Uncertain Fractional-order Systems Research Articles

Robust H∞ adaptive model following control for fractional-order systems with polytopic and bounded uncertainties subject to input saturation

This paper investigates a type of robust observer-based tracking control for the well-known inverted pendulum as an uncertain fractional-order system with two-norm bounded uncertainties subject to input saturation along with external disturbance that focuses on the case of a fractional-order α such that [Formula: see text]. The main goal is to design a robust model following tracking control servomechanism by the side of stability analysis. This paper addresses the well-known quadratic Lyapunov function in addition to the Gronwall–Bellman lemma and the sector condition of the saturation function stability synthesis. This paper proposes an outstanding strategy to achieve the best model following robust observer in the cast of output-feedback control subject to input saturation in the framework of an uncertain fractional-order system based on the linear matrix inequality (LMI) solution. The LMI procedure can be used to achieve static output-feedback tracking control and observer gain. Several valuable theorems support this approach, and various simulation scenarios are available to showcase its effectiveness.

Event-triggered $$H_{\infty }$$ controller design for uncertain fractional-order systems with time-varying delays

Event-triggered $$H_{\infty }$$ controller design for uncertain fractional-order systems with time-varying delays

Vibration control of giant electrorheological damper combining nonlinear fractional-order controller and extended state observer

As a typical smart material, giant electrorheological fluid (GERF) has a greater yield stress than electrorheological fluid for vibration isolation. However, as rheological material, the modeling precision is severely degraded by its innate rate-dependent hysteretic nonlinearity and uncertainty. In this paper, a novel control method is proposed, which requires little information about the damper based on GERF. The proposed method combines nonlinear fractional-order control (FC) and extended state observer (ESO) by constructing damper as a second-order disturbance-based structure to handle hysteretic nonlinearities, dynamic uncertainties and unknown disturbances. In contrast to the prevalent model-free control (MFC) that neglects hysteresis nonlinearity, the proposed control algorithm considers it as a general disturbance and eliminates it. In addition, compared with linear FC with complex fractional-order ESO, where the order needs to be known in advance, nonlinear FC has improved robustness for uncertain fractional-order systems only via pure ESO. Simulations and experimental results demonstrate that the nonlinear controller outperforms the linear counterpart and the proposed method exhibits superior control performance compared to the existing MFC, with an improvement of 26.9%.

Composite adaptive fuzzy bipartite consensus of fractional-order multiagent systems with a switched event-triggered mechanism

This paper focuses on online recorded-data-based composite adaptive fuzzy bipartite consensus control for uncertain fractional-order multiagent systems with interconnected terms and external disturbances by employing a switched-threshold-based event-triggered mechanism (ETM) under the backstepping structure. Fuzzy logic system is used as a universal function approximation to deal with function uncertainties that are not prone to model in the system. A new composite learning adaptive parameter design scheme that synthesizes both prediction error and tracking error is developed to enhance the tracking performance, where the prediction error is raised from the utilization of online recorded data and instantaneous data. A unique switched-threshold-based ETM is introduced, in which the information transmission between the sensor and the controller is imposed on one of the individuals. One merit of this work consists in that it can automatically and rapidly switch and adjust between the fixed threshold and relative threshold ETM according to the amplitude of input signals to balance the network resources and impede the occurrence of pulse phenomenon. In addition, it is theoretically proven that the proposed scheme can ensure that all internal signals of the closed-loop system are bounded and achieve local bipartite consistent errors through the fractional Lyapunov stability criterion. Finally, a numerical example is provided to confirm the feasibility of the proposed approach.

Study on Stability for Interconnected Uncertain Fractional-Order Systems Based on Vector-Bounded Technique

Study on Stability for Interconnected Uncertain Fractional-Order Systems Based on Vector-Bounded Technique

A composite fractional order active disturbance rejection control method for a class of uncertain systems with multiple disturbances

SummaryThe existing active disturbance rejection control (ADRC) method may not provide sufficient disturbance rejection to multiple mismatched disturbances for the fractional order systems. In this paper, a composite disturbance rejection approach is developed for a class of fractional order uncertain systems, by synthesizing the fractional order ADRC (FOADRC) approach and a disturbance observer (DO)‐based compensation scheme. Taking advantage of more disturbance information and a filter structure, an improved DO is developed to achieve precise estimation of disturbances in the presence of sensor noises. In addition, a state transformation is developed to convert the system into a simple integral chain model with only matched disturbances. Then a composite control law is designed to compensate the disturbances and provide satisfying dynamic performance. The efficiency of the proposed method is demonstrated by a numerical simulation and an actual servo control simulation, as well as the comparison with two kinds of the existing ADRC methods and the commonly used integral sliding mode control (I‐SMC) method.

