Fractional Order PD Controller Research Articles

Spotted Hyena Optimizer enhances the performance of Fractional-Order PD controller for Tri-copter drone

AbstractIn this study, nonlinear control design is presented for trajectory tracking of Tricopter system. A Fractional Order Proportional Derivative (FOPD) controller has been developed. The performance of controlled Tri-copter system can be enhanced by suggesting modern optimization technique to optimally tune the design parameters of FOPD controller. The Spotted Hyena Optimizer (SHO) is proposed as an optimization method for optimal tuning of FOPD's parameters. To verify the performance of controlled Tricopter system based on optimal SHO-based FOPD controller, computer simulation is implemented via MATLAB codes. Moreover, a comparison study between SHO and Particle Swarm Optimization (PSO) has been made in terms of robustness and transient behavior characteristics of FOPD controller.

Fractional order PD control of the Hopf bifurcation of HBV viral systems with multiple time delays

This paper will explore a fractional-order hepatitis B virus (HBV) infection model that takes into account cell-cell and virus-cell transmission and multiple treatment modalities. The desired control strategy is realized by means of a fractional order PD controller. Firstly, we calculated the basic regeneration number and equilibrium point of the HBV model. Afterwards, for the uncontrolled HBV virus system, the adequate conditions for both stability and Hopf bifurcation are systematically investigated via choosing the appropriate time delay as a parameter for bifurcation. Subsequently, under fractional order PD controller, the effect of a proposed controller on system stability and Hopf bifurcation is studied. The desired dynamic characteristics can be obtained afterwards. Finally, numerical simulations show that all three treatments significantly reduce R0. The onset of oscillations can be delayed by decreasing the order of the fractional order. There are three control pathways for fractional order PD control, and the generation of bifurcation can also be delayed by changing the gain parameter. Using the above methods, the diffusion of HBV virus particles in the body can be effectively controlled. The conclusions drawn in this paper are extremely novel and have potential theoretical value for the future treatment of hepatitis B illness.

Fractional Control of a Class of Underdamped Fractional Systems with Time Delay—Application to a Teleoperated Robot with a Flexible Link

This work addresses the robust control of processes of the form G(s)=K·e−τ·s/(1+T·sλ) with 1<λ≤2. A new method for tuning fractional-order PI and PD controllers is developed. The stability is assessed based on the frequency domain tuning of the regulators to control such delayed fractional-order underdamped processes. In order to analyze the closed-loop stability and robustness, the new concept of Robust High-Frequency Condition is introduced. The analysis based on that demonstrates that each controller has a different region of feasible frequency specifications, and, in all cases, they depend on their fractional integral or derivative actions. Finally, an application example, the position control of a teleoperated manipulator with a flexible link, is presented. Simulations and experiments illustrate that the region of feasible frequency specifications defined at low and high frequencies allows us to obtain robust controllers that fulfill frequency requirements.

Model references frequency-domain design based fractional order PD controller for quadrotor control: attitude, altitude, and position studies

This article proposes a control strategy based on a Fractional Order Proportional Derivative (FOPD) controller to enhance the performance and robustness of quadrotors operating in trajectory tracking mode. The tuning method proposed for the FOPD controller is based on an ideal Bode transfer function in a closed loop as a fractional reference model. The latter represents the quadrotor desired closed-loop response, and its specification is the key design decision. In this study, we first appropriately set model reference parameters to realistic desired time response characteristics (i.e. overshoot and time constant). Then, the parameters of the FOPD controller are computed using the direct synthesis design approach. The dynamic model of the quadrotor is developed using Lagrange formulation. The model is used to test the performance of the proposed FOPD-based control strategy for the attitude, position, and altitude of the flying robot. The results show that the proposed controller achieved accurately set points tracking with transient responses (i.e. overshoot and time constant) that satisfactorily meet those assigned for their respective reference models. These results validate the effectiveness of our proposed control strategy. Moreover, our comparative studies of the proposed FOPD controller with the conventional PD controller show that the former satisfies better-desired design requirements.

