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.     

Analytical design of fractional order PID controllers based on the fractional set-point weighted structure: Case study in twin rotor helicopter

In this paper, an analytical method for tuning the parameters of the set-point weighted fractional order PID (SWFOPID) controller is proposed. The studied control scheme is the filtered fractional set-point weighted (FFSW) structure. Also to achieve a desired closed-loop performance, a fractional order pre-filter is employed. The proposed method is applicable to stable plants describable by a simple three-parameter fractional order model. Such a model can be considered as the fractional order counterpart of a first order transfer function without time delay. Finally, the proposed method is implemented on a laboratory scale CE 150 helicopter platform and the results are compared with those of applying a filtered fractional order PI (FFOPI) controller in a similar structure. The practical results show the effectiveness of the proposed method.     

Hybridized ABC-GA optimized fractional order fuzzy pre-compensated FOPID control design for 2-DOF robot manipulator

The robotic manipulator is an extremely nonlinear, multi-input multi-output (MIMO), highly coupled, and complex system wherein the parameter uncertainties and external disturbances adversely affect the performance of this system. From this, it necessitates that the controllers designed for such system must overcome these complexities. In this paper, we develop a novel fractional order fuzzy pre-compensated fractional order PID (FOFP-FOPID) controller for 2-degree of freedom (2-DOF) manipulator dealing with trajectory tracking problem. In order to optimize the controller’s parameters while minimizing integral of time absolute error (ITAE), a metaheuristic optimization technique, viz., artificial bee colony-genetic algorithm (ABC-GA) is presented. The efficacy of our proposed controller is demonstrated by comparing it with some existing controllers, such as integer order fuzzy pre-compensated PID (IOFP-PID), fuzzy PID (FPID), and conventional PID controllers. Furthermore, the robustness analysis for proposed controllers is also investigated for parameter variations and external disturbances. The simulation results indicate that FOFP-FOPID controller can not only guarantee the best trajectory tracking but also ameliorate the system robustness for parameter variations as well as external disturbances.     

Robustness evaluation of fractional order control for varying time delay processes

In this paper, we investigate the robustness of a methodology to design fractional order PI controllers combined with Smith Predictors, for varying time delay processes. To overcome the drawback of possible instability associated with Smith Predictor control structures, mainly due to the changes in the time delay, the design focuses on ensuring robustness of the closed loop system against time delay uncertainties. The proposed method is based on time-domain performance specifications??more accessible to the process engineer, rather than the more abstract notions related to the frequency domain. A second advantage of the proposed method relies on additional robustness to plant uncertainties, achieved by maximizing open-loop gain margin. The convergence problems associated with optimization techniques, previously used in fractional order controller designs, are eliminated by an iterative procedure in computing the gain margin. The simulation example provided demonstrates the efficiency of the proposed method, in comparison to classical integer order PI controller.     

Development and implementation of an FPGA based fractional order controller for a DC motor

Fractional calculus has been gaining more and more popularity in control engineering in numerous fields, including mechatronic applications. One of the most common applications in all mechatronic domains is the control of DC motors. Several control algorithms have been proposed for such motors, ranging from traditional PID algorithms, to the more sophisticated advanced methods, including fractional order controllers. Nevertheless, very little information regarding the implementation problems of such fractional algorithms exists today. The paper proposes a simple approach for designing a fractional order PI controller for controlling the speed of a DC motor. The resulting controller is implemented on an FPGA target and its performance is compared to other possible benchmarks. The experimental results show the efficiency of the designed fractional order PI controller. Beside the initial DC motor, two other different DC motors are also used in the experiments to demonstrate the robustness of the controller.     

Fractional calculus in 13C separation column control

Controller design for an isotope separation column is recognized as a difficult and challenging problem. The dynamics of the isotope separation process is difficult to model precisely using integer order transfer functions; thus, a fractional order approach is preferred. The objective of this work is to design two different PI controllers??a classical one and a fractional order one??and test their closed loop performance under nominal conditions as well as gain uncertainties. Since the process is represented by a fractional order mathematical model, the simplest approach to design both controllers is based on a frequency specification. For the fractional order of the PI controller and its parameters, the authors solve a system of equations that includes a robust performance specification to gain uncertainties. For the classical PI controller, a traditional tuning algorithm based on phase margin specification is implemented. The simulation results show that both controllers meet the design specifications, with the fractional order PI controller behaving more robustly to plant gain variations.     

