Tuning of PID Controllers for time Delay Unstable Systems with Two Unstable Poles

Tuning of PID Controllers for time Delay Unstable Systems with Two Unstable Poles

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4th International Conference on Advances in Control and Optimization of Dynamical Systems 4th International Conference on in and Optimization Dynamical Systems 4th International Conference on Advances Advances in Control Control February 1-5, of 2016. NIT Tiruchirappalli, India Available onlineand at www.sciencedirect.com 4th International Conference on Advances in Control and Optimization of Dynamical Systems February 1-5, 2016. NIT Tiruchirappalli, India Optimization of Dynamical Systems Optimization of Dynamical Systems February 1-5, 2016. NIT Tiruchirappalli, India February 1-5, 2016. NIT Tiruchirappalli, India February 1-5, 2016. NIT Tiruchirappalli, India

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TUNING OF PID CONTROLLERS FOR TIME DELAY UNSTABLE SYSTEMS IFAC-PapersOnLine 49-1 (2016) 801–806 TUNING OF PID CONTROLLERS FOR TIME DELAY UNSTABLE SYSTEMS TUNING OF PID CONTROLLERS TIME DELAY WITH TWOFOR UNSTABLE POLESUNSTABLE TUNING FOR TIME UNSTABLE SYSTEMS SYSTEMS TUNING OF OF PID PID CONTROLLERS CONTROLLERS TIME DELAY DELAY WITH TWOFOR UNSTABLE POLESUNSTABLE SYSTEMS WITH TWO UNSTABLE POLES WITH TWO UNSTABLE POLES * WITH TWO POLES Saxena NikitaUNSTABLE and M. Chidambaram

