Fractional Order PIλDμ Controller Research Articles

Fractional-Order PIλDμ Control to Enhance the Driving Smoothness of Active Vehicle Suspension in Electric Vehicles

The suspension system is a crucial part of an electric vehicle, which directly affects its handling performance, driving comfort, and driving safety. The dynamics of the 8-DoF full-vehicle suspension with seat active control are established based on rigid-body dynamics, and the time-domain stochastic excitation model of four tires is constructed by the filtered white noise method. The suspension dynamics model and road surface model are constructed on the Matlab/Simulink simulation software platform, and the simulation study of the dynamic characteristics of active suspension based on the fractional-order PIλDμ control strategy is carried out. The three performance indicators of acceleration, suspension dynamic deflection, and tire dynamic displacement are selected to construct the fitness function of the genetic algorithm, and the structural parameters of the fractional-order PIλDμ controller are optimized using the genetic algorithm. The control effect of the optimized fractional-order PIλDμ controller based on the genetic algorithm is analyzed by comparing the integer-order PID control suspension and passive suspension. The simulation results show that for optimized fractional-order PID control suspension, compared with passive suspension, the average optimization of the root mean square (RMS) of acceleration under random road conditions reaches over 25%, the average optimization of suspension dynamic deflection exceeds 30%, and the average optimization of tire dynamic displacement is 5%. However, compared to the integer-order PID control suspension, the average optimization of the root mean square (RMS) of acceleration under random road conditions decreased by 5%, the average optimization of suspension dynamic deflection increased by 3%, and the average optimization of tire dynamic displacement increased by 2%.

Motion Strategy of Drilling Anchor Manipulator Based on Intelligent Optimization Algorithm

The drilling and anchoring robot is an important equipment to realize the intellectualization of fully mechanized coal mining. The control effect of its manipulator directly affects the safety and efficiency of the support operation of the drilling and anchoring robot. The control of drilling anchor manipulator usually adopts PID controller, but due to the limitation of integer order PID control algorithm and traditional empirical parameter adjustment, it is difficult to find a group of parameters with the best control effect in a short time, resulting in the failure of timely and accurate positioning of the end of the manipulator. In this paper, the combination of numerical modeling and simulation analysis is used to adjust the parameters of fractional order PIλDμ controller by using search algorithm (GPS) and applied to the motion control of manipulator; Based on the independent joint control theory, the single input single output system model of hydraulic cylinder at the joint of manipulator is established by using Matlab-Simulink software and fractional PIλDμ control technology. The fractional order control system is regulated by input shaping feedforward control technology(IS). The step response effect of the hydraulic system is further analyzed by setting input feedforward controllers with different gain ratios. The numerical simulation results show that the control strategy of “Input Shaping(IS) + Intelligent optimization algorithm (GPS) parameter adjustment + Fractional PIλDμ control technology” can effectively improve the spatial accurate positioning of mining manipulator. The research expand the application of intelligent control technology in the field of mining equipment.

Approximation of Fractional Order PIλDμ-Controller Transfer Function Using Chain Fractions

The approximation of a fractional order PIλDμ-controller transfer function using a chain fraction theory is considered. Analytical expressions for the approximation of s±α components of the transfer functions of PIλDμ-controllers were obtained through the application of the chain fraction theory. Graphs of transition functions and frequency characteristics of Dμ (α = μ = 0.5) and Iλ (α = λ = −0.5) parts for five different decomposition orders were obtained and analyzed. The results showed the possibility of applying the approximation of the PIλDμ-controller transfer function by the method of chain fractions with different valuesof λ and μ. For comparison, the transfer functions with the same order polynomials, obtained by the methods of Oustaloup transformation and chain fractions, were approximated for α = ±0.5. The analysis proved the advantages of using the chain fraction method to approximate the transfer function of the PIλDμ-controller. The performed approximation opens up the possibility of developing engineering methods for the technical implementation of PIλDμ-controllers. The accuracy of the same order transfer function approximation is higher when the method of chain fractions is used. It has been established that the adequacy of the frequency characteristics of the transfer functions obtained by the chain fraction method also depends on the approximation order.

