Volume 16 Number 3
June 2019
Article Contents
Gargi Baruah, Somanath Majhi and Chitralekha Mahanta. Design of FOPI Controller for Time Delay Systems and Its Experimental Validation. International Journal of Automation and Computing, vol. 16, no. 3, pp. 310-328, 2019. doi: 10.1007/s11633-018-1165-4
Cite as: Gargi Baruah, Somanath Majhi and Chitralekha Mahanta. Design of FOPI Controller for Time Delay Systems and Its Experimental Validation. International Journal of Automation and Computing, vol. 16, no. 3, pp. 310-328, 2019. doi: 10.1007/s11633-018-1165-4

Design of FOPI Controller for Time Delay Systems and Its Experimental Validation

Author Biography:
  • Gargi Baruah received the B. Eng. degree in electronics and communication engineering from Sri Venkateshwara College of Engineering and Technology, India in 2009, and M. Eng. degree in electronics design and technology from Tezpur University, India in 2012. Currently, she is a Ph. D. degree candidate at Indian Institute of Technology, India with focus on fractional order control system. Her research interests include relay based process identification and controller design.E-mail: gargi.baruah@iitg.ac.in (Corresponding author) ORCID iD: 0000-0001-9393-7430

    Somanath Majhi received the Ph. D. degree in control systems engineering from University of Sussex, UK in 1999, and has been with Indian Institute of Technology (IIT) Guwahati, India since 1999. He has published several books/book chapters, conference proceedings, journal papers apart from holding a few responsible positions in his institute. His research interests include relay based identification and auto-tuning, control systems, and control theory applications. E-mail: smajhi@iitg.ac.in

    Chitralekha Mahanta received the Ph. D. degree in control from IIT Delhi, India in 2000. She joined as an assistant professor in Department of Electronics and Communication Engineering (ECE), IIT Guwahati, India in 2000. Since then she has been involved in active research in the area of control theory and its applications. She has offered a variety of courses in undergraduate and post graduate studies in the field of control systems at IIT Guwahati, India. She has been a full time professor in the Department of Electronics and Electrical Engineering (EEE), IITGuwahati since April 2012, starting her research at IIT Guwahati, India in the field of intelligent control. Currently, she is involved in the areas of robust and adaptive control with applications in robotics and flight control. She is a senior member of IEEE and a Fellow of the Institution of Electronics and Telecommunication Engineers (IETE). Her research interests include control of nonlinear uncertain systems, sliding mode control of underactuated systems specific to humanoid robot arm, actuator failure tolerant control design for nonlinear systems with application in aircraft control.E-mail: chitra@iitg.ac.in

  • Received: 2018-04-26
  • Accepted: 2018-11-14
  • Published Online: 2019-02-22
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Design of FOPI Controller for Time Delay Systems and Its Experimental Validation

Abstract: In this paper, we report on the identification and modeling of unknown and higher order processes into first order plus dead time (FOPDT) plants based on the limit cycle information obtained from a single relay feedback test with an online fractional order proportional integral (FOPI) controller. The parameters of the test processes are accurately determined by the state space method while the FOPI controller settings are re-tuned to achieve enhanced performance based on the identified model parameters based on the balanced-tuning method. A new performance index, integral time fractional order absolute error (ITFIAE) is introduced in this paper for balanced tuning of fractional order (FO) controllers. It requires minimum design specifications without a-priori knowledge of gain and phase crossover frequencies and is done non-iteratively without disrupting the closed loop. Four test processes and experimental analysis on a coupled tank system (CTS) validate the theory proposed.

Gargi Baruah, Somanath Majhi and Chitralekha Mahanta. Design of FOPI Controller for Time Delay Systems and Its Experimental Validation. International Journal of Automation and Computing, vol. 16, no. 3, pp. 310-328, 2019. doi: 10.1007/s11633-018-1165-4
Citation: Gargi Baruah, Somanath Majhi and Chitralekha Mahanta. Design of FOPI Controller for Time Delay Systems and Its Experimental Validation. International Journal of Automation and Computing, vol. 16, no. 3, pp. 310-328, 2019. doi: 10.1007/s11633-018-1165-4
    • The relay feedback auto-tuning method was introduced by Astrom and Hagglund and modified relay feedback methods were proposed in [1, 2]. Relay based auto-tuning methods of which Hang et al.[3] are pioneers, can be split into online and offline approaches. The latter usually affects the operational process regulations as the controller has to be in or out of the loop depending upon the conditions required for tuning which disrupts the system performance. For tuning of the fractional order (FO) controllers, Lanusse et al.[4] used the frequency-domain approach to design proportional integral and derivative (PID) controllers, fractional order proportional integral and derivative (FOPID) controllers and first generation commande robuste d′ordre nonentier (CRONE) controllers for a class of plant models without dead time. The phase and gain margin-based analytical method was used in [58] to do the necessary tuning of the FOPID. Maione and Lino[9] designed an fractional order proportional integral (FOPI) controller based upon the symmetrical optimum method while Chen et al.[10] designed a FOPI controller by using fractional maximum load disturbance to-output sensitivity, Ms constrained integral gain optimization (F-MIGO) algorithm. Beschi et al.[11] used a new generalised iso-damping property to design FOPID controllers while Muresan et al.[12] used internal model control (IMC) based FO controllers for a class of time delay processes. Meanwhile, the design of two different types of FOPI controllers for FO systems was proposed by Luo et al.[13]. Recently, FOPI controllers have also been used for the auto-tuning of two input two output processes[14]. However, these manual tuning methods designed for processes known a priori have a significant disadvantage since their performance depends on the process knowledge of the system which are typically unknown. This leads to the necessity of investigating automatic tuning in more detail. Monje et al.[15] proposed a relay based auto-tuning method for FOPID controllers while Malek et al.[16] designed PID and FOPI controllers for first order plus dead time (FOPDT) plants. Caponeto et al.[17], Roy et al.[18], Tepljakov et al.[19], Belikov and Petlenkov[20], Katal et al.[21] derived a relay-based auto-tuning approach using describing function (DF) and employed analytical methods such as those involving the concepts of gain margin, phase margin and iso-damping properties which are similar to those reported in [22]. These methods used the describing function (DF) method to approximate the non-linearity of the relay while various other tuning methodologies were used by [23, 24].

      The proposed algorithm seeks to model unknown and higher order processes into FOPDT plants and subsequently, to design simple FOPI controller subsequently. The motivation behind using an FOPI controller is that it can achieve more accurate and robust performance as compared to a conventional proportional integral (PI) controller. Also the controller is always in the loop along with the relay and neither needs to be replaced while the other is functioning, thus making it an auto-tuning algorithm which in turn makes the tuning process self sustainable. Identification of the unknown plant and performance of the controller is interdependent. A proper identification leads to better controller response and smooth functioning of the system as a whole. The basic relay auto-tuning method can be improved significantly by accurately estimating the process parameters by using exact analysis of the limit cycle. Most of the tuning rules provide a very poor damping factor due to the high proportional action of the controller. This can cause variations or spikes in the controller output which would lead to faster response and settling time but at the cost of harming the actuators in physical systems. Thus, a new tuning principle is considered wherein a critically damped control loop and/or minimized fractional order absolute error is achieved by the balanced tuning method. The motivation behind using balanced tuning method is that the balanced tuning between the proportional and integral actions helps in attaining a proper transient response with smooth control action that is perfectly suitable for controlling physical systems. The idea of balanced tuning, already used for PID controllers as reported by Klan et al.[2527], is being used on FOPI controllers in this work in order to add the advantages of the balanced tuning method and the extra degree of freedom provided by the FOPI controller to perform better than the classical controller. One of the highlights of this work is that we introduce a new performance index integral time fractional order absolute error ($ {\rm ITF_IAE}$) to enable the achievement of balanced control for the FO controllers.

