Fractional-order Observer Research Articles

STATE ESTIMATION FOR A CLASS OF FRACTIONAL-ORDER UNCERTAIN NONLINEAR SYSTEMS

The robust state estimation problem for a class of fractional-order uncertain non- linear systems is investigated in this paper. The class of fractional-order uncertain nonlinear systems is more general than the existing one in the literature. For the first time, a method based on a new fractional-order state observer is proposed to give robust state estimation of the fractional-order uncertain nonlinear systems. A new condition for the existence of the fractional-order observer in terms of an optimization problem is proposed. The effectiveness of the proposed method is demonstrated through two examples.

A novel approach to state estimation of HIV infection dynamics using fixed-time fractional order observer

This paper presents a novel approach to designing a fixed-time fractional order observer for estimating the states of the dynamic model of human immunodeficiency virus (HIV) infection. The proposed approach combines output injection terminal sliding mode and RBF neural network strategies to achieve a robust and efficient estimation of the states of the HIV model within a fixed time frame. The main contributions of this work are the introduction of an output injection observer that ensures the stability of the error system along with a novel nonlinear sliding surface that guarantees the fixed-time error convergence to the neighborhood of zero. Moreover, the closed-loop scheme of the observer design is proven to be bounded, and the fixed-time stability of the observer error is obtained using the fractional Lyapunov stability approach. Simulation results show that the proposed fixed-time fractional order observer design provides accurate and efficient estimation of the states of the HIV model.

Fractional order PI observer for a class of fractional order systems

Currently the study of fractional order systems has become of great research interest, in particular the state estimation stands out within the lines of studies for this type of systems. Different methodologies have been proposed in order to solve this problem, however most of the techniques involve complete information of the system. Thus, this work presents a methodology for state estimation in a class of fractional order systems based on a fractional order observer which is constructed through an algebraic technique. This observer presents some significant properties, for instance, only variables of interest are estimated, i.e., it is a non redundant observer, it does not need complete information of the system and the initial conditions for the observer can be freely assigned. The stability of this observer is analyzed with the global Mittag-Leffler boundedness approach. Finally, the effectiveness and accuracy of the proposed methodology is verified through state estimation of a fractional-order Duffing chaotic system.

Trajectory planning and prescribed performance tracking control based on fractional-order observers for unmanned helicopters with unmeasurable states

The trajectory planning and tracking control problem is studied in this paper based on prescribed performance method (PPM) for the small-scale unmanned autonomous helicopter (UAH) with wind-gust disturbances (WGDs) and unmeasurable states. For the purpose, the nonlinear model with flapping dynamics is established, and the transformation performance function is used to ensure that the errors of trajectory tracking satisfy the corresponding performance. A fractional-order observer is designed for unmeasurable states to estimate the flapping angles in actual flight, and the fractional-order extended state observer (ESO) is constructed to estimate the WGDs, respectively. On the basis of PPM and the designed observers, the fractional-order theory-based backstepping trajectory tracking control scheme is developed for the UAH system, and the three-dimensional trajectory is planned by the improved wolf pack algorithm. Then the stability of the entire system is proven through strict theoretical analysis. Finally, the simulation analysis on the UAH is presented to demonstrate the efficiency of the designed method.

Decentralized observer-based event-triggered control for an interconnected fractional-order system with stochastic Cyber-attacks

<abstract> <p>The problem of decentralized observer-based event-triggered stabilization for an interconnected fractional-order system subject to stochastic cyber-attacks is studied. To address this issue, the decentralized event-triggered mechanism is proposed for the interconnected fractional-order system, where the event-triggered schemes are designed based on the states of fractional-order observers, and the stochastic attacks are considered both on control inputs and observer outputs. By combining decentralized observers and decentralized event-triggered controllers, we aim to achieve decentralized control with reduced amplifying error and use local signals to improve overall system performance. By utilizing the diffusive representation of the fractional-order system, the interconnected fractional-order system is transformed into an equivalent integer-order one to simplify the analysis and control design. Employing the Lyapunov indirect approach, a sufficient condition is obtained to guarantee the stochastic asymptotically stability of the augmented system. Additionally, by the singular value decomposition technique, the approach of simultaneously computing the decentralized observer gains and controller gains is presented. Finally, a simulation example is provided to validate the theoretical findings.</p> </abstract>

State-of-charge estimation for lithium-ion batteries based on incommensurate fractional-order observer

This paper presents a new state-of-charge (SOC) estimation method for lithium-ion batteries (LIBs) based on incommensurate fractional-order (FO) observer. First, a FO RC equivalent circuit model is constructed to describe the charging and discharging dynamic characteristics of the LIB. An adaptive FO particle swarm optimization (AFOPSO) algorithm is employed to identify the parameters of the established FO model. The accuracy of the parameterized model is verified, showing its superiority over the available ones. Then, by using the comparison principle of FO systems with multi-order and stability of FO incommensurate linear systems with multi-order, a new order-dependent incommensurate FO observer is designed for battery SOC estimation. Finally, the validity and feasibility of the proposed SOC observer are examined with real time experimental data of LIBs.

