Uncertain Fractional-order Nonlinear Systems Research Articles

Adaptive asymptotic tracking control of uncertain fractional-order nonlinear systems with unknown control coefficients and actuator faults

For uncertain fractional-order nonlinear systems (UFONS) with unknown control coefficients and intermittent actuator faults, the asymptotic tracking control problem is investigated in this paper. Firstly, to weaken the influence of virtual control coefficients and intermittent actuator faults, a smooth fractional-order projection operator-based adaptive compensation mechanism is presented. Additionally, a fractional-order nonlinear filter is constructed to replace the fractional-order derivative of virtual control functions approximately, which not only avoids the issue of complexity explosion existed in backstepping control frame, but fully compensates the effects of boundary errors caused by the employed filter in spite of the unknown virtual control coefficient. By constructing a fractional Lyapunov function from the property of projection operator, it is proved that all signals in the closed-loop system are bounded, and the asymptotic tracking control object is achieved. Definitively, a simulation study is presented to verify the availability of the presented method.

Adaptive Fuzzy Fault-Tolerant Control of Uncertain Fractional-Order Nonlinear Systems with Sensor and Actuator Faults

In this work, an adaptive fuzzy backstepping fault-tolerant control (FTC) issue is tackled for uncertain fractional-order (FO) nonlinear systems with sensor and actuator faults. A fuzzy logic system is exploited to manage unknown nonlinearity. In addition, a novel FO nonlinear filter-based dynamic surface control (DSC) method is constructed, effectively avoiding the inherent complexity explosion problem in the backstepping recursive process, and in the light of the construction of auxiliary functions, compensating the coupling term introduced by faults. On account of certain assumptions, the stability criterion of the FO Lyapunov function is applied to guarantee the stability of the closed-loop system. Finally, the simulation example verifies the validity of the presented control strategy.

Event-triggered finite-time fuzzy control approach for fractional-order nonlinear chaotic systems with input delay

In this paper, a novel fuzzy event-triggered control approach with guaranteed performance of practical finite-time tracking convergence is constructed for uncertain fractional-order nonlinear chaotic systems in the presence of time-varying input delay. Firstly, fuzzy logic systems are employed to deal with immeasurable systematic information. Secondly, fractional-order command filters are incorporated to circumvent the problem of “complexity explosion” in backstepping controller construction. Meanwhile, a sequence of fractional-order finite-time compensation signals are introduced to counteract the adverse influence of time-varying input delay rapidly. Furthermore, a novel practical finite-time chaos control approach based on event-triggered strategy is proposed, which not only facilitates the decrease of sampling frequency and controller burden dynamically, but also ensures the convergence of all tracking errors towards a small region around the origin in finite time. Finally, through the simulation research, the transient response performance and the chaos-suppression ability of fractional-order chaotic systems stabilized via the suggested approach are rigorously testified.

Nonlinear Filter-Based Adaptive Output-Feedback Control for Uncertain Fractional-Order Nonlinear Systems with Unknown External Disturbance

This study is devoted to a nonlinear filter-based adaptive fuzzy output-feedback control scheme for uncertain fractional-order (FO) nonlinear systems with unknown external disturbance. Fuzzy logic systems (FLSs) are applied to estimate unknown nonlinear dynamics, and a new FO fuzzy state observer based on a nonlinear disturbance observer is established for simultaneously estimating the unmeasurable states and mixed disturbance. Then, with the aid of auxiliary functions, a novel FO nonlinear filter is given to approximately replace the virtual control functions, together with the corresponding fractional derivative, which not only erases the inherent complexity explosion problem under the framework of backstepping, but also completely compensates for the effects of the boundary errors induced by the constructed filters compared to the previous FO linear filter method. Under certain assumptions, and in line with the FO stability criterion, the stability of the controlled system is ensured. An FO Chua–Hartley simulation study is presented to verify the validity of the proposed method.