Bifurcation control of a fractional-order PD control strategy for a delayed fractional-order prey–predator system

Bifurcation control of a fractional-order PD control strategy for a delayed fractional-order prey–predator system

Fractional order PD controller design for third order plants ıncluding time delay

Fractional order PD controller design for third order plants ıncluding time delay

Fractional-Order PD Attitude Control for a Type of Spacecraft with Flexible Appendages

As large-sized spacecraft have been developed, they have been equipped with flexible appendages, such as solar cell plates and mechanical flexible arms. The attitude control of spacecraft with flexible appendages has become more complex, with higher requirements. In this paper, a fractional-order PD attitude control method for a type of spacecraft with flexible appendages is presented. Firstly, a lumped parameter model of a spacecraft with flexible appendages is constructed, which provides the transfer function of the attitude angle and external moment. Then, a design method for the fractional-order PD controller for the attitude control of a spacecraft with flexible appendages is provided. Based on the designed steps, a numerical example is provided to compare the control performances between the fractional-order and integer-order PD controllers. Finally, the obtained numerical results are presented to verify the effectiveness of the proposed control method.

Fractional Control of a Lightweight Single Link Flexible Robot Robust to Strain Gauge Sensor Disturbances and Payload Changes

In this paper, a method to control one degree of freedom lightweight flexible manipulators is investigated. These robots have a single low-frequency and high amplitude vibration mode. They hold actuators with high friction, and sensors which are often strain gauges with offset and high-frequency noise. These problems reduce the motion’s performance and the precision of the robot tip positioning. Moreover, since the carried payload changes in the different tasks, that vibration frequency also changes producing underdamped or even unstable time responses of the closed-loop control system. The actuator friction effect is removed by using a robust two degrees of freedom PID control system which feeds back the actuator position. This is called the inner loop. After, an outer loop is closed that removes the link vibrations and is designed based on the combination of the singular perturbation theory and the input-state linearization technique. A new controller is proposed for this outer loop that: (1) removes the strain gauge offset effects, (2) reduces the risk of saturating the actuator due to the high-frequency noise of strain gauges and (3) achieves high robustness to a change in the payload mass. This last feature prompted us to use a fractional-order PD controller. A procedure for tuning this controller is also proposed. Simulated and experimental results are presented that show that its performance overcomes those of PD controllers, which are the controllers usually employed in the input-state linearization of second-order systems.

Modelling and Simulation of Nonlinear Jump Phenomena of a Non-ideal Rotor Involving Fractional Order PD Controller

Rotating machinery with high speed powered by industrial motors frequently suffers from instability by exhibiting non-linear jump phenomena, formally known as Sommerfeld effect. The drives whose excitation is a function of the system responses, referred to as non-ideal. The system dynamics of such systems exhibit a couple of complex and interesting features when the input power exceeds a critical value. The present research suggests a novel approach to study the efficacy of active magnetic bearing with fractional PD controller to suppress the instability caused by the Sommerfeld effect. The steady-state results obtained by solving the system characteristic equation numerically is compared with the transient analysis. Finally, root locus method is introduced to obtain the bifurcation points at which this kind of instability completely disappears.

Fractional Order PD Control of Friction-Induced Vibrations in a Continuous System

Fractional-order PD (PD$^{\lambda })$ control of friction-induced vibrations in a beam-mass model is analysed in this paper. Mathematical modelling of the system with a piezo-patch actuator bonded to the beam surface is presented. Linear stability analysis is performed to determine the stability boundary corresponding to the Hopf bifurcation points. The nature of bifurcation is found to be supercritical from the non-linear analysis by the method of averaging. The role of fractional-order on the effectiveness of the controller in quenching friction-induced vibrations is thoroughly investigated. The efficacy of the controller with varying size and locations of the piezo-patch is also studied.