      The rest of the paper is organized as follows. The method for process identification and modelling is explained in Section 2. Section 3 explains the design procedure of the FOPI controller. The tuning rule evaluation is analysed in Section 4. Simulation results are shown in Section 5. Section 6 explains the experimental results on a coupled tank system (CTS) setup. The conclusion is drawn in Section 7.

    • Let the unknown process be modelled into an FOPDT model which is given by (1):

      $ G_m(s) =\dfrac{k{\rm e}^{-\theta s}}{\tau s +1} $

      (1)

      where $ k $, $ \tau $ and $ \theta $ are the steady state gain, time constant and time delay of the process model, respectively, and the FOPI controller to be tuned is defined in (2).

      $ G_c(s)=\eta K_c\left(1 + \dfrac{1}{T_i s^\lambda}\right). $

      (2)

      Here, $ K_c $ is the proportional gain, $ T_i $ is the integral gain, $ \lambda $ is the fractional order and $ \eta $ is the tuning factor which can increase or decrease the damping of the control loop according to the requirement of the response needed. In this Section, an identification method will be described where an FOPI controller will be in the loop along with the relay. The block diagram of this closed loop system is shown in Fig. 1.

      Figure 1.  Block diagram of on-line tuning scheme

      In online identification, the controller is always in the loop. When a relay is connected in parallel with the controller in the tuning test, the whole structure can be converted into an equivalent form as reported by Majhi[28] (Fig. 2). For identification of the process parameters, the controller has to be set with the initial pre-determined values. Here, the initial FOPI controller at the time of relay feedback test is

      Figure 2.  Process input and output signals for half limit cycle oscillation

      $ G_c(s)=K_{co}\left(1 + \dfrac{1}{T_{io}s^{\lambda_o}}\right) $

      (3)

      where $ K_{co} $ and $ T_{io} $ are its initial gain and time constant and $ \lambda_o $ is the initial fractional order.

      The relay here is considered to have a relay height of $ \pm h $ and hysteresis width of $ \pm \varepsilon $. The relay helps to induce a limit cycle condition and the process input and output data are collected and analysed. Fig. 2 shows the limit cycle signals as the plant produces the typical oscillatory input and output signals.

      The relay in the autonomous control loop is thus subjected to a transfer function given by

      $ \begin{split} \dfrac{Y(s)}{U(s)} =& \dfrac{G_m(s)}{1+G_m(s) G_c(s)} =\\ & \dfrac{k {\rm e}^{-\theta s}}{\tau s+1 + k K_{co} {\rm e}^{-\theta s} + \dfrac{k\dfrac{K_{co}}{T_{io}} {\rm e}^{-\theta s}}{s^{\lambda_o}}} =\\ & \dfrac{k {\rm e}^{-\theta s}}{\tau s+1 + k K_{co} {\rm e}^{-\theta s} + \dfrac{kK_{io} {\rm e}^{-\theta s}}{s^{\lambda_o}}}. \end{split} $

      (4)

      Here, $ K_{io}=\dfrac{K_{co}}{T_{io}} $.

      The transfer function in the above equation when represented in state-space form is given by

      $ \dot{y}(t) = \alpha y(t)+ B u(t-\theta). $

      (5)

      Here, $ B u(t) $ has three parts: $ b_1u_1(t-\theta) $, $ b_2u_2(t-\theta) $, $ b_3 u_3(t-\theta) $. The transfer function in (5) when represented in state-space form, extending Mehta and Majhi′s work[29], is given.

      Thus for (5), the actual state space representation is given by

      $ \;\;\;\dot{y}(t)=\alpha y(t)+b_1u_1(t-\theta) + b_2u_2(t-\theta)+b_3 u_3(t-\theta). $

      (6)

      Here, $ \alpha=-\dfrac{1}{\tau} $, $ b_1=\dfrac{k}{\tau} $, $ b_2=-\dfrac{kK_{co}}{\tau} $, $ b_3=-\dfrac{kK_{io}}{\tau} $, $ u_2(t-\theta)=y(t-\theta) $, $ u_3(t_\theta)=\,_0\!D_t ^{-{\lambda_o}}y(t-\theta) $.

      The solution for (5) has to be determined for two time intervals, $ t_0 \leq t \leq t_1 $ and $ t_1 \leq t \leq t_2 $, to find the unknown plant parameters.

      In the interval, $ t_0\leq t \leq t_1 $, we have, $ u(t-\theta)=h $, $ y(t-\theta) =kh(1-{\rm e}^{\alpha (t-\theta)}) $ and (6) becomes

      $ \begin{split} y(t) = & \, {\rm e}^{\alpha(t-t_0)}C x(t_0)+\alpha^{-1}h[{\rm e}^{\alpha t}-1] C b_1 +\\ & kh[\alpha^{-1}({\rm e}^{\alpha t}-1)-t{\rm e}^{\alpha (t-\theta)}]C b_2 + \\ & \dfrac{kh}{\Gamma (1+ \lambda_o)}[\alpha^{-1}(t-\theta)^{\lambda_o}+ \\ & \alpha ^{-2} (t-\theta)^{(\lambda_o -1)}-{\rm e}^{\alpha(t-\theta)}\alpha ^{-\lambda_o}t] C b_3. \end{split} $

      (7)

      Thus, finally substituting $ \alpha $, $ b_1 $, $ b_2 $, $ b_3 $ and $ C $, $ y(t) $ becomes

      $ \begin{split} y(t)= & \, {\rm e}^{\alpha t}y(t_0)+kh(1-{\rm e}^{\alpha t})[(1 - kK_{co})-\\ & k^2ht\left[{\alpha K_{co} -\dfrac{K_{io}}{\alpha ^{{\lambda_o}+1}\Gamma(1+\lambda_o)}}\right]{\rm e}^{\alpha (t-\theta)}+ \\ & \dfrac{k^2hK_i}{\Gamma (1+\lambda_o)}[(t-\theta)^\lambda - \alpha(t-\theta)^{\lambda_o-1}]. \end{split} $

      (8)

      Similarly, for the interval $ t_1 \leq t \leq t_2 $, the output signal is given by