Saxena Nikita and M. Chidambaram** Saxena Nikita Nikita and and M. M. Chidambaram Chidambaram** Saxena Department ofand Chemical Engineering, Saxena Nikita M. Chidambaram Department of Chemical Engineering, Indian Institute of Technology-Madras, Department of Chemical Engineering, Department of Chemical Engineering, Indian Institute of Technology-Madras, Department of Chemical Engineering, Chennai 600036, India Indian Institute Institute of of Technology-Madras, Technology-Madras, Indian Chennai 600036, India IndianE-mail:[email protected] Institute of Technology-Madras, Chennai 600036, India Chennai 600036, E-mail:[email protected] Chennai 600036, India India E-mail:[email protected] E-mail:[email protected]m.ac.in E-mail:[email protected] ABSTRACT - An improved continuous cycling method is proposed for PID controllers for unstable ABSTRACT - Anunstable improved continuous cyclingThe method is involves proposedthe fordetermination PID controllers forcontroller unstable systems with two poles and time delay. method of the ABSTRACT Anunstable improved continuous cyclingThe method is involves proposedthe fordetermination PID controllers controllers forcontroller unstable ABSTRACT --- An improved continuous cycling method is proposed for PID for unstable systems with two poles and time delay. method ofathe settings by solving the magnitude and the phase angle criteria for the system with proportional ABSTRACT An improved continuous cycling method is proposed for PID controllers for unstable systems with two unstable poles and time delay. The method involves the determination of the controller systems with two unstable poles and time delay. The method involves the determination of the controller settings by solving the magnitude and the phase angle criteria for the system with a proportional controller. Subsequently, incorporating the Proportional-Derivative-Integral (PID) controller transfer systems with two unstable poles and time delay. The method involves the determination of the controller settings by solving the magnitude and the phase angle criteria for the system with aa proportional settings by solving the magnitude and the phase angle criteria for the system with proportional controller. Subsequently, incorporating the Proportional-Derivative-Integral (PID) controller transfer function, with unity proportional gain and pre-determined values of reset time and derivative time in the settings by solving the magnitude and the phase angle criteria for the system with a proportional controller. Subsequently, incorporating Proportional-Derivative-Integral (PID) controller transfer controller. Subsequently, incorporating the Proportional-Derivative-Integral (PID) controller transfer function, with unity proportional gainagain andthe pre-determined values and of reset time and derivative time in the the first step, with the system model and solving the amplitude the phase angle criteria, we get controller. Subsequently, incorporating the Proportional-Derivative-Integral (PID) controller transfer function, with unity proportional gain and pre-determined values of reset time and derivative time in the function, with unity proportional gain and pre-determined values of reset time and derivative time in first step, with the system model and again solving the amplitude and the phase angle criteria, we get the updated gain ofunity thesystem controller. The is applied by values simulation on (i) a second order system function, with proportional gainmethod and pre-determined of reset time and derivative time inwith the first step, with the model and again solving the amplitude and the phase angle criteria, we get the first step, with system model and again solving the amplitude and the phase angle criteria, we get the updated gain ofthe the controller. The method isaapplied by simulation on (i) a second order system with time delay and two unstable poles and (ii) non-linear model of a CSTR with complex conjugate first step, with the system model and again solving the amplitude and the phase angle criteria, we get the updated gainand of the the controller. The method method isaapplied applied by simulation simulation on (i) aa second second order system system with updated gain of controller. The by (i) order time delay two unstable poles and (ii)is non-linear model enhance of on a CSTR with complex unstable poles time delay. The significantly performances ofconjugate bothwith the updated gain ofand the controller. Thecontroller method issettings applied by simulation on (i) the a second order system with time delay and two unstable poles and (ii) a non-linear model of a CSTR with complex conjugate time delay and two unstable poles and (ii) a non-linear model of a CSTR with complex conjugate unstable poles and time delay. The controller settings significantly enhance the performances of both the servo and the regulatory problems and give robust performances. time delay andand two unstable polescontroller and (ii) settings a non-linear model enhance of a CSTR with complexofconjugate unstable poles time delay. The significantly the performances performances both the the unstable and delay. The controller settings significantly enhance the of servo andpoles the regulatory problems and give robust performances. unstable poles and time time delay. The controller settings significantly enhance the performancesmethod of both both the Keywords: improve dPID, time delay unstable systems, Two unstable poles, Ziegler-Nichols servo and the regulatory problems and give robust performances. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. servo and the regulatory problems and give robust performances. Keywords: improve dPID, time delay systems, Two unstable poles, Ziegler-Nichols method servo and the regulatory problems andunstable give robust performances. Keywords: Keywords: improve improve dPID, dPID, time time delay delay unstable unstable systems, systems, Two Two unstable unstable poles, poles, Ziegler-Nichols Ziegler-Nichols method method Keywords: improve dPID, time delay unstable systems, Two unstable poles, Ziegler-Nichols method stable processes. Controller design for the unstable process I. INTRODUCTION stable processes. design for of thethe unstable process I. INTRODUCTION with time delay Controller is difficult. Some methods are The use of Proportional-Integral-Derivative (PID) controllers with stable processes. Controller design for the unstable process I. INTRODUCTION stable processes. Controller design for the unstable process I. INTRODUCTION time delay is difficult. Some of the methods are modified based on achieving the desired closed loop response The use of Proportional-Integral-Derivative (PID) controllers stable processes. Controller design for the unstable process I. INTRODUCTION for industrial process control is the most popular technique. modified with time delay is Some of the are with Many time methods delay is difficult. difficult. Some of the methods methods are on achieving theuse desired closed response The use of Proportional-Integral-Derivative (PID) controllers [24]. of PID controller in series Theindustrial use of Proportional-Integral-Derivative (PID) controllers for process control is the most popular technique. with timebased delay is involving difficult. Some of the loop methods are With the invention of PID controller in 1910 [1] and the Z-N modified based on achieving the desired closed loop response The use of Proportional-Integral-Derivative (PID) controllers modified based on achieving the desired closed loop response [24]. Many methods involving use of PID controller in series for industrial process control is the most popular technique. with the lead lag are also proposed. Nikita and for industrial process control is the most popular technique. With the invention of[2] PID controller in of 1910 [1] and the Z-N modified based oncompensator achieving theuse desired closed loop response tuning rules inprocess 1942of thecontroller popularity the PIDand controllers [24]. Many methods involving PID controller in for industrial control is the most popular technique. [24]. Many methods involving use of PID controller in series series with the lead lag compensator areaof also proposed. Nikita and With the invention PID in 1910 [1] the Z-N With the invention of PID controller in 1910 [1] and the Z-N Chidambaram [25,26] proposed method to improve the tuning rules in 1942 [2] the popularity of the PID controllers [24]. Many methods involving use of PID controller in series increased immensely. The widespread of [1] PID controllers lag compensator are proposed. Nikita and With the invention of[2] PID controller inuse 1910 the Z-N with with the the lead lead of lag compensator areaalso also proposed. Nikita and Chidambaram [25,26] proposed method toorder improve the tuning rules in 1942 the popularity of the PIDand controllers tuning rules in 1942 [2] the popularity of the performance a PID controller for First unstable increased immensely. The widespread use of PID controllers with the lead lag compensator are also proposed. Nikita and owe to rules its simple structure the ease of on-line aa for method to improve the tuning in 1942 [2] theand popularity the PIDretuning. controllers Chidambaram Chidambaram [25,26] proposed method to systems improve the of[25,26] PID proposed controller unstable increased The widespread of controllers systems and for aaasecond order plus timeFirst delay with increased immensely. The and widespread use of PID PIDretuning. controllers performance owe to its immensely. simple structure the ease use of on-line Chidambaram [25,26] proposed a for method toorder improve the performance of PID controller First order unstable increased immensely. The widespread use of PID controllers performance of a PID controller for First order unstable To determine the parameters of the controller, many design systems and for a second order plus time delay systems with owe to its simple structure and the ease of on-line retuning. one stable and one unstable poles. owe to its simple structure and the ease of on-line retuning. performance of aasecond PID controller for First order unstable systems and plus To determine the parameters controller, many design owe to its such simple andof thethe ease of on-line retuning. systems and forone a second order plus time time delay delay systems systems with with methods, asstructure gain margin/phase margin method [3-6], one stable andfor unstableorder poles. systems and for a second order plus time delay systems with To determine the parameters of the controller, many design In the present work, this method is extended to unstable time To determine the parameters of the controller, many design one