Fractional Order PIλDμ Research Articles

Fuzzy Fractional Order PIλDμ Controller Design Based on Correction Projectile System

Fuzzy Fractional Order PIλDμ Controller Design Based on Correction Projectile System

A comparative study of fractional order PIλ/PIλDμ tuning rules for stable first order plus time delay processes

Conventional PID tuning methods may not be sufficient to deal with complex processes of modern industry. For better control, fractional orderPIλDμ controller was introduced as the generalization of classical PID controller with the help of non-integer order (fractional order) calculus. Thefractional calculus uses integration and differentiation with a fractional order or complex order. The major advantage of fractional derivative is theability to inherit the nature of the processes. In general, the control loop includes both fractional order process model and fractional order controller.However, the processes to be controlled are usually modeled as integer order models and controlled using fractional order controllers. But if theplant model is obtained as fractional model, it is converted into integer order model by approximating the fractional terms using differentapproximations proposed in the literature. With all the above mentioned advantages, several fractional order PIλ/PIλDμ tuning rules are proposedin the literature for integer order systems and researchers are still proposing the new rules. The main aim of this paper is to compare fractional orderPI/PID tuning methods based on Integral of Absolute Error (IAE), Total Variation (TV) and Maximum Sensitivity (Ms). The main reason forchoosing fractional order PIλ/PIλDμ controllers is their additional degrees of freedom that result in better control performance. These tuning ruleswere applied on several first order plus time delay processes subjected to step change in setpoint and disturbance.Six recent tuning methods, three for fractional order PIλ and the remaining for fractional order PIλDμ, were considered. Finally, from thesimulation results the optimal tuning method is recommended based on the control objective of the process and the process dead time (L) to timeconstant (T) ratio. It is observed that the performance of tuning methods vary with the nature of the process like lag dominant, balanced and delaysignificant processes. The FOPTD processes were checked for robustness with increasing L/T ratio with respect to IAE, TV and Ms.

A comparative study of fractional order PIλ/PIλDµ tuning rules for stable first order plus time delay processes

Conventional PID tuning methods may not be sufficient to deal with complex processes of modern industry. For better control, fractional order PIλDµ controller was introduced as the generalization of classical PID controller with the help of non-integer order (fractional order) calculus. The fractional calculus uses integration and differentiation with a fractional order or complex order. The major advantage of fractional derivative is the ability to inherit the nature of the processes. In general, the control loop includes both fractional order process model and fractional order controller. However, the processes to be controlled are usually modeled as integer order models and controlled using fractional order controllers. But if the plant model is obtained as fractional model, it is converted into integer order model by approximating the fractional terms using different approximations proposed in the literature. With all the above mentioned advantages, several fractional order PIλ/PIλDµ tuning rules are proposed in the literature for integer order systems and researchers are still proposing the new rules. The main aim of this paper is to compare fractional order PI/PID tuning methods based on Integral of Absolute Error (IAE), Total Variation (TV) and Maximum Sensitivity (Ms). The main reason for choosing fractional order PIλ/PIλDµ controllers is their additional degrees of freedom that result in better control performance. These tuning rules were applied on several first order plus time delay processes subjected to step change in setpoint and disturbance. Six recent tuning methods, three for fractional order PIλ and the remaining for fractional order PIλDµ, were considered. Finally, from the simulation results the optimal tuning method is recommended based on the control objective of the process and the process dead time (L) to time constant (T) ratio. It is observed that the performance of tuning methods vary with the nature of the process like lag dominant, balanced and delay significant processes. The FOPTD processes were checked for robustness with increasing L/T ratio with respect to IAE, TV and Ms.