      $ \begin{split} y(t) = & {{\rm e}^{\alpha (t - \theta )}}y({t_0}) - kh(1 - {{\rm e}^\alpha }(t - \theta )) - \\ & \dfrac{{{k^2}h{K_{co}}}}{\tau }\left[ {(t - \theta )(1 - {{\rm e}^{\alpha t}}) + \dfrac{{1 - {{\rm e}^\alpha }(t - \theta )}}{\alpha }} \right] - \\ & \dfrac{{{k^2}h{K_{io}}}}{\tau }\bigg( {{\alpha ^{ - {\lambda _o}}}(t - \theta )(2{{\rm e}^{\alpha (t - \theta )}} - 1)} \bigg. - \\ & \left. {\dfrac{{{t^{{\lambda _o}}}}}{\alpha } - \dfrac{{{t^{{\lambda _o} - 1}}}}{{{\alpha ^2}}} + {{\rm e}^{\alpha (t - \theta )}}\left[ {\dfrac{{{\theta ^{{\lambda _o}}}}}{\alpha } + \dfrac{{{\theta ^{{\lambda _o} - 1}}}}{{{\alpha ^2}}}} \right]} \right). \end{split} $

      (9)

      If $ T $ is the half time period of the limit cycle whose time period is $ T_u $, due to the symmetrical nature of the output waveform, $ y(t_o)=-y(T+t_o) $ and $ y(t_o)=\varepsilon $. Also, the peak amplitude $ A $ of the output waveform occurs at $ t_1 $.

      From the limit cycle input and output waveforms, it can be clearly seen that

      $ \theta=t_1-t_o. $

      (10)

      The two other parameters can be obtained by solving (11) and (12) at $ t_o $ and $ t_1 $ with the help of the technical computing software Mathematica.

      $ \begin{split} \varepsilon = & \,{{\rm e}^{\alpha {t_o}}}\varepsilon \!+\! kh(1 - {{\rm e}^{\alpha {t_o}}})\bigg( {(1 - k{K_{co}}) - } \bigg.\\ & \left. {{k^2}h{t_o}\left[ {\alpha {K_{co}} - \dfrac{{{K_{io}}}}{{{\alpha ^{{\lambda _o} \!+\! 1}}\Gamma (1 \!+\! {\lambda _o})}}} \right]} \right){{\rm e}^{\alpha ({t_o} - \theta )}}+\\ & \dfrac{{{k^2}h{K_i}}}{{\Gamma (1 \!+\! {\lambda _o})}}\left[ {{{({t_o} - \theta )}^\lambda } -\! \alpha {{({t_o} -\! \theta )}^{{\lambda _o} - 1}}} \right]\!\!\! \end{split} $

      (11)

      $ \begin{split} \quad A = & \, {{\rm e}^{\alpha ({t_1} - \theta )}}\varepsilon - kh\left( {1 - {{\rm e}^\alpha }({t_1} - \theta )} \right) - \\ & \dfrac{{{k^2}h{K_{co}}}}{\tau }\left[ {({t_1} - \theta )(1 - {{\rm e}^{\alpha {t_1}}}) + \dfrac{{1 - {{\rm e}^\alpha }({t_1} - \theta )}}{\alpha }} \right]-\\ & \dfrac{{{k^2}h{K_{io}}}}{\tau }\bigg( {{\alpha ^{ - {\lambda _o}}}({t_1} - \theta )(2{{\rm e}^{\alpha ({t_1} - \theta )}} - 1) - } \bigg.\\ & \left. {\dfrac{{t_1^{{\lambda _o}}}}{\alpha } - \dfrac{{t_1^{{\lambda _o} - 1}}}{{{\alpha ^2}}} + {{\rm e}^{\alpha ({t_1} - \theta )}}\left[ {\dfrac{{{\theta ^{{\lambda _o}}}}}{\alpha } + \dfrac{{{\theta ^{{\lambda _o} - 1}}}}{{{\alpha ^2}}}} \right]} \right). \end{split} $

      (12)
    • In order to recover the smooth signal from the noisy outputs due to measurement noise, the wavelet denoising technique is used. This technique decomposes the signal with noise into wavelet coefficients followed by the application of thresholding technique, and the signal is finally reconstructed using inverse DWT.

    • Following the identification of the process parameters, the three parameters of the FOPI controller namely the proportional gain ($ K_c $), the integral time constant ($ T_i $) and the fractional order of the integral part ($ \lambda $) of the controller are required to be tuned (or designed) as part of tuning the FOPI controller. The tuning factor $ \eta $ is taken as 1 while determining the parameters. Its value can be adjusted according to the needs of the designer to obtain the perfect damping for the control loop. Balanced tuning[25] where proportional and integral actions are equal in average, is used here to obtain one of the expressions to determine the three unknown parameters of the FOPI controller.

      The FOPI control equation is given by

      $ u(t)=K_c\left( e(t)+ \dfrac{1}{T_i} D_t^{-\lambda}e(t)\right) $

      (13)

      where $ e(t)=r(t)-y(t) $ represents the error between the set point value $ r(t) $ and the output $ y(t) $ at time $ t $. The velocity form of control law is

      $ \dot{u}(t)=K_c \left(\dot{e}(t)+ \dfrac{1}{T_i} \dfrac{{\rm d}}{{\rm d}t}\big(D^{-\lambda}e(t)\big)\right). $

      (14)

      If the controller is now subjected to a unit step input at $ t=0 $, the proportional control term $ K_ce(t) $ immediately attains the maximum value. This term however progressively reduces as $ t\rightarrow\infty $ while the integral term slowly starts from 0 and reaches steady state value as $ t\rightarrow\infty $. To maintain a balance between the two control terms, the method of balanced tuning was proposed.

      For an ideal controller, $ u(t)=u(\infty) $ for all $ t>0 $ which makes $ {\dot u}(t)=0 $. Thus, we get the following constraint:

      $ K_c\dot{e}(t)=-\dfrac{K_c}{T_i}\dfrac{{\rm d}}{{\rm d}t}\left(D^{-\lambda}e(t)\right). $

      (15)

      This constraint however cannot be satisfied at any time $ t $ except for the most simple first order process. Also with real values of error, there is a huge possibility of oscillatory signals. Hence, in order to do away with these issues, the real values of the error and its derivative are substituted with the absolute values over the average time.