Fractional-Order PIλDμ Controller Using Adaptive Neural Fuzzy Model for Course Control of Underactuated Ships

For the uncertainty caused by the time-varying modeling parameters with the sailing speed in the course control of underactuated ships, a novel identification method based on an adaptive neural fuzzy model (ANFM) is proposed to approximate the inverse dynamic characteristics of the ship in this paper. This model adjusts both its own structure and parameters as it learns, and is able to automatically partition the input space, determine the number of membership functions and the number of fuzzy rules. The trained ANFM is used as an inverse controller, in parallel with a fractional-order PIλDμ controller for the course control of underactuated ships. Meanwhile, the sine wave curve and the sawtooth wave curve are considered as the input learning samples of ANFM, respectively, and the inverse dynamics simulation experiments of the ship are carried out. Two different ANFM structures are obtained, which are connected in parallel with the fractional-order PIλDμ controller respectively to control the course of ship. The simulation results show that the proposed method can effectively overcome the influence of uncertainty of ship modeling parameters, track the desired course quickly and effectively, and has a good control effect. Finally, comparative experiments of four different controllers are carried out, and the results show that the FO PIλDμ controller using ANFM has the advantages of small overshoot, short adjustment time, and precise control.

Enhanced fractional-order PI[formula omitted]D[formula omitted] control for a forced circulation evaporator system via advanced disturbance observer

Strong disturbances, time delays, and coupling effects complicate the control of evaporator systems. In this paper, to overcome these difficulties a three-stage controller design is given and as a first step a disturbance observer (DOB) is designed. Then, fractional order PIλDμ controllers (PIλDμ FOC) with inverted decoupling structure are realized. Finally, PIλDμ FOC is enhanced via DOB and applied to control of a considered forced circulation evaporator system. Controlling the evaporator system with the DOB-enhanced PIλDμ FOC is showed more accurate and precise control results when compared with the optimally tuned PID and fuzzy logic controller (FLC). The DOB-enhanced PIλDμ FOC method has remarkable superiority in disturbance rejection and the proposed control method is improved the tracking ability of the system for reference changes.

An analytical synthesis of fractional order PI[formula omitted]D[formula omitted] controller design

This paper proposed a comprehensive synthesis of fractional order PIλDμ (FOPIλDμ) controller analytical design, which is illustrated through the typical first order plus normalized time delay (FOPNTD) systems, with fulfilling five frequency-domain specifications simultaneously: phase margin (ϕm), gain margin (Am), phase crossover frequency (ωpc), gain crossover frequency (ωgc) and a “flat phase” constraint. The control loop shape can be adjusted with wide freedom according to the five design specifications, which can all be beneficial on optimizing the control system: ωgc represents the system control bandwidth and response speed, ϕm and Am guarantee the stability, and flat phase constraint keeps the system with iso-damping property on robustness of loop gain variations. The impact of ωpc adjustment is thoroughly discovered via frequency-domain analysis and also time-domain analysis with low-frequency disturbance and high-frequency noise. The frequency response functions are presented to show the loop-shaping advantages of the proposed synthesis scheme. A further in-depth study on designing guideline is also presented: the feasible region of four specifications, e.g. ωgc, ωpc, ϕm and Am, can all be collected and visualized in the multi-dimensional graphics. This feasible region gives users prior information and great flexibility before the controller design. Simulation results using the designed FOPIλDμ controller are carried out to demonstrate the performance advantages over the optimized integer-order PID, three-parameter FOPID, fractional filter-fractional order PID and Ziegler–Nichols FOPID controllers.