Real time implementation of fractional order PID controllers for a magnetic levitation plant

Fractional calculus has been a topic of great interest for the last few decades. The applications of fractional calculus can be found in the area of viscoelastic and chaotic systems, whose dynamics is expressed in the form of fractional differential equations. The ongoing research work is based on the design of 1-Degree of Freedom (1-DOF) and 2-Degrees of Freedom (2-DOF) Fractional Order PID (FOPID) controllers for a Magnetic levitation (Maglev) plant and the performance has been compared with that of 1-DOF and 2-DOF Integer Order PID (IOPID) controllers in both simulation and real time. The Degree of Freedom (DOF) represents the number of feed-forward control loops in a closed loop system. A 2-DOF controller configuration comprises of a serial compensator and a feed-forward compensator in a closed loop structure. An FOPID controller has a structure similar to that of a conventional IOPID controller, except that its derivative and integral orders are fractional numbers. The design of such a controller requires the determination of five parameters: K_p, K_i, K_d, α and β, where α and β are the derivative and integral orders of the FOPID controller. The controller design problem has been framed as an optimization problem, in which the cost function is formulated from the characteristic equation of the closed loop system at dominant poles that are identified from the given performance specifications. The closed loop response shows that the proposed2-DOF FOPID controller exhibits superior response and robustness with respect to its integer order counterpart.     

Fractional order modeling and control for permanent magnet synchronous motor velocity servo system

This paper presents the application of fractional order system on modeling the permanent magnet synchronous motor (PMSM) velocity servo system. The traditional integer order model of the PMSM velocity system is extended to fractional order one in this work. In order to identify the parameters of the proposed fractional order model, an integer order approximation of the fractional order operator is applied and a state-space structure is presented for using the output-error identification algorithm. In real-time PMSM velocity servo plant, the fractional order model is identified according to some experimental tests using the presented algorithm. Two proportional integral (PI) controllers are designed for velocity servo using a simple scheme according to the identified fractional order model and the traditional integer order one, respectively. The experimental test performance using these two designed PI controllers is compared to demonstrate the advantage of the proposed fractional order model of the PMSM velocity system.     

Order and pole locator estimation in fractional order systems using bode diagram

This paper deals with estimation of fractional order and pole locator in fractional order systems. The estimation is based on Bode diagram of the system that is obtained using input and output measurements. Here the magnitude diagram is approximated with number of straight lines depending on the level of complexity and in consequence a very good estimation of fractional order and acceptable approximations of pole locators are determined. Relying on the proposed method, complexity of fractional order system identification which is mostly due to the estimation of fractional order is substantially resolved. Some example simulation results are provided to explain the work and show its effectiveness.     

分数阶控制在位置随动系统中的应用研究

位置随动系统对响应速度、定位精度、抗扰动性能等要求越来越高,整数阶PID 控制实现容易,但性能有限,很难满足高要求。分数阶PID(FOPID)控制由于引入了积分和微分两个阶次参数,可控参数更多更灵活。文中针对位置随动系统引入分数阶PI 控制,首先,利用分数阶微积分的数值计算方法(irid_fod 法)给出了差分方程;然后,在MATLAB 环境下利用粒子群寻优算法,分别整定随动系统整数阶PI 及分数阶PI 最佳控制参数;接着,进行了定位控制及抗扰动性能的仿真与实物试验。结果表明:位置随动系统中,分数阶PI 控制较整数阶PI 控制定位快、抗扰动性能强,具有一定的工程应用价值。     

模糊PID控制系统的设计与研究

为了解决工程中二阶系统的控制问题,在此对PID控制与模糊控制的原理进行了研究,并将二者的优势相互结合,设计了一种具有参数自整定功能的模糊PID控制系统。对PID参数初值的确定,隶属度函数的选取,模糊控制规则表的设计做了较为深入的研究。并利用Matlab/Simulink软件对控制系统进行了仿真研究。对阶跃输入下PID控制系统与该文设计的模糊PID控制系统的响应情况做出了定量的比较。结果表明对于二阶延迟系统,模糊PID控制器的超调量与调节时间均小于传统的PID控制,能显著提高控制效果。     

Robust loop shaping-fuzzy gain scheduling control of a servo-pneumatic system using particle swarm optimization approach

In this paper, a new technique called robust loop shaping-fuzzy gain scheduled control (RLS-FGS) is proposed to design an effective nonlinear controller for a long stroke pneumatic servo system. In our technique, a nonlinear dynamic model of a long stroke pneumatic servo plant is identified by the fuzzy identification method and is used as the plant for our design. The structure of local controllers is selected as PID control which is proven by many research works that this type of control has many advantages such as simple structure, well understanding, and high performance. The proposed technique uses particle swarm optimization (PSO) to find the optimal local controllers which maximize the average stability margin. In addition, performance weighting function which is normally difficult to obtain is automatically determined by PSO. By the proposed technique, the RLS-FGS controller can be designed, and the structure of local controllers is still not complicated. As seen in the simulation and experimental results, our proposed technique is better than the classical gain scheduled PID controller tuned by pole placement and the conventional fuzzy PID controller tuned by ISE method in terms of robust performance.     

