PDμ Controller Research Articles

Design of PIλ-PDμ controller for high-order systems based on model order reduction using BB-BC and time moment matching

In this paper, a practical fractional proportional–integral and proportional–derivative (PIλ-PDμ) controller tuning method and a model reduction method are proposed for high-order systems with or without time delay. First, the Big Bang–Big Crunch (BB-BC) algorithm and the time moment matching method are used to reduce the order of the high-order system function, and the approximate low-order system function is obtained. Second, the parameter region of fractional-order PIλ-PDμ controller is obtained according to the D partition method, and the improved weighted geometric center (WGC) method is introduced to select the special points in the stable region as the control parameters. Then, the fractional-order PIλ controller is discretized by Muir method, and the fractional-order PDμ controller is discretized by continued fraction expansion (CFE). Finally, the BB-BC algorithm is used to optimize the control parameters, and the parameters with better control performance are obtained. The simulation results show that the fractional PIλ-PDμ parameter setting method based on BB-BC algorithm can improve the control performance of the system, and the model reduction scheme based on the combination of moment matching method and BB-BC algorithm is effective in both time domain and frequency domain analysis.

A novel design methodology on cascaded fractional order (FO) PI-PD control and its real time implementation to Cart-Inverted Pendulum System

The tune up of fractional order (FO)-PIλDμ controllers is largely advocated in available literature which mostly requires evolutionary optimization techniques or solution of complex non-linear equations thereby leading to lack of practical solutions in terms of selecting the controller parameters. It may be quite feasible if one can consider PID control in the form of cascaded PI−PD control with which the task of attainment of the control objectives can be accomplished methodically. In extension to this, studies related to FO PIλ−PDμ controller are not too exhaustive. Generally, graphical methods are adopted to compute the controller parameters. There is still paucity of comprehensive design algorithms for these controllers to which end this work comes up with a novel and simple specification driven design methodology of the cascaded structures of FO PIλ−PDμ and [PI]λ−[PD]μ controllers following the classical control theory Keeping away from the complex and implicit design strategies they assist with an exact and unique solution to design these FO controllers in frequency domain with satisfactory dynamic performance in time domain which may not be possible with the FO-PIλDμ controllers. Illustrative examples and experimental validations aid to substantiate the reliability of the proposed method.

A Look-Up Table Based Fractional Order Composite Controller Synthesis Method for the PMSM Speed Servo System

Considering the performance requirements in actual applications, a look-up table based fractional order composite control scheme for the permanent magnet synchronous motor speed servo system is proposed. Firstly, an extended state observer based compensation scheme was adopted to suppress the motor parametric uncertainties and convert the speed servo plant into a double-integrator model. Then, a fractional order proportional-derivative (PDμ) controller was adopted as the speed controller to provide the optimal step response performance for the servo system. A universal look-up table was established to estimate the derivative order of the PDμ controller, according to the optimal samples collected by an improved differential evolution algorithm. With the look-up table, the optimal PDμ controller can be tuned analytically. Simulation and experimental results show that the servo system using the composite control scheme can achieve optimal tracking performance and has robustness to the motor parametric uncertainties and disturbance torques.

Robust Motion Control Using Combined Centralized Non-Integer Pre-Filter of Type FBLFD and Fractional Order Pdμ Controller

Designing and automation of a robust controller and pre-filter for multi-variable processes are a very challenging task. In this study, an automated and simple conception of multi- variable QFT (QFT: Quantitative Feedback Theory) is proposed using fractional order controllers design. Contrary to the traditional manual QFT design, our methodology consists of obtaining all required performances in QFT without going to the QFT loop shaping process.A non-integer order proportional derivative controller PDμ is conceived for a robust control of MIMO (Multi Input Multi Out- put) systems through multi-objective optimization based genetic algorithm . In the designed approach an implicit fractional order pre-filter FBLFD (FBLFD : Frequency Band Limited Fractional Differentiator) is adopted and its parameters are optimized. A diagonal version of this fractional order pre-filter and a non diagonal one are proposed. The problem of loop interactions is eliminated by the non diagonal designed pre-filter. Consequently, the procedure of fractional controller and pre-filter parameters optimization is illustrated. A comparative study and the efficiency of the developed methodologies are considered through SCARA robot.

