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  • 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME 115 TUNING OF AN I-PD CONTROLLER USED WITH A HIGHLY OSCILLATING SECOND-ORDER PROCESS Galal A. Hassaan Emeritus Professor, Department of Mechanical Design & Production, Faculty of Engineering, Cairo University, Giza, EGYPT ABSTRACT High oscillation in industrial processes is something undesired and controller tuning has to solve this problems. I-PD is a controller type of the PID-family which is suggested to overcome this problem with improved performance regarding the spike characteristics associated with certain types of controllers. This research work has proven that using the I-PD is capable of solving the dynamic problems of highly oscillating processes. A second order process of 85.45 % maximum overshoot and 8 seconds settling time is controlled using an I-PD controller (through simulation). The controller is tuned by minimizing the sum of square of error (ISE) of the control system using MATLAB. The MATLAB optimization toolbox is used assuming that the tuning problem is an unconstrained one. The result was cancelling completely the 85.45 % overshoot and decreasing the settling time from 8 seconds to only 1.46 seconds without any undershoot. The performance of the control system using an I-PD controller using the present tuning technique is compared with that using the ITAE standard forms tuning technique. Keywords: Highly oscillating processes – I-PD controller - Optimal controller tuning – Improving control system performance. 1. INTRODUCTION Highly oscillating response is present in a number of industrial processes incorporating low damping levels. Conventionally, the PID controller is used and tuned for better performance of the control system. The I-PD controller is one of the next generation of PID controllers where research and application is required to investigate its effectiveness compared with PID controllers. INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2014): 7.5377 (Calculated by GISI) www.jifactor.com IJMET © I A E M E
  • 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME 116 Katsumata (1994) described a method for the auto-tuning of an I-PD controller for processes with long dead time using neural networks [1]. Seraji (1998) introduced a class of simple nonlinear PID-type controllers comprising a sector-bounded nonlinear gain in cascade with a linear fixed-gain P, PD, PI or PID controller [2]. Lelic (1999) extracted the essence of the most recent development of PID control during the 1990’s based on a survey of 333 papers published in various journals [3]. Hamdan and Gao (2000) developed a modified PID (MPID) controller to control and minimize the hysteresis effect in pneumatic proportional valves. The modified controller showed better command following and disturbance rejection qualities than other types [4]. El-Sousi (2001) presented the analysis, design and simulation of 1DOF and 2 DOF controllers for the indirect field orientated control of an induction motor drive system [5]. Tagami and Ikeda (2002) proposed a design method of robust I-PD controllers based on a genetic algorithm [6]. Araki and Taguchi (2003) surveyed the important results about two degree of freedom PID controllers including equivalent transformations, the effect of 2DOF structure and relation to the preceded derivative PID and the I-PD controllers [7]. Astrom and Hagglund (2004) presented a design method used to maximize the integral gain subject to a robust constraint giving the best reduction of load disturbance. They revised tuning of PID controllers in the spirit of Ziegler and Nichols technique [8]. Su, Sun and Duan (2005) proposed an enhanced nonlinear PID controller with improved performance than the conventional linear fixed- gain PID controller. They incorporated a sector-bounded nonlinear gain in cascade with the conventional PID controller [9]. Li, Ang and Chong (2006) presented remedies for problems involving the integral and derivative terms of the PID controller. They studied the PI-D and I-PD control structures [10]. Arvanitis, Pasgianos and Kalogeropoulos (2007) investigated the control of unstable second order plus dead-time process using PID-type controllers. They proposed tuning rules based on the satisfaction of gain and phase margin specifications [11]. Madady (2008) proposed a PID type with iterative learning control update law to control discrete-time SISO linear time- invariant systems performing repetitive tasks. He proposed an optimal design method to determine the PID parameters [12]. Coelho (2009) proposed a tuning method to determine the parameters of PID control for an automatic regulator voltage system using chaotic optimization approach based on Lozi map [13]. Khare (2010) developed an internal model mode based PID controller to control the temperature of outlet fluid of the heat exchanger system. His controller demonstrated 84 % improvement in overshoot and 44 % improvement in settling time compared to the classical controller [14]. Ntogramatzidis and Ferrante (2011) introduced a range of techniques for the exact design of PID controllers for feedback control problems involving requirements on the steady-state performance and standard frequency domain specifications. The control parameters had to be calculated on-line meaning that their techniques appear convenient with adaptive and self-tuning control strategies [15]. Shiota and Ohmari (2012) proposed an adaptive I-PD controller using augmented error method for SISO systems [16]. Prasad, Varghese and Malakrishnan (2012) designed I-PD controller and optimized its parameters using particle swarm intelligence for a first order lag integrating plus time delayed model [17]. Rajinikanth and Latha (2012) proposed a method to tune the I-PD controller for the time-delayed unstable process using bacterial foraging optimization algorithm [18]. Agalya and Nagaraj (2013) studied using a nonlinear feedback controller to control continuous stirred tank reactors. They used PID and I-PD controller structures with integral square error (ISE) criterion to tune the controllers [19]. Shiota and Ohmori (2013) proposed an adaptive I- PD control scheme with variable reference model as one of the control schemes which overcome model uncertainty. They showed the validity of their scheme through numerical simulation [20]. Hassaan (2014) presented a simple tuning approach to tune PID controllers used with both underdamped and overdamped equivalent processes based on the integral square error (ISE) criterion [21,22].