Synchronization of mutual coupled fractional order one-sided lipschitz systems

Up to now, the problem of synchronization of mutual coupled integer order systems has been tackled by several researchers. However, few works have been done to investigate such context on fractional order systems. In this context, this paper presents a complete methodology to solve this problem. On the other hand, to the best of our knowledge, among all the existing works dealing with the synchronization of mutual coupled fractional order systems, no paper has treated the special class of one-sided Lipschitz systems. In order to validate the theoretical results, two numerical examples are studied in the simulation section.     

On the distributed order PID controller

This paper proposes a generalized fractional controller that covers the complete range of differintegrators of order −1 to order 1. This controller is named distributed order PID controller (DOPID) due to the fact that it broadens the structure of the conventional PID controller. Furthermore, a new method for implementation of distributed order PID controller (DOPID), as well as analysis of its properties in both time and frequency domain are proposed. As DOPID controller represents generalized classical PID controller, it is approximated by a compound fractional controller with multiple fractional differintegrators, with multiple gains, connected in parallel. Orders of these differintegrators have been equally spaced, with the first one being the first order classical integrator, and the last one being the first order classical differentiator. Noise cancellation filter is considered explicitly in the controller’s structure.     

Stability preservation analysis in direct discretization of fractional order transfer functions

In this paper, a class of the direct methods for discretization of fractional order transfer functions is studied in the sense of stability preservation. The stability boundary curve is exactly determined for these discretization methods. Having this boundary helps us to recognize whether the original system and its discretized model are the same in the sense of stability. Finally, some illustrative examples are presented to evaluate achievements of the paper.     

Random-order fractional differential equation models

This paper proposes a new concept of random-order fractional differential equation model, in which a noise term is included in the fractional order. We investigate both a random-order anomalous relaxation model and a random-order time fractional anomalous diffusion model to demonstrate the advantages and the distinguishing features of the proposed models. From numerical simulation results, it is observed that the scale parameter and the frequency of the noise play a crucial role in the evolution behaviors of these systems. In addition, some potential applications of the new models are presented.     

Adaptive minimum energy cognitive lighting control: Integer order vs fractional order strategies in sliding mode based extremum seeking

To achieve comfortable illumination while minimizing the energy consumption in hybrid lighting, a minimum energy point tracking algorithm is developed to achieve the minimized energy usage despite of environmental variations in this paper. A hardware-in-the-loop prototype of an adaptive minimum energy cognitive lighting control is proposed, designed and built. A sliding mode based extremum seeking controller (SM–ESC) including integer order (IO) and fractional-order (FO) strategies is firstly employed to minimize energy usage in the lights, while a PID controller is applied to maintain a light level. The performance of the designed controller is compared with both fractional order and integer order controllers which are designed based the proposed lighting system. The hardware-in-the-loop experimental results are presented to demonstrate the practicality and effectiveness of the proposed cognitive lighting control scheme.     

Fuzzy Controllers With Maximum Sensitivity for Servosystems

In this paper, new Takagi-Sugeno proportional-integral-fuzzy controllers (PI-FCs) to control a class of servosystems are proposed. The controlled plants in these control systems (CSs) are of integral type. In the first phase, there are designed linear PI controllers tuned in terms of the extended symmetrical optimum method to ensure the imposed overshoot and settling time with respect to the set point and to three possible types of load disturbance inputs. The connections between the two design parameters of the linear controllers and the desired maximum sensitivity and complementary sensitivity considering one of the disturbance inputs are derived. Then, accepting the approximate equivalence between the fuzzy controllers and the linear ones in certain conditions and using the modal equivalence principle, an attractive design method for the PI-FCs is proposed. With this respect, the design method guarantees maximum imposed sensitivity and complementary sensitivity for the CSs and, therefore, good responses with respect to modifications of the set point and of the disturbance inputs, and robustness with respect to model uncertainties. An application in speed control of a nonlinear servosystem with variable load, accompanied by experimental results, is provided to validate the new results, the fuzzy controllers, and a design method     

Application of the Positive Reality Principle to Metal Electrode Linear Polarization Phenomena

The linear electrode-electrolyte polarization immittance (impedance or admittance) can be analyzed by the fractional power pole (FPP) model [38] based on the modification of Bode's [4] method to include functions with poles and/or zeros of fractional power. This is, in essence, an extension of the Davidson-Cole model [12] to the entire frequency spectrum. In this paper, the minimum phase property of the FPP model is verified and the positive reality principle [5] is used to identify the specific type of immittance function for which the FPP model is applicable. This condition is an extension of the "universal" law of dielectric response as advanced by Jonscher [26].     

Self-tuning PID controllers for dead time processes

A number of proportional-integral-derivative (PID) based self-tuning controllers exist for the control of difficult processes. A common weakness of these self-tuning PID controllers is their inability to cope with dead-time processes. Here, self-tuning controllers based on a pole-assignment approach, which can overcome fractional dead time, constant and known dead time, plus time-varying dead time, are presented. It is shown using the simulation and experimental results that the controllers work well in handling dead-time processes