Fractional-Order Controller for Course-Keeping of Underactuated Surface Vessels Based on Frequency Domain Specification and Improved Particle Swarm Optimization Algorithm

In this paper, a new fractional-order (FO) PIλDµ controller is designed with the desired gain and phase margin for the automatic rudder of underactuated surface vessels (USVs). The integral order λ and the differential order μ are introduced in the controller, and the two additional adjustable factors make the FO PIλDµ controller have better accuracy and robustness. Simulations are carried out for comparison with a ship’s digital PID autopilot. The results show that the FO PIλDµ controller has the advantages of a small overshoot, short adjustment time, and precise control. Due to the uncertainty of the model parameters of USVs and two extra parameters, it is difficult to compute the parameters of an FO PIλDµ controller. Secondly, this paper proposes a novel particle swarm optimization (PSO) algorithm for dynamic adjustment of the FO PIλDµ controller parameters. By dynamically changing the learning factor, the particles carefully search in their own neighborhoods at the early stage of the algorithm to prevent them from missing the global optimum and converging on the local optimum, while at the later stage of evolution, the particles converge on the global optimal solution quickly and accurately to speed up PSO convergence. Finally, comparative experiments of four different controllers under different sailing conditions are carried out, and the results show that the FO PIλDµ controller based on the IPSO algorithm has the advantages of a small overshoot, short adjustment time, precise control, and strong anti-disturbance control.

Fractional Order Piλdµ Controller for Microgrid Power System Using Cohort Intelligence Optimization

Fractional Order Piλdµ Controller for Microgrid Power System Using Cohort Intelligence Optimization

Research on nonlinear model and fuzzy fractional order PIλDμ control of air suspension system

To improve the ride comfort of wheeled armored vehicles, air springs are used. To describe the vehicle motion more accurately, a nine-degree-of-freedom air suspension system for the whole vehicle was established, and its equations of motion were derived. Through theoretical analysis of the stiffness characteristics and forces on the air springs, the nonlinear restoring force was obtained as a cubic polynomial of the air spring displacement. The simulation results obtained by fitting the polynomial and radial basis function curves with MATLAB/Simulink software are consistent with the actual test results, thus verifying the correctness of the nonlinear air spring polynomial model. Finally, a fuzzy fractional order PIλDμ controller is designed and simulated for the vehicle-seat-body model in terms of wheel dynamic load, suspension dynamic deflection, body acceleration, and other indicators. The simulation results show that the fuzzy fractional order PIλDμ Proportion Integral Differential (PID) control strategy has better overall performance than the PID control strategy, fuzzy control strategy, and fuzzy PID control strategy.

Experimental analysis of fractional order PI λ D μ controller for improvement of power quality in smart grid environment

Power electronics load is considered to be a strengthening factor that leads the system operators more concerned about power quality issues. Poor power quality can lead to distinctive operation of electrical components and devices, which may cause heavy economic loss to the customers and power network operators. An application of fractional order PIλDμ (FOPIλDμ) controller for power quality improvement in smart power distribution systems is presented here. Analytical design steps are presented for the FOPIλDμ controller. A genetic algorithm is used for tuning the control parameters of the FOPIλDμ controller optimally. The FOPIλDμ controller is employed and evaluated for shunt active power filter (APF) to filter out harmonics and improve reactive power compensation for non-stationary, non-linear load conditions for a smart power distribution system. Effectiveness of FOPIλDμ controller for instantaneous reactive power theory based PQ control algorithm applied to shunt APF is validated using a small-scale laboratory experimental setup. The experimental results confirm the excellent performance of the FOPIλDμ controller in terms of its superior transient response, significant reduction in harmonic distortion, and improved reactive power compensation in a smart grid environment.