  • 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME 117 1. ANALYSIS Process: The process is a second order process having the parameters: Natural frequency: ωn = 10 rad/s Damping ratio: ζ = 0.05 The process has the transfer function: Mp(s) = ωn 2 / (s2 + 2ζωn s + ωn 2 ) (1) The time response of this process to a unit step input is shown in Fig.1 as generated by MATLAB: Fig.1 Step response of the uncontrolled process. The severity of the process oscillations is measured by its maximum percentage overshoot. It has a maximum overshoot of 85.4 % and an 6 seconds settling time. Controller: The controller used in this study is an integral (I) - proportional derivative (PD) controller. In this controller, the integral part acts only on the error e of the closed loop control system incorporating the controller and the controlled process. The proportional and derivative parts act on the system output y. By this it is possible to get red of the kick following a reference input change (set-point kick) as quoted by Prasad, Varghese and Balakrishnan [17]. The block diagram of the closed-loop control system incorporating the I-PD controller is shown in Fig.2 [17]. controller Fig.2 I-PD controller.
  • 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME 118 The controller model is function of the control system three variables: - Control system error, E(s). - Controller output, U(s). - Control system output, Y(s). Using the block diagram of Fig.2, the 3 variables are related through the model: U(s) = (Ki/s)R(s) – (Kpc + Kds + Ki/s)Y(s) (2) Where:Kpc = Proportional gain Ki = Integral gain Kd = Derivative gain i.e. the controller has 3 parameters to be identified to control the process and produce a satisfactory performance. Control System Transfer Function: U(s) is related to Y(s) through the process transfer function, Gp(s). That is: Gp(s) = Y(s) / U(s) (3) Combining Eqs.2 and 3 gives the transfer function of the closed-loop system as: M(s) = b0 / {a0s3 + a1s2 + a2s + a3} (4) where: b0 = ωn 2 Ki , a0 = 1 a1= 2ζωn + ωn 2 Kd , a2 = ωn 2 (1 + Kpc) a3 = ωn 2 Ki System Step Response: A unit step response is generated by MATLAB using the numerator and deniminator of Eq. 3 providing the system response c(t) as function of time [23]. 2. CONTROLLER TUNING The sum of square of error (ISE) is used an objective function, F of the optimization process. Thus: F = ∫ [c(t) – css]2 dt (5) where css = steady state response of the system. The performance of the control system is judged using two time-based specifications: (a) Maximum percentage overshoot, OSmax (b) Settling time, Ts Tuning Results: The MATLAB command "fminunc" is used to minimize the optimization objective function given by Eq.5 without any parameters ot functional constraints [24]. The results are as follows: Controller parameters:
  • 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME 119 Kpc = 1.7523 Ki = 5.3314 Kd = 0.1113 The time response of the closed-loop control system to a unit step input is shown in Fig.3. Fig.3 Step response of the I-PD controlled second order process. Characteristics of the control system using the tuned I-PD controller: - Maximum percentage overshoot: 0 % - Maximum percentage undershoot: 0 % - Settling time: 1.46 s 3. COMPARISON WITH STANDARD FORMS TUNING The control system in terms of its transfer function is a fourth order one. The optimal characteristic equation of such a system with a second-order numerator is given using an ITAE critertion by [25]: s3 + 1.75ωo s2 + 2.15ωo 2 s + ωo 3 (6) Comparing Eq.6 with the corresponding one in Eq.4 we get 3 equations in ωo, Kpc, Ki and Kd i.e. 4 unknowns and 3 equations. To be able to get the controller parameters using this tuning technique, one of the parameters has to be assumed. It was reasonable from the equations to assign Ki (it was taken as 5.3314 as obtained in the present tuning technique using the ISE criterion). The rest of the controller parameters were calculated as: Kpc = 0.4136 Ki = 5.3314 Kd = 0.1319 The time response of the control system using this standard forms tuning technique is shown in Fig.4:
  • 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME 120 Fig.4 Step response of the I-PD controlled second order process using the ITAE standard forms. Characteristics of the control system using the standard forms tuning technique: - Maximum percentage overshoot: 1.98 % - Maximum percentage undershoot: 2.35 % - Settling time: 0.44 s 4. CONCLUSIONS - It is possible to suppress completely the higher oscillations in processes through using the I-PD controller. - It was possible to overcome the set-point kick problem associated with the standard PID. - Through using an I-PD controller it was possible reduce the settling time from about 8 seconds to about 1.46 seconds indicating the fast settlement of the controlled process. - The maximum and minimum overshoot were cancelled completely . - Tuning the controller using standard forms produced a time response of the closed loop system having more overshoot and undershoot (1.98 % and 2.35 % respectively). - However, the settling time was better in the standard forms technique than the present technique based on ISE error criterion. REFERENCES 1. K. Katsumata (1994), “A predictive I-PD controller for processes with long deadtime using neural networks”, IEEE World Congress on Computational Intelligence, 27 June – 2 July , pp.3154-3157. 2. H. Seraji (1998), “A new class of nonlinear PID controllers”, J. of Robotic Systems, Vol.15, No.3, March 1998, pp.161-181. 3. M. Lelic 1999), “PID controllers in nineties”, Corning Incorporation, Science & Technology Division, Corning, N.Y., 12 July. 4. M. Hamdan and Z. Gao (2000), “A novel PID controller for pneumatic proportional valves with hysterisis”, IEEE Industry Applications Conference, October , Vol.2, pp.1198-1201. 5. F. El-Sousy (2001), “Design and implementation of 2DOF I-PD controller for indirect field orientation control induction machine drive system”, IEEE Int. Symposium on Industrial Electronics, Vol.2, pp.1112-1118.
