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

The 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 PI^λD^μ (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.     

Multi‐objective robust fuzzy fractional order proportional–integral–derivative controller design for nonlinear hydraulic turbine governing system using evolutionary computation techniques

The present paper proposes a novel multi‐objective robust fuzzy fractional order proportional–integral–derivative (PID) controller design for nonlinear hydraulic turbine governing system (HTGS) by using evolutionary computation techniques. The fuzzy fractional order PID (FOPID) controller takes closed loop error and its fractional derivative as inputs and performs fuzzy logic operations. Then, it produces the output through the fractional order integrator. The predominant advantages of the proposed controller are its capability to handle complex nonlinear processes like HTGS in heuristic manner, due to fuzzy incorporation and extending an additional flexibility in tuning the order of fractional derivative/integral terms to enhance the closed loop performance. The present work formulates the optimal tuning problem of fuzzy FOPID controller for HTGS as a multi‐objective one instead of a traditional single‐objective one towards satisfying the conflicting criteria such as less settling time and minimum damped oscillations simultaneously to ensure the improved dynamic performance of HTGS. The multi‐objective evolutionary computation techniques such as non‐dominated sorting genetic algorithm‐II (NSGA‐II) and modified NSGA‐II have been utilized to find the optimal input/output scaling factors of the proposed controller along with the order of fractional derivative/integral terms for HTGS system under no load and load turbulence conditions. The performance of the proposed fuzzy FOPID controller is compared with PID and FOPID controllers. The simulations have been conducted to test the tracking capability and robust performance of HTGS during dynamic set point changes for a wide range of operating conditions and model parameter variations, respectively. The proposed robust fuzzy FOPID controller has ensured better fitness value and better time domain specifications than the PID and FOPID controllers, during optimization towards satisfying the conflicting objectives such as less settling time and minimum damped oscillations simultaneously, due to its special inheritance of fuzzy and FOPID properties.     

Control quality enhancement using fractional PIλDμ controller

The proportional–integral–derivative (PID) controllers have remained, by far, the most commonly and practically used in all industrial feedback control applications; therefore, there is a continuous effort to improve the system control quality performances. More recently Podlubny has proposed the fractional PI^λD^μ controller, a generalisation of the classical PID controller, involving an integration action of order λ and differentiation action of order μ. Since then, many researchers have been interested in the use and tuning of this type of controller. In this article, a new conception method of this fractional PI^λD^μ controller is considered. The basic ideas of this new tuning method are based, in the first place, on the classical Ziegler–Nichols tuning method for setting the parameters of the fractional PI^λD^μ controller for λ = μ = 1, which means setting the parameters of the classical PID controller, and on the minimum integral squared error criterion by using the Hall–Sartorius method for setting the fractional integration action order λ and the fractional differentiation action order μ. Illustrative examples and simulation results are presented to show the control quality enhancement of this proposed fractional PI^λD^μ controller conception method compared to the PID controller conception using Ziegler–Nichols tuning method.     

Robust adaptive control of a bio-inspired robot manipulator using bat algorithm

This paper proposes a novel adaptive fractional order PID sliding mode controller (AFOPIDSMC) using a Bat algorithm to control of a Caterpillar robot manipulator. A fractional order PID (FOPID) control is applied to improve both trajectory tracking and robustness. Sliding mode controller (SMC) is one of the control methods which provides high robustness and low tracking error. Using hybridization, a new combined control law is proposed for chattering reduction by means of FOPID controller and high trajectory tracking through using SMC. Then, an adaptive controller design motivated from the SMC is applied for updating FOPID parameters. A metaheuristic approach, the Bat search algorithm based on the echolocation behavior of bats is applied for optimal design of the Caterpillar robot in order to tune the parameter AFOPIDSMC controllers (BA-AFOPIDSMC). To study the effectiveness of Bat algorithm, its performance is compared with five other controllers such as PID, FOPID, SMC, AFOPIDSMC and PSO-AFOPIDSMC. The stability of the AFOPIDSMC controller is proved by Lyapunov theory. Numerical simulation results completely indicate the advantage of BA-AFOPIDSMC for trajectory tracking and chattering reduction.     

Robust Controller Design And Pid Tuning For Multivariable Processes

In this paper, a robust controller design method is first formulated to deal with both performance and robust stability specifications for multivariable processes. The optimum problem is then dealt with using a loop‐shaping H_∞ approach, which gives a sub‐optimal solution. Then a PID approximation method is proposed to reduce a high‐order controller. The whole procedure involves selecting several parameters and the computation is simple, so it serves as a PID tuning method for multivariable processes. Examples show that the method is easy to use and the resulting PID settings have good time‐domain performance and robustness.     

Design of a fractional order PID controller for an AVR using particle swarm optimization

Application of fractional order PID (FOPID) controller to an automatic voltage regulator (AVR) is presented and studied in this paper. An FOPID is a PID whose derivative and integral orders are fractional numbers rather than integers. Design stage of such a controller consists of determining five parameters. This paper employs particle swarm optimization (PSO) algorithm to carry out the aforementioned design procedure. PSO is an advanced search procedure that has proved to have very high efficiency. A novel cost function is defined to facilitate the control strategy over both the time-domain and the frequency-domain specifications. Comparisons are made with a PID controller and it is shown that the proposed FOPID controller can highly improve the system robustness with respect to model uncertainties.     

