Optimal tuning of decentralized fractional order PID controllers for TITO process using equivalent transfer function
Introduction
Controlling the multi input multi output interconnected process is the most challenging problems for control system engineer. Several authors have been proposed the control designing procedures for multivariable systems, but still, the researchers working on this problem to enhance the control performance for Multi Input Multi Output (MIMO) process. There are three major control schemes such as decentralized (multiloop), decoupled control, centralized control have been presented for MIMO process.
In general, decentralized (multiloop) control scheme has been widely used in the industrial process because of its advantages in failure tolerance and easy implementation. However, Model predictive control (MPC) is capable of handling MIIMO control problem effectively; But MPC is used mostly used on higher level to provide setpoints to the lower level PID controllers (Xiong, Cai, & He, 2007). The MPC is employed in supervisory level, where the sampling time of MPC is higher than the lower level PID control loops. Managing the coupling problem using MPC can be troublesome due to the limited bandwidth of MPC (Garrido, Vázquez, & Morilla, 2012). Hence, the PI/PID based controllers are used to control the process at the lower level. The improvement in the lower level control system increases the overall performance of multivariable control. The multiloop PI/PID controllers generally utilized technique in the multivariable process. The purpose of utilizing the PI/PID in multi loop because of its simple structure, failure tolerance and capacity of meeting user specification. But the tuning of PI/PID is difficult due to the interaction effect between input and output, because the design of one loop depends on another loop. There has been many designing procedure proposed in the literature such as detuning (Besta & Chidambaram, 2016), sequential loop closing, independent loop (Vu & Lee, 2010b) and relay auto tuning method. In the detuning method, the controller parameters are found for most important loop transfer function without considering interaction effect and then the controller gains are detuned by considering interaction effect to meet some user control specification. But the performance and stability of closed loop system has not been discussed properly in the detuning procedures. In the sequential tuning methods, controllers are tuned while closing the loop one after other, but the final controller design completely depends on the order of other controller.
The multiloop PI/PID Controller provides better control performance for the process with modest interaction. It fails to provide reasonable control performance when the interaction effects between loops are significant. In such a case, the decoupler based control scheme is preferred for MIMO process. Three types of decouplers are available in the literature such as ideal, inverted, and simplified decoupler (Cai, Ni, He, & Ni, 2008). The ideal decoupler formed using the inversion of process transfer function which may result in complex dynamics. The simplified decoupler is widely used to develop an ETF model and decentralized controller is designed for corresponding ETF model.
Vu and Lee (2010a) have designed independent IMC based PI/PID controller using Effective Open Loop Transfer Function (EOTF) model, where the higher order EOTF model is approximated into reduced order model using maclaurin series. The EOTF resulted in a higher order model, it requires model reduction techniques to form reduced order model which make the controller design easier. Hajare and Patre (2015) have approximated the higher order EOTF model to reduced order model using frequency response fitting. The formulation of EOTF for higher order process is complex and decoupler design also makes controller complicated control structure. Xiong and Cai (2006) has demonstrated a method of designing decentralized PI/PID controller using effective transfer function. The ETF model is developed using the information of effective relative gain array, relative gain array (RGA), relative frequency of open loop transfer function model. Vijaykumar, Rao, and Chidambaram (2012) has designed ETF model using normalized relative gain array (RNGA) and relative average time array (RARTA) to form ETF model and then controller is tuned using maclaurin series for corresponding EFT model. Wang, Huang, and Guo (2000) has developed systematic design for full dimensional PID control for higher order process using approximated decoupler but the robustness of closed loop control system cannot be guaranteed.
Shen, Sun, and Xu (2014) expanded the normalized decoupling method for higher order process. Rajapandiyan and Chidambaram (2012) have proposed new method of decoupled process approximation for higher order process, where the simplified decoupler with ETF approximation method is combined to form new approximation procedure for decoupled process, and then the diagonal controller is designed using simplified internal model control (SIMC) method for approximated decoupled process.