DE-based tuning of [formula omitted] controllers

A new method that relies on evolutionary computation concepts is proposed in this paper to tune the parameters of fractional order PIλDμ controllers, in which the orders of the integral and derivative parts, λ and μ, respectively, are fractional. The main advantage of the fractional order controllers is that the increase in the number of parameters in the controller allows an increase in the number of control specifications that can be met. A Differential Evolution (DE) algorithm is proposed to make the controlled system fulfill different design specifications in time and frequency domains. This method is based on the minimization of a fitness function. Experiments have been carried out in simulation and in a real DC motor platform. The results illustrate the effectiveness of this method.

Particle Swarm Optimization of PI<sup>λ</sup>D<sup>μ</sup> Control of Heating Furnace Temperature Control System

Due to the complexity of the heat transfer for heating furnace, some characteristics are caused such as big inertia, great lag. In the temperature control system for heating furnace, the traditional PID controller can not get satisfactory effect, that dynamic is instability and control accuracy is bad, which is very detrimental to the system to achieve optimum efficiency. A fractional order PIλDμ controller based on particle swarm optimization method was designed, at the same time compared with PID control. Simulation results show that, fractional order PIλDμ control based on particle swarm optimization has better convergence stability, faster response times and higher accuracy value. Fractional order PIλDμ controller has better dynamic performance, compared with traditional PID controller, greatly improves the quality control system.

A New Tuning Method for Stabilization Time Delay Systems Using PIλDμ Controllers

AbstractThis paper presents a new design procedure to tune the fractional order PIλDμ controller that stabilizes a first order plant with time delay. The procedure is based on a suitable version of the Hermite–Biehler Theorem and the Pontryagin Theorem. A Theorem and a Lemma are developed to compute the global stability region of the PIλDμ controller in the (kp,ki,kd) space. Hence, this Theorem and Lemma allow us to develop an algorithm for solving the PIλDμ stabilization problem of the closed loop plant. The proposed approach has been verified by numerical simulation that confirms the effectiveness of the procedure.

Trajectory Tracking Control for a 3-DOF Parallel Manipulator Using Fractional-Order $\hbox{PI}^{\lambda}\hbox{D}^{\mu}$ Control

In this paper, a 3-degrees-of-freedom parallel manipulator developed by Tsai and Stamper known as the Maryland manipulator is considered. In order to provide dynamic analysis, three different sequential trajectories are taken into account. Two different control approaches such as the classical proportional-integral-derivative (PID) and fractional-order PID control are used to improve the tracking performance of the examined manipulator. Parameters of the controllers are determined by using pattern search algorithm and mathematical methods for the classical PID and fractional-order PID controllers, respectively. Design procedures for both controllers are given in detail. Finally, the corresponding results are compared. Performance analysis for both of the proposed controllers is confirmed by simulation results. It is observed that not only transient but also steady-state error values have been reduced with the aid of the PIλDμ controller for tracking control purpose. According to the obtained results, the fractional-order PIλDμ controller is more powerful than the optimally tuned PID for the Maryland manipulator tracking control. The main contribution of this paper is to determine the control action with the aid of the fractional-order PI λDμ controller different from previously defined controller structures. The determination of correct and accurate control action has great importance when high speed, high acceleration, and high accuracy needed for the trajectory tracking control of parallel mechanisms present unique challenges.

Application of fractional-order control for vibration suppression of viscoelastic beams

There are several parameters required for viscoelastic material modeling to describe the material behavior. On the other hand, the parameter identification of viscoelastic materials often leads to difficult experimental procedures. Since the fractional controllers are robust to parameter uncertainties in the plant model, in this paper, this type of controller is proposed to avoid the difficulties of parameters identification in viscoelastic materials. As a prototype, a viscoelastic beam covered by piezoelectric patches is modeled for suppressing vibrations arising from transient excitations of viscoelastic materials. Different fractional-order PIλDμ controllers are used to illustrate the effect of fractional integral and fractional derivative orders separately. The results show that the fractional-order controllers are successful and effective for vibration suppression of viscoelastic beams.