      Thus,

      $ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \dfrac{1}{T_i}\int_0^{\infty}t \dfrac{{\rm d}}{{\rm d}t}\big(D_t^{-\lambda}|e(t)|\big){\rm d}t= \int_0^{\infty}t|\dot{e}(t)|{\rm d}t \Rightarrow $

      (16)

      $ \int_0^{\infty}t \dfrac{{\rm d}}{{\rm d}t}\left(D_t^{-\lambda}|e(t)|\right){\rm d}t= T_i\int_0^{\infty}t|\dot{e}(t)|{\rm d}t. $

      (17)

      The term $\displaystyle\int_0^{\infty}t|\dot{e}(t)|{\rm d}t $ is known as ITAD (integral of time multiplied by absolute value of time derivative of the error signal) and was introduced by Klan and Gorez[25] as a performance index to characterize the proportional control action. And the cost minimization performance index is introduced as follows:

      $ ITF_IAE=\int_0^{\infty}t \dfrac{{\rm d}}{{\rm d}t}\left(D_t^{-\lambda}|e(t)|\right){\rm d}t. $

      (18)

      It is the generalised version of the more well known integral of time multiplied by absolute value the error signal (ITAE). Klan and Gorez[25] proved that the balanced tuning provided a response close to the dynamics of the actual process to be controlled. For the FOPDT plant $ \dfrac{k{\rm e}^{-\theta s}}{(\tau s+1)} $, the control error when a step input is applied at the controller set point is

      $ e(t) = \left\{ {\begin{array}{*{20}{l}} {k,} & {0 \le t \le \theta }\\ {k{\rm e}^{\frac{-(t - \theta )}{\tau }}}, & {t \ge \theta}. \end{array}} \right. $

      (19)

      Equation (17), i.e., $ ITF_IAE=T_i\;ITAD $ for FOPI control thus leads to an explicit design relation from which we get the relation between $ T_i $ and $ \lambda $ as

      $ T_i=\dfrac{\tau}{\Gamma(\lambda)}+\dfrac{\theta^\lambda \zeta}{(\lambda+1)\Gamma(\lambda+1)} $

      (20)

      where

      $ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \zeta = \dfrac{\theta}{\theta+\tau} $

      (21)

      $ \frac{T_i}{\tau} = \frac{1}{\Gamma(\lambda)}+\frac{\theta^\lambda \zeta\tau^{-1}}{(\lambda+1)\Gamma(\lambda+1)}. $

      (22)

      For tuning the proportional gain of the controller, the explicit relation between the average residence time of an open loop process to that of the closed loop process is used. This relation shows the sluggish response of the open loop response compared to the closed loop response. The average residence time of the open loop process $ G({\rm j}\omega) $ is given by

      $ -\dfrac{{G'(0)}}{G(0)}=\tau+ \theta $

      (23)

      where $ {G'({\rm j}\omega)}=\dfrac{{\rm d}G({\rm j}\omega)}{{\rm d}\omega} $. On the other hand, the closed loop residence response is given by

      $ -\dfrac{D^\lambda T(s)}{T(s)}\bigg|_{s=0} $

      (24)

      where $ T(s) $ is the closed loop transfer function given by

      $ \begin{split} T(s) = &\, \dfrac{G_m(s)G_c(s)}{1+G_m(s) G_c(s)}=\\ & \dfrac{k K_c (T_i s^\lambda +1) {\rm e}^{-\theta s}}{\tau T_i s^{\lambda+1}+T_is^\lambda(1 + k K_{c} {\rm e}^{-\theta s}) + kK_{c} {\rm e}^{-\theta s}}. \end{split} $

      (25)

      Thus,

      $ -\dfrac{D^\lambda T(s)}{T(s)}\bigg|_{s=0}=-\dfrac{{G'}(0)}{G(0)}. $

      (26)

      Solving (26), the proportional gain $ K_c $ is obtained as follows:

      $ K_c = T_i\dfrac{\Gamma (\lambda+1)}{k(\tau+\theta)} $

      (27)

      where $ \Gamma (\;) $ is the gamma function.

      To obtain $ \lambda $, cost minimization of the performance index $ ITF_IAE $ is used, i.e.,

      $ {\rm min}|J_{ITF_IAE}(\lambda, K_c, T_i)|. $

      (28)

      The search space for $ \lambda $ is between 0.7 to 1.4 based on the ambiguous region reported by Chen et al.[10] Other integer order cost functions cannot be used for FO controllers because the fraction integral order $\dfrac{1}{s^\lambda} $ produces a strong singularity at origin when step input is applied especially in the search space $ 0< \lambda < 1 $ and is convergent only for $ \lambda=1 $.

      The tuning factor $ \eta $ is used in order to maintain a balance between the transient response characteristics and steady state response characteristics. It is either increased or decreased in order to reduce the rise time ($ t_r $) of the system such that the cost function integral of absolute error (IAE) is minimum and also peak overshoot $M_p < 1\%$. Here, $\eta$ is increased in integral multiples or reduced in integral fractions such that $\eta$ or $\dfrac{1}{\eta}$ is an integer. Generally, the search space is kept between 1 and 5.

      $ {\rm min}|J_{IAE}(\eta)| $

      (29)

      and

      $ M_p<0.01. $

      (30)

      Thus with its help, a perfect damping in the control loop can be obtained as per the requirements of the designer.

    • Thus, the whole design algorithm can be precisely described by the following steps:

      Step 1. The plant parameters are first identified by exact analysis or state space method. The controller is always in the loop and the initial values are set as described in the sections above.

      Step 2. The unknown plant is then modelled to a FOPDT plant model that mimics the actual plant and their accuracy is shown in Table 1 by the error values (exact analysis (EA)/describing function (DF)).

      Initial
      controller $ { G_{co}(s) }$
      $ { t_0 }$ $ { t_1 }$ $ { t_2 }$ $ { A }$ $ { y(t_0) }$ Process
      model $ { G_p(s) }$
      New controller Errors
      EA DF
      1) $ {2+ \dfrac{0.015}{s^{0.99}}}$ 82.89 83.875 87.87 0.057 0.05 $ { \dfrac{0.53{\rm e}^{-0.98s}}{63.5s+1} }$ $ { 14.4\left(1+\dfrac{1}{63.8 s^{0.88}}\right) }$ 0.002 0.04
      2) $ { 3.75 + \dfrac{2.03}{s^{0.99}} }$ 2.177 2.281 2.389 0.088 0.05 $ { \dfrac{{\rm e}^{-0.104s}}{1.09s+1} }$ $ { 3.262\left(1+\dfrac{1}{1.09 s^{1.19}}\right) }$ 0.035 0.08
      3) $ {0.484+ \dfrac{1.65}{s^{0.99}} }$ 5.41 6.28 6.49 2.15 0.05 $ { \dfrac{1.02{\rm e}^{-0.87s}}{0.10s+1} }$ $ { 0.25\left(1+\dfrac{1}{0.554 s^{1.201}}\right) }$ 0.087 0.204
      4) $ {0.95+ \dfrac{10.42}{s^{0.99}} }$ 53.23 69.73 72.79 1.31 0.05 $ { \dfrac{{\rm e}^{-16.23s}}{1.76s+1} }$ $ { 2.407\left(1+\dfrac{1}{8.77 s^{0.89}}\right) }$ 0.097 0.17

      Table 1.  Results from a single relay test and new controller setting

      Step 3. Depending on the values of the process parameters identified by the relay test, the FOPI controller is designed by using the balanced tuning method based on the obtained model parameters.

       1) $ITF_IAE =T_i $ ITAD for FOPI control thus leads to an explicit design relation from which we get the relation between $T_i$ and $\lambda$ as given in (20).

       2) The sluggish response of the open loop response compared to the closed loop response gives the other parameter $K_c$ as shown in (27).

       3) To obtain $\lambda$, cost minimization of the performance index $ITF_IAE$ is used as given by (28).

       4) The tuning factor $\eta$ is either increased or decreased in order to reduce the rise time ($t_r$) of the system such that the cost function IAE is minimum and also peak overshoot $M_p < 1\%$.

