On selection of improved fractional model and control of different systems with experimental validation

Highlights

• This paper proposes an algorithm to detect more accurate fractional order model of a system.

• This paper also compares the performance merit of the algorithm with existing fractional model.

• Experimental validation of the proposed study is applied to robotic manipulators.

Abstract

A major challenge in fractional order calculus applied to control systems is the selection of fractional values to satisfy the required set of dynamical characteristics pertaining to controllers or model. The research work of this paper is divided into three levels. The first level is centered around deriving an accurate fractional model corresponding to real systems using a novel algorithm. The second level is to select a dynamical system model from literature whose fractional model is already in existence and then apply the proposed algorithm to derive an accurate fractional model for improved system performance. The third level comprises of experimental validation of the proposed study. Two robotic manipulators are considered: serial flexible link robotic manipulator, and serial flexible joint robotic manipulator, both with two degrees of freedom. Their fractional equivalent model is derived using the proposed algorithm, and then a controller is designed for three different scenarios i.e fractional model and integer Controller, integer model and fractional controller, fractional model and fractional controller. The simulated results are compared with the experimentally examined results and it is found that the proposed fractional model of these robotic manipulators are highly accurate and more reliable. The examined results show significantly improved performance using the fractional controller and fractional model and this shows the strong evidence of reliability of the fractional model in control system design.

Introduction

Under certain conditions, approximation based control performs effectively [1]. Most of the dynamic systems are better characterized using a fractional order dynamic model based on fractional calculus [2]. When dealing with fractional order systems, a major challenge is to select the fractional values that generally termed as α. Many researchers have proposed different medium to choose appropriate α values whether it belongs to fractional model or fractional controller. A ranking based on various researchers working in the field of fractional calculus is reported in [3]. Monje et al. delineate an idea of tuning the fractional value for fractional order PI^α controller [4] and recommend tuning of the controller using an iterative optimization method based on a nonlinear function minimization [5]. In [6], fractional order controller is designed for fractional order plant which supervises heating furnace system to a general modified fractional order model and PID controller is designed. In [7], a method for tuning fractional PI^λD^μ controller is proposed to fulfill five different design specifications, and in [8], a new tuning method for fractional order proportional and derivative controller is proposed for a class of second-order plants, and the plant model is an integer order model. In [9], fractional order controller is designed for a class of fractional order systems. In article [10] a set of tuning rules is presented for fractional order controllers based on a first-order-plus-dead-time model. A fractional order sliding mode control is presented in [11] to control the velocity of the permanent magnet synchronous motor. In [12], a tuning method of fractional order controller is presented for a class of fractional order systems. Recently in [13], a fractional order PID controller is designed for fractional order systems. A new practical tuning method development for fractional order PI controller is presented in [14] which is also valid for general class of plants, and another algorithm stabilizes fractional-Order time-delay systems using fractional-order PID controllers [15].

There are also other studies which have derived fractional order models and its control [16], [17], [18]. In [19], AboBakr et al. reported different integer and fractional order models from the electrical point of view. Modeling of aging materials within the limitation of fractional calculus consisting of variable order is presented [20]. A Fractional order methodology of capacitor quality control for porous electrode behavior is observed [21]. The advantage of such fractional order modeling is the absence of high-order polynomials in transfer functions. In [22], a detailed tutorial is presented for the representation of a fractional order system in MATLAB. More recently, a study of fractional modeling and fractional controller design for inverted pendulum system has been provided [23]. An intuitive study of Variable order fractional systems with an inverse Laplace transform is presented [24]. A fraction model of semiconductor pn junction diode is proposed [25]. A detailed review of literature related to fractional derivatives is also presented [26]. An interesting to describe the dynamics of political systems is presented [27].

Various efforts have taken place to tune/find the appropriate α value. Most of the papers above have taken a system model (which is integer or fractional) and tuned a fractional PID controller to get the desired response. There is no unique way to find or to choose the appropriate value of α (for a system’s fractional model). Fractional Calculus has been a powerful new tool that is recently opted to model different dynamical systems whose modeling equations are in the form of differential equation to study the fractional dynamical behavior (accurate behavior) of these systems and control. In this manuscript, the modeling equations of the dynamical systems which are taken as an example and for experimentally validating the results obtained can be written in the form of differential equations. Hence, fractional calculus concept is chosen for further investigation applied on robotic manipulators. This paper contributes to the area of choosing an appropriate value of ‘α’, detecting the best fractional model and designing controller for the derived fractional model. The contributions of this paper are summarized below:

• An algorithm to select the best suitable fractional model is proposed.

• Proposed an improved fractional model of a system. Different sets of second-order systems are considered, and the corresponding suitable fractional model is reported. The performance merit of the algorithm is evaluated against some pre-existing models via a simulation experiment. The fractional order model obtained by the proposed algorithm performs significantly better. Improvement of 85% in settling time is achieved against example 2 of Zhao et al. [6] with zero overshoot and an improvement of 50.2%, 93.8% respectively are achieved against the fractional order model of heating furnace [28].

• Proposed fractional model of two robotic manipulators of two degrees of freedom: (i) serial flexible link, and (ii) serial flexible joint.

• Experimental Contribution. Simulated results are compared with practical examined results and the proposed fractional model of these robotic manipulators are proven to be highly accurate and reliable.