Devising a procedure for the parametric synthesis of fractional order controllers and their implementation in the FC–IM system

The modern industry is dominated by electric drives of the system frequency converter‒ induction motor, which employ integer-order controllers. Allowing the implementation and adjustment of fractional order controllers in the converter itself greatly expand their capabilities and, therefore, is relevant. This paper reports a procedure for the parametric synthesis of PIλDμ-controllers of fractional order, providing for the use of the desired forms of fractional order, as well as their practical implementation in the system frequency converter–induction motor. In this case, the control object is described by a transfer function of the fractional or integer order, derived on the basis of experimental results. The study results demonstrate the possibility of constructing new, as well as modernizing existing, electromechanical systems involving the fractional-order PIλDμ-controllers with an expanded range of dynamic properties that correspond to the desired forms of fractional order. The procedure for the parametric synthesis of a PIλDμ-controller has been theoretically substantiated, which was confirmed by applying the simulation and during field experiments concerning the system frequency converter–induction motor. The reported procedure is universal because it makes it possible to synthesize the PIλDμ-controller for standard forms of both the integer and fractional orders. It is clear that the range of the desired standard forms in the synthesis process can include all possible known forms, including those of fractional order. The result of this study allows us to argue that it is possible to apply the developed algorithm of actions for those engineering tasks that aim to build such systems for various industrial mechanisms. At the same time, no restrictions are imposed on the transfer function of the control object

SBW System Control Strategy for Vehicle Based on Fractional-order PIλ Dμ Controller

SBW System Control Strategy for Vehicle Based on Fractional-order PIλ Dμ Controller

Precision Displacement Control of a Diamond-shaped Amplifying Mechanism Driven by Piezoelectric Actuator Based on Fuzzy Fractionalorder PIλDμ Controller

Background: As a high-performance functional material, stacked piezoelectric actuator can produce a displacement under the effect of changing voltage. Its advantages of fast response and easy operation make it to be widely applied in the precision structure field. However, its small displacement stroke and hysteresis nonlinearity affect the accuracy of the output. : In the next step, some experiments were undertaken based on the constructed platform. Methods: In order to enlarge the displacement of piezoelectric actuator and reduce the influence of hysteresis, this study designs a diamond-shaped amplifying mechanism to amplify the output of the piezoelectric actuator, and then develops a self-tuning fuzzy fractional-order PIλDμ controller for the high precision displacement control of the proposed amplifying mechanism. After analyzing the working principle and modeling the amplifying mechanism, the fractional-order PIλDμ control model of the proposed mechanism was built and discretized according to the theoretical base of the fractional calculus in the time domain. Moreover, the fuzzy control algorithm was also introduced to achieve self-turning of parameters. Besides, the amplifying mechanism was also adopted for a microdroplet jetting dispenser to verify the practicability of the mechanism and control strategy. In the next step, some experiments were undertaken based on the constructed platform. Results: Experiments show that the displacement overshoots, the times of reaching a steady state of the traditional integer-order controller and the fractional-order controller are 5.08%, 1.17% and 17.25 s, 12.00 s, respectively. However, the fuzzy PIλDμ controller lowers the overshoot and the time of reaching a steady state to 0.95% and 9.00 s, respectively. The control algorithm can not only improve the follow-ability of the output displacement of the proposed mechanism, but also maintain the deviation within the range of 0.4% after the displacement stroke is stable and reduce the entering time of the mechanism up to 47.8%. In actual application, the droplet volume of micro-droplet jetting dispenser under fuzzy fractional-order PID control method is more stable, and its repeatability accuracy can reach up to 1.6475%. Conclusion: Experimental results indicate that the self-tuning fuzzy fractional-order PIλDμ controller can significantly improve the tracking performances of the PID and the integer-order PID with regard to the amplifying mechanism with the advantages of good dynamic character and regulation precision. Furthermore, the diamond-shaped amplification mechanism and control strategy can be applied for some micro-droplet jetting dispensers used in microelectronic packaging, life science and 3D printing fields.