Dynamics of a Modified Leslie-Gower Model with Crowley-Martin Functional Response and Prey Harvesting

This paper considers a mariner under PDμ controller and analyses the effects of controller parameters on the yaw rate by using the Nomoto model. The Nomoto model describing the time evolution of the yaw rate of the steering dynamics of a mariner is reduced to an asymmetric Duffing oscillator with fractional order derivative. Under the approximation of calm water, the steady behavior of the mariner shows an “imperfect” supercritical pitchfork bifurcation. Region of safe behavior is identified and strategy to reduce the yaw rate by an appropriated selection of controller parameters are discussed. The frequency analysis of the mariner shows the prominence of hysteresis is reduced for small order of the fractional derivative as well as the amplitude of the yaw rate. Evidence of chaotic response is illustrated using robust chaotic indicators such as the Lyapunov exponent and the fast Fourier transform.

Geometrical design of fractional PDμ controllers for linear time-invariant fractional-order systems with time delay

This article presents a simple procedure that allows a practical design of fractional –[Formula: see text] controllers for single-input single-output linear time-invariant fractional-order systems subject to a constant time delay. The methodology is based on a geometric approach, which provides practical guidelines to design stabilizing and non-fragile PDμ controllers. The simplicity of the proposed approach is illustrated by considering several numerical examples encountered in the control literature. Moreover, with the aim of showing the performance of the PDμ over a classical PD controller, both controllers were implemented at the end of the article in an experimental test-bench consisting a teleoperated robotic system.

Study on the Control Method of Mine-Used Bolter Manipulator Based on Fractional Order Algorithm and Input Shaping Technology

Mine-used bolter is the main equipment to solve the imbalance of excavation and anchor in well mining, and the manipulator is the main working mechanism of mine bolt drilling rig. The manipulator positioning requires high rapidity and stability. For this reason, this paper proposes a composite control method of “input shaping + fractional order PDμ control”. According to the mathematical model of the valve-controlled cylinder, the fractional-order controller PDμ is developed. At the same time, the input shaping is used to feed forward the accurate positioning and path planning of the manipulator, which not only improves the robustness of the system, but also shortens the stability time of the system and restrains the maximum amplitude of the system vibration. In this paper, the control effects of fractional order PDμ controller and integer order PD controller are compared. The results show that the maximum amplitude of the control system is reduced by 75% and the stabilization time is reduced by 60% after using the fractional order PDμ controller, which fully reflects the superiority of the fractional order controller in response speed, adjusting time, and steady-state accuracy. Finally, the control effects of “input shaping + fractional order PDμ control” and fractional order PDμ controller on the stability of the system are compared. The maximum amplitude of the system was reduced by 50% by using “input shaping + fractional order PDμ control”. Numerical simulation confirms the feasibility and effectiveness of the composite control method. This composite control method provides theoretical support for the precise positioning of the manipulator, and the high stability and high safety of the manipulator also expand the application scope and depth of the composite control method.

Analytical criterion on stabilization of fractional-order plants with interval uncertainties using fractional-order PD[formula omitted] controllers with a filter

This paper proposes a stabilization criterion for a fractional-order plant involving interval uncertainties in the coefficients using fractional-order PDμ controller with a filter. The vertex functions of the value set corresponding to the characteristic function of the closed loop system are determined by the law of Minkowski sum. The finite test frequency interval is offered to simplify the calculation of the test for the stabilization. The analytical method to test the position relationship between the origin and the value set is investigated within the finite test frequency interval. Then, the stabilization criterion with an analytical method is provided based on the zero exclusion principle. Finally, three examples are given to verify the effectiveness of this criterion.

