PID controllers for inverse response processes

5th International Conference on Advances in Control and 5th Conference on Optimization of Dynamical Systems 5th International International Conference on Advances Advances in in Control Control and and Optimization of Systems 5th International Conference on Advances in Control February 18-22, 2018. Hyderabad, India Optimization of Dynamical Dynamical Systems Available onlineand at www.sciencedirect.com 5th International Conference on Advances in Control and February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India

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IFAC PapersOnLine 51-1 (2018) 413–418

Optimal Optimal tuning tuning rules rules for for PI/PID PI/PID Optimal tuning rules for PI/PID controllers for inverse response processes controllers fortuning inverse response processes Optimal rules for PI/PID controllers for inverse ∗response ∗∗processes Mohammad Irshad response Ahmad Ali processes controllers for inverse ∗∗ Mohammad Irshad ∗∗ Ahmad Ali ∗∗

Mohammad Irshad ∗ Ahmad Ali ∗∗ Mohammad Irshad Ahmad Ali ∗ Department of Electrical Engineering, Indian Institute Irshad ∗ Ahmad Ali ∗∗ of Technology ∗ ∗ Department ofMohammad Engineering, Institute of Electrical Electrical India Engineering, Indian Institute of of Technology Technology Patna, Bihta-801106 (e-mail: Indian [email protected]). ∗ Department of Electrical Engineering, Indian Institute of Technology Patna, Bihta-801106 India (e-mail: [email protected]). ∗∗Department Patna, Bihta-801106 India (e-mail: [email protected]). ∗ Department of Electrical Engineering, Indian Institute of Technology ∗∗ of Electrical Engineering, Indian Institute of Technology Patna, Bihta-801106 (e-mail: [email protected]). ∗∗Department Department of Electrical Engineering, Indian Institute of Bihta-801106 ElectricalIndia Engineering, Indian Institute of of Technology Technology Patna, India (e-mail: [email protected]). ∗∗ Department Patna, Bihta-801106 India (e-mail: [email protected]). Department of Electrical Engineering, Indian Institute of Technology Patna, Bihta-801106 India (e-mail: [email protected]). Patna, Bihta-801106 India (e-mail: [email protected]). ∗∗