Fractional Order PID Controller Design for Supply Manifold Pressure Control of Proton Exchange Membrane Fuel Cell

Abstract In this work, fractional order PIλDµ (FOPID) controller designed to enhance the dynamic performance of the Proton Exchange Membrane (PEM) fuel cell. The control objective is to regulate the supply manifold pressure on cathode side to maintain oxygen excess ratio of the PEM fuel cell. The higher order PEM fuel cell model is approximated to First order plus time delay (FOPTD) model for controller design and analysis. The proposed FOPID controller is designed based on minimization of Integral Absolute Error (IAE) with pre specified maximum sensitivity (Ms) as a constraint. Uncertainty and measurement noise analysis is carried out to verify the robustness of the designed controller. The simulation results of proposed FOPID controller is compared with other designing methods. Based on minimization of IAE value, the SP 1.4 FOPID controller produces IAE value of 0.255 where as AMIGO 1.4 tuning method and ZN based FOPID tuning methods produces 0.263 and 3.817 respectively for perfect case. Based on maximum sensitivity Ms is 1.4, the SP 1.4 FOPID controller produces Ms of 1.4 where as AMIGO 1.4 PID and ZN based FOPID tuning methods produces Ms of 1.5 and 1.25 respectively for perfect case, which indicates that the proposed SP 1.4 FOPID controller is robust. The proposed SP 1.4 FOPID provides better values (rise time of 0.331 sec, settling time of 0.692 sec and percentage of peak overshoot of 0.797 for perfect case) when compared with other methods. From simulation results, for the control of supply manifold pressure of PEM fuel cell, the proposed fractional-order PID controllers improves the closed loop performance in terms of rise time, settling time and percentage of peak overshoot when compared to the integer-order PID controllers.

A new approach to optimal design of digital fractional-order PIλDμ controller

Fractional-order PIλDμ controller has been under intensive investigations for better control performance in recent years. To facilitate the implementation of digital controller, the PIλDμ controller is usually firstly optimally designed in continuous domain, and then discretized to obtain a digital PIλDμ controller. However, the aforementioned methodology may lead to performance deterioration and other undesirable results. To tackle the issues identified above, a new approach is proposed to directly design the optimal digital PIλDμ controller, which is formulated to be a controller optimization problem with six tunable parameters by using the parameterized Al-Alaoui operator for discretization. To optimize the digital PIλDμ controller, an improved PSO-SADV algorithm with switching adaptive delayed velocities has been developed to deal with some frequently encountered obstacle phenomena in the optimization problem, and the superiorities of the PSO-SADV are verified by several comparative simulation experiments. Finally, the PSO-SADV is employed to handle the parameter optimization issue of the digital PIλDμ controller based on the ITSE index of step response. Several illustrative simulation experiments are carried out upon some typical controlled objects for the comparison with the PIλDμ controller and its discretizations, and the effectiveness and superiority of the new approach can be confirmed by the simulation results.

Synthesis of a fractional-order PIλDμ-controller for a closed system of switched reluctance motor control

The relevance of creating high-quality control systems for electric drives with a switched reluctance motor (SRM) was substantiated. Using methods of mathematical modeling, transient characteristics of the process of turn-on of SRMs with various moments of inertia were obtained. Based on analysis of the obtained transient characteristics, features of the SRM turn-on process determined by dynamic change of parameters of the SRM during its turn-on were shown. Low accuracy of SRM identification using a fractionally rational function of rat34 class was shown. Regression coefficient of the resulting model was 85 %. Based on analysis of transient characteristics of the SRM turn-on process, a hypothesis was put forward about the possibility of identifying the SRM by means of a fractional-order transfer function. Using the methods of mathematical modeling, transient characteristics of the process of turning-on the SRMs with various moments of inertia were obtained. Using the FOMCON MATLAB Toolbox, identification of the SRM turn-on process with the help of a fractional-order transfer function of second order was performed. Regression coefficient of the resulting model was 93–96 %. For the obtained fractional-order transfer functions, a method of synthesis of a fractional-order PI λ D μ controller optimized in terms of minimum integral square error of the transition function of the closed system of fractional-order control of objects was implemented. The FOMCON MATLAB Toolbox was used for synthesis of the PI λ D μ controller. Comparative analysis of the SRM turn-on processes in both open and closed control systems with a classical integer-order PID controller and with a fractional-order PI λ D μ controller was made. Use of the fractional-order PI λ D μ controller in comparison with the classical integer-order regulator makes it possible to reduce overshoot from 13.3 % to 2.64 %, increase speed of the closed ACS, decrease regulation time from 1.48 s to 0.53 s while reducing variability of transient characteristics. The study results can be used to improve performance of closed systems for controlling angular velocity of the SRM