stable and one unstable poles. methods, such as gain margin/phase margin method [3-6], one stable and one unstable poles. pole placement technique [7-8], optimization technique [9To determine the parameters of the controller, many design In the present thisunstable method is extended to unstable time one stable and work, one unstable poles.poles. methods, such as gain margin/phase margin method [3-6], delay systems with two The method is simple, methods, such as gain margin/phase margin method [3-6], pole placement technique [7-8], optimization technique [911], direct synthesis method [12-15], internal model control In the this method is to time methods, such as gain margin/phase margin method [3-6], In the present present work, work, this method is extended extended to unstable unstable time delay with two The mostly method is simple, pole placement technique [7-8], optimization technique [9analytically andunstable can bepoles. to alltime the pole placement technique [7-8], optimization technique [9- In 11], direct synthesis method [12-15], internal[18] model control the systems presentderived work, this method isapplied extended to unstable method [16-17], equating coefficient method and robust delay systems with two unstable poles. The method is simple, pole placement technique [7-8], optimization technique [9delay systems with two unstable poles. The method is simple, analytically derived and can be applied mostly to all the 11], direct synthesis method [12-15], internal model control different classes of processes. Maximum sensitivity, phase 11], direct synthesis method [12-15], internal model control method [16-17], equating coefficient method [18] and robust delay systems with two unstable poles. The method is simple, loop direct shaping [19], have been reported in the last few decades. analytically derived and applied mostly to the 11], synthesis method [12-15],method internal model control analytically derived and can can be be are applied mostly to all all the different classes of processes. sensitivity, phase method [16-17], coefficient [18] and robust method [16-17], equating coefficient method [18] and robust margin and gain margin used for the robustness loop shaping [19],equating have been reported inPID the controllers last few decades. analytically derived and criteria can beMaximum applied mostly to all the One of the earliest methods of tuning is the different classes of processes. Maximum sensitivity, phase method [16-17], equating coefficient method [18] and robust different classes of processes. Maximum sensitivity, phase margin and gain margin criteria are used for the robustness loop shaping [19], have been reported in the last few decades. loop shaping [19], have been reported inPID the controllers last few decades. analysisand ofclasses the proposed method. One of the earliest methods of is the different of processes. Maximum sensitivity, phase Ziegler-Nichols [2].of Ittuning is ain heuristic method of analysis margin gain margin criteria loop shaping [19],method have been reported the controllers last few decades. margin and gain marginmethod. criteria are are used used for for the the robustness robustness One of methods PID is of the proposed One of the the earliest earliest methods ofIttuning tuning PID controllers is the the Ziegler-Nichols method [2]. is a heuristic method of margin and gain margin criteria are used for the robustness determining the ultimate values of the controller. At the One of the earliest methods ofIttuning PID controllers is of analysis analysis of of the the proposed proposed method. method. Ziegler-Nichols [2]. is aathe heuristic method Ziegler-Nichols method [2]. It at isof the heuristic method of analysis determining the method ultimate values controller. At the of the method. DESIGN METHOD II. proposed CONTROLLER ultimate value, the system is point of marginal Ziegler-Nichols method [2]. It is a heuristic method of determining the ultimate values the controller. At determining the ultimate values of the the point controller. At the the ultimate value, the system isoscillations atof ofoutput. marginal II. CONTROLLER DESIGN METHOD instabilityvalue, andthe gives sustained in theof determining ultimate values of the the point controller. At The the II. CONTROLLER DESIGN METHOD ultimate the system is at marginal II. DESIGN METHOD ultimate gain value, the system is at the point of marginal instability and gives sustained oscillations in the output. The In this paper, single loop feedback controller structure is and ultimate frequency are used toofoutput. getmarginal the PID II.the CONTROLLER CONTROLLER DESIGN METHOD ultimate value, the system isoscillations at the point instability and gives sustained in the The instability and gives sustained oscillations in the output. The ultimate gain and ultimate frequency are used to get the PID In this paper, the single loop feedback controller structure is is the process transfer used as shown in Figure(1), where G controller settings. The PID settings proposed by Ziegler – instability and gives sustained oscillations in the output. The p In this paper, the single loop feedback controller structure is ultimate gain and ultimate frequency are used to get the PID In this paper, the single loop feedback controller structure is ultimate gain and ultimate frequency are used to get the PID controller settings. The PID settings proposed by Ziegler – is the process transfer used as shown in Figure(1), where G p the PIDloop controller transfer function of the Gc isthe Nichols gain results a large overshoot and an oscillatory In thisaspaper, single feedback controller structure is ultimate andinultimate frequency are used tobyget the PID– function, is the process transfer used shown in Figure(1), where G controller settings. The PID settings proposed Ziegler p is the process transfer used as shown in Figure(1), where G controller settings. The PID settings proposed by Ziegler – 𝐾𝐾𝐼𝐼 Nichols results in a large overshoot and an oscillatory p is the PID controller transfer function of the function, G c response. The correlation between the ultimate period, the is the process transfer used as shown in Figure(1), where G controller settings. The PID settings proposed by Ziegler – p (𝑠𝑠) = 𝐾𝐾 + + 𝐾𝐾 𝑠𝑠 form given by 𝐺𝐺 𝐷𝐷 PID 𝑐𝑐controller transfer function of the function, G Nichols results in overshoot and oscillatory 𝐾𝐾 cc is is 𝑐𝑐the the function Nichols results in aa large large overshoot and an an period, oscillatory response. The correlation between thebased ultimate the (𝑠𝑠) PID = 𝐾𝐾𝑐𝑐controller + 𝐾𝐾𝐾𝐾𝑠𝑠𝐼𝐼𝐼𝐼 + 𝐾𝐾𝐷𝐷transfer 𝑠𝑠 form givenG reset timeresults and the time was of function, PID controller transfer function of of the the function, Gby Nichols inderivative a large overshoot andonansimulation oscillatory c is𝐺𝐺𝑐𝑐the response. The correlation between the ultimate period, the 𝑠𝑠 𝐼𝐼 𝑘𝑘 (𝑠𝑠) = 𝐾𝐾 + + 𝐾𝐾 𝑠𝑠 form given by 𝐺𝐺 𝑐𝑐 response. The the correlation between thebased ultimate period, the reset time and derivative time was on simulation of 𝐾𝐾 𝑐𝑐 𝑐𝑐 𝐷𝐷 (𝑠𝑠) = 𝐾𝐾 + + 𝐾𝐾 𝑠𝑠 form given by 𝐺𝐺 𝐼𝐼 aresponse. large number of processes. The key criterion is a quarter where, 𝐾𝐾 = 𝑎𝑎𝑛𝑛𝑑𝑑 𝐾𝐾 𝑘𝑘 𝑐𝑐 (𝑠𝑠) 𝑐𝑐= 𝑠𝑠𝑐𝑐 ∗+𝜏𝜏𝐷𝐷𝐾𝐾𝐷𝐷 𝑠𝑠 The correlation between the ultimate period, the I 𝐷𝐷 = 𝐾𝐾 + form given by 𝐺𝐺 𝑠𝑠 𝑘𝑘 reset time and time based on of time and the theofderivative derivative time was based on simulation simulation of areset large number processes. Thewas keyhave criterion is a the quarter 𝐾𝐾I = 𝜏𝜏𝑘𝑘𝑘𝑘𝐼𝐼,𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎𝑛𝑛𝑑𝑑 𝐾𝐾𝐷𝐷 𝑐𝑐= 𝑘𝑘𝑠𝑠𝑐𝑐 ∗ 𝜏𝜏𝐷𝐷 𝐷𝐷 decay ratio. other researchers modified ZN reset time andMany theof derivative time was based on simulation of where, adecay large number processes. The key criterion is aa the quarter where, 𝐾𝐾 𝐾𝐾I = 𝜏𝜏𝑘𝑘𝐼𝐼,𝑐𝑐𝑐𝑐 𝑎𝑎𝑛𝑛𝑑𝑑 𝑎𝑎𝑛𝑛𝑑𝑑 𝐾𝐾𝐷𝐷𝐷𝐷 = = 𝑘𝑘𝑐𝑐𝑐𝑐 ∗∗ 𝜏𝜏𝜏𝜏𝐷𝐷𝐷𝐷 amethod largeratio. number of processes. The key criterion is quarter where, Many other researchers have modified ZN to obtain significant performance improvement. adecay largeratio. number of processes. The keyhave criterion is a the quarter where, 𝐾𝐾II = = 𝜏𝜏𝜏𝜏𝐼𝐼,𝐼𝐼, 𝑎𝑎𝑛𝑛𝑑𝑑 𝐾𝐾 𝐾𝐾𝐷𝐷 = 𝑘𝑘 𝑘𝑘𝑐𝑐 ∗ 𝜏𝜏𝐷𝐷 Many other researchers modified ZN 𝜏𝜏 𝐼𝐼, decay ratio. Many other researchers have modified the ZN method to obtain significant performance improvement. Dwyer ratio. [20] has reviewed the methods. Tyreus-Luyben decay Many other researchers have modified the [21] ZN method to obtain significant performance improvement. method to obtain significant performance improvement. Dwyer [20] has reviewed thePID methods. Tyreus-Luyben [21] proposed settings forsignificant PI and controllers, but the method method to obtain performance improvement. Dwyer [20] has reviewed the methods. Tyreus-Luyben [21] Dwyer [20] has reviewed the methods. Tyreus-Luyben [21] proposed settings for PI and PID controllers, but the method results [20] in a has longreviewed settling time. Smith Tyreus-Luyben [22]but and Yu [23] Dwyer the methods. [21] proposed for PI and PID controllers, the method proposedinsettings settings for PI and PID controllers, but the method results a long settling time. Smith [22] and Yu [23] modification in the tuning formulae based on the proposed settings for PI and PID controllers, but the method results aa long time. Smith [22] results in invalues. long settling settling time. Smith [22] and and Yuon[23] [23] proposed modification in the tuning formulae basedYu the ultimate results in a long settling time. Smith [22] and Yu [23] proposed modification proposedvalues. modification in in the the tuning tuning formulae formulae based based on on the the ultimate proposed modification the proposed tuning formulae based on the Furthermore, most of inthe methods, based on ultimate values. ultimate values. Furthermore, most of the proposed methods, mostly based on on Figure 1: Feedback controller structure ultimate values. ultimate values of controllers, are implemented Furthermore, of methods, based on Figure 1: Feedback controller structure Furthermore, most of the the proposed proposed methods, mostly based on ultimate valuesmost of controllers, are implemented Furthermore, most of the proposed methods, mostly based on on Figure Figure 1: 1: Feedback Feedback controller controller structure structure ultimate values of controllers, are implemented ultimate values of controllers, are implemented mostly on Copyright © 2016 IFAC 801 Figure 1: Feedback controller structure ultimate values of controllers, are implemented mostly on 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 801Control. Peer review© under of International Federation of Automatic Copyright © 801 Copyright © 2016 2016 IFAC IFAC 801 10.1016/j.ifacol.2016.03.155 Copyright © 2016 IFAC 801