The integer order PI/PID controller still dominated in the industries. Recently, the fractional order controller has received extensive consideration in industries and academia. The three hundred years old fractional calculus has some interesting history, however last few decades the fractional calculus has been gain popularity in control system engineering and other engineering application. The researcher have demonstrated that the FOPID controller outer performs than the integer order PID controller for many application. However there is lack of tuning methods available compared to PI/PID controller. The analytical and numerical based tuning method for FOPI/FOPID is reported in the literature. The complete review of FOPID tuning methods are discussed in Shah and Agashe (2016). Generally, the FOPID controller tuned to meet user defined specification (time, frequency domain) by analytical method and optimization method. Mostly, the optimization based tuning methods are used to obtain the FOPI/FOPID controller parameters. In Agababa (2015), the particle swarm optimization, differential evolutionary optimization, bat optimization used to tune the fractional order PI/PID controller for SISO process. The design of multiloop FOPID controller is difficult for MIMO process because of its coupling effect between input and output.
The main objective of this paper is to make the student to understand about the real time industrial control problem. Also, this papers helps control system student community to understand about the fractional order control system in simple manner. The design of coupled system is difficult due to its coupling effect; hence, the coupled interconnected system is separated into equivalent single input single output system to make design of control system easier. The design of one loop controller depends on the other loop, so it is always complicated to design a FOPID controller for multivariable system. By using independent loop method, the MIMO process is decomposed into independent single input single output (SISO) for tuning of controller easily. In this paper, the FOPID controller is used as a diagonal controller with simplified decoupler. The diagonal controller parameters are obtained using bat optimization algorithm for minimum values of time weighted integral absolute error.
Section snippets
Literature survey
Chen, Tang, Li, and Lu (2018) proposed a synthesis tuning method for PIλ to satisfying the frequency domain specification such as gain crossover frequency and phase margin. In general, the synthesis methods for tuning controllers are proposed for reduced order model. The optimal PIλ controllers are tuned for higher order process according to the specified phase margin and crossover frequency (Chen et al., 2018).
Bingul and Karahan (2018) compared the PID and FOPID controller performance where the
Basic fractional calculus
Fractional calculus is an emerging technique in engineering and sciences, it has some unique features and it represents system completely. Non integer order differentiator and Integrator are represented by differ-integral operators. The fractional derivative and fractional integral combined and expressed in generalized form,where q is a fractional order and ‘a’ is an initial conditions. Many definitions are proposed for fractional differ- integral. The Riemann and
Equivalent transfer function
The TITO model of this process is defined by the G(s),
The multiloop controller transfer function matrix is,
The effective open loop transfer function is developed by incorporating decouplers in the TITO process. The open loop transfer function between input and output is developed while other loop is closed. It is not easy to develop an EOTF model by closing other loops, because closing other loops requires controller, so the EOTF model is
Fractional order PI/PID control
The FOPID is a special case of PID controller where it is less sensitive to parameter variation of controller
The fractional order PID controller provides more flexibility in tuning of controller with additional two tuning parameters.
The transfer function of FOPID controller Gfopid(s) is given below,where Kp is the proportional gain, ti is the integral time constant, λ is the order of integrator, td is the derivative time and µ order of derivative. The performance FOPID
Bat optimization algorithm
In evolutionary computation techniques, Bat optimization has gained popularity in all engineering nonlinear multimodal problems (Yang & Gandomi, 2012). The natural forging strategies of bat is inspired and mimicked into bat optimization algorithm. The bat emits sounds pulses and receive back to identify the prey. The pulse emission rate varies depending on the location of species. Initially bats flies randomly with velocity Vi at position xi with fixed frequency fmin to search prey. It updates
Case 1: VL column system
The bench mark example of VL column given by luyben (1986) is a TITO process with higher interaction between input and output. The Transfer function model of VL column is given below,
The normalized gain matrix, RGA, RNGA, RARTA are found using Eqs. (16), (17), (18) and then the ETF model is obtained using Eq. (18).
The EOTF/ETF models are
Conclusion
Generally, it is quite difficult to design a fractional order PI/PID controller for multivariable process. In this paper, the independent design of FOPI/FOPID controller is designed with decoupler for TITO process. The ETF model is developed for TITO system using the information of RGA, RNGA and then the FOPID controller parameters are independently designed for ETF model. The FOPID controller parameters are tuned using bat optimization algorithm to achieve minimum value of ITAE. The proposed
Declaration of Competing Interest
The authors declared that there is no conflict of interest.
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