlinear Uncertainty

The stabilization task for uncertain integer-order systems has been widely and extensively investigated in the literature. However, stabilizing uncertain fractional-order systems (despite the recent great interest given by researchers to this research axis) is still considered as a fertile area of research. In this paper, an original adaptive scheme to handle this particular problem, under the nonlinear uncertainty modeling, is suggested. The approach consists of estimating the upper bound of uncertainties and designing an adaptive output feedback controller, using the Lyapunov direct method. The concept of uniform practical Mittag–Leffler stability is used throughout the paper. The feasibility and effectiveness of the theoretical results are shown through simulations via two numerical examples.

A fractional-order hyper-chaotic economic system with transient chaos

We introduce for the first time a fractional-order hyperchaotic economic system. In this system, chaos generation depends upon the value of fractional-order. For certain fractional orders, a sustained regime of chaos is obtained. Also, the transient chaos phenomenon is detected for lower fractional orders. The dynamical behavior of the system is numerically investigated using bifurcations diagrams, basins of attraction, and Lyapunov exponents. Next, an adaptive terminal sliding mode control (ATSMC) with neural network estimator for finite-time stabilization and synchronization of the fractional-order system has been proposed. The radial basis function (RBF) neural network is used to achieve the estimation of the unknown function of the system. Also, the effects of external disturbances are fully taken into account with neural network estimator. The weights of the RBF neural network are updated based on the appropriate adaptation law. Using the fractional version of the Lyapunov stability theorem, the finite-time convergence of the closed-loop system has been proven. Finally, the new control technique is used for control and synchronization of the fractional-order hyperchaotic economic system. Simulation results illustrate the effectiveness of the proposed control scheme for uncertain fractional-order systems in the presence of external disturbances.

Stabilization of uncertain fractional order system with time‐varying delay using BMI approach

AbstractThis paper considers the systematic design of robust stabilizing state feedback controllers for fractional order nonlinear systems with time‐varying delay being possibly unbounded. By using the fractional Halanay inequality and the Caputo fractional derivative of a quadratic function, stabilizability conditions expressed in terms of bilinear matrix inequalities are derived. The controllers can then be obtained by computing the gain matrices. In order to derive the gain matrices, two algorithms are proposed by using the existing computationally linear matrix inequality techniques. Two numerical examples with simulation results are provided to demonstrate the effectiveness of the obtained results.

Robust passivity and feedback passification of a class of uncertain fractional-order linear systems

ABSTRACTTheoretical results on robust passivity and feedback passification of a class of uncertain fractional-order (FO) linear systems are presented in the paper. The system under consideration is subject to time-varying norm-bounded parameter uncertainties in both the state and controlled output matrices. Firstly, some suitable notions of passivity and dissipativity for FO systems are proposed, and the relationship between passivity and stability is obtained. Then, a sufficient condition in the form of linear matrix inequality (LMI) for such system to be robustly passive is given. Based on this condition, the design method of state feedback controller is proposed when the states are available. Moreover, by using matrix singular value decomposition and LMI techniques, the existing condition and method of designing a robust observer-based passive controller for such systems are derived. Numerical simulations demonstrate the effectiveness of the theoretical formulation.

Sliding mode control of uncertain fractional order systems with delay

ABSTRACT This paper presents a sliding mode control (SMC) based framework to design a stabilizing controller for uncertain fractional order time-delay systems (FOTDS) using the Lambert W function technique. This technique is exploited to design the fractional order (FO) sliding manifold which provides a constructive method to design an FO sliding manifold as compared to the other methods. A stabilizing switching controller is derived, that ensures the finite time reachability of the system trajectories to the sliding manifold for all subsequent time. Moreover, the conventional SMC technique can be readily applied to a particular class of FOTDS with delay free sliding mode dynamics obtained through a specific transformation. A Duffing oscillator is considered here to illustrate the proficiency of the proposed approach. Simulation results demonstrate the efficiency of the proposed method in the presence of time delay and uncertainty.

Robust Control and Stability Bound Analysis for a Class of LTI Fractional Order Uncertain Systems With 0 &lt; $\alpha$ &lt; 1

This paper concentrates on the robust control and maximal bound analysis of uncertainty for the LTI fractional order system (FOS), which is subjected to poly-topic and H-infinity bounded uncertainties with 0 <; α <; 1. Firstly, two problems including robust stability analysis and stabilization are investigated. Subsequently, the method of how to determine the maximal uncertainty bound of such system is discussed, and the corresponding linear state feedback stabilizing controller is obtained together. The conditions in terms of linear matrix inequalities (LMI) for these problems mentioned above are concluded as four theorems. Finally, the advantage of the proposed methods is illustrated by two numerical examples.

Robust Stability Analysis of Uncertain Linear Fractional-Order Systems With Time-Varying Uncertainty for 0 &lt; α &lt; 2

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0&lt;α&lt;2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0&lt;α&lt;1 and 1&lt;α&lt;2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.