Class Of Fractional-order Nonlinear Systems Research Articles

Deviation analysis in transient response of fractional-order systems: An elementary function-based lower bound

In this paper, transient response of commensurate fractional-order systems with non-zero initial conditions is investigated in the viewpoint of the existence of deviation from initial condition. In particular, firstly peak effect for a class of linear fractional order systems is inspected by presenting a lower bound for their responses. Such a lower bound is described by an elementary function, where the commensurate order of the system is considered as a rational number. Furthermore, it has been demonstrated that under particular circumstances the derived lower bound can be extended to apply for deviation analysis in response of a class of fractional-order nonlinear systems. Moreover, included are various illustrative examples intended to assess the applicability of the obtained lower bound in prediction of the deviation value and the time instance at which the peak effect occurs.

A New Observer Design for the Joint Estimation of States and Unknown Inputs for a Class of Nonlinear Fractional-Order Systems

This article designs an observer for the joint estimation of the state and the unknown input for a class of nonlinear fractional-order systems (FOSs) such that one portion satisfies the Lipschitz condition and the other does not necessarily satisfy such a condition. Firstly, by reconstructing system dynamics, the observer design is transformed equivalently into the tracking problem between the original nonlinear FOSs and the designed observer. Secondly, the parameterized matrices of the desired observer are derived by use of the property of the generalized inverse matrices and the linear matrix inequality (LMI) technique combined with the Schur complement lemma. Moreover, an algorithm is presented to determine the desired observer for the nonlinear FOSs effectively. Finally, a numerical example is reported to verify the correctness and efficiency of the proposed algorithm.

Finite-time stability of equilibrium point of a class of fractional-order nonlinear systems

This paper studies the finite-time stability (FTS) of equilibrium point of a class of fractional-order nonlinear systems (FONSs) described by Caputo fractional-order derivative (CFOD). The definition of finite-time equilibrium point (FTEP) is proposed for the FONSs, and it is proved that the CFOD of the FTEP is not constantly equal to 0. A sufficient and necessary condition on the FTEP is proposed. Based on the asymptotical stability and the method of the integral of unbounded integrand, two sufficient conditions on the FTS of the equilibrium point are provided. Consequently, the existence of finite-time stable equilibrium points in the FONSs is confirmed. Two examples are presented to illustrate the proposed results.

State estimation-based parameter identification for a class of nonlinear fractional-order systems

Parametric identification is an important part of system theory since knowledge of the parameters allows the analysis and control of the system. The aim of this paper is to propose a novel robust (against measurement noise) parameter identification method for a class of nonlinear fractional-order systems. In order to solve the parametric identification we carry out this problem to a state estimation problem, we introduce a Fractional Algebraic Identifiability (FAI) property which allows to represent the system parameters as a function of the inputs and outputs of the system, this parameter identification method provides an on-line identification process (while the system is operating), we also propose a fractional-order differentiator which allows to reduce the effect of measurement noise as well as to provide the estimation of a fractional-order derivative of the system output. Moreover, we use the Mittag–Leffler boundedness to demonstrate the convergence of this method, a different approach for this stability analysis method is given in this paper. Finally, we illustrate the accuracy and robustness of our proposed method by means of the parametric identification of two nonlinear fractional-order systems: a time-varying nonlinear fractional-order system and a nonlinear fractional-order mathematical model of a simple pendulum.

Uncertainty Observer-Based Control for a Class of Fractional- Order Non-Linear Systems with Non-Linear Control Inputs

This paper presents a novel control strategy based on an uncertainty estimator for a class of fractional-order nonlinear systems characterized by a polynomial input. The proposed strategy allows the system to be controlled without resorting to transformations or approximate linearization. This is accomplished by using a fractional-order sliding-mode observer, whose task is to estimate certain portions of the state of the nonlinear system of a non-integer order, thus allowing the control law to counteract these elements to steer the system towards a desired behavior. To validate the performance of the proposed strategy, it was implemented, both in simulation and experimentally, to regulate the temperature of the cold side of a thermoelectric module fed by a DC/DC electronic power converter of the step-down type, a system that is known to have a nonlinear polynomial-type control input.

Disturbance Observer-Based Event-Triggered Adaptive Command Filtered Backstepping Control for Fractional-Order Nonlinear Systems and Its Application

This paper considers the disturbance observer-based event-triggered adaptive fuzzy tracking control issue for a class of fractional-order nonlinear systems (FONSs) with quantized signals and unknown disturbances. To improve the disturbance rejection ability, a fractional-order nonlinear disturbance observer (FONDO) is designed to estimate the unknown composite disturbances. Furthermore, by combining an improved fractional-order command-filtered backstepping control technique and an event-triggered control mechanism, an event-triggered adaptive fuzzy quantized control scheme is established, which guarantees the desired tracking performance can be achieved even in the presence of network constraint. Finally, the validity and superiority of the theoretic results are verified by a fractional-order horizontal platform system.

