Fractional Order Proportional Derivative Controller Research Articles

Robust Hybrid Fractional Order Proportional Derivative Sliding Mode Controller for Robot Manipulator Based on Extended Grey Wolf Optimizer

SUMMARYThis paper presents a novel robust hybrid fractional order proportional derivative sliding mode controller (HFOPDSMC) for 2-degree of freedom (2-DOF) robot manipulator based on extended grey wolf optimizer (EGWO). Sliding mode controller (SMC) is remarkably robust against the uncertainties and external disturbances and shows the valuable properties of accuracy. In this paper, a new fractional order sliding surface (FOSS) is defined. Integrating the fractional order proportional derivative controller (FOPDC) and a new sliding mode controller (FOSMC), a novel robust controller based on HFOPDSMC is proposed. The bounded model uncertainties are considered in the dynamics of the robot, and then the robustness of the controller is verified. The Lyapunov theory is utilized in order to show the stability of the proposed controller. In this paper, the EGWO is developed by adding the emphasis coefficients to the typical grey wolf optimizer (GWO). The GWO and EGWO, then, are applied to optimize the proposed control parameters which result in the optimized GWO-HFOPDSMC and EGWO-HFOPDSMC, respectively. The effectivenesses of the optimized controllers (GWO-HFOPDSMC and EGWO-HFOPDSMC) are completely verified by comparing the simulation results of the optimized controllers with the typical FOSMC and HFOPDSMC.

Robust stabilisation of rotary inverted pendulum using intelligently optimised nonlinear self-adaptive dual fractional-order PD controllers

ABSTRACTThis article presents an intelligently optimised self-tuning fractional-order control scheme to improve the attitude-stabilisation of an inverted pendulum. Primarily, the scheme employs two Fractional-order Proportional-Derivative (FPD) controllers acting concurrently on the system to minimise the deviations in its state-trajectories. Wherein, one FPD controller compensates the variations in pendulum-angle and its fractional-order derivative to vertically balance the pendulum, where as the other FPD controller acts as a position controller and regulates the variations in arm-angle and its fractional-order derivative. The integration of fractional calculus with conventional PD controllers optimises the reference-tracking performance of the control scheme by increasing its degrees-of-freedom and design flexibility. In order to further improve the system’s immunity against exogenous disturbances, the PD gains of each controller are dynamically adjusted after each sampling interval using piecewise nonlinear functions of their respective state-variations. The hyper-parameters of the nonlinear gain-adjustment functions as well as the fractional-number power of the derivative-operator of each controller are selected via Particle-Swarm-optimisation (PSO) algorithm. The proposed adaptive control scheme is tested on the QNET Rotary Inverted Pendulum setup via ‘hardware-in-the-loop’ experiments. The optimality and robustness of the proposed control scheme are validated by comparing its performance with PSO-based fixed-gain dual-PD and dual-FPD control schemes.

HVDC tie-link modeling for restructured AGC using a novel fractional order cascade controller

This paper deals with the Automatic Generation Control (AGC) strategy for multi-source interconnected power system in deregulated environment. Physical constraints like Generation Rate Constraints (GRC), Governor Dead Band (GDB) and Boiler Dynamics (BD) are incorporated in the system for the realistic approach. In this paper, an accurate model of HVDC tie-link is considered for the proposed AGC mechanism. Inertia Emulation based Control (INEC) technique is adopted to utilize the stored energy of HVDC tie-link of the interconnected power system. Distributed Generation (DG) system is also considered in the proposed AGC structure. The DG system comprises of Solar Photovoltaic (SPV), Wind Turbine System (WTS), Diesel Engine Generator (DEG), Aqua Electrolyzer (AE) and Fuel Cell (FC). Finally, a novel application of Volleyball Premier League (VPL) algorithm has been introduced to obtain the optimized gains of the proposed controller. The proposed controller includes the cascade structure of 2-Degree of Freedom Proportional-Integral controller and Fractional Order Proportional-Derivative controller with filter (2-DOF-PI-FOPDN). The robustness and effectiveness of VPL algorithm based proposed controller has been tested by comparing with the published literatures on same platform.