IFAC ACODS 2016 802 February 1-5, 2016. NIT Tiruchirappalli, IndiaSaxena Nikita et al. / IFAC-PapersOnLine 49-1 (2016) 801–806

𝐴𝐴𝑟𝑟 =

It should be noted that for an unstable system, along with the maximum value of controller gain (𝐾𝐾𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 ), the minimum value of controller gain (𝐾𝐾𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 ) also plays an important role. Ziegler and Nichols [2] proposed a tuning rule for the PID controllers based on the ultimate values of the system. However, the method gives oscillatory responses particularly for the control of the unstable systems. In the present work, a method is proposed for updating the value of the controller gain once integral and derivative actions are put into effect. The amplitude ratio (Ar) and cross over frequency (ωc) for a system are obtained by solving the amplitude and the phase angle criteria. To make the overall loop gain equal to unity at phase lag equals to 180 deg, the controller gain is set equal to 1/Ar. At this value, the closed loop system will become marginally stable and will give sustained oscillations with frequency equal to 𝜔𝜔𝑚𝑚𝑎𝑎𝑥𝑥 . To eliminate the offset we must include the integral mode to the controller. To speed up the response of the system, a derivative mode is added to the controller.

𝜑𝜑 = −𝜔𝜔 + 2tan

𝐼𝐼𝑇𝑇𝐴𝐴𝐸𝐸 =



𝑡𝑡 𝜖𝜖 𝑡𝑡 𝑑𝑑𝑡𝑡

∞ 0

III.

where, 𝐾𝐾𝑝𝑝 = 1

10𝑠𝑠−1 5𝑠𝑠−1

(3a) (3b)

𝐴𝐴𝑟𝑟 𝐴𝐴𝑟𝑟

(4a) (4b)

𝑚𝑚𝑖𝑖𝑛𝑛

= 0.2420 = 0.0878

𝑚𝑚𝑎𝑎𝑥𝑥

𝐾𝐾𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 = 4.1322 𝐾𝐾𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 = 11.3792

1

𝐴𝐴𝑟𝑟

, the minimum and maximum values (5a) (5b)

For unstable systems, the minimum value of controller gain plays an important role. It is seen that if the minimum value of controller gain is ignored while designing the PID controller, many systems cannot be stabilized. The ultimate period of oscillation is calculated based on the frequency [eq.(3b)]. The reset time and derivative time are determined using the Ziegler Nichols tuning rule. 𝑃𝑃𝑢𝑢 =

𝜏𝜏𝐼𝐼 =

𝜏𝜏𝐷𝐷 =

2𝜋𝜋

ω 𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 𝑃𝑃𝑢𝑢 2 𝑃𝑃𝑢𝑢 8

= 5.5431

(6a)

= 2.7716

(6b)

= 0.6929

(6c)

The PID controller parameters for the ZN continuous cycling method are given in Table (1). Here, Pu = 2π/ωc,max.

𝜖𝜖 𝑡𝑡 𝑑𝑑𝑡𝑡 ;

For the proposed method, the values of reset time and derivative time are calculated using the formula given by 𝜏𝜏𝐼𝐼′ =

5

ω 𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥

; 𝜏𝜏𝐷𝐷′ =

0.8

ω 𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥

(7)

The proposed method includes determining the new ultimate value of the controller once the integral and derivative mode are put into effect.