Nonasymptotic Fractional Derivative Estimation of the Pseudo-State for a Class of Fractional-Order Partial Unknown Nonlinear Systems.

This work is devoted to the nonasymptotic and robust fractional derivative estimation of the pseudo-state for a class of fractional-order nonlinear systems with partial unknown terms in noisy environments. In particular, the estimation for the pseudo-state can be obtained by setting the fractional derivative's order to zero. For this purpose, the fractional derivative estimation of the pseudo-state is achieved by estimating both the initial values and the fractional derivatives of the output, thanks to the additive index law of fractional derivatives. The corresponding algorithms are established in terms of integrals by employing the classical and generalized modulating functions methods. Meanwhile, the unknown part is fitted via an innovative sliding window strategy. Moreover, error analysis in discrete noisy cases is discussed. Finally, two numerical examples are presented to verify the correctness of the theoretical results and the noise reduction efficiency.

Unified neuroadaptive fault-tolerant control of fractional-order systems with or without state constraints

The problem of controlling fractional-order nonlinear systems remains interesting and challenging due to the powerful and hereditary memory features of such systems. In this paper, we investigate the adaptive tracking control problem for a class of fractional-order nonlinear systems with or without state constraints. The tan-type barrier Lyapunov function (BLF) is adopted for the first time in fractional Lyapunov direct method. Such technique allows for the accommodation of the situations with and without state constraints in a unified framework. Moreover, neural network (NN) is utilized to reconstruct the sophisticated nonlinear functions arising from the fractional-order differentiation of the virtual controller, resulting in a neuroadaptive fault-tolerant control scheme. It is shown that, with the proposed unified control, all the closed-loop signals are semiglobally ultimately uniformly bounded whether state constraints are imposed or not. Both theoretical analysis and numerical simulation confirm the effectiveness of the developed approach.

Reduced-Order State Estimation for a Class of Nonlinear Fractional-Order Systems

Reduced-Order State Estimation for a Class of Nonlinear Fractional-Order Systems

Non-asymptotic and robust estimation for a class of nonlinear fractional-order systems

This paper aims to deal with the non-asymptotic and robust estimation problem for nonlinear fractional-order systems. The motivation starts with a physical system which develops into a group of system models including linear and nonlinear, commensurate and noncommensurate, integer-order and noninteger-order ones. Different from the existing modulating functions method, the proposed method breaks through the limitation of transforming the original system into the input–output differential equation, which makes the estimation for more types of systems possible. By introducing a new class of fractional order modulating functions, the state is exactly expressed by algebraic integral formulas without knowing the initial conditions of the studied system, despite the corrupting noises in the output measurement. Finally, it is illustrated in the simulation examples that the proposed estimator not only proves to be effective in the state estimation of noninteger-order systems, but also shows its advantages in that of integer-order systems compared to the high-gain observer.

Event-triggered adaptive fuzzy tracking control for a class of fractional-order uncertain nonlinear systems with external disturbance

In this paper, we are dedicated to addressing the event-triggered adaptive fuzzy tracking control problem for a class of fractional-order nonlinear systems (FONSs) with uncertainty and external disturbance. A novel condition is presented for estimating the fractional derivative of event-triggered strategy measurement error signal. To avoid the “expansion of complexity” problem caused by repeatedly solving the fractional derivatives of the virtual control signals in the backstepping method, an event-triggered-based adaptive fuzzy controller is constructed by taking advantage of the fractional dynamic surface control (DSC) scheme and fractional Lyapunov direct method. The designed controller can not only guarantee the tracking error finally converges to a small neighborhood around the origin, but also ensure all signals included in closed-loop FONSs are semiglobally uniformly ultimate bounded (SUUB) and the Zeno behavior is favorably avoided. In addition, the fuzzy logic systems (FLSs) tend to be applied in estimating the unknown nonlinear function, and an auxiliary function is also adopted to compensate for the approximation error of FLSs and the external disturbance. Finally, a numerical example indicates that the event-triggered adaptive fuzzy controller designed in this paper for FONSs is feasible and effective.

Dynamic Event-Triggered Control for a Class of Nonlinear Fractional-Order Systems

In this brief, we contribute to adaptive neural network (NN) backstepping control design for the nonlinear double-integrator fractional-order (FO) systems comprising unknown dynamics and disturbances by the event-triggered procedure. To this end, a new triggering rule with a dynamical variable is first introduced, including some available static triggering rules as its particular form. Then, by using the contradiction and some properties of the Mittag-Leffler function, it is proved for the first time that the applied dynamical variable is positive and bounded for the FO systems. Hence, the inter-event time between any two successive triggering moments can be enlarged than static event-triggered results. By integrating the backstepping approach into the NN, an adaptive NN controller is constructed, which introduces a new analysis viewpoint on the Zeno phenomenon avoidance for the FO controllers. Under this controller, the tracking error converges to a small compact set including the origin. Finally, an example is given to show the feasibility of the proposed method.