Stochastic Hopf Bifurcation of MDOF Quasi-Integrable Hamiltonian Systems with Delayed Feedback Fractional-Order PD Controller

Hopf bifurcation, as the most representative dynamic bifurcation, is closely related to the stability of many engineering structures. In this work, the stochastic Hopf bifurcation (SHB) of a controlled quasi-integrable Hamiltonian system (H.S.) of multi-degree-of-freedom (MDOF) is investigated, where the system is subjected to wide-band noise and controlled by a Fractional-order Proportional-Derivative (FOPD) controller with time delay. By decoupling FOPD control force and simplifying it without time delay, the averaged Itô differential equations of the approximated system are derived with the technique of stochastic averaging. Then, the average bifurcation parameter expression of system is obtained, which can determine the criterion of the SHB deduced by the FOPD control force. Last, an illustration of coupled Rayleigh oscillators is given to demonstrate the validity of the procedure. The influences of time delay, noise intensities and fractional order on the system SHB are discussed.

Synthesised fractional‐order PD controller design for fractional‐order time‐delay systems based on improved robust stability surface analysis

An improved robust stability surface analysis method is proposed in this study for fractional-order time-delay systems. Through the stabilisation process, a synthesised fractional-order PD controller can be designed with guaranteed robustness specifications. Firstly, the specification for improved robustness requirements is proposed. In order to find selectable robust controller design parameter combinations for the controlled system, stability region is discussed based on the stability boundary locus, and the robust stability surface is derived straight after. Secondly, the selectable parameter combinations are checked to find the one that best fulfils all the proposed robustness specifications. Finally, numerical simulations are given to demonstrate the effectiveness and flexibility of the presented control algorithm.

Design of new fractional order PI–fractional order PD cascade controller through dragonfly search algorithm for advanced load frequency control of power systems

Owing to integrating the dense range of distinct electric power sources, high volume of power generation units, abrupt and continuous changes in load demand, and rising utilization of power electronics, the electric power system (EPS) is striving for high-performance control schemes to counterwork the concerns depicted above. Additionally, it is highly creditable to have the controller structure as simple as possible from a viewpoint of practical implementation. Thus, this paper describes a virgin application of fractional order proportional integral–fractional order proportional derivative (FOPI–FOPD) cascade controller for load frequency control (LFC) of electric power generating systems. The proposed controller includes fractional order PI and fractional order PD controllers connected in cascade wherein orders of integrator ( $$\lambda$$ ) and differentiator ( $$\mu$$ ) may be fractional. The gains and fractional order parameters of the controller are concurrently tuned using recently proposed dragonfly search algorithm (DSA) by minimizing the integral time absolute error (ITAE) of frequency and tie-line power deviations. DSA is the mathematical model and computer simulation of static and dynamic swarming behaviors of dragonflies in nature, and its implementation in LFC studies is very rare, unveiling additional research gap to be bridged. Performance of the advocated approach is first explored on popular two-area thermal PS with/without governor dead band (GDB) nonlinearity and then on three-area hydrothermal PS with suitable generation rate constraints. To highlight the prominence and universality of our proposal, the work is extended to single-/multi-area multi-source EPSs. Several comparisons with DSA optimized FOPID controller and the relevant recent works for each test system indicate the contribution of proposed DSA optimized FOPI–FOPD cascade controller in alleviating settling time/undershoot/overshoot of frequency and tie-line power oscillations.

Design, Implementation, and Validation of Robust Fractional-Order PD Controller for Wheeled Mobile Robot Trajectory Tracking

In this paper, a trajectory tracking control algorithm is proposed based on the fractional-order PD (FOPD) controller for a Wheeled Mobile Robot (WMR). Firstly, an improved flat phase property is put forward as a robust controller tuning specification. This specification is capable of guaranteeing the flatness of the phase curve in a frequency interval, so the controlled system robustness can be improved. Then, the stabilization process is discussed with respect to the parameters of the FOPD controller through a visualized 3-dimensional surface, so both the stability and robustness of the controlled system can be guaranteed under the proposed controller. Furthermore, the implementation of the proposed robust FOPD controller is presented, which makes the control algorithm easy to be realized. At last, the effectiveness of the proposed trajectory tracking control algorithm is verified by the simulation and experiment results.