A simple tuning method of fractional order PIλ-PDμ controllers for time delay systems

In this paper, a practical tuning technique is presented to obtain all stabilizing fractional order PIλ-PDμ controller parameters ensuring stability for processes with time delay using the stability boundary locus and the weighted geometrical center (WGC) methods. The method is based on obtaining of stability regions plotted by using the stability boundary locus in the (kd,kf)-plane and (kp,ki)-plane, and then computing the weighted geometrical centers of these regions. After obtaining PDμ controller parameters using the WGC method from the stability region, desired PIλ controller parameters are computed by the same procedure. This paper provides a simple and efficient tuning method to obtain stabilizing parameters of PIλ-PDμ controller for time delay systems. The important advantages of the method are both calculating of controller parameters without using any complex solution methods and ensuring the stability of closed loop system. Illustrative examples are given to demonstrate the benefits and the simplicity of the proposed method.

Stabilizing region of PDμ controller for fractional order system with general interval uncertainties and an interval delay

This paper concentrates on computing the stabilizing regionof PDμ controller for fractional order system with general interval uncertainties and an interval delay. The stabilizing region means the complete/approximate set of PDμ controllers that stabilize the given closed-loop control system. General interval uncertainties refer to both coefficients and orders of the fractional system suffer from interval uncertainties. Interval delay indicates that the delay also vary in a specified interval.Firstly, a method is presented to calculate the stabilizing region for general interval fractional system with an intervaltime-constant delay. Based on a novel mapping function and the concept of critical controller parameters, the stabilizing region can be determined numerically. Secondly, the stabilizing region computation problem for general interval fractional system with an interval time-varyingdelay is considered. By applying a revised small-gain theorem, the stabilizing region can be calculated like the time-constant delay case. Thirdly, two alternative methods are proposed to improve the computational efficiency of stabilizing region calculation. Both methods can reduce the number of polynomials which are used to determine the stabilizing region. Examples are followed to illustrate the proposed results.

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

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

A Complex Variable Method to Predict a Range of Arbitrary Shape Ballistic Projectiles

This paper considers a mariner under PDμ controller and analyses the effects of controller parameters on the yaw rate by using the Nomoto model. The Nomoto model describing the time evolution of the yaw rate of the steering dynamics of a mariner is reduced to an asymmetric Duffing oscillator with fractional order derivative. Under the approximation of calm water, the steady behavior of the mariner shows an “imperfect” supercritical pitchfork bifurcation. Region of safe behavior is identified and strategy to reduce the yaw rate by an appropriated selection of controller parameters are discussed. The frequency analysis of the mariner shows the prominence of hysteresis is reduced for small order of the fractional derivative as well as the amplitude of the yaw rate. Evidence of chaotic response is illustrated using robust chaotic indicators such as the Lyapunov exponent and the fast Fourier transform.

Practical Tuning Algorithm of PDµ Controller for Processes with Time Delay

In this paper, a practical tuning algorithm of fractional order PD controller for processes with time delay using the weighted geometrical center (WGC) method is presented. This method is based on calculating of the stabilizing PDµ controller parameters region which is plotted using the stability boundary locus in the (kd,kp) plane and computing the weighted geometrical center of stability region.The important advantages of the proposed method are both calculating of controller parameters without using complex graphical methods and ensuring the stability of closed loop system. From the examples, it can be easily seen that this simple tuning method can perform quite reliable results in that unit step response.

Design and Implementation of Fractional Order Controller for Service Robots

In this paper, the fractional order motion control is designed basing on the Leader IDX service robotic platform, where the implementation of robust fractional order controller is introduced in details. The fractional order PDμ control is analyzed in frequency domain by using an FIR/IIR method. The system model of the motor of Leader I-DX is established in MATLAB so that the influence of μ to the whole control system can be verified, and the comparisons of PD and PDμ controllers are shown as well. Motivated by the better performance of fractional order controllers, a digital implementation of fractional order PDμ controllers using PRONY technique is applied so that the incremental fractional order controllers can be realized. The real experiments on Leader I-DX platform are illustrated to validate the above concepts.

