Dingyu Xue

Fractional order control - A tutorial

Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.

Linear feedback control : analysis and design with MATLAB

Preface 1. Introduction to feedback control 2. Mathematical models of feedback control systems 3. Analysis of Linear control systems 4. Simulation analysis of nonlinear systems 5. Model based controller design 6. PID controller design 7. Robust control systems design 8. Fractional-order controller - an introduction Appendix. CtrlLAB: a feedback control system analysis and design tool Bibliography Index of MATLAB functions Index.

A fractional order PID tuning algorithm for a class of fractional order plants

Fractional order dynamic model could model various real materials more adequately than integer order ones and provide a more adequate description of many actual dynamical processes. Fractional order controller is naturally suitable for these fractional order models. In this paper, a fractional order PID controller design method is proposed for a class of fractional order system models. Better performance using fractional order PID controllers can be achieved and is demonstrated through two examples with a comparison to the classical integer order PID controllers for controlling fractional order systems.

Fractional order PID control of a DC-motor with elastic shaft: a case study

In this paper, a fractional order PID controller is investigated for a position servomechanism control system considering actuator saturation and the shaft torsional flexibility. For actually implementation, we introduced a modified approximation method to realize the designed fractional order PID controller. Numerous simulation comparisons presented in this paper indicate that, the fractional order PID controller, if properly designed and implemented, will outperform the conventional integer order PID controller

A Modified Approximation Method of Fractional Order System

Many real world systems, including smart mechatronics systems, can be better characterized by dynamic systems of non-integer order. Using non-integer order or fractional order calculus, fractional order systems can be modeled more authentically. Due to the nature of the infinite dimensional model, proper approximations to fractional order differentiator (salpha, alpha isin Ropf) are fundamentally important. This paper contributed a new approximation scheme which is an extension of the well-established Oustaloups approximation method. The benefits from using the proposed scheme are illustrated by numerical examples in both time and frequency domains

Robust controllability of interval fractional order linear time invariant systems

We consider uncertain fractional-order linear time invariant (FO-LTI) systems with interval coefficients. Our focus is on the robust controllability issue for interval FO-LTI systems in state-space form. We revisit the controllability problem for the case when there is no interval uncertainty. It turns out that the controllability check for FO-LTI systems amounts to checking the controllability of conventional integer order state space. Based on this fact, we further show that, for interval FO-LTI systems, the key is to check the linear dependency of a set of interval vectors. Illustrative examples are presented.

Digital Fractional Order Savitzky-Golay Differentiator

This brief proposes a design method for a digital fractional order Savitzky-Golay differentiator (DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order for estimating the fractional order derivative of the contaminated signal. The proposed method calculates the moving windows weights using the polynomial least-squares method and the Riemann-Liouville fractional order derivative definition, and then computes the fractional order derivative of the given signal using the convolution between the weights and the signal, instead of the complex mathematical deduction. Hence, the computation time is greatly improved. Frequency-domain analysis reveals that the proposed differentiator is essentially a fractional order low-pass filter. Experiments demonstrate that the proposed DFOSGD can accurately estimate the fractional order derivatives of both noise-free signal and contaminated signal.

Recognition and location of the internal corners of planar checkerboard calibration pattern image

The recognition and location of the internal corners of a planar checkerboard calibration pattern image is very important to camera calibration. An effective approach is proposed to automatically recognize and locate the internal corners of the planar checkerboard calibration pattern image based on the characteristics of local intensity and the grid line architecture of the planar checkerboard pattern image. The proposed procedure consists of the detection of image corners, the recognition of the corners at the intersections of black and white squares and the recognition of the corners at the intersections of two groups of grid lines. Experiments show that compared with the commonly used interactive method, the proposed approach obviously reduces the time cost for camera calibration, speeds up calibration process and is especially adapted for automatic calibration based on multiple images.

Real-time controlling of inverted pendulum by fuzzy logic

Inverted pendulum is a typical multivariable, nonlinear, strong-coupling, instable system. The basic aim of our work was to balance a real pendulum in the position in center of course. For this purpose we used fuzzy logic controller. The fuzzy logic controller designed in the Matlab-Simulink environment. In this paper, the inverted pendulum mathematical model is built. MATLAB based Hardware in Loop simulation system is designed. A novel expert fuzzy control scheme was proposed. The proposed control scheme was implemented in Matlab and showed good performance in the real-time fuzzy control of the inverted pendulum. The results of simulation and experiment indicated that the control method has good control ability.

Design of an optimal fractional-order PID controller using multi-objective GA optimization

The fractional-order PID controller provides more adjustable parameters in the controller optimization than conventional PID controller. Therefore, FOPID is designed to achieve more goals, and multi-objective optimization based genetic algorithm is adopted. To solve this problem, a multi-objective optimization design method is proposed in this paper. Not only the robust performance, but also frequency angle margin, overshoot and rise time are all taken as the objectives to optimize. Then a variant of NSGA-II reach the optimal solution. This method can obtain uniformly distributed Pareto-optimal solutions and have good convergence and excellent robustness. The satisfactory solution is selected in Pareto optimum solution set according to the system requirement, which provides an effective tool for trade-off among the performance of quickness, stability and robustness. Simulation results support the superiority and effectiveness of the proposed method.