Stochastic averaging technique for SDOF strongly nonlinear systems with delayed feedback fractional-order PD controller

A stochastic averaging technique is proposed to study the randomly excited single-degree-of-freedom (SDOF) strongly nonlinear systems with delayed feedback fractional-order proportional-derivative (PD) controller. The delayed feedback fractional-order PD control force is approximated by an equivalent non-delay feedback control force combining with a quasi-linear elastic force and a quasi-linear damping force. The averaged Ito stochastic differential equation for amplitude of the equivalent nonlinear system is derived by the generalized harmonic functions. The analytical stationary probability density function (PDF) is obtained with solving the reduced Fokker-Planck-Kolmogorov (FPK) equation. Two examples of van der Pol oscillator and Rayleigh-Duffing oscillator are studied to illustrate the application and effectiveness of the proposed method. Numerical results display that the proposed method can yield to the high precision, and the time delay could ruin the control effectiveness, but also even amplifies the response of the system more than that of uncontrolled system. Furthermore, the study finds that the parameters of fractional-order α and time delay may cause the stochastic P-bifurcation. It is indicated that the delayed feedback fractional-order PD controller can offer a potentially effective tool for anti-control of stochastic bifurcation

A metaheuristic optimization approach to discretize the fractional order Laplacian operator without employing a discretization operator

A single-step procedure to obtain first and higher order discrete-time models in terms of infinite impulse response templates of the fractional order Laplacian operator sα, where 0<α<1, is proposed in this paper. The Moth Flame Optimization (MFO) algorithm based rational approximations are generated using a discretization operator-free method. Solution accuracy and the convergence performance of MFO are extensively compared with several other advanced evolutionary algorithms. Simulations justify the improved modelling accuracy of the proposed models over the recently published designs. The effects due to the finite word length leading to truncated filter coefficients are also considered, and the design stability robustness is demonstrated. The efficacy of the proposed model as a fractional order proportional-derivative controller is also validated.

A Comparative Study Between Cascaded FOPI–FOPD and IOPI–IOPD Controllers Applied to a Level Control Problem in a Coupled Tank System

This paper presents a novel liquid level control of a coupled two tank SISO system. Physical system is modeled from the open loop experimental response in two stages. A cascade control strategy is adopted. Fractional Order Proportional Integral (FOPI) and Fractional Order Proportional Derivative (FOPD) controllers are designed for the outer and inner loop, respectively, using frequency domain method. Similarly, conventional Integer Order Proportional Integral and Integer Order Proportional Derivative controllers are also designed to compare the performances. Reference signal tracking performance is tested with multiple step, ramp and sinusoidal reference signals. Ability of disturbance rejection is tested by adding more water into the tanks through a disturbance input channel. Experimental results reveal that the proposed FOPI–FOPD control scheme outperforms its integer order counterparts both in terms of reference signal tracking and disturbance rejection.

Structural vibration attenuation using a fractional order PD controller designed for a fractional order process

Structural vibration is a highly studied topic, especially in civil structures. Unwanted earth vibrations during seismic activity endanger life and often destroy buildings. In this paper, a Fractional Order Proportional Derivative controller is designed with the purpose of vibration mitigation in a three-story building. The experimental setup consists of a third floor building equipped with an active pendulum attached to the last floor. The controller is designed for the fractional order mathematical approximation of the structure by imposing frequency domain constraints such as gain crossover frequency, phase margin and robustness to gain variations. The validity of the controller is analyzed considering the simulated behavior of the compensated building to the El Centro earthquake and experimental disturbance rejection performance.

