IJRIT International Journal Of Research In Information Technology, Volume 3, Issue 7, July 2015, Pg. 206-213

International Journal of Research in Information Technology (IJRIT) www.ijrit.com

ISSN 2001-5569

Tuning of a PI-P Controller for use with a highly Oscillating Second-order-like Process Galal Ali Hassaan Emeritus Professor, Department of Mechanical Design & Production, Faculty of Engineering, Cairo University [email protected]

Abstract The purpose of this paper is to investigate using a PI-P controller to control a highly oscillating second-order-like process. The controller is tuned using MATLAB optimization toolbox and four error-based objective functions. The best objective function is assigned. The effect of the time constant of the controller on the control system performance in the time domain is investigated. The performance of the control system using the PI-P controller is compared with that using other different six controllers studied before by the author. The PI-P controller can compete with only one of the six controllers. It provides an oscillatory time response around an increasing trend which is not preferable.

Keywords: PI-P Controller, Second-order-like Process, Controller Tuning, Control System Performance.

1. Introduction Oscillating second-order-like processes are one of the difficult ones to be controlled because of their oscillating nature. It represents a challenge for the control engineer to select a proper controller and tune this controller for optimal performance of the control system . The author investigated using six types of controllers with such a process, and this is the seventh controller. The objective of this research is to help control engineers to sect and tune a reasonable controller for a specific application. Taylor, McCabe, Young and Chotai (2000) applied the PIP methodology to the ALSTOM benchmark challenge representing the gasifier system of an integrated gasification combined cycle power plant. They tuned a discrete-time PIP control system based on a reduced-order backward-shift model of the gasifier [1]. Quanten, Janssens, McKenna, Young and Berckmans (2002) developed a PIP climate controller for a SISO system. They used a second-order model as a good description for the temperature dynamics with 0.985 correlation coefficient [2]. Gu, Taylor and Steward (2004) considered the application of PIP control to the Lancaster University computerized intelligent excavator. They compared the PIP controller with a conventionally tuned PID algorithm showing the feasibility of the proposed PIP controller [3]. In a real time application of the PIP controller the response was oscillating around the steady-state response [3]. Dixon, Taylor and Shaban (2005) developed two different design methods to achieve fast smooth movement of an excavator arm. They identified the advantages and limitations of the two approaches in terms of performance and design effort. They compared the application of P and PIP controllers to control the excavator arm. The PIP control showed more overshoot than the P-control [4]. Al-Hammouri, Liberatore, Brancki and Philips (2006) described a method for finding the stability regions of PI and PIP controllers for TCP AQM. They showed that the PIP controller can be unstable in the presence of delays for the control parameters given in the literature [5]. Exdaktylos, Taylor and Chotai (2006) developed an

Galal Ali Hassaan, IJRIT-206

IJRIT International Journal Of Research In Information Technology, Volume 3, Issue 7, July 2015, Pg. 206-213

approach motivated by the PIP control structure suggesting that the tuning techniques developed can be used for straightforward NMSS/MPC design [6]. Taylor, Shaban, Stables and Ako (2007) considered PIP control of nonlinear system defined by statedependent parameter models. Their approach yielded a PIP control with improved performance and robustness in comparison with conventional linear PIP control [7]. Exadaktylos, Taylor and Chotai (2008) considered PIP control of nonlinear dynamic systems described by state dependent parameter models with constraints. They developed a low level stabilizing SDP/PIP controller to steer the system in the desired direction, whilst a reference governor was introduced to account for constraints in the system variables [8]. Shaban (2012) considered state dependent parameter and PIP control of a wide class of nonlinear systems. He developed an exact linearization by local coordinate transformations returning the closed-loop system to controllable state. He illustrated simulated examples to verify the applicability of his approach [9]. Shaban and Nada (2013) carried out a full identification process for a discrete-time model of an autonomous robot for PIP control. They compared the performance of the PIP control with an optimal PID control methodology using an ITSE index for controller tuning [10]. Shaban, Nada and Taylor (2014) developed an exact linearization by feedback approach for state dependent parameter and PIP control. They demonstrated their method using a simple automated belt drive by a DC motor. An exact linearization step returned a linear model used to design a PIP control algorithm based on linear design strategies. Experimental results demonstrated an acceptable control performance for the nonlinear system [11].

