Fractional order control of conducting polymer artificial muscles

Highlights

• Fractional order control has been designed for conducting polymer artificial muscle actuator.

• The Cuckoo search algorithm has been used for controller optimization.

• A specific cost function considering control signal and maximum overshoot has been used in optimization.

Abstract

This paper proposes a fractional order PID (FOPID) controller to improve the positioning ability of conducting polymer actuators (CPAs), a novel class of smart material based actuators. In the controller design process, the performance requirements and constraints which are crucial in precise positioning of CPAs such as fast settling time, low steady-state error, overshoot and control voltage are considered. In order to obtain the optimal controller parameters, cuckoo search (CS) and particle swarm optimization (PSO) meta-heuristic search methods which utilize a fractional order model of the CPA and a specifically defined fitness function, are used. Both of the algorithms are compared in terms of convergence rate and success of converging to an optimal solution. In order to test the performance of the FOPID controller, a PID controller is also tuned with both algorithms and all controllers are implemented experimentally on the CPA. The results show that the FOPID controller tuned with CSA has provided less overshoot, settling and rise-time than that tuned with PSO. The performance of the PID control is slightly worse than the FOPID controllers in terms of transient and steady-state response. Although both search algorithms have satisfied the control input constraint in FOPID and PID controllers, CSA tuned PID controller has required smallest control signal.

Introduction

Conducting polymer actuators (CPAs) are a novel class smart material based actuators, which change their shape and volume upon implementation of a sufficient electrical potential difference or current. They are also known as artificial muscles as they can mimic the motion of biological muscles very well. Moreover, they have attracted great attention because of their superior properties such as biocompatibility, high force output to weight ratio and elasticity (Kim & Tadokoro, 2007). Selection of the controller parameters in applications of CPAs is of great importance as they are also used in precise positioning systems. As they are in their infancy, there are some challenges in achieving the desired control performance requirements such as fast settling time, minimum steady state error and overshoot out of these actuators. In addition, the control voltage to drive CPAs must be limited below 1–2 V in order to prevent possible damages. Some advanced control methods have been proposed to achieve a significant positioning performance from CPAs: Intelligent control based on fuzzy logic PD + I control and neuro-fuzzy adaptive neural fuzzy inference system (ANFIS) control (Druitt & Alici, 2014), robust adaptive control (Fang, Tan, & Alici, 2008), adaptive sliding mode control (Wang, Alici, & Nguyen, 2013), repetitive control (Itik, 2013), etc. Moreover, Wang, Alici, and Tan (2014) established a hysteresis model for CPAs and controlled the tip displacement of a CPA using inverse feedforward control. Blanchard and Nguyen (2014) proposed a robust controller which is based on quantitative feedback theory to improve the accuracy of the tip displacement positioning of a CPA. Beyhan and İtik (2015) used adaptive fuzzy-Chebyshev network control to identify and control the position of a CPA without utilizing a physical model. The common aim of these works is to improve the positioning accuracy of CPAs. They achieved this aim using very complicated control methods none of which pre-considered the aforementioned important performance requirements for CPAs. Moreover, to show the effectiveness of all these controllers, their responses were compared to those of classical PID controllers which were not tuned optimally by using a desired performance index.

Fractional calculus extends the theory of ordinary differential equations to fractional order differential equations which have non-integer order of integrals and derivatives. Conventional integral operator ‘1/s’ and derivative operator ‘s’ are replaced to ‘$1/s^{\lambda }$’ and ‘$s^{\mu }$’ in fractional calculus where $\lambda$ and $\mu$ are the fractional order parameters of integral and derivative operators, respectively. Fractional calculus has recently become widespread in engineering applications especially in control system design. Several fractional order controllers have been proposed as alternatives to their non-fractional counterparts. One of the most common and practical fractional order control methods for linear systems is the fractional order PID (FOPID) control which is an extension of the classical PID control. Due to easy design and implementation, FOPID controllers have gained attention and found many applications such as robot manipulators (Sharma, Rana, & Kumar, 2014), power electronics (Calderón, Vinagre, & Feliu, 2006), drive systems (Xue, Zhao, & Chen, 2006) and process control (Monje, Vinagre, Feliu, & Chen, 2008). For systems including nonlinearities and uncertainties, nonlinear fractional order controllers such as fractional order sliding mode controller (FOSMC) can be used (Dadras & Momeni, 2012). Aghababa (2013) proposed a fractional order sliding mode controller for vibration suppression of uncertain structures. Fractional order control has also been implemented effectively in synchronization of chaotic systems (Tavazoei and Haeri, 2008, Aghababa, 2012a, Aghababa, 2012b).

A FOPID controller, which is denoted as ${\mathit{PI}}^{\lambda }{D}^{\mu }$, consist of 5 independent parameters: the proportional gain $(K_P)$, integral gain $(K_I)$, derivative gain $(K_D)$, order of integrator $(\lambda )$ and differentiator $(\mu )$. The parameters $\lambda$ and $\mu$ are additional to the traditional PID, which give control engineers more design flexibility to further enhance the control systems performance. Heuristic methods such as genetic algorithms (Copot et al., 2013, Bingul and Karahan, 2012, Machado, 2010), particle swarm optimization (PSO) (Bingul and Karahan, 2012, Zamani et al., 2009), differential evolution (DE) (Biswas, Das, Abraham, & Dasgupta, 2009), chaotic ant swarm method (Li, Yang, Peng, & Wang, 2006) and more recently Cuckoo search algorithm (Sharma et al., 2014) were used to find the optimal parameters for the FOPID control all of which agreed on the improvements in the control systems performance compared to optimized PID controllers. Optimally tuned FOPID control have been applied to many systems such as (Majhi et al., 2015, Pradeepkannan and Sathiyamoorthy, 2015, Zamani et al., 2009). When the parameters of a FOPID controller are optimally tuned based on some pre-defined objectives, they may also improve the positioning performance of CPAs. This may reduce the complexity of control design while satisfying the tracking error, maximum overshoot requirements and control constraints.

The main contribution of this paper is that we design a fractional order controller to improve the positioning performance of CPAs.To the best of author’s knowledge a FOPID controller has not been designed for CPAs yet. As a second contribution, a specific cost function which considers maximum overshoot, control signal, settling time, rise time and tracking error, is used in the controller design process. Such performance requirements are very important for the position control of CPAs and should be given a careful consideration. This has not been considered for CPAs neither. To obtain the best controller parameters based on the defined performance index, we employ Cuckoo search algorithm and compare the results with those obtained by the PSO algorithm. PID controllers are also designed by using both search algorithms and their responses are also compared to those of FOPID controllers.

The rest of this paper is organized as follows. Section 2 presents a brief background about fractional calculus and the fractional PID control. In Section 3, we introduce the trilayer CPA and obtain its fractional order model. Section 4 discuses the meta-heuristic algorithms used in this study and their implementation to FOPID control design. Simulation and experimental results are given in Section 5. Finally, the conclusions are drawn.