Department of Bihta-801106 Electrical Engineering, Indian Institute of Technology Patna, India (e-mail: [email protected]). Patna, Bihta-801106 India Abstract: In this work, optimal tuning rules for (e-mail: PI/PID [email protected]). controllers for stable and integrating Abstract: In work, tuning for for stable integrating Abstract:inverse In this thisresponse work, optimal optimal tuning rules for PI/PID PI/PID controllers forcriteria stable and and integrating first-order processes are rules reported. Integralcontrollers performance (ISTE, IST 22 E Abstract: In this work, optimal tuning rules for PI/PID controllers for stable and integrating first-order inverse response processes are reported. Integral performance criteria (ISTE, IST 3 first-order inverse response processes are reported. IntegralOptimization performance(PSO), criteriaan (ISTE, IST 22 E E and IST 3 E) been using Particle evolutionary Abstract: Inhave thisresponse work,minimized optimal tuning rules for Swarm PI/PID controllers for(PSO), stable an and integrating first-order inverse processes are reported. Integral performance criteria (ISTE, ISTsetE 3 E) and IST have been minimized using Particle Swarm Optimization evolutionary and IST E) have been minimized using Particle Swarm Optimization (PSO), an evolutionary optimization technique, to get optimal controller parameters. For integrating process model, 2 3 first-order inverse response processes are reported. IntegralOptimization performance criteria (ISTE, ISTsetE and IST E) have been minimized using Particle Swarm (PSO), an evolutionary optimization technique, to get optimal controller parameters. For integrating process model, optimization to get optimal controllerFor parameters. For integrating processresults model,show setpoint filter used for reducing largeusing overshoot. nominal conditions, simulation 3 is technique, and IST E) have been minimized Particle Swarm Optimization (PSO), an evolutionary optimization technique, to get optimal controller parameters. For integrating process model, setpoint filter is used for reducing large overshoot. For nominal conditions, simulation results show point filter improvement is used for reducing large overshoot. For nominal conditions, simulation results show significant both servo and regulatory performances . The optimal controllers optimization toin get optimal controller parameters. For integrating process model, setpoint filter is technique, used for reducing large overshoot. For nominal conditions, simulation show significant improvement in servo and regulatory performances .. The optimal controllers significant improvement in both both servo and during regulatory performances The optimalresults controllers also give good robust control performances model-mismatch conditions. point filter is used for reducing large overshoot. For nominal conditions, simulation results show significant improvement in both servo and regulatory performances . The optimal controllers also also give give good good robust robust control control performances performances during during model-mismatch model-mismatch conditions. conditions. significant improvement inFederation both servo and during regulatory performances . The optimal controllers also giveIFAC good robust control performances model-mismatch conditions. © 2018, (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Optimal, inverse response, integral performance also give good robustPI/PID control controllers, performances during model-mismatch conditions. criteria, servo. Keywords: Optimal, Optimal, PI/PID PI/PID controllers, controllers, inverse inverse response, response, integral integral performance performance criteria, criteria, servo. servo. Keywords: Keywords: Optimal, PI/PID controllers, inverse response, integral performance criteria, servo. 1. INTRODUCTION control theory. A performance Smith-type predictor scheme that uses Keywords: Optimal, PI/PID controllers, inverse response, integral criteria, servo. 1. INTRODUCTION INTRODUCTION control theory. A predictor 1. control theory. A Smith-type Smith-type predictor scheme that uses uses H is reported in Alc´antara et al.scheme (2009).that ∞ design 1. INTRODUCTION control theory. A Smith-type predictor scheme H design is reported in Alc´ a ntara et al. (2009). H∞ antara et al. (2009).that uses ∞ design is reported in Alc´ 1. INTRODUCTION control theory. Smith-type predictor scheme thatplane uses Parameters of reported PIAcontroller asafunctions of right-half H∞ design is in Alc´ ntara et al. (2009). of PI controller as of right-half plane Proportional integral (PI) and proportional integral deriva- Parameters H∞ design is reported in Alc´ afunctions ntara et al. (2009). Parameters of PI controller as functions of right-half plane (RHP) zero and time delay are proposed in Luyben (2000) Proportional integral (PI) (PI) and proportional integral derivaderivaPItime controller asdelay functions of plane Proportional integral proportional integral (RHP) zero and proposed in Luyben tive (PID) controllers are and widely used in industrial appli- Parameters (RHP) zero of and time delay are proposed inright-half Luyben (2000) second-order plusdelay timeare inverse response(2000) proProportional integral (PI) and proportional integral derivative (PID) controllers are widely used in industrial appli- for Parameters of PItime controller asdelay functions of right-half plane (RHP) zero and delay are proposed in Luyben (2000) tive (PID) controllers are widely used in industrial applifor second-order plus time inverse response procations because of their simplicity, acceptable cost/benefit for second-order plus time delay inverse response processes. A PID controller is designed using direct synthesis Proportional integral (PI) and proportional integral derivative (PID) controllers are widely used in industrial