2. Process The process is a second-order-like one having an equivalent transfer function, Gp(s) given by: Gp(s) = ωn2 / (s2 + 2ζωns + ωn2) Where: ωn = process natural frequency = 10 ζ = process damping ratio = 0.05

(1) rad/s

The process has the time response to a unit step input shown in Fig.1.

Fig.1 Time response of the highly oscillating second-order process. The process has a 85.446 % maximum percentage overshoot and a 5.975 seconds settling time.

3. PI-P Controller A PI-P controller consists of two parts: one feedforward part which is a PI-sub-controller and a feedback proportional part in a separate loop with the controlled process as shown in Fig.2 [5].

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IJRIT International Journal Of Research In Information Technology, Volume 3, Issue 7, July 2015, Pg. 206-213

Reference input [R(s)]

PI-part [GPI(s)]

Process [Gp(s)]

Output [C(s)]

Proportional part [GP(s)] Fig.2 Structure of a PI-P controller [5]. The PI-part of the PI-P controller has the transfer function, GPI(s) [5]: GPI(s) = Kpc[1 + (Tis)-1] Where: Kpc = proportional gain of the PI-controller. Ti = integral time constant of the PI-controller. The proportional part of the PI-P controller has the transfer function GP(s) [5]: GP(a) = Kf Where: Kf = gain of the feedback proportional part.

(2)

(3)

4. Closed-loop Transfer Function The closed-loop control system incorporating the PI-P control and the process is shown in Fig.2. Using the block diagram in Fig.2 , the process transfer function in Eq.1 and the controller sub-transfer functions in Eqs.2 and 3, the overall transfer function of the closed-loop control system of Fig.2 M(s) is given by: M(s) = (b0s + b1) / (a0s3 + a1s2 + a2s + a3)

(4)

Where: b0 = KpcTi ωn2 b1 = Kpc ωn2 a0 = T i a1 = 2ζωnTi a2 = ωn2(1+Kf)Ti + KpcTiωn2 a3 = Kpc ωn2

5. Controller Tuning The PI-P controller is tuned using an error function (objective function) based on an error function e(t) between the control system output time response c(t) and the steady-state time response of the system which is a unit value for a unit step reference input. That is: e(t) = c(t) -1 (5) Four objective functions are used in the tuning process of the PI-P controller. They are related to the error function of Eq.5 as follows [12,13]: ITAE: ISE: ITSE: ISTSE:

∫ t|e(t)| dt ∫ [e(t)]2 dt ∫ t[e(t)]2 dt ∫ t2[e(t)]2 dt

(6) (7) (8) (9)

The MATLAB optimization toolbox is used to minimize the objective functions in Eqs.6 through 9 through its command ‘fminunc’ [14]. The time response of the closed-loop control system is evaluated using the command ‘step’ of the MATLAB control toolbox [15]. The time based specifications of the control is evaluated using the command ‘stepinfo’ of the control toolbox [15]. The tuning process results of the PI-P three parameters and some of time based specifications of the control system are given in Table 1.

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IJRIT International Journal Of Research In Information Technology, Volume 3, Issue 7, July 2015, Pg. 206-213

Kpc Kf Ti (s) OSmax (%) Ts (s)

ITAE 0.0119 -0.9908 1.9940 9.421 6.2333

Table 1: PI-P controller tuning ISE ITSE 0.0160 0.0113 -0.9613 -0.8566 0.9996 0.1137 0.1017 4.3748 8.5709 6.9030

ISTSE 0.0124 -0.17123 0.0209 2.1221 5.6681

The negative sign of the proportional part can be easily achieved through the electronic design of the controller part using operational amplifiers [16]. The unit step time response of the control system using the four objective function is shown in Fig.3 as simulated by MATLAB.

Fig.3 Control system time response using objective functions. The time response to a unit step reference input has an oscillating nature before settling at a unit steadystate value. The best performance is achieved with the ISTSE objective function where it has less oscillation amplitude. However this oscillation is dynamically undesired since it excites dynamic forces and other physical properties in the controlled process.