Tracking control for a class of fractional order uncertain systems with time-delay based on composite nonlinear feedback control

<abstract><p>In this paper, we dealt with the tracking control problem of a class of fractional-order uncertain systems with time delays. In order to handle the effects brought by the uncertainties, external disturbances, time-delay terms, and to overcome the obstacles caused by inputs saturation, the tracking controller, which consisted of linear control law, nonlinear law, and robust control law proposed in this paper, was designed by combining the composite nonlinear feedback control method and the properties of fractional order operators. Furthermore, the validation of this tracking controller was proved.</p></abstract>

Anti-disturbance observer-based proportional-retarded control design for polytopic uncertain fractional-order systems

This study deals with the stabilisation problem of fractional-order time-delay systems with polytopic uncertainties and multiple disturbances with the use of Lyapunov stability theory. Notably, the multiple disturbances comprising both the known and unknown signals, are respectively characterised by the norm-bounded and exogenous system which signifies the harmonic signals with modelling perturbations. The control protocol is configured by the integration of proportional-retarded controller with gain perturbations and the output of an anti-disturbance observer. Precisely, the disturbance observer is taken into account for attenuating the influence of disturbance signals. Moreover, the implementation of a time-delay term in the feedback loop replicates the dynamic features of a derivative action that relies only on position measurements and its time differencing capabilities make it to be minimally sensitive against noise and provide a smoothing effect. By virtue of these circumstances, the asymptotic stability of the considered system is assured. With the aid of a suitable Lyapunov function, a set of robust order-dependent sufficient conditions is derived and then solved through the MATLAB LMI toolbox. As a result, the controller gain matrices can be computed and two numerical examples, including an electrical circuit model with simulation results are presented to certify the inherent potential of the theoretical outcomes.

Robust stability of fractional-order systems with mixed uncertainties: The [formula omitted] case

This paper addresses the problems of robust stability of fractional-order systems with mixed uncertainties, including multi-parameter uncertainties and norm-bounded uncertainties. The problems are relevant because on the one hand, uncertainties are common in real systems and the uncertainties in different components of systems may be of different types, and on the other hand, the non-convex and decoupling problems for fractional-order systems with α∈(0,1) and mixed uncertainties need to be thoroughly resolved. The core approaches utilized in this paper include the robust nonsingularity analysis of uncertain matrices, error-free determinant transformation and μ-analysis methods. Based on the above approaches, necessary and sufficient conditions for the robust stability of fractional-order systems with mixed uncertainties are proposed. Then, by estimating the spectral radius of the uncertain matrices, novel robust stability conditions for such uncertain fractional-order systems are obtained. After that, the robust stability bounds for such uncertain fractional-order systems are established, based on which an algorithm for solving the robust stability bounds is provided by utilizing the traversal method in the finite frequency interval. The derived results in this paper are more universal and less conservative than the existing works. Finally, two numerical examples including a fractional-order electrical circuit example are provided to demonstrate the applicability and effectiveness of the developed methods.

The stabilization of uncertain dynamic systems involving the generalized Riemann-Liouville fractional derivative via linear state feedback control

The main objective of this paper is to investigate the stabilization problem of uncertain fractional-order dynamic systems (UFDSs) that involve the generalized Riemann-Liouville fractional derivative using a linear feedback controller. We establish several sufficient conditions for achieving Mittag-Leffler stabilization and asymptotic stabilization of UFDSs. The primary methodology employed in our study is Lyapunov's direct method (LDM), along with other inequality techniques, which have demonstrated significant efficacy in studying the stability theory of nonlinear dynamic systems. To demonstrate the effectiveness of the proposed approach, examples and corresponding simulations are provided.

Discrete-time event-triggered finite-time non-fragile control of uncertain fractional-order singular systems

Discrete-time event-triggered finite-time non-fragile control of uncertain fractional-order singular systems

Convergence analysis for iterative learning control of fractional-order nonlinear differential inclusion system

This paper considers a class of finite-time tracking problems of fractional-order uncertain system governed by nonlinear differential inclusion. We introduce a Lipschitz continuous condition into the set-valued mapping described by the closure of the convex hull of a set, and design PDγ-type and PIαDγ-type update laws with initial learning mechanisms in open loop and closed loop, respectively. Then, the Steiner-type selector and mathematical analysis tools are used to complete the convergence analysis. Finally, numerical examples are used to verify the validity of the theoretical results, and the tracking performance of system is compared for different values of γ,α and learning laws.

Fixed-Time Sliding Mode Synchronization of Uncertain Fractional-Order Hyperchaotic Systems by Using a Novel Non-Singleton-Interval Type-2 Probabilistic Fuzzy Neural Network

In this study, we proposed a sliding mode control method based on fixed-time sliding mode surface for the synchronization of uncertain fractional-order hyperchaotic systems. In addition, we proposed a novel self-evolving non-singleton-interval type-2 probabilistic fuzzy neural network (SENSIT2PFNN) to estimate the uncertain dynamics of the system. Moreover, an adaptive compensator was designed to eliminate the influences of random uncertainty and fuzzy uncertainty, thereby yielding an asymptotically stable controlled system. Furthermore, an adaptive law was introduced to optimize the consequence parameters of SENSIT2PFNN. The membership layer and rule base of SENSIT2PFNN were optimized using the self-evolving algorithm and whale optimization algorithm, respectively. The simulation results verified the effectiveness of the proposed methods for the synchronization of uncertain fractional-order hyperchaotic systems.