An innovative modulating functions method for pseudo-state estimation of fractional order systems

In this paper, the objective is to estimate the pseudo-state of fractional order systems defined by the Caputo fractional derivative from discrete noisy output measurement. For this purpose, an innovative modulating functions method is proposed, which can provide non-asymptotic estimation within finite-time and is robust against corrupting noises. First, the proposed method is directly applied to the Brunovsky’s observable canonical form of the considered system. Then, the initial value of the pseudo-state is exactly expressed by an algebraic integral formula, based on which the pseudo-state is estimated. Second, the properties and construction of the required modulating functions are studied. Furthermore, error analysis is provided in discrete noise cases, which is useful for improving the estimation accuracy. In order to show the advantages of the proposed method, two numerical examples are given, where both rational order and irrational order dynamical systems are considered. After selecting the design parameters using the provided noise error bound, the pseudo-states of considered systems are estimated. The fractional order Luenberger-like observer and the fractional order H∞-like observer are also applied. Better than the applied fractional order observers, the proposed method can guarantee the convergence speed and robustness at the same time.

Tracking control of state constrained fractional order nonlinear systems

This article investigates adaptive output-feedback control problems for full-state constrained fractional order uncertain strict-feedback systems with unmeasured states and input saturation. By considering the structure of the systems, a fractional order observer is framed to estimate unmeasurable states. By using the backstepping procedure and barrier Lyapunov function, the adaptive controller with adaptation laws are proposed in each step. With the Lyapunov stability theory for fractional order systems, it proves all the states remain in their constraint bounds and the error system converges to a bounded set containing the origin. In the end, Two examples are presented to show the effectiveness of the designed control scheme.

Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System

The present study uses linear quadratic regulator (LQR) theory to control a vibratory system modeled by a fractional-order differential equation. First, as an example of such a vibratory system, a viscoelastically damped structure is selected. Second, a fractional-order LQR is designed for a system in which fractional-order differential terms are contained in the equation of motion. An iteration-based method for solving the algebraic Riccati equation is proposed in order to obtain the feedback gains for the fractional-order LQR. Third, a fractional-order state observer is constructed in order to estimate the states originating from the fractional-order derivative term. Fourth, numerical simulations are presented using a numerical calculation method corresponding to a fractional-order state equation. Finally, the numerical simulation results demonstrate that the fractional-order LQR control can suppress vibrations occurring in the vibratory system with viscoelastic damping.

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.

Terminal observer and disturbance observer for the class of fractional-order chaotic systems

In this paper, a terminal fractional-order observer and a terminal disturbance observer is proposed to estimate internal states and external disturbances of the class of fractional-order chaotic systems. The estimation of states within fixed time is achieved by employing a nonlinear feedback in terms of the observer error. The fixed convergence time is not relevant to the initial conditions and can be adjusted to any desired values by tuning the designable parameters. Finally, the numerical simulations are performed on fractional-order chaotic Liu, Chen, and Financial systems to validate the theoretical results. Moreover, some numerical simulations are provided to compare the obtained theoretical results with the other methods in the literature.

Fractional-order sliding mode control for deployment of tethered spacecraft system

This paper proposes a fractional-order integral sliding mode control with the order 0 &lt; ν &lt; 1 to stabilize the deployment of tethered spacecraft system with only tension regulation. The work in this paper is partially based on integer-order nonlinear sliding mode controller and improves its performance with fractional-order calculus. The proposed scheme makes use of integral sliding surface to obtain smaller convergence regions of state errors, and the fractional derivative is synthesized to enhance the flexibility of controller design by fining parameters for better dynamic and steady-state performance. Fractional-order observers help to eliminate external disturbances while the adaptive law is presented to remove the adverse effect in stability analyses, and fractional-order uniform ultimate boundedness is proved to guarantee the existence of the proposed sliding surface. According to theoretical analyses, the fractional order will indeed affect the dynamic and steady-state performance of control system, and the proposed method will be verified in numerical simulations compared with the nonlinear sliding mode counterpart.

Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems

Adaptive and non-adaptive minimal realization (MR) fractional order observers (FOO) for linear time-invariant systems (LTIS) of a possibly different derivation order (mixed order observers, MOO) are studied in this paper. Conditions on the convergence and robustness are provided using a general framework which allows observing systems defined with any type of fractional order derivative (FOD). A qualitative discussion is presented to show that the derivation orders of the observer structure and for the parameter adjustment are relevant degrees of freedom for performance optimization. A control problem is developed to illustrate the application of the proposed observers.

Fault tolerant system based on non-integers order observers: Application in a heat exchanger

This work presents a fault-tolerant (FT) scheme based on the application of non-integer order observers also called fractional observers, the case of study is a double pipe countercurrent heat exchanger (HE). The aim of the FT is to detect sensors faults as soon as possible, and to provide a healthy signal in order to replace the faulty sensor signal by the fractional observer estimation. To develop the FT scheme a bank of high gain fractional order observers (HGFOO) is proposed. The Riemann-Liouville (RL) fractional derivative definition is used to solve each fractional observer. Experimental measures from a HE were used to test the performance of the fractional observers and the control scheme. The results show the robustness of the proposed observers.