      Step 4. The step response of the system is monitored. Whenever the system falls below the specified ranges provided by the designer, the controller is automatically tuned because of the feedback and presence of relay and controller in parallel.

    • Robust control with respect to model uncertainties and good rejection of load disturbance effects are the two most important criteria in controller design. Generally, the larger the integral gain of the controller is, the less robust to the presence of load disturbances will the controller be as stated by Astrom and Hagglund[30].

      Fig. 3 (a) shows the relation between normalized internal gain and time delay of the proposed method, integral square error (ISE) and ITAE tuning methods by O′Dwyer[31]. It is observed that the integral gains vary between 1 and 2 whereas that of the other methods have extreme fluctuations. This proves the robustness of the proposed closed loop systems.

      Figure 3.  Evaluation of the tuning method

      The robustness is also measured from the maximum value of the sensitivity function which is as

      $ Ms=\left\| {\dfrac{1}{1+G_p({\rm j} \omega)G_c({\rm j} \omega) }} \right\|. $

      (31)

      It is well known that the reasonable values of $ M\!s $ are from 1.2 to 2 for robust processes. Different processes of the test batch given by Astrom and Hagglund[30] are tested with the proposed method and Fig. 3 (b) shows that the proposed tuning method is truly robust.

    • In this section, the proposed method is verified with the help of four processes reported in literature. The output oscillation amplitude (A) should always be in the prescribed limit as per the process variable tolerance. It is the designer who has to decide the value of the relay height so that the limit cycle is well within the acceptable amplitude level. The relay height here is taken as $ h=\pm 1 $ and the hysteresis width $ \varepsilon=\pm 0.05 $. The proposed method by exact analysis (EA) is compared with the describing function (DF) method with the help of frequency domain performance index $ J_f $. Here, $ J_f $ is the deviation of the $ H_2 $ norm of the original and the identified systems as explained by Xue and Chen[32]. Further, the $ H_2 $ norm shows how much a system amplifies or attenuates its inputs over all frequencies, i.e., it represents the output signal′s energy when excited by an impulse signal. Mathematically, $ H_2 $ norm and $ J_f $ are given by

      $ {\left\| {G_p(s)} \right\|}_2=\sqrt{\dfrac{1}{2 \pi}{\int}^{\infty}_{\infty}{\rm tra}(G_p({\rm j}\omega)\overline{G_p({\rm j}\omega)^{\rm T}}) {\rm d}\omega} $

      (32)

      $ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! J_f(s)=\left\| {G_p(s)-{{\tilde G}_p(s)}} \right\|_2. $

      (33)

      As fractional calculus is used in this method, a fitting frequency range is required in order to approximate the fractional order to its closest integer form. Hence, the nominal range is set to be (10–2 rad/s, 102 rad/s), i.e., $ \omega_b=10^{-2} $, $ \omega_h=10^2 $ rad/s. Here $ N=5 $. Non-integer derivative and Fomcon blocks (Simulink blocks for fractional order calculus) are used here[33]. Table 1 shows the four model parameters obtained, the initial controller, the new controller obtained from the three relay tests respectively as well as the comparison of proposed EA method and DF method for the lag dominant processes $ G_1(s)=\dfrac{0.55{\rm e}^{-s}}{62s+1} $ and $ G_2(s)=\dfrac{1}{(s+1)(0.2s+1)} $ and the delay dominant processes $ G_3(s)=\dfrac{{\rm e}^{-s}}{0.09s+1} $, $ G_4(s)=$$ \dfrac{{\rm e}^{-15s}}{(s+1)^3} $. As seen from Table 1, the error for EA is much smaller than that for DF method.

      In order to evaluate the quantitative performance measure of the tuning method to calculate the controller parameters, integral square error (ISE), integral of absolute error (IAE) and integral of time multiplied absolute error (ITAE) are used.

      The performance of the controller can also be associated with total variation (TV) of the control variable ($ u $) which gives the smoothness of the control output as stated by Skogestad[26]. For actuator preservation, the TV of the controllers, defined by (34), should be minimized. TV is given by

      $ \text{TV}=\sum_{i=1}^{\infty}|u_{i+1}-u_{i}| $

      (34)

      where subscript $ i $ refers to sampled values.

    • A lag dominant plant $ G_1(s)=\dfrac{0.55{\rm e}^{-s}}{62s+1} $ is considered here. The plant is identified by using the online relay tuning method. Additive white Gaussian noise (AWGN) of 20 dB signal to noise ratio (SNR) is added which is denoised by using the discrete wavelet technique (DWT). The recovered and noisy outputs signals with SNR = 20 dB are shown in Fig. 4 (a). Fig. 4 (b) shows the overall plant output during the online identification procedure. The limit cycle oscillation is carried out till t = 400 s after which the relay is switched off. At $ t = 200$ s, a static disturbance of 1 unit is applied but due to the presence of the controller, the limit cycle is not affected by it. At $ t = 500$ s, step response with the new FOPI controller is obtained which is again studied separately by comparing with other contemporary controllers. The estimated FOPDT model is listed in Table 1 and $ J_f $ values show the similarity between the actual and the identified model.

      Figure 4.  Simulation results for identification of the plant $ G_1(s)=\dfrac{0.55{\rm e}^{-s}}{62s+1} $

      Step responses of the proposed FOPI controllers are compared with the FOPID controller and PID controller designed in [6]. A step disturbance of $ L=1 $ is applied at $ t=300$ s in order to study the disturbance rejection capabilities of the controllers. It can be seen from Fig. 5 (a) that the proposed controller with $ \eta=1 $ has the longest response time and least disturbance rejection capability, although the balance between the proportional and the integral action is maintained. Hence, the tuning factor $ \eta $ is increased to 8 without compromising controller output.

      Figure 5.  Simulation results of the plant $ G_1(s)=\dfrac{0.55{\rm e}^{-s}}{62s+1} $

      The controller performance is separately shown in order to compare its performance with other controllers. The controller output as seen from Fig. 5 (b) shows that control energy shows no aggressiveness to achieve the desired response of the system and thus has a balance between the proportional and integral action. The absence of sudden change in control action that can harm the actuators of any physical system is thus avoided. This is the most significant achievement of the proposed method compared to other methods. The response of the system can also be changed according to the requirement of the designer with the help of the tuning factor $ \eta $. Fig. 5 (a) also shows the response of the plant with the proposed FOPI with increasing $ \eta $. Although the response time decreases drastically and the disturbance rejection capability also increases with increasing $ \eta $, the control energy increases as can be seen from the TV value in Table 2. Here, $ \eta=8 $ is chosen for the controller but it can be chosen based on the maximum control energy the actuator in physical systems actually can handle. Table 2 is tabulated with the new FOPI settings obtained by the auto-tuning procedure as well as with the FOPID values given by Monje et al.[6] and the PID values by direct, multiple point on-line tuning method used by Monje et al.[6] Transient response characteristics as well as performance indices ISE, IAE, ITAE and TV are also given in order to show the performance comparison among the FOPI, FOPID and PID controllers.