Observer-Based Adaptive Fuzzy Output Feedback Control for a Class of Fractional-Order Nonlinear Systems with Full-State Constraints

Observer-Based Adaptive Fuzzy Output Feedback Control for a Class of Fractional-Order Nonlinear Systems with Full-State Constraints

Adaptive quantized controller design for synchronization of uncertain fractional-order nonlinear systems satisfying incremental quadratic constraints

This paper proposes the adaptive quantized controller design for the synchronization of a class of fractional-order nonlinear systems satisfying incremental quadratic constraints governed by an incremental multiplier matrix. The incremental quadratic constraints can describe many commonly encountered nonlinearities in existing literature. The adaptive quantized controller are designed and formulated in terms of matrix inequalities to make the error system asymptotically stable. Meanwhile, the sufficient conditions can be obtained via solving linear matrix inequalities. Moreover, an algorithm is presented to illustrate the steps of designing adaptive quantized controllers. Finally, examples about fractional-order Lorenz chaotic system and fractional-order Bloch system are provided to illustrate the effectiveness of the designed controller.

Non-Fragile Fault-Tolerant Control Design for Fractional-Order Nonlinear Systems With Distributed Delays and Fractional Parametric Uncertainties

This work discuss the stabilization issue for a class of fractional-order nonlinear systems together with time delay, parametric uncertainties and actuator faults. Precisely, the considered system comprises of two delays namely distributed delay and time-varying delay. Moreover, the occurrence of the actuator faults and fractional parametric uncertainties may induce poor performance of the systems. To overcome these issue, a non-fragile fault-tolerant controller is designed which makes the system asymptotically stable with the specified mixed <inline-formula> <tex-math notation="LaTeX">$H_{\infty} $ </tex-math></inline-formula> and passive performance index. A fractional Razumikhin theorem is applied to handle the distributed delay term in the stabilization analysis. With the aid of suitable Lyapunov-Krasovskii functional, the sufficient conditions are established in terms of linear matrix inequalities together with Razumikhin stability theorem for getting the required results. By virtue of this, the controller gain matrix is obtained by solving the obtained LMIs and the graphical results are simulated using FOMCON toolbox. Later, the potency of the developed results are validated by virtue of three numerical examples including a rocket motor chamber.

A New Sliding Mode Observer Design for Fault Estimation in a Class of Nonlinear Fractional-Order Systems Based on the Super-Twisting Algorithm

A New Sliding Mode Observer Design for Fault Estimation in a Class of Nonlinear Fractional-Order Systems Based on the Super-Twisting Algorithm

A Proposed High-Gain Observer for a Class of Nonlinear Fractional-Order Systems

This paper proposes a high-gain observer for a class of nonlinear fractional-order systems. Indeed, this approach is based on Caputo derivative to solve the estimation problem for nonlinear systems. The proposed high-gain observer is used to estimate the unknown states of a nonlinear fractional system. The use of Lyapunov convergence functions to establish stability of system is detailed. The influence of different fractional orders on the estimation is presented. Ultimately, numerical simulation examples demonstrate the efficiency of the proposed approach.

Stability Analysis of a Class of Fractional-Order Nonlinear Systems Under Unknown Stochastic Disturbance and Actuator Saturation

This brief investigates the problem of mean-square exponential stability for a class of fractional-order nonlinear systems (FNSs) subject to unknown stochastic disturbance and actuator saturation. By means of theory of stochastic system and fractional calculus theory, and Cauchy-Schwartz inequality, mean-square exponential stability criteria are obtained. The constraint on the order of the system is relaxed. Meantime, the relevant state feedback controller is designed to overcome the influence of the stochastic disturbance and actuator saturation. Finally, effectiveness of the results is demonstrated via the fractional-order brushless DC motor(BLDCM) system.

Correction to: Stability analysis for a class of fractional-order nonlinear systems with time-varying delays

Correction to: Stability analysis for a class of fractional-order nonlinear systems with time-varying delays

Stability analysis for a class of fractional-order nonlinear systems with time-varying delays

This paper presents the stability analysis problem of fractional-order nonlinear systems with time-varying delay. After formulating the problem and selecting the nonlinear model as the system under study, stability analysis and expression of the sufficient conditions for fractional-order nonlinear systems with time-varying delay are obtained using two different methods. In these methods, sufficient conditions for stability of fractional-order nonlinear systems are found in the form of satisfying some inequalities based on norms of nonlinear functions in the system and in terms of linear matrix inequality according fractional-order and nonlinear functions. In each case, despite the presence of time-varying delay, the system stability is ensured by meeting the stability sufficient conditions in terms of an inequality of functions and system parameters. Finally, numerical examples are given to determine the effectiveness of the proposed theorem.