6. Effect of Controller Time Constant Because of the nonlinearity of the optimization problem, there are too many locus minima of the ISTSE objective function representing local minima. The effect of the PI-part of the PI-P controller on the dynamics of the control system is shown in Fig.4 for time constant in the range 0.0193 to 15 s.

Fig.4 Effect of PI time constant of control system step response.

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IJRIT International Journal Of Research In Information Technology, Volume 3, Issue 7, July 2015, Pg. 206-213

The effect of the controller time constant of the optimum ISTSE, maximum percentage overshoot and settling time is shown in Fig.5 for controller time delay up to 17.5 s.

Fig.5 Effect of PI time constant of control system ISTSE, maximum overshoot and settling time. The settling time is almost constant at about 6 s for Ti ≥ 2 s, The maximum percentage overshoot increases to about 55 % as the time constant of the controller increases, then drops to settle at 50 %. The ISTSE objective function value decreases as the time constant of the controller increases.

7. Comparison with other Controllers The effect of using seven different controllers to control the highly oscillating second-order-like process is shown in Fig.6. The comparison covers the I-PD controller [17], the PD-PI controller [18], the PI-PD controller [19], the PID + first-order lag controller [20], the PID controller [21] , the PPI controller [22] and the present PI-P controller. It is clear that the PI-P control can compete only with the PPI controller while the other 5 controllers are better than both the PPI and the PI-P controllers.

Fig.6 Performance comparison using different controllers.

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IJRIT International Journal Of Research In Information Technology, Volume 3, Issue 7, July 2015, Pg. 206-213

A quantitative comparison of the control system performance parameters using the seven controllers is given in Table 2. Table 2: Quantitative comparison of performance parameters.

I-PD controller PD-PI controller PI-PD controller PID + first-order lag controller PID controller PPI controller Present PI-P controller

OSmax (%) 0 0 0 15.924

Ts (s) 1.5000 0 0.3714 0.5643

3.648 0 2.1221

0.8550 16.0403 5.6681

7. Conclusions - The possibility of using a PI-P controller to control a highly oscillating second-order-like process was studied. - The controller was tuned using the MATLAB toolbox. - Four error-based objective functions were used in the tuning process. - The study assigned the ISTSE as the more suitable objective function for the highly oscillating secondorder process. - Because of the nonlinearity of the optimization problem, local minima were expected and different levels of the integral time constant of the PI-P controller were investigated. - The performance of the control system using the PI-P controller was compared with other six controllers (I-PD, PD-PI, PI-PD, PID + first-order-lag, PID and PPI controllers). - The PI-P controller could compete only with the PPI controller. - Other controllers (I-PD, PD-PI and PI-PD) provided much better performance when used with such highly oscillating second-order-like process.

References [1] C. Taylor, A. McCabe, P. Young and A. Chotai, “Proportional-integral-plus (PIP) control of the ALSTOM gasifier problem”, Proceedings of the Institution of Mechanical Engineers, Vol.214, Part I, 2000, pp.469-480. [2] S. Quanten, K. Janssens, P. McKenna, P. Young and D. Berckmans, “Measurement and analysis of the thermal micro-climate in cars”, Proceedings Indoor Air, 2002, pp.701-706. [3] J. Gu. C. Taylor and D. Seward, “Proportional-integral-plus control of an intelligent excavator”, Computer-aided Civil nd Infrastructure Engineering, Vol.19, No.1, 2004, pp.16-27. [4] R. Dixon, C. Taylor and E. Shaba, “Comparison of classical and modern control applied to excavatorarm”, AFAC 16th Triennial World Congress, 2005, 6 pages. A. A. Name, "Conference Paper Title", in Conference Name, Year, Vol. x, pp. xxx-xxx. [5] A. Al-Hammouri, V. Liberatore, M. Branicki and S. Philips, “Parameterizing PI congestion controllers”, Workshop on Feedback Control Implementation and Design in Computing Systems and Networks, Vancouver, Canada, April 2006. [6] V. Exadaktylos, C. Taylor and A. Chotai, “Model predictive control using a non-minimal state space form with an integral-of-error state variable”, UKACC International Conference (Control 06), Glasgow, UK, August 2006. [7] C. Taylor, E. Shaban, M. Stables and S. Ako, “Proportional-integral-plus control application of statedependent parameter models”, Proceedings of the Institution of Mechanical Engineers, Vol.22`, Part I, Journal of System Dynamics and Control Engineering, 2007, pp.1019-1032. [8] V. Exadaktylos, C. Taylor and A. Chotai, “Constraint handling for state dependent parameter models”, UKACC International Conference on Control, Manchester, UK, September 2008, 6 pages.