      Tuning methods Controller Transient response Performance indices
      Set point Whole system
      $ { t_r }$ (s) $ { t_s }$ (s) $ { M_p }$ (%) ISE IAE ITAE TV ISE ITAE
      C1 $ { 1.8\left(1+\dfrac{1}{63.8 s^{0.88}}\right) }$ 146.9 265 0 33.4 67.8 4 660 0.17 38.2 1 877
      C2 $ { 14.4\left(1+\dfrac{1}{63.8 s^{0.88}}\right) }$ 14.6 45 0 4.51 8.69 137.2 1.65 4.66 1 889
      C3 $ { 45\left(1+\dfrac{1}{63.8 s^{0.88}}\right) }$ 2.7 7.3 0 1.9 2.66 17.01 7.12 1.92 544.9
      C4 $ { 54\left(1+\dfrac{1}{1.11 s^{0.88}}\right) }$ 2 90 2.4 1.73 2.42 22.82 9.42 1.74 483.9
      C5 $ { 5.47+\dfrac{0.082}{s} }$ 29.2 181 8.5 15.7 27.4 898.9 0.58 16.23 5 231
      C6 $ { 7.961\,9+\dfrac{0.229\,9}{s^{0.964\,6}} +0.150\,4s^{0.015\,0}}$ 20.6 34.9 3.9 6.97 13.9 304.7 1.008 7.22 2 095
      C1 = proposed FOPI with $ { \eta=1 }$, C2 = proposed FOPI with $ { \eta=8 }$, C3 = proposed FOPI with $ { \eta=25 }$, C4 = proposed FOPI with $ { \eta=30 }$, C5 = PID by [6] and C6 = FOPID by [6]

      Table 2.  FOPI/IOPI controller parameters

      The performance indices ISE, IAE and ITAE and the transient responses for the three methods given in Table 2 and the step and controller responses observed from Figs. 5 (a) and 5 (b), respectively, show the better performance feature of the proposed method as compared to the other two. The three parameters $k$, $\tau$ and $\theta$ are perturbed by ±20% of the nominal process values and the corresponding step responses as well as the control outputs for the proposed FOPI controllers with $\eta=1$ & 8 and the other controllers are shown in Figs. 6 (a)6 (d). Thus, the robustness of the proposed tuning method is established.

      Figure 6.  Simulation results of the plant $ G_1(s)=\dfrac{0.55{\rm e}^{-s}}{62s+1} $ with ±20% perturbations in the process parameters

    • A second order process $ G_2(s)=\dfrac{1}{(s+1)(0.2s+1)} $ studied by Monje et al.[5] is considered. It can be modelled into an equivalent FOPDT model, the transfer function of which is listed in Table 1. The model as can be seen from Table 1 is a lag dominant plant. The $ J_f $ values in Table 1 indicate the similarity between the actual plant and the estimated model. AWGN of 20 dB SNR is added which is denoised using DWT. The noisy and the denoised signal is shown in Fig. 7 (a). Fig. 7 (b) shows the plant output to the online auto-tuning method. The limit cycle oscillation of the plant is obtained in the first 50 s. At $ t=30 $ s, the relay is perturbed by static disturbance of 1 unit, but due to the presence of the controller, the limit cycle is not affected. At $ t=50 $ s, the relay is switched off and from $ t=60 $ s, the step response of the plant is recorded with the newly designed FOPI controller. The step response of the plant is studied separately by comparing it with other controllers.

      Figure 7.  Simulation results for the identification of plant $ G_2(s)=\dfrac{1}{(s+1)(0.2s+1)} $

      Since the model obtained is a lag dominant one, the proposed FOPI controller will give a very sluggish response. Hence, tuning factor $ \eta= 7 $ is used to get the best response without harming the actuators of the plant. Table 3 is tabulated with the new FOPI settings obtained by the auto-tuning procedure. The proposed controller with the tuning factor shows superior controller performance as is clear from Fig. 8 (a). Fig. 8 (b) highlights the controller output which shows that the control energy of the proposed controller (FOPI with $ \eta=1 $) shows no sudden change to achieve the desired response due to the balanced tuning method. The tuning factor is so chosen that the controller does not harm the actuators involved. An aggressive control action can harm the actuators of any physical system. Fig. 8 (b) shows that even with the tuning factor, the control energy is not aggressive compared to the other controllers. The transfer functions of the three different controllers, transient response characteristics like rise time ($ t_r $), settling time ($ t_s $), peak overshoot ($ M_p $) and peak undershoot ($ M_u $) and the performance indices ISE, IAE and ITAE and TV for the three methods are also listed in Table 3. The controller performance is separately shown in order to compare its performance with other controllers. The flexibility of the proposed method is that the response of the system can also be changed according to the requirement of the designer with the help of the tuning factor $ \eta $.

      Tuning methods Controller Transient response Performance indices
      Set point Whole system
      $ { t_r }$ (s) $ { t_s }$ (s) $ { M_p }$ (%) $ { M_u }$ (%) ISE IAE ITAE TV ISE ITAE
      C1 $ { 0.466\left(1+\dfrac{1}{1.09 s^{1.19}}\right) }$ 5.25 12 0 0 1.25 2.26 4.25 0.013 1.206 7.73
      C2 $ { 3.262\left(1+\dfrac{1}{1.09 s^{1.19}}\right) }$ 0.43 1.45 0 0 0.13 0.36 0.154 0.153 0.132 0.136
      C3 $ { 6\left(1+\dfrac{1}{0.925 s^{0.7}}\right) }$ 0.14 1.45 24.1 10 0.166 0.50 0.165 0.626 0.166 0.163
      C4 $ { 3.43\left(1+\dfrac{1}{0.449 s^{0.7}}\right) }$ 0.25 2.8 29.8 10 0.22 0.57 0.18 0.243 0.221 0.253
      C5 $ { 3.1\left(1+\dfrac{1}{0.402 s}\right) }$ 0.266 2.8 35.5 9.5 0.245 0.602 0.23 0.214 0.245 0.231
      C1 = proposed FOPI, C2 = proposed FOPI with $ { \eta=7 }$, C3 = FOPI by [34], C4 = FOPI by F-MIGO method given in [34], C5 = PI by AMIGO method in [34]

      Table 3.  FOPI/PI controller parameters

      Figure 8.  Simulation results of the plant $ G_2(s)=\dfrac{1}{(s+1)(0.2s+1)} $

      The performance indices ISE, IAE and ITAE for the three methods given in Table 3 and the step and controller responses observed from Figs. 8 (a) and 8 (b), respectively, show improved performance of the proposed method as compared to the other two. The three parameters $k$, $\tau$ and $\theta$ are perturbed by –20% of the nominal process values and from Figs. 8 (c) and 8 (d), it is observed that the proposed tuning method is really robust.

