Fractional Order Sliding Mode Observer Research Articles

Fractional-Order Sliding Mode Observer for Actuator Fault Estimation in a Quadrotor UAV

In this paper, we present the design of a fractional-order sliding mode observer (FO-SMO) for actuator fault estimation in a quadrotor unmanned aerial vehicle (QUAV) system. Actuator faults can significantly compromise the stability and performance of QUAV systems; therefore, early detection and compensation are crucial. Sliding mode observers (SMOs) have recently demonstrated their accuracy in estimating faults in QUAV systems under matched uncertainties. However, existing SMOs encounter difficulties associated with chattering and sensitivity to initial conditions and noise. These challenges significantly impact the precision of fault estimation and may even render fault estimation impossible depending on the magnitude of the fault. To address these challenges, we propose a new fractional-order SMO structure based on the Caputo derivative definition. To demonstrate the effectiveness of the proposed FO-SMO in overcoming the limitations associated with classical SMOs, we assess the robustness of the FO-SMO under three distinct scenarios. First, we examined its performance in estimating actuator faults under varying initial conditions. Second, we evaluated its ability to handle significant chattering phenomena during fault estimation. Finally, we analyzed its performance in fault estimation under noisy conditions. For comparison purposes, we assess the performance of both observers using the Normalized Root-Mean-Square Error (NRMSE) criterion. The results demonstrate that our approach enables more accurate actuator fault estimation, particularly in scenarios involving chattering phenomena and noise. In contrast, the performance of classical (non-fractional) SMO suffers significantly under these conditions. We concluded that our FO-SMO is more robust to initial conditions, chattering phenomena, and noise than the classical SMO.

Anti-disturbance speed control of permanent magnet synchronous motor based on fractional order sliding mode load observer

Based on the fractional order sliding mode load observer, this paper presents an anti-disturbance speed control method of permanent magnet semi-direct drive transmission system for overhead manned equipment. According to the Lagrange equation, the dynamic model of semi-direct drive transmission system is established. Furthermore, the mathematical model of the permanent magnet synchronous motor (PMSM) is established based on the coordinate transformation theory. Subsequently, the fractional order sliding mode observer (FOSMO), which combines nonsingular terminal sliding mode control and fractional order theory is designed to observe the load disturbance changes in the permanent magnet semi-direct drive transmission system. Then, the concept of “active damping” is used to design the speed loop PI controller, and combined with the designed FOSMO to construct the composite speed controller. The designed composite speed controller is further improved by the anti-saturation design and parameter optimization. The results demonstrate that the designed FOSMO can precisely estimate the load disturbance on the PMSM output shaft. Additionally, the designed composite speed controller can realize stable speed control in the case of complex load disturbance, which meets the anti-disturbance speed control requirements of permanent magnet semi-direct drive transmission system, and the robustness of the control system is further improved by adding the anti-saturation link.

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.

UMV control via FOSMO

In this work, the speed and direction angle of an unmanned vehicle (UMV) are controlled by using a sliding mode controller (SMC). However, no mechanical speed or position sensor is used to control the speed and direction angle of the UMV. The speed and direction angle of the UMV are estimated using the fractional-order sliding mode observer (FOSMO), wherein the real UMV motor currents and voltages are measured by sensors and the kinematic modal of the UMV is used. The angular speeds of the UMV are approximated with the help of the estimated and real UMV motor currents. Then, the estimated speed and direction angle of the UMV are approximated by estimated angular speed of the UMV and those produced by the FOSMO are used in the SMC, which controls the speed and direction angle of a UMV. Given that fractional-order differential equations and algorithms are used in FOSMO, a more precise control than that in other controllers that use full-order differential equations is achieved. Lyapunov stability theory is also used to establish stability conditions for the observer. Experimental results show that the proposed observer can be perfectly implemented without a mechanical or position sensor and robustly follows different complex reference trajectories.

Fractional order predictive sliding-mode control for a class of nonlinear input-delay systems: singular and non-singular approach

ABSTRACTNonlinear models of physical systems usually suffer from input delay and external disturbances. Moreover, when a delayed state is in the input signal gain, it can be non-singular or singular. So, designing a robust controller in a nonlinear system with input and state delay, suitable for non-singular and singular input signal gain, is imperative. The main contribution of our study is to design a new state feedback fractional order predictive sliding mode control (FOPSMC) procedure which not only guarantees the stability of a nonlinear system with known constant input and state delay but also controls the output signal to the desired value. Firstly, a predictor is designed for the system to achieve an input-delay-free one. Then, a state feedback FOPSMC is proposed based on a fractional order sliding signal for a nonlinear system with non-singular control gain. Also, a state feedback FOPSMC and a fractional order sliding mode observer (FOSMO) for the virtual disturbance are designed for singular control gain situation. It is proved analytically, through the Lyapunov stability criteria, that both control procedures can stabilise the system and can control the output signal to the desired value, effectively. Finally, the simulation results verify the effectiveness of the analytical achievements.