  • 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 115-121 © IAEME 121 6. T. Tagami and K. Ikeda (2002), “A design of robust I-PD controller based on genetic algorithm”, Modelling, Identification and Control, February 18-21. 7. M. Araki and H. Taguchi (2003), “Two-degree of freedom PID controllers”, Int. J. of Control, Automation and Systems, Vol.1, No.4, pp.401-411. 8. K. Astrom and T. Hagglund (2004), “Revisiting the Ziegler-Nichols step response method for PID control”, J. of Process Control, Vol.14, pp.635-650. 9. Y. Su, D. Sun and B. Duan (2005), “Design of an enhanced nonlinear PID controller”, Mechatronics, Vol.15, pp.1005-1024. 10. Y. Li, K. Ang and G. Chong (2006), “PID control system analysis and design”, IEEE Control Systems Magazine, Vol.26, No.1, pp.32-41. 11. K. Arvanitis, G. Pasgianos and G. Kalloeropoulos (2007), “Tuning PID controllers for a class of unstable dead-time processes based on stability margins specifications”, Proceedings of the 15th Mediterranean Conference on Control and Automation, July 27-29, Athena, Greece. 12. A. Madady (2008), “PID-type iterative learning control with optimal gains”, Int. J. of Control, Automation and Systems, Vol.6, No.2, April , pp.194-203. 13. L. Coelho (2009), “Tuning of PID controller for an automatic regulator voltage system using chaoticn optimization approach”, Chaos, Solitons and Fractals, Vol.39, pp.1504-1514. 14. Y. Khare (2009), “PID control of heat exchanger systems”, Int. J. of Computer Applications, Vol.8, No.6, October, pp.22-27. 15. L. Ntogramatzidis and A. Ferrante (2011), “Exact tuning of PID controllers in control feedback design”, Proceedings of the 18th IFAC World Congress, Milano, Italy, August 28-September 2, pp.5759-5764. 16. T. Shiota and H. Ohmori (2012), “Design of adaptive I-PD controller using augmented error method”, IFAC Conference on Advances in PID Control, Brescia, Italy, March 28-30. 17. S. Prasad, S. Varghese and P. Balakrishnan (2012), “Optimization of I-PD controller for a FOLIPD model using particle swarm intelligence”, Int. J. of Computer Applications, Vol.43, No.9, April 2012, pp.23-26. 18. V. Rajinikanth and K. Latha (2012), “I-PD controller tuning for unstable system using bacterial foraging algorithm”, Applied Computational Intelligence and Soft Computing, Vol.2012, Article ID 329389. 19. A. Agalya and B. Nagaraj (2013), “Certain investigation on concentration control of CSTR- A comparative approach”, International Journal of Advances in Soft Computing and its Application, Vol.5, No.2, July, pp.1-14. 20. T. Shiota and H. Ohmori (2013), “Design of adaptive I-PD control system with variable reference model”, 3rd Australian Control Conference (AUCC), 4-5 November, pp.115-120. 21. G.A. Hassaan (2014), “Simple tuning of PID-controllers used with underdamped second-order processes”, International Journal of Mechanical and Production Engineering Research and Development, Vol.4, No.2, Accepted for Publication. 22. G.A. Hassaan (2014), “Simple tuning of PID-controllers used with overdamped second-order processes”, International Journal of Research in Engineering and Technology, Vol.2, No.4, pp.87-96. 23. ------ (2012), "Control system toolbox", MathWorks, September. 24. ------ (1998-2012), “The MOSEK optimization toolbox for MATLAB”, MOSEK, Denmark. 25. D. Graham and R. Lathrop (1953), “The synthesis of optimal response: criteria and standard forms”, Transactions of AIEE, Vol.72, November. 26. Galal A. Hassaan, “Optimal Design of Machinery Shallow Foundations With Clay Soils” International Journal of Mechanical Engineering & Technology (IJMET), Volume 5, Issue 3, 2014, pp. 91 - 103, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.