Self-tuning fractional order PI<SUP align="right">λ</SUP>D<SUP align="right">µ</SUP> controller based on extremum seeking approach

Fractional PID (PIλDµ) controllers are not commonly used in industry when compared to classical PID controllers. The main reason is that the fractional controller parameters tuning remains an open problem. In this work we propose to generalise a classical PID tuning method based on extremum seeking (ES) approach to PIλDµ control parameters tuning in order to achieve optimal performance. ES is a non-model-based method which searches online for the parameters that minimise a cost function which is representative of the controller’s performance. The main advantage of this method is that it can be applied to a plant in which there is no knowledge of the model. Comparative simulation examples show the effectiveness of the proposed strategy for fractional order PIλDµ tuning.

The Frequency Research of Fractional Order PIλDμ Controller

Since most of the control systems in real industrial production are fractional，there is necessery to propose fractional order PIλDμ controller, which extend the traditional integer-order PID controller to fractional order, it has increased two free degree: Integral-order λ and differential-order μ, to more accurately control those complex systems. At the same time analysis the performance of fractional order PIλDμ controller in frequency domain. Especially,the two degrees’ effect to controller.

Chaotic multi-objective optimization based design of fractional order PIλDμ controller in AVR system

In this paper, a fractional order (FO) PIλDμ controller is designed to take care of various contradictory objective functions for an automatic voltage regulator (AVR) system. An improved evolutionary non-dominated sorting genetic algorithm II (NSGA II), which is augmented with a chaotic map for greater effectiveness, is used for the multi-objective optimization problem. The Pareto fronts showing the trade-off between different design criteria are obtained for the PIλDμ and PID controller. A comparative analysis is done with respect to the standard PID controller to demonstrate the merits and demerits of the fractional order PIλDμ controller.

Master-slave chaos synchronization via optimal fractional order PI λ D μ controller with bacterial foraging algorithm

Chaos synchronization in a master-slave configuration has been studied in this paper with a fractional order (FO) Proportional-Integral-Derivative (PID) controller using an intelligent Bacterial Foraging Optimization (BFO) algorithm. A comparative study has been made to highlight the advantage of using a fractional order PIλDμ controller over the conventional PID controller for chaos synchronization using two Lu systems as a representative example. Simulation results are presented to show the effectiveness of the proposed chaos synchronization technique over the existing methodologies.

Optimum design of fractional order PIλDμ controller for AVR system using chaotic ant swarm

Fractional-order PID (FOPID) 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 challenge problem. This paper focuses on the design of FOPID controller using chaotic ant swarm (CAS) optimization method. The tuning of FOPID controller is formulated as a nonlinear optimization problem, in which the objective function is composed of overshoot, steady-state error, raising time and settling time. CAS algorithm, a newly developed evolutionary algorithm inspired by the chaotic behavior of individual ant and the self-organization of ant swarm, is used as the optimizer to search the best parameters of FOPID controller. The designed CAS-FOPID controller is applied to an automatic regulator voltage (AVR) system. Numerous numerical simulations and comparisons with other FOPID/PID controllers show that the CAS-FOPID controller can not only ensure good control performance with respect to reference input but also improve the system robustness with respect to model uncertainties.

Fractional-Order Control of a Nonlinear Time-Delay System: Case Study in Oxygen Regulation in the Heart-Lung Machine