Event-Triggered Adaptive Fuzzy PI Control of Uncertain Fractional-Order Nonlinear Systems With Full-State Constraints

This article is devoted to the adaptive fuzzy proportional-integral (PI) control of uncertain fractional-order (FO) strict-feedback nonlinear systems subject to full-state constraints (FSCs) and external disturbances via an event-triggered (ET) control technique. First, fuzzy logic systems are introduced to approximate unknown nonlinear functions. Second, a novel adaptive FOPI controller with a simple structure and low calculation cost is developed, and it is able to determine its PI gains adaptively and automatically without involving manual operation or trial and error procedures, which is an improvement compared with the traditional PI control scheme. Third, a relative-threshold-based ET controller is constructed so that the communication load from the controller to the actuator can be significantly decreased, and there is no Zeno behavior. Then, a barrier Lyapunov function is introduced into the backstepping recursive process to avoid FSC violations. Moreover, all closed-loop signals are bounded, and the output is driven to follow the desired output through the FO Lyapunov stability analysis. In particular, the proposed approach does not require a priori knowledge of the FO derivative of the reference signal. Finally, the feasibility of the proposed scheme is verified by numerical simulations.

Adaptive neural backstepping control of nonlinear fractional-order systems with input quantization

This article addresses the tracking control problem of uncertain fractional-order nonlinear systems in the presence of input quantization and external disturbance. An adaptive backstepping scheme is proposed by combining with radial basis function (RBF) neural networks (NNs), fractional-order disturbance observer (FODO), and backstepping method. The RBF NNs are used to approximate the unknown nonlinearities of fractional-order systems. The FODO is designed to compensate for disturbance and uncertain parameters. The hysteresis quantizer is used to avoid chattering that possibly appears in actual application. The stability of the proposed controller is proved by fractional-order Lyapunov method. In addition, all the signals in the closed-loop system are bounded. The effectiveness of the proposed method is confirmed by the simulation results.

Command Filter-Based Adaptive Fuzzy Finite-Time Tracking Control for Uncertain Fractional-Order Nonlinear Systems

In this article, we focus on the issue of adaptive fuzzy finite-time tracking control for a class of uncertain fractional-order nonlinear systems with external disturbance. A new finite-time fractional-order command filtered implementation scheme is presented for the adaptive backstepping method. In the command filtered implementation approach, analytical computation of the fractional derivatives of the stabilizing functions is not necessary. Therefore, the controller and adaptive law of the systems are easier to derive and implement. Based on the proposed command filter, adaptive backstepping technique, and fractional Lyapunov’s direct method, a novel adaptive fuzzy finite-time controller is designed, which can guarantee that the tracking error converges to a small neighborhood of the origin in finite time, and other signals of the closed-loop systems are semiglobal uniform ultimate bounded. In the design process of the controller, fuzzy logic systems (FLSs) are employed to approximate the unknown nonlinear functions, and an auxiliary function is adopted to compensate for the unknown external disturbance and approximation errors of FLSs. Finally, a simulation example is given to show the effectiveness and availability of the proposed control strategy.

ADAPTIVE TRACKING CONTROL FOR FRACTIONAL-ORDER NONLINEAR UNCERTAIN SYSTEMS WITH STATE CONSTRAINTS VIA COMMAND-FILTERING AND DISTURBANCE OBSERVER

In this paper, the state-constrained adaptive control problem for a class of uncertain fractional-order nonlinear systems with unknown external disturbances is studied. Using the fuzzy approximation and backstepping control layout, an adaptive disturbance-observer-based control method is proposed, which can effectively estimate the unknown external disturbances through the designed disturbance observers. As we know, the standard backstepping control has inherent computational complexity. Therefore, a design approach of fractional-order command filter is introduced, which takes advantage of fractional-order command filter to pre-estimate the virtual input signal and its fractional derivative. In addition, barrier Lyapunov functions are exploited to cope with the fractional states-constrained control problem. The combination of barrier Lyapunov function technique and backstepping design not only ensures excellent tracking performance, but also enhances the robustness of the system. Furthermore, based on the fractional Lyapunov approach, the relevant stability analysis is established, which demonstrates that all states of the system remain within their constrained scope, and all signals of the closed-loop system are bounded. Finally, the effectiveness of the proposed scheme is verified by numerical simulation examples of RLC circuit and horizontal platform system.