Estimation and Fault Diagnosis of Lithium-Ion Batteries: A Fractional-Order System Approach

This study investigates estimation and fault diagnosis of fractional-order Lithium-ion battery system. Two simple and common types of observers are designed to address the design of fault diagnosis and estimation for the fractional-order systems. Fractional-order Luenberger observers are employed to generate residuals which are then used to investigate the feasibility of model based fault detection and isolation. Once a fault is detected and isolated, a fractional-order sliding mode observer is constructed to provide an estimate of the isolated fault. The paper presents some theoretical results for designing stable observers and fault estimators. In particular, the notion of stability in the sense of Mittag-Leffler is first introduced to discuss the state estimation error dynamics. Overall, the design of the Luenberger observer as well as the sliding mode observer can accomplish fault detection, fault isolation, and estimation. The effectiveness of the proposed strategy on a three-cell battery string system is demonstrated.

State of charge estimation of lithium-ion batteries using fractional order sliding mode observer

This paper presents a state of charge (SOC) estimation method based on fractional order sliding mode observer (SMO) for lithium-ion batteries. A fractional order RC equivalent circuit model (FORCECM) is firstly constructed to describe the charging and discharging dynamic characteristics of the battery. Then, based on the differential equations of the FORCECM, fractional order SMOs for SOC, polarization voltage and terminal voltage estimation are designed. After that, convergence of the proposed observers is analyzed by Lyapunov’s stability theory method. The framework of the designed observer system is simple and easy to implement. The SMOs can overcome the uncertainties of parameters, modeling and measurement errors, and present good robustness. Simulation results show that the presented estimation method is effective, and the designed observers have good performance.

State estimation based on fractional order sliding mode observer method for a class of uncertain fractional-order nonlinear systems

State estimation of dynamic systems is quite significant in many research areas, such as state-based control and stabilization, state-based monitoring and fault detection. This paper is concerned with the problem of observer-based state estimation for a class of fractional-order (FO) nonlinear dynamic systems. The objective is to consider the state estimation of FO nonlinear system with the sliding mode control (SMC) technique, and conduct the asymptotic stability analysis for the estimate error dynamic system. Firstly, the considered FO nonlinear system models with constant and uncertain parameters are both presented. Fractional order sliding mode observer (FOSMO) structures are established for the FO nonlinear system models. Then, the asymptotic stability of the estimate error dynamic systems are analyzed via employing the Lyapunov stability analysis method for FO systems, and the sufficient conditions of asymptotic stability are derived. FOSMOs design for FO nonlinear systems of Caputo׳s and Riemann–Liouville׳s differential operators are both investigated, and the corresponding asymptotic stability sufficient conditions of the error dynamic systems are presented to insure the estimation accuracy of FOSMOs. Finally, multiple simulation examples are provided to demonstrate the effectiveness of the presented FOSMOs.

H ∞ and Sliding Mode Observers for Linear Time-Invariant Fractional-Order Dynamic Systems With Initial Memory Effect

The H∞ and sliding mode observers are important in integer-order dynamic systems. However, these observers are not well explored in the field of fractional-order dynamic systems. In this paper, the H∞ filter and the fractional-order sliding mode unknown input observer are developed to estimate state of the linear time-invariant fractional-order dynamic systems with consideration of proper initial memory effect. As the first result, the fractional-order H∞ filter is introduced, and it is shown that the gain from the noise to the estimation error is bounded in the sense of the H∞ norm. Based on the extended bounded real lemma, the H∞ filter design is formulated in a linear matrix inequality form, and it will be seen that numerical methods to solve convex optimization problems are feasible in fractional-order systems (FOSs). As the second result of this paper, not only state but also unknown input disturbance are estimated by fractional-order sliding-mode unknown input observer, simultaneously. In this paper, it is shown that the design and stability analysis of the two estimation techniques are not related with the initial history. Through two numerical examples, the performance of the fractional-order H∞ filter and the fractional-order sliding-mode observer is illustrated with consideration of the initialization functions.

Research on PMSM Control for Position Sensorless Based on Fractional PD Sliding Mode Observer

According to the phase delaying question of estimated rotor position angle in traditional sliding mode observer, using fractional order calculus theory and singular perturbation two-time-scale decomposition theory, making up the parameter designing rules of fractional order PD sliding mode observer, proposed position sensorless control strategy for permanent magnet synchronous motor based on fractional order PD sliding mode observer. On one side, taking fast time-varying subsystem of singular perturbation two-time-scale decomposition theory to establish fractional order sliding mode observer; on the other side, taking slow time-varying subsystem of singular perturbation two-time-scale decomposition theory to obtain parameter of fractional order sliding mode observer. The results show that the method could not only making estimated rotor position angle follow actual rotor angle, but also establishing a global compensating control strategy of estimated rotor position angle effectively.