A fractional-order controller will be proposed to regulate the inlet oxygen into the heart-lung machine. An analytical approach will be explained to satisfy some requirements together with practical implementation of some restrictions for the first time. Primarily a nonlinear single-input single-output (SISO) time-delay model which was obtained previously in the literature is introduced for the oxygen generation process in the heart-lung machine system and we will complete it by adding some new states to control it. Thereafter, the system is linearized using the state feedback linearization approach to find a third-order time-delay dynamics. Consequently classical <i >PID</i> and fractional order <svg style="vertical-align:-0.0pt;width:44.724998px;" id="M1" height="13.6875" version="1.1" viewBox="0 0 44.724998 13.6875" width="44.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6875)"> <g transform="translate(72,-61.05)"> <text transform="matrix(1,0,0,-1,-71.95,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑃</tspan> <tspan style="font-size: 12.50px; " x="7.4267821" y="0">𝐼</tspan> </text> <text transform="matrix(1,0,0,-1,-57.4,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝜆</tspan> </text> <text transform="matrix(1,0,0,-1,-52.16,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐷</tspan> </text> <text transform="matrix(1,0,0,-1,-42.1,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝜇</tspan> </text> </g> </g> </svg> controllers are gained to assess the quality of the proposed technique. A set of optimal parameters of those controllers are achieved through the genetic algorithm optimization procedure through minimizing a cost function. Our design method focuses on minimizing some famous performance criterions such as IAE, ISE, and ITSE. In the genetic algorithm, the controller parameters are chosen as a random population. The best relevant values are achieved by reducing the cost function. A time-domain simulation signifies the performance of <svg style="vertical-align:-0.0pt;width:44.724998px;" id="M2" height="13.6875" version="1.1" viewBox="0 0 44.724998 13.6875" width="44.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6875)"> <g transform="translate(72,-61.05)"> <text transform="matrix(1,0,0,-1,-71.95,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑃</tspan> <tspan style="font-size: 12.50px; " x="7.4267821" y="0">𝐼</tspan> </text> <text transform="matrix(1,0,0,-1,-57.4,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝜆</tspan> </text> <text transform="matrix(1,0,0,-1,-52.16,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐷</tspan> </text> <text transform="matrix(1,0,0,-1,-42.1,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝜇</tspan> </text> </g> </g> </svg> controller with respect to a traditional optimized <i >PID</i> controller.

Improved model reduction and tuning of fractional-order [formula omitted] controllers for analytical rule extraction with genetic programming

Genetic algorithm (GA) has been used in this study for a new approach of suboptimal model reduction in the Nyquist plane and optimal time domain tuning of proportional–integral–derivative (PID) and fractional-order (FO) PIλDμ controllers. Simulation studies show that the new Nyquist-based model reduction technique outperforms the conventional H2-norm-based reduced parameter modeling technique. With the tuned controller parameters and reduced-order model parameter dataset, optimum tuning rules have been developed with a test-bench of higher-order processes via genetic programming (GP). The GP performs a symbolic regression on the reduced process parameters to evolve a tuning rule which provides the best analytical expression to map the data. The tuning rules are developed for a minimum time domain integral performance index described by a weighted sum of error index and controller effort. From the reported Pareto optimal front of the GP-based optimal rule extraction technique, a trade-off can be made between the complexity of the tuning formulae and the control performance. The efficacy of the single-gene and multi-gene GP-based tuning rules has been compared with the original GA-based control performance for the PID and PIλDμ controllers, handling four different classes of representative higher-order processes. These rules are very useful for process control engineers, as they inherit the power of the GA-based tuning methodology, but can be easily calculated without the requirement for running the computationally intensive GA every time. Three-dimensional plots of the required variation in PID/fractional-order PID (FOPID) controller parameters with reduced process parameters have been shown as a guideline for the operator. Parametric robustness of the reported GP-based tuning rules has also been shown with credible simulation examples.

Algorithm of robust stability region for interval plant with time delay using fractional order PIλDμ controller

By using PI λ D μ controller, we investigate the problem of computing the robust stability region for interval plant with time delay. The fractional order interval quasi-polynomial is decomposed into several vertex characteristic quasi-polynomials by the lower and upper bounds, in which the value set of the characteristic quasi-polynomial for vertex quasi-polynomials in the complex plane is a polygon. The D-decomposition technique is used to characterize the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters. We investigate how the fractional integrator order λ and the derivative order μ in the range (0, 2) affect the stabilizability of each vertex characteristic quasi-polynomial. The stability region of interval characteristic quasi-polynomial is determined by intersecting the stability region of each quasi-polynomial. The parameters of PI λ D μ controller are obtained by selecting the control parameters from the stability region. Using the value set together with zero exclusion principle, the robust stability is tested and the algorithm of robust stability region is also proposed. The algorithm proposed here is useful in analyzing and designing the robust PI λ D μ controller for interval plant. An example is given to show how the presented algorithm can be used to compute all the parameters of a PI λ D μ controller which stabilize a interval plant family.