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[9] E. Shaban, “Deadbeat response of nonlinear systems described by discrete-time state dependent parameter using exact linearization by local coordinate transformation”, Journal of American Science, Vol.8, No.10, 2012, pp.355-366. [10] E, Shaban and A. Nada, “Proportional integral derivative versus proportional integral plus control applied to mobile robotic system”, Journal of American Science, Vol.9, No.12, 2013, pp.583-591. [11] E. Shaban, A. Nada and C. Taylor, “Exact linearization by feedback of state dependent governor models applied to a Mechatronics demonstrator”, UKACC International Conference on Control, Loughborough, UK, 9-11 July 2014, pp.609-614. [12] C. Mishra, J. Jebakumar and B. Mishra, “Controller selection and sensitivity check on the basis od performance index calculation”, International Journal of Electrical, Electronics and Data Communication, Vol.2, No.1, 2014, pp.91-93. [13] S. Das, I. Pan, S. Das and A. Gupta, “A novl fractional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices”, Journal of Engineering Application of Artificial Intelligence, Vol.25, No.2, March 2012, pp.430-442. [14] P. Venkataraman, Applied optimization with MATLAB programming, J. Wiley, 2009. [15] C. Houpis and S. Sheldon, Linear control system analysis and design with MATLAB, CRC Press, 2013. [16] W. Jung, Op amp application handbook (analog devices series), Newnes, 2005. [17] G. A. Hassaan, “Tuning for an I-PD controller used with a highly oscillating second-order process”, International Journal of Mechanical Engineering and Technology, Vol.5, No.5, 2014, pp.115-121. [18] G. A. Hassaan, “Tuning for an PD-PI controller used with a highly oscillating second-order process”, International Journal of Scientific and Technology Research, Vol.3, No.7, 2014, pp.145-147. [19] G. A. Hassaan, “Tuning for an PI-PD controller used with a highly oscillating second-order process”, International Journal of Research and Innovation Technology, Vol.1, No.3, 2014, pp.42-45. [20] G. A. Hassaan, “Tuning for an PID with a first-order-lag controller used with a highly oscillating second-order process”, International Journal of Scientific and Technology Research, Vol.3, No.9, 2014, pp.314-317. [21] G. A. Hassaan, “On simple tuning of PID controllers for underdamped second-order processes”, International Journal of Mechanical and Production Engineering Research and Development, Vol.4, No.3, 2014, pp.61-68. [22] G. A. Hassaan, “PPI controller tuning for use with a highly oscillating second-order-like process”, International Journal of Science and Engineering, (under publication).

BIOGRAPHY

Galal Ali Hassaan

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Emeritus Professor of System Dynamics and Automatic Control. Has got his B.Sc. and M.Sc. from Cairo University in 1970 and 1974. Has got his Ph.D. in 1979 from Bradford University, UK under the supervision of Late Prof. John Parnaby. Galal Ali Hassaan, IJRIT-212

IJRIT International Journal Of Research In Information Technology, Volume 3, Issue 7, July 2015, Pg. 206-213

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Now with the Faculty of Engineering, Cairo University, EGYPT. Research on Automatic Control, Mechanical Vibrations , Mechanism Synthesis and History of Mechanical Engineering. Published more than 100 research papers in international journals and conferences. Author of books on Experimental Systems Control, Experimental Vibrations and Evolution of Mechanical Engineering. Chief Justice of International Journal of Computer Techniques. Member of the Editorial Board of some international journals including the International Journal of Research in Information Technology. Reviewer in some international journals. Scholars interested in the author publications can visit: http://scholar.cu.edu.eg/galal

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Galal Ali Hassaan, IJRIT-213