SIMULATION STUDIES

𝐾𝐾𝑝𝑝 5𝑠𝑠+1 exp −𝑠𝑠

(3)

of controller gain can be determined as

For determination of new ultimate values, the controller transfer function, with proportional gain equal to 1 and with the calculated values of reset time and derivative time, are added to the system. The value of 𝜏𝜏𝐼𝐼′ and 𝜏𝜏𝐷𝐷′ calculated in eq. (7) are for the series form PID controller. These values are then transformed to parallel form for PID [23] by: (8a) 𝜏𝜏𝐼𝐼𝑝𝑝 = 𝜏𝜏𝐼𝐼 ′ + 𝜏𝜏𝐷𝐷 ′

Example 1: Consider the open loop unstable second order plus time delay system with two unstable poles and a negative zero 𝐺𝐺𝑝𝑝 𝑠𝑠 =

10𝜔𝜔 − 2𝜋𝜋

ω𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 = 0.4009 ω𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 = 1.1335

As stated above, 𝐾𝐾𝑐𝑐 =

𝑤𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒, 𝜖𝜖 𝑡𝑡 = 𝑦𝑦𝑟𝑟 𝑡𝑡 − 𝑦𝑦(𝑡𝑡) is the deviation of response from desired set point. 0

5𝜔𝜔 + tan

(2)

On substituting the values of 𝜔𝜔𝑚𝑚𝑖𝑖𝑛𝑛 and 𝜔𝜔𝑚𝑚𝑎𝑎𝑥𝑥 in eq.(2), we get

To compare the performance of the system based on proposed method over the other methods present in the literature, Integral of the Square Error (ISE), Integral of the Time weighted Absolute Error (ITAE) and Integral of the Absolute error (IAE) values are considered for both the cases of unit step change in the input and unit step change in the disturbance. The criteria are defined as 𝜖𝜖 2 𝑡𝑡 𝑑𝑑𝑡𝑡; 𝐼𝐼𝐴𝐴𝐸𝐸 =

−1

Solving numerically eq.(3b) with 𝜑𝜑 = −𝜋𝜋, we get,

A. Time integral performance

∞ 0

−1

To determine the ultimate values of the controller, the system is made to approach the marginal instability at cross over frequency. The phase angle criterion [eq.(3) with 𝜑𝜑 = −𝜋𝜋 ] is numerically solved to obtain the minimum value of frequency (ω𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 ) and maximum value of frequency (ω𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 ) corresponding to which minimum and maximum amplitude ratio of the system are calculated using eq.(2). To make the overall gain of the system unity, the ultimate value of controller gain (Kc) set equal to inverse of amplitude ratio (Ar).

It is observed that on addition of the integral and derivative modes, the value of controller gain changes and its new value is to be calculated. For stable systems, Douglas [26] proposed a trial and error procedure to determine the values of controller settings based on the Bode plots. He proposed that the best results are obtained when the phase lag of approximately 10 deg (by the integral mode) and phase lead of approximately 45 deg (by the derivative mode) are added at the original cross over frequency. For this, the integral and derivative times are considered respectively as: 5/ω c and 1/ωc. Around these values, the parameters are tuned to get a satisfactory performance. However, the examples considered are stable processes.

𝐼𝐼𝑆𝑆𝐸𝐸 =

1

10 2 𝜔𝜔 2 +1

(1)

𝜏𝜏𝐷𝐷 𝑝𝑝 =

The amplitude ratio and the phase angle of the system can be written as

802

𝜏𝜏 𝐼𝐼 ′ ∗ 𝜏𝜏 𝐷𝐷 ′ 𝜏𝜏 𝐼𝐼 ′ +𝜏𝜏 𝐷𝐷 ′

(8b)

IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, IndiaSaxena Nikita et al. / IFAC-PapersOnLine 49-1 (2016) 801–806

803

The updated system is given by 𝐺𝐺𝑝𝑝 𝑠𝑠 = 1 +

1

𝜏𝜏 𝐼𝐼 𝑝𝑝 𝑠𝑠

+ 𝜏𝜏𝐷𝐷𝑝𝑝 𝑠𝑠

𝐾𝐾𝑝𝑝 5𝑠𝑠+1 exp −𝑠𝑠 10𝑠𝑠−1 5𝑠𝑠−1

(9a)

where, Kp = 1. For calculating the new ultimate values, the amplitude ratio and the phase angle criteria of the new system [eq. (9a)] are written as: 𝐴𝐴𝑟𝑟 =

𝜏𝜏 𝐷𝐷 𝑝𝑝 ω−

1 𝜏𝜏 𝐼𝐼 𝑝𝑝 ω

10 2 ω 2 +1

2

𝜑𝜑 = tan−1 (𝜏𝜏𝐷𝐷𝑝𝑝 ω − 2𝜋𝜋

+1

(9b)

1

𝜏𝜏 𝐼𝐼 p ω

) − ω + 2tan−1 5ω + tan−1 10ω −

(9c) Figure 2(a): Servo response of the system (example 1) Legend: Dash – ZN method; Solid: Proposed method

The phase angle criterion is solved numerically to get the new value of maximum frequency. At the crossover frequency, the phase lag of the system is π and the system generates a sustained oscillation.The minimum and maximum values of the frequency and the corresponding amplitude ratio are obtained as:

ω𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 = 0.4882 ω𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 = 2.2613 𝐴𝐴𝑟𝑟 𝑚𝑚𝑖𝑖𝑛𝑛 = 0.2017 𝐴𝐴𝑟𝑟 𝑚𝑚𝑎𝑎𝑥𝑥 = 0.0720

Table 1:Controller Settings Controller parameters

(10b) (10c) (11a) (11b)

𝐾𝐾𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 = 13.8717

𝜏𝜏𝐼𝐼

𝜏𝜏𝐷𝐷

𝐾𝐾𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 = 4.9569

𝜏𝜏𝐼𝐼

(12b)

𝜏𝜏𝐷𝐷

The design value of the proportional gain of the controller is calculated as the average of the minimum and the maximum values of the controller. The controller parameters calculated are given in Table 1 for both the Ziegler Nichols method (eq. 6) and the proposed method.