Adaptive Neuro Fuzzy Inference Controller for Full Vehicle Nonlinear Active Suspension Systems

The main objective of designed the controller for a vehicle suspension system is to reduce the discomfort sensed by passengers which arises from road roughness and to increase the ride handling associated with the pitching and rolling movements. This necessitates a very fast and accurate controller to meet as much control objectives, as possible. Therefore, this paper deals with an artificial intelligence Neuro-Fuzzy (NF) technique to design a robust controller to meet the control objectives. The advantage of this controller is that it can handle the nonlinearities faster than other conventional controllers. The approach of the proposed controller is to minimize the vibrations on each corner of vehicle by supplying control forces to suspension system when travelling on rough road. The other purpose for using the NF controller for vehicle model is to reduce the body inclinations that are made during intensive manoeuvres including braking and cornering. A full vehicle nonlinear active suspension system is introduced and tested. The robustness of the proposed controller is being assessed by comparing with an optimal Fractional Order PIλDμ (FOPID) controller. The results show that the intelligent NF controller has improved the dynamic response measured by decreasing the cost function.

FPGA Based Adaptive Neuro Fuzzy Inference Controller for Full Vehicle Nonlinear Active Suspension Systems

The main objective of designed the controller for a vehicle suspension system is to reduce the discomfort sensed by passengers which arises from road roughness and to increase the ride handling associated with the pitching and rolling movements. This necessitates a very fast and accurate controller to meet as much control objectives, as possible. Therefore, this paper deals with an artificial intelligence Neuro-Fuzzy (NF) technique to design a robust controller to meet the control objectives. The advantage of this controller is that it can handle the nonlinearities faster than other conventional controllers. The approach of the proposed controller is to minimize the vibrations on each corner of vehicle by supplying control forces to suspension system when travelling on rough road. The other purpose for using the NF controller for vehicle model is to reduce the body inclinations that are made during intensive manoeuvres including braking and cornering. A full vehicle nonlinear active suspension system is introduced and tested. The robustness of the proposed controller is being assessed by comparing with an optimal Fractional Order PIλDμ (FOPID) controller. The results show that the intelligent NF controller has improved the dynamic response measured by decreasing the cost function.

New results on the synthesis of FO-PID controllers

In this paper a new procedure that allows to define the parameters of fractional order PIλDμ controller, designed to stabilize a first-order plant with delay-time, is proposed. Using a version of the Hermite–Biehler theorem applicable to quasipolynomials, the complete set of stabilizing PIλDμ parameters is determined. The widespread industrial use of PID controllers and the potentiality of their non-integer order representation justifies a timely interest to PIλDμ tuning techniques.

Evaluation of a vehicle directional control with a fractional order PDμ controller

In this paper, a novel controller, the fractional order PDμ controller, is designed to improve the performance of the driver-vehicle system. First, fractional calculus and fractional order PIλDμ controller are introduced. A control algorithm for vehicle directional control using the fractional order PDμ controller is then presented. Based on preview-follower theory, the on-line tuning method of the fractional order PDμ controller is designed. By comparing simulated and experimental results, the validity and robustness of the proposed fractional order PDμ controller in the closed loop system are verified. Finally, comprehensive evaluations are performed between the systems with the fractional order PDμ controller and with an integer PD controller. The results demonstrate that the use of the fractional order controller leads to an improvement of the performance of the driver-vehicle system.