    • A delay dominant plant $ G_3(s)=\dfrac{{\rm e}^{-s}}{0.09s+1} $ studied in [5] is considered. AWGN of 20 dB SNR is added which is denoised using DWT. Fig. 9 (a) shows the noisy and the denoised signals during limit cycle oscillation. A static load disturbance of $ L=1 $ is applied at $ t=30$ s during the relay test. Due to the presence of the FOPI controller during the online test, the effect of the disturbance is discarded as shown in Fig. 9 (b). It can be seen from Fig. 9 (b) that at $ t=60$ s the new FOPI controller comes into action and produces a smooth response to the step input. It is also able to reject the step disturbance of 1 unit which occurs at $ t=80 $ s. Table 1 shows the FOPDT plant and the new FOPI controller parameters. The $ J_f $ values in Table 1 shows the accuracy of the identification method.

      Figure 9.  Simulation results for identification of the plant $ G_3(s)=\dfrac{1.2 {\rm e}^{-s}}{0.09s+1} $

      In order to establish the superiority of the proposed tuning method, the proposed FOPI controller is compared with the FOPI controller (designed by Vu and Lee[34]), PI by F-MIGO method (mentioned by Vu andLee[34]) as well as with the PI controller cascaded with a first orderlag filter (designed by Lee et al.[35]). Table 4 is listed with the new FOPI controller with $ \eta=\dfrac{1}{2} $ obtained by the auto-tuning procedure as well as with the controller parameters of the other three controllers. Transient characteristics like rise time ($ t_r $), settling time ($ t_s $), peak overshoot ($ M_p $) and peak undershoot ($ M_u $) and the performance indices ISE, IAE and ITAE and TV for the three methods given in the table show that the proposed method has the minimum values. The step response that can be observed from Fig. 10 (a), is a smooth response without any overshoot and without affecting other transient parameters. The controller output as can be seen from Fig. 10 (b) shows that control energy shows no aggressiveness to achieve the desired response of the system and thus has a balance between the proportional and integral action. Consequently, there is minimum possibility of harming the system as a whole.

      Tuning methods Controller Transient response Performance indices
      Set point Whole system
      ${ t_r }$ (s) ${ t_s }$ (s) ${ M_p }$ (%) ${ M_u }$ (%) ISE IAE ITAE TV ISE ITAE
      C1 ${ 0.25\left(1+\dfrac{1}{0.554 s^{1.201}}\right) }$ 1.545 3 0 0 1.44 1.97 3.57 0.2 2.87 36
      C2 ${ 0.45\left(1+\dfrac{1}{0.74 s^{1.201}}\right) }$ 0.785 10 5.1 10 1.18 1.79 4.62 0.25 2.37 36.8
      C3 ${ 0.32\left(1+\dfrac{1}{0.604 s^{1.201}}\right) }$ 1.71 11 5.4 0 1.32 2.26 5.64 0.25 2.67 43.3
      C4 ${ 0.138\left(1+\dfrac{1}{0.324 s}\right) \left(\dfrac{1}{0.021\,5s+1}\right) }$ 2.83 8 0 0 1.67 2.35 3.66 0.202 3.3 45.1
      C1 = proposed FOPI with ${ \eta=\dfrac{1}{2}}$, C2 = FOPI by Vu and Lee[34], C3 = FOPI by F-MIGO method mentioned in Vu and Lee[34], C4 = PI by Lee et al.[35]

      Table 4.  FOPI/IOPI controller parameters

      Figure 10.  Simulation results of the plant $G_3(s)=\dfrac{{\rm e}^{-s}}{0.09s+1}$

      The three parameters $ k $, $ \tau $ and $ \theta $ are also perturbed by $ \pm20\% $ of the nominal process values and corresponding step responses and controller outputs are shown in Figs. 11(a)11(d). All the figures testify to the robustness of the controller design.

      Figure 11.  Simulation results showing responses for $ \pm20 \% $ perturbation in $ k $, $ \tau $ and $ \theta $

    • The higher order process $ G_4(s)=\dfrac{{\rm e}^{-15s}}{(s+1)^{3}} $ studied by Monje et al.[5] can be modeled into an equivalent FOPDT model by the online relay auto-tuning method. The parameters of the identified model are tabulated in Table 1 and the model is clearly a delay dominant model. The $ J_f $ values given in Table 1 indicate the similarity of the actual plant and the model that is obtained by the proposed method. Measurement noise of 20 dB SNR is added which is then denoised using DWT. The noisy and the denoised signals are shown in Fig. 12 (a). Like the previous examples, the online relay output is shown in Fig. 12 (b). At $ t=200 $ s, a static disturbance of $ L=0.5 $ is applied and from Fig. 12 (b), it is again observed that the disturbance does not affect the limit cycle condition. After obtaining the information of the plant, at $ t=500 $ s, the step response is observed with the new FOPI controller and a step disturbance of 1 unit is applied at $ t=700 $ s. The step response with the proposed controller with $ \eta=\dfrac{1}{2} $ is further studied separately and compared with three other controllers and is shown in Fig. 13 (a).

      Figure 12.  Simulation results for identification of the plant $G_4(s)=\dfrac{{\rm e}^{-15s}}{(s+1)^{3}}$

      Figure 13.  Simulation results of the plant $G_4(s)=\dfrac{{\rm e}^{-15s}}{(s+1)^{3}}$

      The proposed controller shows superior controller performance as is clear from Fig. 13 (a). Fig. 13 (b) highlights the controller output which shows that the control energy of the proposed controller shows no sudden change to achieve the desired response, thus maintaining a balance between the proportional and integral action. The performance indices tabulated in Table 5 reflect the better features of the proposed method compared to the other controllers. The three parameters $ k $, $ \tau $ and $ \theta $ are perturbed by $ \pm20\% $ of the nominal process values and the corresponding responses are shown in Figs. 14 (a)14 (d). Figs. 14 (a)14 (d) clearly show the superior performance of the controller in terms of robustness.

      Tuning methods Controller Transient response Performance indices
      Set point Whole system
      $ { t_r }$ (s) $ { t_s }$ (s) $ { M_p }$ (%) ISE IAE ITAE TV ISE ITAE
      C1 $ { 2.407\left(1+\dfrac{1}{8.77 s^{0.89}}\right) }$ 37.62 90 0 29.94 38.7 87 0.207 58.96 1 444
      C2 $ { 0.386\left(1+\dfrac{1}{13.156 s^{1.1}}\right) }$ 14.3 202.5 4.1 20.92 29.96 710 0.33 46.52 1 993
      C3 $ { 0.33\left(1+\dfrac{1}{10.313 s^{1.1}}\right) }$ 14 188 10 22.67 31.82 902.6 0.332 48.76 1 975
      C4 $ { 0.14\left(1+\dfrac{1}{5.33 s}\right) \left(\dfrac{1}{0.135\,6s+1}\right) }$ 45.5 122 0 29 38.17 840 0.213 53.34 1 403
      C1 = proposed FOPI with $ {\eta=\dfrac{1}{2}}$ , C2 = FOPI by [34], C3 = FOPI by F-MIGO method mentioned in [34], C4 = PI by [35]

      Table 5.  FOPI/IOPI controller parameters

      Figure 14.  Simulation results with $ \pm20 \% $ perturbation of the plant $ G_4(s)=\dfrac{{\rm e}^{-15s}}{(s+1)^{3}} $

    • The CTS apparatus from feedback, shown in Fig. 15 (a), consists of of four small towers each having a maximum height of 30 cm and one reservoir tank at the bottom to store water. Two submersed pumps in this reservoir pump water to the tanks. The four tanks can be coupled in a number of ways with the help of the manual valves (MV). The water pressure level is amplified and converted to analog signal by power supply unit amplifier (PSUPA) and is passed to the PCI1711 card. The control signal is maintained through Matlab/Simulink and can be sent from the PC through the PCI1711 card and PSUPA unit.