Design and Robust Performance Evaluation of a Fractional Order PID Controller Applied to a DC Motor

This paper proposes a methodology for the quantitative robustness evaluation of PID controllers employed in a DC motor. The robustness analysis is performed employing a 23 factorial experimental design for a fractional order proportional integral and derivative controller (FOPID), integer order proportional integral and derivative controller (IOPID) and the Skogestad internal model control controller (SIMC). The factors assumed in experiment are the presence of random noise, external disturbances in the system input and variable load. As output variables, the experimental design employs the system step response and the controller action. Practical implementation of FOPID and IOPID controllers uses the MATLAB stateflow toolbox and a NI data acquisition system. Results of the robustness analysis show that the FOPID controller has a better performance and robust stability against the experiment factors.      

Optimum Design of Fractional Order Pid Controller Using Chaotic Firefly Algorithms for a Control CSTR System

A fractional‐order PID controller is a generalization of a standard PID controller using fractional calculus. Compared with the standard PID controller, two adjustable variables, “differential order” and “integral order”, are added to the PID controller. Fractional‐order PID is more flexible, has better responses, and the precise adjustment closed‐loop system stability region is larger than that of a classic PID controller. But the design and stability analysis is more complicated than for the PID controller. Therefore, the optimal setting of parameters is very important. A firefly algorithm in standard mode has only local optimization and accuracy is low. In order to fix this flaw an improved chaotic algorithm firefly is proposed for a design controller FOPID. To evaluate the performance of the proposed controller, it has been used in the control of a CSTR system with a variety of fitness functions. Simulations confirm the optimal performance of the proposed controller.     

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

This paper presents a new design procedure to tune the fractional order PI^λD^μ 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 PI^λD^μ controller in the (k_p,k_i,k_d) space. Hence, this Theorem and Lemma allow us to develop an algorithm for solving the PI^λD^μ stabilization problem of the closed loop plant. The proposed approach has been verified by numerical simulation that confirms the effectiveness of the procedure.     

Optimum design of fractional order PID 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.     

A novel fractional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices

A novel fractional order (FO) fuzzy Proportional-Integral-Derivative (PID) controller has been proposed in this paper which works on the closed loop error and its fractional derivative as the input and has a fractional integrator in its output. The fractional order differ-integrations in the proposed fuzzy logic controller (FLC) are kept as design variables along with the input–output scaling factors (SF) and are optimized with Genetic Algorithm (GA) while minimizing several integral error indices along with the control signal as the objective function. Simulations studies are carried out to control a delayed nonlinear process and an open loop unstable process with time delay. The closed loop performances and controller efforts in each case are compared with conventional PID, fuzzy PID and PI^λD^μ controller subjected to different integral performance indices. Simulation results show that the proposed fractional order fuzzy PID controller outperforms the others in most cases.     

PI‐PD controller design for time delay systems via the weighted geometrical center method

In this study, a PI‐PD controller tuning method is presented using the weighted geometrical center method, which is based on the calculation of the weighted geometric center of the stability region obtained by the stability boundary locus method. The proposed method for tuning of PI‐PD controller parameters (k_d,k_f,k_p and k_i ) is performed in three steps. In the first step, the (k_d,k_f) parameter region for the inner loop with PD controller is obtained, and then the weighted geometric center of this region is calculated. In the second step, the inner PD loop is reduced to a single block using the numerical values of (k_d,k_f) that are obtained in the first step. Then, the (k_p,k_i) values of the external loop with PI controller are determined by the same procedure. This tuning method has some advantages over other tuning methods in terms of simplicity and robustness. The simulation examples show that a PI‐PD controller designed using the proposed method provides good performance results when compared to other tuning methods presented in the literature.     

A graphical tuning method of fractional order proportional integral derivative controllers for interval fractional order plant

In this paper, a novel graphical tuning method of fractional order proportional integral derivative (FOPID) controllers is proposed for a given interval fractional order plant family. Firstly, an approach is presented to solve the problem of robustly stabilizing the interval fractional order plant using FOPID controller. Moreover, some alternative methods are developed to reduce the computational burden of the presented approach. The results obtained here are general and strict proofs are given on these results. Secondly, a new approach is presented to calculate the complete sets of FOPID controller parameters which guarantee the specified H_∞-norm constraint for the interval fractional order plant. The developed approach is convenient and flexible. Finally, a unified design framework is proposed. The aim of the unified design is to compute the biggest region which can simultaneously provide internal stability, maintain the classical gain and phase margin and guarantee the modern H_∞-norm constraint for the interval fractional order plant. Examples are followed to illustrate the design procedure.     

