Applied Fractional Calculus in Identification and Control

Table of Contents

1. Frontmatter

2. Chapter 1. Fractional Calculus for Multivariate Vector-Valued Function and Fractal Function

   C. Kavitha, T. M. C. Priyanka, Cristina Serpa, A. Gowrisankar

   Abstract

   This chapter explores the Katugampola fractional integral of a multivariate vector-valued function defined on \(\mathbb {R}^n\). Alongside, it is shown that the prescribed fractional operator preserves some analytical properties of the original function like continuity and boundedness. Further, this chapter discusses applying one of the fractional calculus formulations called the Weyl-Marchaud fractional derivative on the quadratic fractal interpolation function. The quadratic fractal interpolation function with variable scaling factors has been chosen for the study to elucidate the influence of scaling factors as variables on the fractal functions. By prescribing the initial conditions to the quadratic fractal interpolation function, its fractional derivative is analyzed.

3. Chapter 2. Synchronization of Stochastic Fractional Chaotic Systems

   T. Sathiyaraj, K. Ratnavelu, P. Balasubramaniam

   Abstract

   This chapter is involved with the synchronization of fractional-order (SFO) stochastic system in a finite-dimensional space, and we have tested its time reaction and stochastic chaotic behaviors. Firstly, we give a representation of the solution for a stochastic fractional-order chaotic system (CS). Secondly, some sufficient beneficial conditions are investigated using matrix-type Mittag-Leffler (M-L) function, Jacobian matrix via stochastic process, stability analysis, and feedback control technique to assure the synchronization of stochastic error system (ES). After that, numerical illustrations are provided to verify the theoretical parts.

4. Chapter 3. Fractional-Order Comb Filter Design For Power-Line Interference Removal

   Lipan Ehmes, Eric Vonseu, Sunil Narayan

   Abstract

   Signal filtering has gained much importance in our electronic run world. Every electronic device has some form of filtering component inside it. One important type of filter is the comb filter which is essential in several applications. This chapter introduces the fractional-order comb filter (FOCF) that can be utilized to filter power line interference specifically the harmonics that come with non-linear loads. The method of designing the FOCF is a non-conventional, but ingenious method. Instead of combining a low-pass with a high-pass filter to form the notches of the filter, the general second-order transfer function of the low-pass filter is tweaked and stabilized to create the individual notch filters. These notch filters are then converted to fractional form and cascaded in series to form the FOCF. The realized FOCF is essentially compared with the integer-order comb filter (IOCF). In addition, the chapter gives a sufficient discussion on the parameters selection and how they affect the response of the FOCF. The obtained results from the realization successfully demonstrated increased performance in the FOCF compared to the IOCF especially in the selectivity of frequencies.

5. Chapter 4. Practical Realization of Fractional-Order Notch Filter with Asymmetric Slopes and Optimized Quality Factor

   Sunil Narayan, Varian Akwai, Steven Weago, Kajal Kothari

   Abstract

   The chapter presents the implementation of a fractional-order notch filter (FONF). The design is presented by approximating the fractional capacitor and inductor in the realization of the filter circuit. The quality factors and the asymmetric slope magnitude responses of the FONF have been optimized using a well-known particle swarm optimization technique. After the optimized fractional-order \(\alpha \), the filter was verified in simulation and then verified practically on a configurable analogue module. Both results have shown an excellent result with magnitude slopes and quality factors of the filter. It can be seen from the design and implementation techniques that the proposed FONF filter can be physically built with less time and financial burden.

6. Chapter 5. Fractional Order Modified IMC for Integrating Processes for Given Stability Margins and Increased Closed Loop Bandwidth

   Pushkar Prakash Arya

   Abstract

   This chapter discusses the design of fractional order (FO) modified IMC (m-IMC) for integrating time delay processes. The m-IMC constitutes an additional controller in the basic IMC control loop, which broadens the applicability of IMC and can be used to enhance the performance and robustness of the close loop system. In this work, the additional controller is used to enhance the bandwidth of the close loop system. The IMC controller constitutes the minimum phase part of the plant model and a filter. In the proposed controller, a FO filter is chosen with two tuning parameters \(\lambda \) and \(\beta \), which are used to achieve desired gain margin (\(A_m\)) and phase margin (\(\phi _m\)) of the close loop system. The additional controller is chosen as PI with only one tuning parameter \(\chi \) and used to enhance the close loop bandwidth. Two types of integrating process models are considered, first, integrating plus time delay (IPTD), and second, second-order integrating plus time delay (SOIPTD) process. Simulation studies are carried out for nominal and uncertain plant models with 30% uncertainty for set-point tracking and disturbance rejection. The performance of the controller for set-point tracking is measured in terms of rise time \((T_r)\), settling time \((T_s)\), integral absolute error (IAE), and integral of time-weighted absolute error (ITAE). Whereas the disturbance rejection performance is measured in terms of IAE and ITAE and the control effort is measured in terms of time variance (TV).