Optimal Robust Filter of Uncertain Fractional Order Systems: A Penalized Deterministic Approach

Developing accurate and optimum state estimation methods for fractional order systems is highly relevant since it provides vital information related to memory effects. The optimum estimation of these systems can be guaranteed using the Kalman filter (KF) when all parameter matrices are not subject to uncertainties. Nevertheless, this fundamental principle of the filter is violated, and its performance can be degraded when the model is uncertain. In this light and to limit this deterioration, the present study introduces an optimal solution for filtering uncertain fractional order systems, which operates as follows. First, using the robust regularized least-squares (RLSs) problem combined with penalty functions, a robust penalty game approach is proposed. Then, in an independent framework of any auxiliary parameters, unified recursive Riccati equation and optimal robust filter are derived subject to norm-bounded uncertainties of parameter matrices. To accomplish this step, the stability and convergence analysis of the filter are illustrated based on the singular theory conception.

Static and Dynamic Output Feedback Control for Polytopic Uncertain Fractional Order Systems with 0 < μ < 1

Static and Dynamic Output Feedback Control for Polytopic Uncertain Fractional Order Systems with 0 < μ < 1

LMI-Based Delayed Output Feedback Controller Design for a Class of Fractional-Order Neutral-Type Delay Systems Using Guaranteed Cost Control Approach

In this research work, we deal with the stabilization of uncertain fractional-order neutral systems with delayed input. To tackle this problem, the guaranteed cost control method is considered. The purpose is to design a proportional–differential output feedback controller to obtain a satisfactory performance. The stability of the overall system is described in terms of matrix inequalities, and the corresponding analysis is performed in the perspective of Lyapunov’s theory. Two application examples verify the analytic findings.

Enhanced fractional order sliding mode control for a class of fractional order uncertain systems with multiple mismatched disturbances

To improve the control performance of the fractional order uncertain systems with multiple mismatched disturbances, an enhanced fractional order sliding mode control (FOSMC) method is developed in this paper. The multiple disturbances and uncertainties are estimated by the finite-time disturbance observers (FTDO) and a fractional order extended state observer (FOESO), respectively. A fractional order switching law is designed to provide a fast convergence mode for the system states. Then a novel FOSMC law is developed by incorporating the feedforward compensation, the fractional order switching law, and the auxiliary state for input saturation. The proposed method is applied to numerical examples and to a motor speed control problem. The effectiveness of the proposed method is demonstrated by the performance comparisons with some existing control methods.

Global Optimization in Robust Fractional Control of Uncertain Fractional Order Systems: A Thermal Application Using the STM32 Microcontroller

In this paper, we suggest an improvement to our previously undertaken approach. Briefly, this approach consisted of applying the robust fractional predictive control (RFPC) for a class of constrained fractional systems implementing the min–max optimization technique. The RFPC controller requires resolution of a non-convex min–max optimization problem. The resolution of this problem, however, can only conduce to local solutions. The reason is simple: the objective function to be optimized is non-convex due to the presence of uncertainties. In the present work, we propose a global optimization-based RFPC controller for an uncertain fractional order system. A determinist global optimization method, namely, generalized geometric programming (GGP), is proposed to solve this problem for the uncertain fractional order system. The GGP method consists of converting a non-convex problem into a convex one via the application of variable changes. The technique of the convexification of this method is applied in line with the objective function to be optimized. Consequently, we obtained a new convex criterion and a convex problem. From an experimental point of view, we applied the proposed RFPC to a real thermal system using an STM32 microcontroller in order to control our thermal system.

Formula omitted] adaptive controller design for a class of fractional order uncertain systems

ℒ1 adaptive control method is a new and effective technique to perform a robust tracking response with fast adaptation, its applications were generalized for both integer order affine and non-affine nonlinear systems. However, ℒ1 adaptive control scheme has not been yet investigated to control fractional order systems. In this work, an ℒ1 adaptive controller is designed based on a fractional order sliding surface to control a class of fractional order systems with matched uncertainties and external disturbances. The developed controller is able to achieve closed-loop stability with uniform performance bounds for system signals. Such that, increasing the adaptation gain can dramatically enhance the performances for both transient and steady state responses. Finally, the validity and the effectiveness of the controller is tested using a numerical simulation.

Event-triggered robust tracking control for fractional-order uncertain systems

In this paper, based on event-triggered (ET) mechanisms, the problem of output tracking for a class of fractional-order uncertain systems with order [Formula: see text] is investigated. Owing to the difficulties of measuring system full-state in practice, output tracking error is firstly used to construct an ET condition, which eventually decides whether the current signal should be transmitted. Then, by utilizing the designed error-based feedback controller, some sufficient conditions are presented to ensure that the controlled system output asymptotically tracks the reference signal. Finally, numerical simulations are performed to validate the effectiveness of the theoretical formulation.