7.7557

9.4143

2.7716

5.1169

0.6929

0.6084

EXAMPLE 2

𝐾𝐾𝑐𝑐,𝑑𝑑𝑒𝑒𝑠𝑠

(12a)

Proposed method

EXAMPLE 1

𝐾𝐾𝑐𝑐,𝑑𝑑𝑒𝑒𝑠𝑠

To make the overall system gain equal to 1,the controller gain 1 is calculated (𝐾𝐾𝑐𝑐 = ) as: 𝐴𝐴𝑟𝑟

ZN method

-6.5021

-9.0913

0.5237

0.9668

0.1309

0.1150

*𝜏𝜏𝐼𝐼 = reset time; *𝜏𝜏𝐷𝐷 = derivative time *𝐾𝐾𝑐𝑐,𝑑𝑑𝑒𝑒𝑠𝑠 =

Based on the controller settings (given in Table 1), the performance of the process is evaluated for a unit step change in the set point. An improved performance is obtained for the proposed method as seen in Figure 2(a). The improved performance is supported by the time integral performance analysis given in Table (2). For servo response, reduction of 59%, 83% and 59% are obtained in ISE, ITAE and IAE values for proposed method, in comparison with the ZN method. The performance of the process for a unit step change in the load (regulatory response) of the process is also shown in Figure 2(b). The transfer function model for the load is assumed to be same as that of the process transfer function. For the regulatory response, reduction of 50%, 81% and 59% in ISE, ITAE and IAE values are realized (Table 2).

𝐾𝐾𝑢𝑢 ,𝑚𝑚𝑖𝑖𝑛𝑛 + 𝐾𝐾𝑢𝑢 ,𝑚𝑚𝑎𝑎𝑥𝑥 2

= Proportional Gain

Table 2:Time Integral Performance Response

Method

ISE

ITAE

IAE

Example 1 Servo

Regulatory

ZN

7.745

171.80

13.14

Proposed

3.151

27.99

5.29

ZN

0.118

20.65

1.531

Proposed

0.058

3.77

0.627

Example 2 (Non-Linear CSTR) Servo

Regulatory (10% disturbance added)

803

ZN

0.187

1.357

0.751

Proposed

0.284

0.746

0.701

ZN

0.004

0.130

0.092

Proposed

0.002

0.056

0.055

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𝐺𝐺𝑝𝑝 𝑠𝑠 = −

1.77 0.3186 𝑠𝑠+1 exp −0.15𝑠𝑠

0.028 𝑠𝑠 3 +1.324 𝑠𝑠 2 −0.479𝑠𝑠+1

(15)

The same procedure, for the design of PID controller, is followed as in example 1. The amplitude ratio and the phase angle criteria equations are formulated as: 𝐴𝐴𝑟𝑟 =

1.77 (0.3186 2 ω 2 +1 1.324 ω 2 −1 2 + 0.028 ω 3 +0.479ω 2

𝜑𝜑 = −0.15ω + tan−1 0.3186ω − 𝜋𝜋 − tan−1

Eq.(17) can be written as:

−0.15ω + tan−1 0.3186ω − tan−1

(13)

𝐴𝐴𝑚𝑚 ≥

𝑀𝑀𝑠𝑠−1

, and 𝜑𝜑𝑚𝑚 ≥ 2 sin (1/Ms) -1

Maximum sensitivity

5.5782

3.1582

Phase margin

20.6545

36.9196

Gain margin

1.2184

1.4634

(17b)

(18a) (18b)

𝐴𝐴𝑟𝑟

(19a)

𝑚𝑚𝑖𝑖𝑛𝑛

𝑚𝑚𝑎𝑎𝑥𝑥

= −1.7699

= −0.0804

(19b)

As mentioned above, in order to make the overall system gain unity, the minimum and maximum value of controller gain are set equal to the inverse of the amplitude ratio. The calculated values and the ultimate period of oscillation are (20a) given as: 𝐾𝐾𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 = −0.5650

𝐾𝐾𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 = −12.4392 𝑃𝑃𝑢𝑢 = 1.0474

(20b) (21)

𝜏𝜏𝐼𝐼 =

= 0.5237

(22a)

= 0.1309

(22b)

Based on which the reset time and the derivative time are calculated as:

𝜏𝜏𝐷𝐷 =

Table 3:Maximum Sensitivity, Gain Margin and Phase Margin Comparison (Example 1) Proposed method

=0

ω𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 = 5.9990

𝐴𝐴𝑟𝑟

The proposed method is compared with ZN method based on the maximum sensitivity, phase margin and gain margin as given in Table 3. It can be clearly seen in the Table 3 that the proposed method is more robust than the ZN method. The values for Maximum sensitivity (3.1582), phase margin (36.9196) and gain margin (1.4634) obtained for proposed method are better than ZN method indicating that the closed loop system is less sensitive to variations in the process dynamics.

ZN method

1.324 ω 2 −1

The values are substituted in eq. (16). The values of amplitude ratio at minimum and maximum frequencies are calculated as:

(14)

Performance measure

0.028 ω 3 +0.479ω

ω𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 = 0

The shortest distance from the Nyquist curve of the loop transfer function to the critical point (-1, 0) is being equal to 1/Ms. In order to obtain a high robustness, small values of Ms are of interest. The maximum sensitivity can be related to the phase margin and gain margin as it simultaneously ensures both the following constraints [28]: 𝑀𝑀𝑠𝑠

(17a)

Eq.(17b) is solved numerically to get the minimum value of frequency and maximum value of frequency:

The maximum sensitivity is defined as [27]: 1

(17)

At the cross over frequency; φ = −π

The controller is designed based on the transfer function of the process, but there are always some chances of uncertainty in the process parameters. It is important that the controller parameters are designed in such a way that parameters are least affected by the change in the process dynamics. To ensure the robustness of the controller, the maximum sensitivity is determined. 1+𝑃𝑃 𝑗𝑗𝜔𝜔 𝐶𝐶(𝑗𝑗𝜔𝜔 )

1.324 ω 2 −1

To obtain a sustained oscillation, the system is brought on the verge of stability. On analysing the sustained oscillations, ultimate value of the proportional gain and ultimate frequency can be determined.