Robust disturbance rejection for a fractional-order system based on equivalent-input-disturbance approach

This paper presents a disturbance rejection method that is based on the equivalent-input-disturbance approach for uncertain fractional-order (FO) systems. An FO observer is used to reconstruct the plant states and estimate the disturbances. A disturbance estimator is incorporated in the construction of the FO control system to actively compensate for the entire disturbance. A robust stability condition for the control system and the parameters of the controller are derived using an indirect Lyapunov method. The presented method effectively rejects disturbances and handles modeling uncertainties without requiring prior knowledge of the disturbance. Comparison simulations on both numerical and practical examples demonstrate the effectiveness of the proposed method.

Fractional-order follower observer design for tracking consensus in second-order leader multi-agent systems: Periodic sampled-based event-triggered control

This paper investigates the tracking consensus problem for the second-order leader systems by designing fractional-order observer, where a periodic sampled-based data event-triggered control is employed. In order to track the position information of leader, observers for followers are designed by fractional-order system, where only the relative position information is available. Furthermore, in the process of observers design, a sampled-based event-triggered strategy is proposed so that observers use the event-triggered sampled-data, to reduce the overall load of the network. In our proposed event-triggered strategy, the event detection only works at every sampling time instant which determines whether the sampled-data should be discarded or used. Under this control strategy, the Zeno-behavior is absolutely excluded since the minimum of inter-event times is inherently lower bounded by one sampling period. It is found that the followers can track state of the leader if fractional-order observers are appropriately designed and relevant parameters are properly selected. By using the generalized Nyquist stability criterion, a necessary and sufficient condition for the observer tracking consensus of the second-order leader systems is derived. The results show that the real and imaginary parts of the eigenvalues of the augmented Laplacian matrix, and fractional-order α of observer play a vital role in reaching consensus.

Robust disturbance rejection for uncertain fractional-order systems

This paper describes a disturbance rejection scheme that adopts equivalent-input-disturbance (EID) approach for uncertain fractional-order (FO) systems. An EID estimator that contains an FO observer is designed to actively compensate for the disturbances and process modeling uncertainties without requiring their prior knowledge. Under the construction of the FO control system, a robust stability condition and the parameters of the controller are derived using a linear matrix inequality based method. Finally, numerical and practical examples are illustrated to demonstrate the validity and superiority of the method.

A General fractional-order heat transfer model for Photovoltaic/Thermal hybrid systems and its observer design

This paper presents the results of heat transfer process modelling for Photovoltaic/Thermal PV/T hybrid system in state-space. The heat transfer process in solid materials can be described by an integer-order partial differential equation. However, in Photovoltaic (PV/T) hybrid systems, which is characterized by heterogeneous media due to the multilayers that make up the system, it can be described by a fractional-order partial differential equation. The major objectives of this study were to establish a new mathematical model of the Photovoltaic/Thermal (PV/T) hybrid systems and their fractional-order observers in time domain. This model has been constructed in the context of a new consideration of heat conduction with a time fractional-order derivative. On the other hand, after obtained the general model of the system in state-space, the existence conditions of the fractional-order observer of such systems are given. Then, a necessary and sufficient condition for the asymptotic stability of the estimation error is given in a Linear Matrix Inequality (LMI) formulation.

Fractional fault-tolerant dynamical controller for a class of commensurate-order fractional systems

ABSTRACTA fault-tolerant control scheme is proposed for a class of commensurate-order fractional nonlinear systems that consists of two fractional-order observers (hybrid scheme). The diagnosis of the faults is performed by means of a model-free fractional proportional integral reduced-order observer that uses the fractional algebraic observability property. A fractional dynamical controller obtained in a natural way from the dynamics of a fractional high-gain observer is designed, which is constructed from a fractional generalised observability canonical form; the controller performs output tracking, thus eliminating the effects of the faults. A stability analysis on the overall system demonstrates that the origin is Mittag–Leffler stable. The proposed methodology is assessed by means of simulations on the fractional models of the Van der Pol oscillator and a DC motor.

Observer design for a class of nonlinear fractional-order systems with unknown input

A full order fractional-order observer is designed for a class of Lipschitz continuous-time nonlinear fractional-order systems with unknown input. Sufficient conditions of existence for the designed observer and stability of state estimation error system are developed by reconstructing state and using general quadratic Lyapunov function. By applying fractional-order extension of Lyapunov direct method, the stability of the fractional-order state estimation error system is analyzed. Due to the conditions involving a nonlinear matrix inequality, a new sufficient condition with linear matrix inequality (LMI) is reformulated, which makes the full order fractional-order observer implemented easily by using Matlab LMI toolbox. Examples are taken to show the effectiveness of the proposed approach by numerical simulations.