7. Chapter 6. Internal Model Control-Based Fractional Order Controller Design for Process Plants Satisfying Desired Gain Margin and Phase Margin

   Pushkar Prakash Arya, Sohom Chakrabarty

   Abstract

   This chapter considers the design of a fractional order (FO) internal model controller (IMC) for first order plus time delay (FOPTD) processes to satisfy a given set of desired stability margins in terms of gain margin \((A_m)\) and phase margin \((\phi _{m})\). The highlight of the design is the choice of a fractional order (FO) filter in the IMC structure which has two parameters (\(\lambda \) and \(\beta \)) to tune as compared to only one tuning parameter (\(\lambda \)) for traditionally used integer order (IO) filter. These parameters are evaluated for the controller so that \(A_m\) and \(\phi _{m}\) can be chosen independently. This is the first time when the IMC controller is designed without any approximation of the delay in the IMC which has always been approximated using Taylor’s series approximation or Pade approximation. Simulation studies are carried out for three different FOPTD processes and the proposed methodology is validated experimentally on a DC servo-motor system. The results are compared with different well-known IMC techniques and the performance of controllers is evaluated in terms of rise time (\(T_r\)), settling time (\(T_s\)), maximum overshoot (\(\% M_p\)) and integral square error (ISE).

8. Chapter 7. Novel Hybrid Iterative Learning–Fractional Predicative PI Controller for Time-Delay Systems

   P. Arun Mozhi Devan, Fawnizu Azmadi Hussin, Rosdiazli Ibrahim, M. Nagarajapandian, Maher Assaad

   Abstract

   Iterative learning control (ILC) is well known for producing faster convergence using continuous output tracking based on the given reference plant model. This paper proposes a hybridization of the ILC with the fractional-order predictive PI (FOPPI) for time-delay processes. First, the design of L- and Q-filters for the ILC controller will be obtained using the process model. Then, the obtained filter coefficients are combined and fed to the feed-forward path after the error reduction and control signal conditioning. Finally, the performance of the proposed hybrid ILC-FOPPI controller is evaluated with the conventional PI, fractional-order PI (FOPI), and predictive PI (PPI) controllers over the time-delay processes. Simulation results obtained show the proposed technique’s effective improvement in faster settling and peak overshoot minimization.

9. Chapter 8. Design of Robust Model Predictive Controller for DC Motor Using Fractional Calculus

   Abhaya Pal Singh, Srikanth Yerra, Ahmad Athif Mohd Faudzi

   Abstract

   This chapter designs a Robust Fractional Model Predictive Controller (R-FMPC) to control the DC motor’s speed. The proposed controller is developed in two stages. First, the DC motor’s fractional-order model is derived by adopting the fractional definitions such as Oustaloup and fractional Laplacian. The exponent value in the developed DC motors fractional model is chosen by comparing the performances with various orders/values. Then, the model predictive controller is designed by utilizing the developed DC motors’ fractional model. Further, by varying the system dynamics, the robustness of the developed fractional-order model is also verified. Finally, a simulation study is carried out for evaluating the performance of fractional model predictive controller (FMPC) and R-FMPC for the DC motor control system.

10. Chapter 9. Studying Fractional-Order Controller Structures for Load Frequency Control of Interconnected Multiple Source Power System

    Vadan Padiachy, Muhammed Hafeez, Luvkesh Naidu

    Abstract

    This chapter exploits the merits of fractional-order controllers for the Load Frequency Control (LFC) problem. In particular, the slight improvement in stability is remarkable when the fast-reacting system like interconnected power generation control. The idea of ever-improving fractional calculus is being incorporated into the control aspect of the whole power system. With the proven practical viability of proportional integral (PI) control, two simple yet effective structures are verified in the two area interconnected power system model. A fractional integrator achieves the extra degree of freedom to upsurge the flexibility in control. The controller parameters for all forms are optimized from the same metaheuristic algorithm in conjunction with the improved dual performance objective function. A numerical study with analyzed results can be seen for adequate robustness and disturbance rejections.

11. Chapter 10. Hardware Implementation of the Fractional Controller on Quadrotor Aircraft

    Sheikh Izzal Azid, Vivek Pawan Shankaran

    Abstract

    The chapter focuses on the experimentation setup and implementation of the fractional controller on the nonlinear quadrotor aircraft. Firstly, the mathematical model of a quadrotor is derived, which is more suitable for acrobatic manoeuvring. In order to handle such an unstable system, a fraction PI (PI\(^\lambda \)) was designed and implemented for x, y, z and yaw movements of the quadrotor aircraft. Detailed simulation results are presented to test the controller. Moreover, the information is provided on configuring hardware for real-time experimentation. A method is verified using the designed controller on the quadrotor in MATLAB Simulink and parrot rolling spider minidrone. Finally, the hardware results are analysed and present interesting applications of fractional-order control.

12. Chapter 11. Optimum Fractional-Order PID for Active Suspension of Quarter Car Model Control

    Sandeep D. Hanwate, Rohini K. Bhalerao

    Abstract

    The goal of this study is to use active suspension to reduce the vehicle’s vertical oscillations for a quarter car model whose dynamics are considered in a comprehensive automobile model. The nominal system matrices are controlled using an optimum fractional-order PID controller, while the effect of road profile disturbances, parametric uncertainty, and various road profiles are compared with the standard PID controller. The controller is global asymptotic stable with minimum integral performance indices. To test the controller’s performance under parametric uncertainties and external road disturbances, simulations are run in MATLAB/Simulink.