Composite adaptive fuzzy backstepping control of uncertain fractional-order nonlinear systems with quantized input

Composite adaptive fuzzy backstepping control of uncertain fractional-order nonlinear systems with quantized input

Composite Learning Control of Uncertain Fractional-Order Nonlinear Systems with Actuator Faults Based on Command Filtering and Fuzzy Approximation

Composite Learning Control of Uncertain Fractional-Order Nonlinear Systems with Actuator Faults Based on Command Filtering and Fuzzy Approximation

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.

Robust adaptive backstepping control of uncertain fractional-order nonlinear systems with input time delay

In the present work, a fractional adaptive backstepping control approach is proposed for controlling a class of uncertain fractional-order nonlinear systems subjected to unknown external disturbance and input delay. In control design, by introducing fractional command-filter, the key issue of the common backstepping method called the “explosion of complexity” is addressed. Furthermore, the Szász–Mirakyan operator is used as a simple but efficient universal approximator to approximate uncertain or unknown terms and reduce the design complexity. In addition, to enhance the capability of the proposed controller in dealing with the approximation error and the unknown disturbance, an efficient robust control term is also incorporated into it. One advantage of the suggested control approach is that the input-delay fractional-order system is effectively treated by introducing a new variable as a non-delayed fractional-order system. The other advantage is that the number of tuning variables is considerably low; specifically, for an n-order fractional nonlinear system, only n adaptive parameters are adjusted online. This makes the implementation of the proposed approach simpler. The Lyapunov stability theorem is used to prove that all of the closed-loop system’s signals are semi-globally uniformly ultimately bounded. The feasibility of the suggested control approach is shown by comparing the simulation results obtained with the fuzzy system.

Compound FAT-Based Learning Control of Uncertain Fractional-Order Nonlinear Systems With Disturbance

This letter studies a function approximation technique (FAT)-based fractional-order (FO) backstepping compound learning control for uncertain FO strict-feedback nonlinear systems with unknown external disturbance. A FAT-based learning is used to approximate unknown dynamic terms. The proposed control algorithm takes into account the accuracy of FAT approximation by defining a prediction error obtained from a FO serial-parallel identifier (FOSPI). Furthermore, the FO command-filtered approach is adopted to reduce the complexity explosion of the backstepping-based design. New FO compound adaptation laws are constructed by integrating effective feedbacks derived from compensated tracking error and the accuracy of FAT learning. The stability of the overall system is analyzed by the Lyapunov stability concept. Simulations validate the theoretical results.

Neural network [formula omitted] adaptive control for a class of uncertain fractional order nonlinear systems

This research work presents an ℒ1 adaptive control scheme based on a fractional-order sliding surface and a Radial Basis Function (RBF) neural network, for a general class of uncertain fractional-order nonlinear systems. The structure of the proposed controller is composed of a predictor, a control law, an adaptive mechanism and an RBF neural network. The latter is designed as an estimator in the controller architecture to handle and approximate nonlinear uncertain dynamics. The estimation loop in the proposed Neural Network (NN) fractional-order ℒ1 adaptive controller is decoupled from the control loop thanks to the use of a filter in the input channel, which makes it possible to preserve the robustness while enhancing transient performances. Besides, the suggested controller guarantees the closed-loop system’s stability with bounded transient and tracking performances. Finally, the effectiveness and efficiency of the proposed controller are put on test using numerical simulations and comparative studies.

Passivity-based non-fragile control of a class of uncertain fractional-order nonlinear systems

This article investigates the issues of passivity and feedback passification of a class of fractional-order (FO) nonlinear systems with time-varying norm-bounded parameter uncertainties in both the state and output matrices. Firstly, new passivity conditions presented by linear matrix inequalities (LMIs) are established based on new notions of passivity of FO systems. Secondly, considering inaccuracies or uncertainties may occur during controller implementation, non-fragile state feedback passification controller is proposed when the states are available. The obtained results extend and improve some recent results. Finally, the applicability and effectiveness of the developed results are examined via numerical examples with simulation results.