Stability region of a simplified multirotor motor–rotor model with time delay and fractional-order PD controller

ABSTRACTThe main aim of this paper is to present stability region analysis for a closed-loop system with the second-order model with a time delay and continuous-time fractional-order proportional-derivative (PD) controller. The model of the plant used in the paper approximates the dynamics of a simplified motor–rotor model of multirotor's propulsion system. The controller tuning method is based on Hermite–Biehler and Pontryagin theorems. The tracking performance is also analysed in the paper by observing the integral of absolute error and integral of squared error indices. The presented results are expected to be useful in future when comparing simulation with experimental results.

Fractional-order PD control at Hopf bifurcations in delayed fractional-order small-world networks

Bifurcation and control of fractional-order systems are still an outstanding problem. In this paper, a novel delayed fractional-order model of small-world networks is introduced and several topics related to the dynamics and control of such a network are investigated, such as the stability, Hopf bifurcations, and bifurcation control. The nonlinear interactive strength is chosen as the bifurcation parameter to analyze the impact of the interactive strength parameter on the dynamics of the fractional-order small-world network model. Firstly, the stability domain of the equilibrium is completely characterized with respect to network parameters, delays and orders, and some explicit conditions for the existence of Hopf bifurcations are established for the delayed fractional-order model. Then, a fractional-order Proportional-Derivative (PD) feedback controller is first put forward to successfully control the Hopf bifurcation which inherently happens due to the change of the interactive parameter. It is demonstrated that the onset of Hopf bifurcations can be delayed or advanced via the proposed fractional-order PD controller by setting proper control parameters. Meanwhile, the conditions of the stability and Hopf bifurcations are obtained for the controlled fractional-order small-world network model. Finally, illustrative examples are provided to justify the validity of the control strategy in controlling the Hopf bifurcation generated from the delayed fractional-order small-world network model.

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

Fractional Order PI‐PD Control of Liquid Level in Coupled Two Tank System and its Experimental Validation

AbstractThis paper presents a level control problem of a coupled two tank single input single output (SISO) system. A cascade control strategy is adopted having a fractional order proportional integral (FOPI) controller and fractional order proportional derivative (FOPD) controller in the outer and the inner loops, respectively. Cascaded integer order proportional integral (IOPI) and integer order proportional derivative (IOPD) controllers are also designed to compare the performances. A frequency domain approach is followed to design all the controllers. It is mathematically shown that the FOPI and FOPD controllers can achieve less steady state error and consume less energy than that of the IOPI and IOPD controllers while meeting the same phase margin and gain crossover frequency. All propositions are validated on an experimental setup.

Comparative analysis and exprimental results of advanced control strategies for vibration suppression in aircraft wings

The smart beam is widely used as a means of studying the dynamics and active vibration suppression possibilities in aircraft wings. The advantages obtained through this approach are numerous, among them being aircraft stability and manoeuvrability, turbulence immunity, passenger safety and reduced fatigue damage. The paper presents the tuning of two controllers: Linear Quadratic Regulator and Fractional Order Proportional Derivative controller. The active vibration control methods were tested on a smart beam, vibrations being mitigated through piezoelectric patches. The obtained experimental results are compared in terms of settling time and control effort, experimentally proving that both types of controllers can be successfully used to reduce oscillations. The analysis in this paper provides for a necessary premise regarding the tuning of a fractional order enhanced Linear Quadratic Regulator, by combining the advantages of both control strategies.

Extended state observer–based fractional order proportional–derivative controller for precision trajectory tracking control of a novel linear motor

An extended state observer–based fractional order proportional–derivative controller is presented to realize precision trajectory tracking control of a novel linear motor in this article. Structure, working principle, and mathematical model of the linear motor are demonstrated. Fractional order calculus, extended state observer, proportional–derivative control, and feedforward control are combined to accomplish high trajectory tracking accuracy. Comparative simulations and experimental results indicate that the proposed controller is robust to parameter variations and uncertain disturbances, and high tracking performance of the novel linear motor for different desired sinusoidal trajectories has been obtained.

