Fractional-Order Optimal Control of Fractional-Order Linear Vibration Systems with Time Delay

Fractional-Order Optimal Control of Fractional-Order Linear Vibration Systems with Time Delay

Saeed Balochian (Gonabad Branch, Islamic Azad University, Gonabad, Khorasan Razavi, Iran) and Nahid Rajaee (Gonabad Branch, Islamic Azad University, Gonabad, Khorasan Razavi, Iran)
Copyright: © 2018 |Pages: 22
DOI: 10.4018/IJSDA.2018070104
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Vibration control of fractional-order linear systems in the presence of time delays has been dealt in this article. Considering a delayed n-degree-of freedom linear structure that is modeled by fractional order equations, a fractional-order optimal control is provided to minimize both control input and output of delayed system via quadratic objective function. To do this, first the fractional order model of system that is subject to time delay is rewritten into a non-delay form through a particular transformation. Then, a fractional order optimal controller is provided using the classical optimal control theory to find an optimal input control. A delayed viscose system is then presented as a practical worked-out example. Numerical simulation results are given to confirm the efficiency of the proposed control method.
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1. Introduction

The problem of time-delay systems in control theory has been discussed over many years. Time delay is very often encountered in different technical systems and is believed to have a negative impact on the control system performance. The existence of time delay may cause undesirable system transient response, or even instability. Due to the time delay, when unsynchronized control force is applied to a system, it may result in decrease in the system efficiency (Yao & Rohman, 1985; Cao et al., 2009). Till now, so many methodologies have been proposed to deal with the control problem of systems with time delay (Cepeda-Gomez & Olgac, 2010; Ergenc et al., 2007; Bozorg & Davison, 2006; Moulaya et al., 2008; Azar & Serrano, 2015; Azar & Serrano 2016; Kermani & Sakly, 2015; Thanh Pham et al., 2017; Mahmoud & Sunni, 2012). Delay in control inputs of dynamic systems exists in most of the real applications and is one of the important and notable topics in the recent researches.

On the other hand, vibration systems are examples of industrial applications that are exposed to time delay. Vibration control is an important concern in practical systems in industry and plays a critical role in recent researches. Vibration can be controlled through proper process controller designing and by using some special equipment (Kim et al., 2016; Sharma 2017). In vibration control, the technique of time-delay compensation is commonly used to eliminate or reduce the effect of time delay (Chen & Cai, 2009; Shao et al., 2013; Chatterjee, 2008).

Different methodologies have been presented to deal with the delay but in most of them an optimal controller that both overcome delay and stabilizes the system output has not been obtained. Referring to the related reference books in the literature, finding an explicit form of optimal controllers in vibration systems in the presence of time delay might still remain difficult (Dion, 2001; Oguztoreli, 1966; Eller et al., 1969). Optimal controllers have been widely used in the recent researches (Mousa et al., 2015; Gharsellaoui et al., 2012, Uchinda et al., 2006; Shi et al., 2003). The optimal regulator control problem for linear systems with delays is a continuing study and deals with the different challenges such as delay type, system equations, cost function and etc (Uchinda et al., 2006; Shi et al., 2003).

In some cases, vibration systems are presented via fractional order dynamic equations. Fractional order system models have been widely studied over the past two decades (see e.g., Hartley & Lorenzo, 2002; Chen & Ahn, 2006; Ahn et al., 2007; Lin & Kuo, 2012; Azar et al., 2017; Nabavi et al., 2018 and their references). Nowadays it has been shown that the exact model of many systems is denoted more accurate by fractional-order differential equations such as viscoelastic systems (katsikadelis & babouskos, 2010; Jarbouh, 2012), Electromagnetic theory (Baleanu & Golmankhaneh, 2010; Das, 2011), economics (Wang et al., 2012), bioengineering (Magin, 2004), mechatronics (Martin et al., 2008), physics and engineering (West et al., 2003). It has been shown that fractional-order differential equations can describe the behavior of most real systems such as viscoelastic systems in a more precisely way.

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