Joint Position Control Based on Fractional-Order PD and PI Controllers for the Arm of the Humanoid Robot TEO

This paper presents a control scheme for the humanoid robot TEO’s elbow joint based on a novel tuning method for fractional-order PD and PI controllers. Due to the graphical nature of the proposed method, a few basic operations are enough to tune the controllers, offering very competitive results compared to classic methods. The experiments show a robust performance of the system to mass changes at the tip of the humanoid arm.

Saturation Based Nonlinear FOPD Motion Control Algorithm Design for Autonomous Underwater Vehicle

The application of Autonomous Underwater Vehicle (AUV) is expanding rapidly, which drives the urgent need of its autonomy improvement. Motion control system is one of the keys to improve the control and decision-making ability of AUVs. In this paper, a saturation based nonlinear fractional-order PD (FOPD) controller is proposed for AUV motion control. The proposed controller is can achieve better dynamic performance as well as robustness compared with traditional PID type controller. It also has the advantages of simple structure, easy adjustment and easy implementation. The stability of the AUV motion control system with the proposed controller is analyzed through Lyapunov method. Moreover, the controlled performance can also be adjusted to satisfy different control requirements. The outperformed dynamic control performance of AUV yaw and depth systems with the proposed controller is shown by the set-point regulation and trajectory tracking simulation examples.

Design of Non-overshooting Fractional-Order PD and PID Controllers for Special Case of Fractional-Order Plants

This study focuses on shaping the transient response of special case of fractional-order systems by using fractional-order PD (FOPD) and fractional-order PID (FOPID) controllers. For a plant with a fractional-order pole, a FOPID controller is designed in which the orders of its derivative and integral terms are considered equal to the commensurate order of the plant while for an integrating plant with a fractional-order pole, a FOPD controller is designed. The region of controller parameters is extracted to obtain a closed-loop system with a monotonically decreasing magnitude–frequency response. This leads to a non-overshooting or low-overshoot step response. The gain crossover frequency and phase isodamping conditions are employed to select appropriate controller parameters among the mentioned region. The numerical examples are provided to show the efficiency of the proposed FOPD and FOPID controllers.

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.

Analytical method on stabilisation of fractional-order plants with interval uncertainties using fractional-order PIλ Dμ controllers

ABSTRACTThis paper proposes an analytical criterion to determine the stabilisation of a fractional-order plant with interval uncertainties in the coefficients using the fractional-order PID controller. First, the nominal function and the disturbance function are defined according to the characteristic function of the closed loop system. The vertex functions of the value set with respect to the disturbance function are given by using Minkowski sum. The test frequency interval is divided into several intervals, and the vertex functions are unchanged in these intervals determined by the switching frequencies. Therefore, the calculation method of these frequencies is given. Second, the calculation method on the finite test frequency interval is proposed to simplify the number of frequency intervals. Third, the analytical condition that the origin is located on the edge of the value set is investigated. Based on this condition, the sufficient condition and necessary condition for the stabilisation of fractional-order plant using the PID controller are offered. Moreover, the analytical methods on a fractional-order plant with interval uncertainties using the fractional-order PI controller and the fractional-order PD controller are also discussed. Finally, three examples are given to validate the effectiveness of the proposed methods.

Extending the limits of stabilizability of systems with feedback delay via fractional-order PD controllers

In the last few decades the advantages of fractional-order control was demonstrated with several examples in comparison with integer-order control. In this paper, stabilizability of a second-order unstable system subject to a delayed PDµ and PDµDρ controller is investigated in terms of the critical delay. Stabilizability diagrams are determined as a function of the order of the fractional derivatives. It is shown that the critical delay for the PDµ controller is larger by 12% than that of the PD1 (or simply proportional-derivative, PD) controller and the critical delay for the PDµDρ controller is larger by 3.8% than the critical delay of the PD1D2 (or simply proportional-derivative-acceleration, PDA) controller.