      Figure 15.  Coupled tank system from feedback

      The first column of the CTS system, the block diagram of which is shown in Fig. 15 (b), is used for the experimental analysis of the auto-tuning method. All valves except MVB and MVE are kept open. MVE is kept open for safety reasons. Table 6 shows the physical parameters of the coupled tank.

      Physical parameters Values Physical parameters Values
      Cross-sectional area of tanks ($ { {A} }$) 0.138 9 m2 Tank 1 outlet area ($ { a_1 }$) $ { 5.027\times10^{-5} }$ m2
      Tank 2 outlet area ($ { a_2 }$) $ { 5.027\times10^{-5} }$ m2 Acceleration due to gravity ($ { g }$) 9.8 m/s2
      Constant relating the control voltage with water flow from the pump $ { \tilde{\eta} }$ 0.002 4

      Table 6.  Physical parameters of the coupled tank

      The relay based system is run in Simulink. The water levels of the two tanks are recorded. The sampling time used is $ T_s=0.1 $ s. For single tank identification, the limit cycle output is obtained in the first 200 s. The relay is switched off at $ t=200 $ s but the initial controller is still in action. At $ t=300 $ s, the controller designed in this work is used and its step response is recorded. Fig. 16 (a) clearly shows the plant output during this online auto-tuning method. The step response with the proposed FOPI is studied separately in the controller design subsection. The values obtained from the limit cycle, $ h= $3.5 cm, $ A=0.775 $ cm, $ t_1=48.8 $ s and $ t_0=46.8 $ s, are used to determine the model parameters for $ G_{m1T}(s) $ of Tank 1. For the two coupled tank system, the limit cycle output occurs for 300 s and the relay is switched off after that. At $ t=600 $ s, step response to the new FOPI controller is recorded and Fig. 16 (c) shows the output data. The values of $ h=3.5 $ cm, $ A=2.5 $ cm, $ t_0=99.5 $ s, $ t_1=104 $ s and $ t_2=111 $ s from the limit cycle output are used to determine the equivalent FOPDT model $ G_{m2T}(s) $ of the CTS system. Thus, the mathematical models of both the single tank $ G_{m1T}(s) $ and the 2 coupled tank system $ G_{m2T}(s) $ are given by (35) and (36).

      Figure 16.  Experimental results for the control of water level for Tanks 1 and 2

      $ G_{m1T}(s)=\dfrac{4.55 {\rm e}^{-0.9s}}{50s+1} $

      (35)

      $ G_{m2T}(s)=\dfrac{5.15{\rm e}^{-4.5s}}{109s+1}. $

      (36)

      The step responses of the identified models are compared with the experimental step response data of the two tanks. From Figs. 16 (b) and 16 (d), it can be seen that the FOPDT models give similar results as that of the two tanks.

    • The controllers were designed both for the single tank as well as the two coupled system and their transfer functions are given as

      $ G_{c1T}(s)=0.88\left(1+\dfrac{1}{49.6 s^{1.01}}\right) $

      (37)

      $ G_{c2T}(s)= 0.163\left(1+\dfrac{1}{100.05 s^{1.01}}\right). $

      (38)

      The FOPI controllers for the CTS setup have an anti-windup block to prevent integration wind-up that can lead to the motor in the pump to saturate. Figs. 17 (a) and 17 (b) show the block diagrams of Tanks 1 and 2 water level control. The FOPI controller $ G_{c1T}(s) $ designed for Tank 1 level control can reject disturbance. It also provides a smooth water level control and also can provide a good set point tracking performance. It uses a tuning factor of $ \eta=4 $. Step disturbance of 1.5 cm and –1.5 cm are applied at $ t=250 $ s and $ t=450 $ s, respectively. It can be seen from Fig. 18 (a) that the FOPI controller can easily reject the disturbance and maintains the desired water level. Fig. 18 (b) shows the set point tracking data which clearly shows that there is no overshoot in the response of the proposed FOPI controller.

      Figure 17.  Block diagram of CTS

      Figure 18.  Experimental results for the control of water level for Tanks 1 and 2

      The control of water level in Tank 2 is rather more difficult than controlling the water level in Tank 1 because the control voltage doesn′t directly control the water level in Tank 2 but has to be controlled through the outflow from Tank 1. Tank 2 adds a little more non-linearity to the system due to the presence of the extra inlet valve which was the outlet valve for Tank 1. Hence, the water level is somehow maintained in the proper level with some noise. Step disturbance of 1 cm and –1 cm are applied at $ t=600 $ s and $ t=1\,300 $ s, respectively. Figs. 18 (c) and 18 (d) show the good performance of the proposed FOPI $ G_{c2T}(s) $ controller. Figs. 18 (e) and 18 (f) show the controller output of the two tanks. It can be clearly seen that the control signals do not have any abrupt step changes and have balanced proportional and integral action. This leads to a smooth step response without any overshoot and also the pump motors are not harmed by any sudden spike in control action. A few transient response parameters of the system are listed in Table 7.

      Tuning methods Transient analysis Performance indices (set point)
      ${t_r} $ (s) $ {t_s} $ (s) $ {M_p }$ (%) $ {M_u} $ (%) IAE ISE ITAE TV
      FOPI controlled Tank 1 24 50 0 0 230.2 1 894 5 512 0.245 4
      PI controlled Tank 1 71 350 0 0 350 2 100 5 799 0.298 9
      FOPI controlled Tank 2 120.5 200 5 0 847.7 8 518 $ {6\times10^4} $ 0.270 8
      PI controlled Tank 2 24.5 455 15 40 1 173 9 324 $ {1.26\times10^5} $ 0.438 0

      Table 7.  The response parameters of the CTS system

    • This work demonstrates that auto-tuning using an FOPI controller is highly promising. Also, this method shows commendable results with minimum experiment time using a single relay while all other identification tests require repetitive tests to design the optimal PI controller within acceptable gain margins. This method uses a state space method rather than the describing function analysis used in most of the relay identification methods and hence accurate values are obtained. The controller designed produces a smooth response and the control action has no sudden change due to the imbalance in proportional and integral gain. This absence of aggressive control action is perfect for physical systems as the actuators never come into harm. The tuning factor helps to maintain a good trade off between response time and stability and the designer can get the best response according to the requirements of the designer just by adjusting the tuning factor. Four test processes of industrial relevance found in the literature are simulated and they validate these points. Further, experimental analysis on a CTS setup is done and the results obtained show excellent performance of the FOPI controllers. However, the approximation of fractional order to its closest integer form is the challenging issue which determines performance of the controller. Thus a search for better approximation methods with minimum complexity is required in future.

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