Robust fractional-order proportional-derivative control of unified chaotic systems with parametric uncertainties

In this paper, a robust fractional-order proportional-derivative (PDμ) control is designed for controling in integer-order unified chaotic systems with parametric uncertainties. Equivalent plant is obtained by transforming the controlled dynamic system, and then the PDμ controller as an equivalent controller is applied to the equivalent plant. In the uncertain controlled unified chaotic systems, one equation is certain, and the other two equations are uncertain . The equivalent controller for the certain system is then designed based on a fractional-order proportional-derivative controller, in which three specifications for phase margin, gain crossover frequency, and robustness should be met. On the other hand, the robustness of uncertain systems is achieved by an improved Monje-Vinagre tuning method, however, the pre-specified frequency band should be replaced by the gain crossover frequency in order to reduce the complexity in determining the controllers for the uncertain systems. Specifications related to phase margin for the lower bound of the phase, gain crossover frequency for the upper bound of the gain, and robustness for the lower bound of the phase constraints are satisfied by the uncertain system. Parameters of the equivalent controller are determined based on a graphical method. Origins of the unstable equilibrium can be asymptotically stabilized by the proposed strategy for the integer-order unified chaotic systems with parametric uncertainties. Numerical simulation examples for Chen system, L system, and Lorenz system, are given to show the effectiveness of the proposed method.

Fractional-order system identification and proportional-derivative control of a solid-core magnetic bearing

This paper presents the application of fractional-order system identification (FOSI) and proportional-derivative (PDµ) control to a solid-core magnetic bearing (MB). A practical strategy for closed-loop incommensurate FOSI along with a modified error criterion is utilized to model the MB system and a corresponding, verification experiment is carried out. Based on the identified model, integer-order (IO) PD and fractional-order (FO) PDµ controllers are designed and compared with the same specifications. Besides, the relation between the two categories of controllers is discussed by their feasible control zones. Final simulation and experimental results show that the FO PDµ controller can significantly improve the transient and steady-state performance of the MB system comparing with the IO PD controller.

Robust stabilizing regions of fractional-order PDμ controllers of time-delay fractional-order systems

This study investigates the robust stabilizing regions with stability degrees of fractional-order PDμ controllers for time-delay fractional-order systems. By the D-decomposition technology, we identify the stabilizing regions by three types of curves, i.e., the real root boundary (RRB) curves, complex root boundary (CRB) curves and infinite root boundary (IRB) lines. The existence conditions and computing methods of RRB curves, CRB curves and IRB lines are proposed to determine the boundaries of the potential stabilizing regions. The Test Lines and the principle of the identifying the stabilizing regions are presented to find the real stabilizing regions with a given stability degree. To deal with noises existing in the feedback signals, fractional-order PDμ controllers involving filers are adopted. Meanwhile, the robust stabilizing regions are also analyzed via IRB curves, CRB curves and IRB lines with stability degrees. Finally, some illustrative examples are offered to verify the effectiveness of depicting algorithms of the robust stabilizing regions for PDμ controllers with no filer or filers, respectively.

Formula omitted] design with fractional-order [formula omitted] controllers

This article focuses on H∞ performance design for fractional-order systems with time-delay, using fractional-order proportional-derivative (PDμ) controllers. First, the stabilizing parameters region in proportional-derivative plane of PDμ controller, for a fixed derivative-order, is determined in terms of a graphical stability criterion applicable to fractional-order time-delay systems. Then, in the stabilizing region, the pairs of proportional and derivative gains of PDμ controller are calculated for a range of frequencies, which satisfy the H∞-norm constraint of complementary sensitivity function and define the H∞ boundary curve. Finally, by changing the derivative-order of PDμ controller, we observe the relationship between the H∞ curve and the derivative-order. Examples are followed to illustrate the design procedure.

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.