A genetic-multivariable fractional order PID control to multi-input multi-output processes

A multivariable fractional order PID controller is designed and to get suitable coefficients for the controller, a genetic algorithm with a new topology to generate a new population is proposed. The three parts of the genetic algorithm such as reproduction, mutation, and crossover are employed and some variations in the methods are fulfilled so that a better performance is gained. The genetic algorithm is applied to design FOPID controllers for a multivariable process and the results are compared with the responses of a H_∞ based multivariable FOPID controller. The simulation responses show that in all cases, the genetic-multivariable FOPID controller has suitable performance, and the output of the system has a smaller error. Also, in the proposed method, variations in one output have a smaller effect on another output which is shown the ability of the proposed method to overcome the interaction in the multivariable processes.     

PID Stabilization of Retarded‐Type Time‐Delay System

An analytical method is proposed to construct the stabilizing PID region of a retarded‐type time‐delay system, based on Pontryagin's results and a generalization of the Hermite‐Biehler theorem. It is shown that the stable region in the (k_i, k_d)‐plane is made up of some convex polygons for a fixed k_p, and the whole region in the (k_p, k_i, k_d)‐space is comprised of some polyhedrons, each of which is mapped onto a real used string. Additionally, a method for determining the feasible k_p‐intervals is given in this paper. Two examples are employed to illustrate and verify the construction procedure of the stabilizing PID region in detail.     

Robust nonlinear ship course‐keeping control by H∞ I/O linearization and μ‐synthesis

In this paper, the H_∞ input/output (I/O) linearization formulation is applied to design an inner‐loop nonlinear controller for a nonlinear ship course‐keeping control problem. Due to the ship motion dynamics are non‐minimum phase, it is impossible to use the ordinary feedback I/O linearization to resolve. Hence, the technique of H_∞ I/O linearization is proposed to obtain a nonlinear H_∞ controller such that the compensated nonlinear system approximates the linear reference model in I/O behaviour. Then a μ‐synthesis method is employed to design an outer‐loop robust controller to address tracking, regulation, and robustness issues. The time responses of the tracking signals for the closed‐loop system reveal that the overall robust nonlinear controller is able to provide robust stability and robust performance for the plant uncertainties and state measurement errors. Copyright © 2002 John Wiley & Sons, Ltd.     

Particle swarm optimization-based fixed-structure <Emphasis Type="Italic">ℋ</Emphasis><Subscript>∞</Subscript> control design

In this paper, a new robust fixed-structure ℋ _∞ controller design based on the Particle Swarm Optimization (PSO) technique is proposed. The optimization-based structured synthesis problem is formulated and solved by a constrained PSO algorithm. In the proposed approach, the ℋ _∞ controller’s structure is selectable. PI and PID controller structures are especially adopted. The case study of an electrical DC drive benchmark is adopted to illustrate the efficiency and viability of the proposed control approach. A comparison to another similar evolutionary algorithm, such as Genetic Algorithm Optimization (GAO), shows the superiority of the PSO-based method to solve the formulated optimization problem. Simulations and experimental results show the advantages of simple structure, lower order and robustness of the proposed controller.     

基于BF-PSO的船舶电站柴油机分数阶控制器

针对船舶电力系统的频率稳定性问题,对船舶电站柴油机调速系统设计了分数阶PIλDμ控制器。采用细菌觅食-粒子群混合优化(BF-PSO)算法对分数阶PIλDμ控制器参数进行优化整定,解决了分数阶PIλDμ控制器整定参数多、设计复杂的问题。对分别采用分数阶PIλDμ控制器和传统整数阶PID控制器的柴油机调速系统进行了仿真和对比。结果表明,在同等条件下优化得到的分数阶PIλDμ控制器能够有效抑制模型参数摄动,鲁棒性更强,具有更好的控制效果。     

Fractional-order PID controller tuning using continuous state transition algorithm

Theoretical and applied studies of fractional-order PI\(^{\lambda }\)D\(^{\mu }\) (FOPID) controller in many scientific and engineering fields have shown many advantages compared to the classical PID control. However, the adjustment of FOPID controller becomes more complicated due to two additional parameters. In this study, the FOPID controller adjustment problem is transformed into a nonconvex optimization problem, and then a new metaheuristic method, named state transition algorithm (STA), is introduced to select the optimal FOPID controller parameters. In the meanwhile, the influence of objective criterion and sample size on the performance of FOPID controller design is analyzed. The dominance of the proposed method, especially for tuning FOPID controller parameters, is attested by several simulation cases and the comparisons of STA with other stochastic global optimization algorithms over the same problems.

     

2D Bilinear programming for robust PID/DD controller design

A new design method of PID structured controllers to achieve robust performance is developed. Both robust stabilization and performance conditions are losslessly expressed by bilinear constraints in the proportional‐double derivative variable ( k _P, k _DD) and the integral‐derivative variable ( k _I, k _D). Therefore, the considered control design can be efficiently solved by alternating optimization between ( k _P, k _DD) and ( k _I, k _D), which is a 2D computationally tractable program. The proposed method works equally efficiently whenever even higher order differential or integral terms are included in PID control to improve its robustness and performance. Numerical examples are provided to show the viability of the proposed development. Copyright © 2016 John Wiley & Sons, Ltd.