Figure 2(b): Regulatory response of the system (Example 1) Legend same as Figure 2(a)

𝑀𝑀𝑠𝑠 = max𝜔𝜔 |𝑆𝑆 𝜔𝜔𝑗𝑗 | = max𝜔𝜔

(16)

0.028 ω 3 +0.479ω

𝑃𝑃𝑢𝑢

2 𝑃𝑃𝑢𝑢 8

Using the values obtained in above equations, controller based on ZN method is designed. For the proposed method, the values of reset time and the derivative time are determined using the eq. (7) which are then converted into the corresponding parallel form using the eqs. (8a) and eqs. (8b). For determining the new ultimate value, the PID controller transfer function, with Kc =1 and with the calculated 𝜏𝜏𝐼𝐼𝑝𝑝 and 𝜏𝜏𝐷𝐷𝑝𝑝 is added to the system. The new system thus formed is given by Eq.(23).

Example 2: Consider the following locally linearised transfer function model of non-linear continuous stirred tank reactor (CSTR) derived for measurement of the reactor temperature [28]. The system is unstable with 2 complex conjugate unstable poles (0.1851 ± 0.8457i) and a stable pole (47.6559).

Gp s = 1 +

1

𝜏𝜏 𝐼𝐼 𝑝𝑝 𝑠𝑠

+ 𝜏𝜏𝐷𝐷𝑝𝑝 𝑠𝑠

-1.77 0.3186s+1 exp -0.15s 0.028s3 +1.324s2 -0.479s+1

(23)

The equations for the amplitude and the phase angle criteria are formulated for the new system [eq.(23)] as

804

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𝐴𝐴𝑟𝑟 =

𝜏𝜏𝐷𝐷 ω −

1

𝜏𝜏 𝐼𝐼 ω

+1

𝜑𝜑 = tan−1 (𝜏𝜏𝐷𝐷 ω − tan−1

1.77 (0.3186 2 ω 2 +1

2

1

805

1.324 ω 2 −1 2 + 0.028 ω 3 +0.479ω 2

(24)

) − 0.15ω + tan−1 0.3186ω − 𝜋𝜋 −

𝜏𝜏 𝐼𝐼 ω 0.028ω 3 +0.479ω 1.324 ω 2 −1

(25)

At crossover frequency: 𝜑𝜑 = - π, On solving eq.(25), we get, 𝜔𝜔𝑐𝑐,𝑚𝑚𝑖𝑖𝑛𝑛 = 0.7133

𝜔𝜔𝑐𝑐,𝑚𝑚𝑎𝑎𝑥𝑥 = 13.3988

(25a)

(25b)

corresponding to which we get Ar |min= −6.4097

A r |max= −0.0555

Figure 3(b): Regulatory response of Non Linear CSTR when 10% disturbance is added; Legends same as Figure 2(a).

(26a) (26b)

To make the overall loop gain of the system equal to one, the controller gain is set equal to 1/Ar . This leads to the determination of the updated minimum and maximum controller gain respectively as -0.1560 and -18.0266.

IV. CONCLUSION An improved continuous cycling method for tuning the PID controllers is proposed. The simulation results of unstable SOPTD model with two unstable pole and on a non linear CSTR model with time delay and complex conjugate unstable poles are given. Improved dynamic performances along with robustness are obtained with the proposed method. Significant improvements are obtained in the time integral performance analysis for both the servo and the regulatory responses.

The controller is designed based on the parameters calculated for both the cases as shown in Table 1. The performance of the controller is evaluated for the Non-Linear CSTR equation (Appendix) for a step change in the input (x2 from 3.20 to 3.52) and for a step change in disturbance (hc from 1.5 to 1.65). Figure 3(a) and Figure 3(b) illustrate the appreciable performance of the proposed method over ZN continuous method for the Non-Linear CSTR. The enhanced performance is supported by the time integral performance analysis given in Table (2). Based on Table (2) reduction of approximately 45% and 40% in ITAE and IAE values is obtained for the servo problem. The improved performances with the reduction of 50%, 57% and 40% in ISE, ITAE and IAE values are realized for step change in load disturbance. For Non-linear system, the overshoot is more (Figure 3a) so there is a need to add set point filter or slightly detune the parameter. However, for linearised system (eq. 15) reduction of 76% in ISE value and 15% in overshoot are obtained for a step change in input.

REFERENCES [1] Bennett, S. A Brief History of Automatic Control . IEEE Control Systems,1996,17-25. [2] Ziegler, J.G.; Nichols,N.B. Optimum Settings for Automatic Controllers. Trans. ASME, 1942,64, 759765. [3] Wang, Y.G.; Cai,W.J. Advanced Proportional-IntegralDerivative Tuning for Integrating and Unstable Processes with Gain and Phase Margin Specifications. Ind. Eng. Chem. Res., 2002,41, 2910-2914, 2002. [4] Ho, W. K.; Xu, W. PID Tuning for Unstable Processes Based on Gain Margin and Phase Margin Specifications. Proc IEE, CTA, 1998, 145,392-396. [5] DePaor, A.M.; O’Malley, M. Controllers of ZieglerNichols Type for Unstable System with Time Delay. Int. J. Control, 1989, 41, 1025-1036. [6] Venkatashankar, V.; Chidambaram, M. Design of P and PI Controllers for Unstable First Order Plus Time Delay Systems. International Journal of Control, 1994, 60,132-144. [7] Clement, C.V.; Chidambaram, M. PID control of Unstable FOPTD Systems. Chem. Eng. Commun. , 1997b, 162, 63-74. [8] Shafie, Z.; Shenton, A. T. Tuning of PID Type Controllers for Stable and Unstable Systems with Time Delay. Automatica, 1994,30, 1609-1615. [9] Manoj, K.J.; Chidambaram, M. PID Controller Tuning for Unstable Systems by Optimization Method. Chem. Engg. Comm., 2001,185, 91-113.