Optimal fractional order PIλDμ controller for stabilization of cart‐inverted pendulum system: Experimental results

AbstractThe cart‐inverted pendulum is a non‐minimum phase system having right half s‐plane pole and zero in close vicinity to each other. Linear time invariant (LTI) classical controllers cannot achieve satisfactory loop robustness for such systems. Therefore, in the present work the fractional order PIλDμ (FOPID) controller is addressed for robust stabilization of the system, since fractional order controller design allows more degrees of freedom compared to its integer order counterparts by virtue of its two parameters λ and μ. The controller parameters are tuned by three evolutionary optimization techniques. In order to select the controller parameters optimally, a novel non‐linear fitness function using integral time square error (ITSE), settling‐time, and rise time is proposed here. The control algorithm is implemented successfully in real‐time. Moreover, stability analysis of the system compensated with a fractional order controller is presented using Riemann surface. Robustness of the physical cart‐inverted pendulum system towards multiplicative gain variations and plant parameter variations is verified. In this regard, it is shown that the fractional order controller provides satisfactory robust performance in both simulation and real‐time system.

Whale optimisation algorithm tuned fractional order PI<SUP align="right">λ</SUP>D<SUP align="right">μ</SUP> controller for load frequency control of multi-source power system

This paper deals with the design and optimisation of a fractional order PIλDμ (FOPID) controller implemented for load frequency control (LFC) of a multi-area multi-source power system having the nonlinearities of time delays, generation rate constraint (GRC) and governor dead band (GDB). Whale optimisation algorithm (WOA), arguably for the first time in this area, is applied for optimal tuning of the parameters of FOPID controller. The comparative analysis of the performance of WOA vis-a-vis other recent evolutionary optimisation approaches is carried out from which it is observed that WOA optimised FOPID controller gives the best dynamic performance in terms of settling time and maximum overshoot in frequency and tie-line power deviations. Further, investigations carried out against system parametric variations within the range of ± 25% reveal that WOA tuned FOPID controller is the most effective and robust against wide variations in system parameters.

Optimal design of fractional order PIλDμ controller for an AVR system using Ant Lion Optimizer

Abstract Application of fractional order PID (FOPID) controller to an automatic voltage regulator (AVR) is presented and studied in this paper. Fractional-order PIλDμ controller is a generalization of standard PID controller using fractional calculus. Compared to PID controller, the tuning of FOPID is more complex and remains a challenging problem. Indeed, design stage of such a controller consists of determining eleven parameters. This paper proposes a novel population based optimization algorithm called Ant Lion Optimizer algorithm (ALO) for solving the FOPID controller tuning problem. ALO has proved to have very high efficiency. Comparisons are made with Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Cultural algorithms (CAs) and multi-objective evolutionary optimizers (MOEO) techniques and it is shown that the proposed FOPID controller can highly improve the system robustness.

Fractional PD-IλDμ Error Manifolds for Robust Tracking Control of Robotic Manipulators

Linear proportional-integral-derivative (PID) controller stands for the most widespread technique in industrial applications due to its simple structure and easy tuning rules. Recently, considering fractional orders λ and μ, there has been studied the fractional-order PIλDμ (FPID) controller to provide salient advantages in comparison to the conventional integer-order PID, such as, a more flexible structure and a preciser performance. In addition, proportional and derivative (PD) and PID error manifolds have been classically proposed; however, there remains the question on how FPID-like error manifolds perform for the control of nonlinear plants, such as robots. In this paper, this problem is addressed by proposing a PD-IλDμ error manifold for novel vector saturated control. The stability analysis shows convergence into a small vicinity of the origin, wherein, such hybrid combination of integer- and fractional-order error manifolds provides further insights into the closed-loop response of the nonlinear plant. Simulations studies are carried out to illustrate the feasibility of the proposed scheme.