Adaptive asymptotic tracking control of uncertain fractional-order nonlinear systems with unknown quantized input and control directions subject to actuator failures

This article addresses an adaptive backstepping control design for uncertain fractional-order nonlinear systems in the strict-feedback form subject to unknown input quantization, unknown state-dependent control directions, and unknown actuator failure. The system order can be commensurate or noncommensurate. The total number of failures is allowed to be infinite. The Nussbaum function is used to deal with the problem of unknown control directions. Compared with the existing results, the control gains can be functions of states and the knowledge of quantization parameters and characteristics of the actuator failure are unknown. By applying the backstepping control approach based on the frequency-distributed model, it is proved that all the closed-loop signals remain bounded and the output tracking error converges to the origin asymptotically. Finally, the effectiveness of the proposed controller is demonstrated by two simulation examples.

Adaptive sliding mode output tracking control based-FODOB for a class of uncertain fractional-order nonlinear time-delayed systems

An adaptive sliding mode control (ASMC) method, based on fractional-order disturbance-observer (FODOB), is presented for a class of fractional-order nonlinear time-delay systems (FONTDS) with uncertainties to solve the target output tracking problem. The external disturbances are estimated by FODOB, and the unknown internal perturbations of the system are adaptively estimated by sliding mode control (SMC). Furthermore, Gronwalls inequality approach is used to ensure that the output tracking error is uniformly bounded for FONTDS. Firstly, a fractional-order sliding mode control (FOSMC) based FODOB is proposed for a fractional-order linear time-delay system (FOLTDS). Secondly, combined with adaptive estimation, the ASMC of FONTDS is studied. Finally, a numerical example of FONTDS is used to verify the effectiveness of the proposed methods.

Adaptive Backstepping Hybrid Fuzzy Sliding Mode Control for Uncertain Fractional-Order Nonlinear Systems Based on Finite-Time Scheme

A fractional-order integral fuzzy sliding mode control scheme is proposed for a class of uncertain fractional-order nonlinear systems subject to uncertainties and external disturbances. First, in each step, a neuro-fuzzy network system is developed to approximate the uncertain nonlinear function existing in fractional-subsystem and a fractional sliding mode surface is presented. Second, based on the fractional Lyapunov stability theory and the finite-time stability theory, a fractional adaptive backstepping neuro-fuzzy sliding mode controller is designed to drive the state trajectories of fractional-order systems to the prescribed sliding mode surface. Meanwhile, the finite-time stability of the fractional-order closed-loop system is proved. At last, three numerical examples are given to illustrate the effectiveness of the proposed control method.

Adaptive Fuzzy Tracking Control Design for a Class of Uncertain Nonstrict-Feedback Fractional-Order Nonlinear SISO Systems.

In this article, a class of uncertain nonstrict-feedback fractional-order nonlinear single-input-single-output (SISO) systems is investigated. Fuzzy-logic systems (FLSs) are employed to approximate the unknown nonlinear functions and model the uncertain fractional-order nonlinear systems. For the states measurable case, an adaptive fuzzy state-feedback control scheme is developed under the framework of the backstepping technique. For the states unmeasurable case, an observer-based output-feedback control design is proposed by introducing a serial-parallel estimation model and using the dynamic surface control (DSC) technique. Under the drive of the reference signals, the semiglobally uniformly ultimate boundedness for all signals and the tracking errors converging to a small neighborhood of the origin are proved based on the Lyapunov function theory by choosing appropriate design parameters. Two examples with numerical simulations are presented to illustrate the availability of the proposed control approaches.

Composite Learning Adaptive Dynamic Surface Control of Fractional-Order Nonlinear Systems.

Adaptive dynamic surface control (ADSC) is effective for solving the complexity problem in adaptive backstepping control of integer-order nonlinear systems. This article focuses on the ADSC design for parametric uncertain fractional-order nonlinear systems (FONSs). In each backstepping step, the virtual controller is driven to pass through a fractional dynamic surface whose fractional-order derivative can be calculated easily. An ADSC law that ensure tracking error convergence is designed. The proposed ADSC requires a stringent condition called persistent excitation (PE) to achieve parameter convergence. To relax this limitation, a prediction error is defined by using online recorded data and instantaneous data, and a composite learning law is proposed to utilize both the prediction error and the tracking error. Then, a composite learning ADSC (CLADSC) method is developed to guarantee tracking error convergence and accurate parameter estimation under an interval excitation condition that is weaker than the PE one. Finally, an illustrative example is presented to show the performance of our methods.