Robust stability and stabilization of uncertain fractional-order descriptor nonlinear system

This paper considers the robust stability for uncertain fractional-order (FO) descriptor nonlinear systems. A key analysis technique is enabled by proposed a fundamental boundedness lemma, for the first time. It is used for rigorous robust stability analysis of FO systems, especially for Mittag-Leffler stability analysis of FO nonlinear systems. More importantly, how to obtain a more accurate bound is analyzed to reduce conservative. An FO proportional-derivative controller is utilized to normalize the descriptor system. Furthermore, a criterion for stability of the normalized FO nonlinear system is provided by utilizing linear matrix inequality (LMI). Finally, two illustrative examples show the effectiveness of the proposed stability notion.

Fractional-order [formula omitted] controller design

This paper introduces a new design method of fractional-order proportional–derivative (FOPD) and fractional-order proportional–integral–derivative (FOPID) controllers. A biquadratic approximation of a fractional-order differential operator is used to introduce a new structure of finite-order FOPID controllers. Using the new FOPD controllers, the controlled systems can achieve the desired phase margins without migrating the gain crossover frequency of the uncontrolled system. This may not be guaranteed when using FOPID controllers. The proposed FOPID controller has a smaller number of parameters to tune than its existing counterparts. A systematic design procedure is identified in terms of the desired phase and the gain margins of the controlled systems. The viability of the design methods is verified using a simple numerical example.

Stabilizing fractional-order PI and PD controllers: An integer-order implemented system approach

This paper proposes a scheme for computing the stable regions from which the parameters of fractional-order proportional–integral (PI) and proportional–derivative (PD) controllers can be selected. In particular, a previously investigated method for plotting the stability regions of PI λ and PD μ controllers is extended to their integer-order approximations. Next, the two stability regions are compared and their overlapping sections, referred to as the implementable stability region, are derived. In order for the final closed-loop system to be stable, the designer must choose the controller parameters from the overlapping section of the two stability regions. Simulation results for two example systems are provided to illustrate the effectiveness of the proposed method. The authors believe that the devised scheme can facilitate the design procedure for fractional-order PI and PD controllers.

Experimental study of fractional order proportional derivative controller synthesis for fractional order systems

In recent years, studies on real systems have revealed inherent fractional order dynamic behavior, and fractional order systems have attracted more and more attentions. It is intuitively true that these fractional order models require the corresponding fractional order controllers to achieve desired performance. In this paper, an experimental study of the fractional order proportional and derivative (FO-PD) controller systematic design is presented, to validate the control performance for the fractional order systems with generalized fractional capacitor membrane model. The performance of the designed FO-PD controller is compared with both the integer order and fractional order controllers which are designed based on the approximate integer order system. This comparison results are presented both in the simulation and the hardware-in-the-loop experiment.

Fractional order [proportional derivative] controller for a class of fractional order systems

Recently, fractional order systems (FOS) have attracted more and more attention in various fields. But the control design techniques available for the FOS suffer from the lack of direct systematic approaches. In this paper, we focus on a given type of simple model of FOS. A fractional order [proportional derivative] (FO-[PD]) controller is proposed for this class of FOS, and a practical and systematic tuning procedure has been developed for the proposed FO-[PD] controller synthesis. The fairness issue in comparing with other controllers such as the traditional integer order PID (IO-PID) controller and the fractional order proportional derivative (FO-PD) controller has been addressed under the same number of design parameters and the same specifications. Fair comparisons of the three controllers (i.e., IO-PID, FO-PD and FO-[PD]) via the simulation tests illustrate that, the IO-PID controller designed may not always be stabilizing to achieve flat-phase specification while both FO-PD and FO-[PD] controllers designed are always stabilizing. Furthermore, the proposed FO-[PD] controller outperforms FO-PD controller for the class of fractional order systems.