Figure 3(a): Servo response of Non-linear CSTR (example 2); Legends same as Figure 2(a)

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IFAC ACODS 2016 806 February 1-5, 2016. NIT Tiruchirappalli, IndiaSaxena Nikita et al. / IFAC-PapersOnLine 49-1 (2016) 801–806

[10] Poulin, E.; Pomerlaeu, A. A PID Tuning for Integrating and Unstable Processes. in IEE Proceedings - CTA, 1996, 143, 429-435. [11] Visioli, A. Optimal Tuning of PID Controllers for Integral and Unstable Processes. IEE Proc. Control Theory Appln., 2001, 148, 180. [12] Padmasree, R.; Chidambaram, M. Control of Unstable Systems, Narosa Pub, New Delhi, 2006. [13] Chandrashekar, R.; Padmasree, R.; Chidambaram, M. Design of PI/PID Controllers for Unstable Systems with Time Delay by Synthesis Method. Indian Chemical Engineer, 2002, 44, 82-88. [14] Cho W., Lee J.; Edgar T.F., Simple Analytical Proportional Integral Derivative (PID) Controller Tuning Rules for Unstable Processes. Ind. Eng. Chem. Res. 2014,53, 5048 -5064. [15] Panda, R.C. Synthesis of PID Controllers for Unstable and Integrating Processes. Chemical Eng Science. 2009, 64, 2807-2816. [16] Rotstein, G.E.; Lewin, D.R. Simple PI and PID System Controller for Unstable Systems. Ind. Eng. Chem. Res. 1991,30, 1864-1869. [17] Lee, Y.; Lee, J.; Park, S. PID Controller Tuning for Integrating and Unstable Processes with Time Delay. Chem. Eng. Sci, 2000, 55, 3481-3493. [18] Padmasree, R.; Srinivas, M.N.; Chidambaram, M. A Simple Method of Tuning PID Controllers for Stable and Unstable FOPTD Systems. Comp. & Chem. Engg.2004, 28, 2201-2218. [19] Astrom, K.J.; Hagglund,T. Revisiting the ZieglerNichols Step Response Method for PID Control. Journal of process control. 2004, 14, 635-650. [20] O'Dwyer, A. Handbook of PI/PID controller Tuning Rules, 3rd Ed.: Imperial College Press, 2009. [21] Tyreus, B.D.; Luyben,W.L. Tuning PI Controllers for Integrator / Dead-Time Processes. Ind. Eng. Chem. Res., 1992,31,2625. [22] Smith, C. L. Intelligently tune PID controllers, Chemical enginnering, pp. 56-62, january 2003. [23] Yu, C.C. Auto Tuning of PID Controllers. Berlin: Springer-Verlag, 1999. [24] Shamsuzzoha, M.; Moonyong Lee, Design of advanced PID controller for enhanced disturbance rejection of second order processes with time delay, AIChE, Vol. 54, No. 6, 1526-1536, june 2008 [25] Nikita, S.; Chidambaram, M., Tuning of PID Controllers for First Order Unstable Systems, Paper presented in International Conference on Advances in Chemical Engineering, NITK Surathkal [26] Nikita, S.; Chidambaram, M., Improved Continuous Cycling Method of Tuning PID Controllers for Unstable Systems, Indian Chemical Engineer (accepted for publication) [27] Douglas, J. M. Process dynamics and control, 2 Control system synthesis. NJ: Prentice Hall, Englewood Cliffs, 1972.

[28] Vilanova, R.; Visioli, A. PID Control in Third Millennium,. London: Springer verlag Ltd., London, 2012. [29] Chidambaram, M.; Vivek Sathe, S. (2014) Relay autotuning for identification and control, Cambridge university Press, Delhi, pp 162-164. [30] Wang, L-W.; S-H. Hwang (2005) Identification and control for unstable process of three dynamics type, Chem. Eng. Comm., 192,34-61 Appendix: The Non-Linear equation of CSTR [29,30] is given by

𝑑𝑑𝑥𝑥 1 𝑑𝑑𝑡𝑡

= −𝐷𝐷𝑎𝑎 𝑥𝑥1 exp

𝑥𝑥 2

1+

𝑥𝑥 2 𝐸𝐸 𝑎𝑎

𝑑𝑑𝑥𝑥2 𝑥𝑥2 = 𝑄𝑄𝐷𝐷𝑎𝑎 𝑥𝑥1 exp 𝑥𝑥 𝑑𝑑𝑡𝑡 1+ 2 𝐸𝐸𝑎𝑎

+ 1 − 𝑥𝑥1

(A1)

− 1 + ℎ𝑐𝑐 𝑥𝑥2 + ℎ𝑐𝑐 𝑥𝑥3

𝑑𝑑𝑥𝑥3 = 10 𝑞𝑞𝑐𝑐 −1 − 𝑥𝑥3 + ℎ𝑐𝑐 𝑥𝑥2 − 𝑥𝑥3 𝑑𝑑𝑡𝑡

(A2)

(A3) 𝑥𝑥1 , 𝑥𝑥2 and 𝑥𝑥3 are the dimensionless concentration, reactor temperature and cooling jacket temperature. The other parameter values are heat transfer coefficient (ℎ𝑐𝑐 ) = 1.5, Damkohler number 𝐷𝐷𝑎𝑎 = 0.135, Heat of reaction 𝑄𝑄 = 11, Coolant flow rate 𝑞𝑞𝑐𝑐 = 3.2 and Activation Energy 𝐸𝐸𝑎𝑎 = 20. Using these parameter values and taking x2 as the controller variable and qc as the manipulated variable, the three equilibrium points (steady state) are (A4) 𝑥𝑥1𝑠𝑠 𝑥𝑥2𝑠𝑠 𝑥𝑥3𝑠𝑠 = [0.6861 1.3 − 0.1355] 𝑥𝑥1𝑠𝑠 𝑥𝑥2𝑠𝑠 𝑥𝑥3𝑠𝑠 = [0.5460 2.0 0.0036] (A5) (A6) 𝑥𝑥1𝑠𝑠 𝑥𝑥2𝑠𝑠 𝑥𝑥3𝑠𝑠 = [0.3195 3.2 0.3429] The system is considered to be at steady state condition (A6) at t = 0. For the given condition of operating point, the locally linearised model is given by eq. (15). Nomenclature: Ar amplitude ratio Kc controller gain KP process gain Pu period of oscillation τ time constant τI, τD integral and derivative time (parallel form) τI',τD' integral and derivative time (series form) ω frequency ωc cross over frequency φ phase angle subscript: min minimumvalue max maximum value

806