applications because of their theirperformance simplicity, acceptable acceptable cost/benefit (RHP) zero and time delay are proposed indirect Luyben (2000) for second-order plus time delay inverse response procations because of simplicity, cost/benefit cesses. A PID controller is designed using synthesis ratio and satisfactory for a class of process cesses. A PID controller is designed using direct synthesis and IMC principles in Chien et al. (2003) and tive (PID) controllers are widely used ina industrial appli- approach cations because of their simplicity, acceptable cost/benefit ratio and satisfactory performance for class of process for second-order plus time delay inverse response processes. A PID controller is designed using direct synthesis ratio andPIsatisfactory performance for a class ofPID process approach and IMC principles in Chien et al. (2003) and models. controllers are more preferred than conapproach and IMC respectively. principles in Jeng Chienand et al. Chen etA al. (2006), Lin(2003) (Jeng and cations because of theirperformance simplicity, acceptable cost/benefit ratio and satisfactory for a class of process models. PI controllers are more preferred than PID concesses. PID controller is designed using direct synthesis and IMC principles in Chien et al. (2003) models. PI controllers aresensitivity. more preferred than PID con- approach Chen et al. (2006), respectively. Jeng and Lin (Jeng and trollers due to less noise PI and PID tuning Chen(2012)) et al. (2006), respectively. Jeng andrules Lin (Jeng have tuning basedand on ratio and performance forPIa and class ofPID process models. PIsatisfactory controllers are more preferred than con- Lin trollers due to less less noise sensitivity. PID tuning approach and IMCproposed principlesPID in Jeng Chien etrules al. (2003) and Chen et al. (2006), respectively. and Lin (Jeng trollers due to noise sensitivity. and PID tuning Lin (2012)) have proposed PID tuning based on rules proposed in the literature for aPI class of processes Lin (2012)) have proposed PID tuning rules based on Smith-type compensator design that gives robust performodels. PI controllers are more preferred than PID controllers due to less noise sensitivity. PI and PID tuning rules proposed proposed in the the literature for aaunstable, class of of etc) processes Chen et al. compensator (2006), respectively. Jeng andrules Lin (Jeng and Lin (2012)) have proposed PID tuning based on rules in literature for class processes Smith-type design that gives robust perfor(stable, integrating, inverse response, have Smith-type compensator design that gives robust performance. trollers due to less noise sensitivity. PI and of PID tuning rules proposed in the literature for a class processes (stable, integrating, inverse response, unstable, etc) have (2012)) compensator have proposed PIDthat tuning rules based on Smith-type design gives robust perfor(stable, integrating, inverse response, unstable, etc) have Lin mance. been summarized in O’Dwyer (2009). rules proposed in inthe literature for aunstable, class of etc) processes (stable, integrating, response, have mance. been summarized summarized O’Dwyer (2009). Smith-type compensator thatdelay givesinverse robustresponse perforFor integrating first-order design plus time mance. been in inverse O’Dwyer (2009). (stable, integrating, inverse response, unstable, etc) have For integrating first-order time delay inverse response A large number of in tuning formulas have been reported for mance. been summarized O’Dwyer (2009). integrating first-order plus timehas delay inverseempirical response processes, Luyben (Luybenplus (2003)) proposed A large number of of in tuning formulas have been reported for For been summarized O’Dwyer (2009). For integrating first-order plus time delay inverseempirical response A large number tuning formulas have reported for processes, Luyben (Luyben (2003)) has proposed stable overdamped processes. As far as been inverse response processes, Luyben (Luyben (2003)) has proposed empirical PI/PID tuning rules based on frequency-domain concept. A largeoverdamped number of tuning formulas have been reported for For stable overdamped processes. As far far as tuning inverse response integrating first-order plus time delay inverseempirical response processes, Luyben (Luyben (2003)) has proposed stable processes. As as inverse response PI/PID tuning rules based on frequency-domain concept. processes are concerned, less number of rules have PI/PID tuning rules based on frequency-domain concept. Based on H optimization and IMC principles, tuning A large number of tuning formulas have been reported for ∞ stable overdamped processes. As far of as inverserules response processes are concerned, concerned, less tuning rules have processes, Luyben (Luyben (2003)) has proposed empirical PI/PID tuning rules based on frequency-domain concept. processes are less number number of tuning have Based on H optimization and IMC principles, tuning been proposed. For first-order inverse response processes ∞ Based on H optimization and IMC principles, tuning rules for PI/PID controllers are reported in Gu et al. ∞ stable overdamped processes. As far as inverse response processes are concerned, less number of tuning rules have been and proposed. Fortime first-order inverse response processes PI/PID tuning rules based onand frequency-domain concept. Based on H∞ optimization IMC principles, tuning been proposed. For first-order inverse response processes rules for PI/PID controllers are reported in Gu et with without delay, PI tuning rules using two rules for PI/PID controllers are reported in Gu et al. al. (2006). A PID controller is designed in Pai et al. (2010) processes are concerned, less number ofresponse tuning rules have been proposed. For first-order inverse processes with and without time delay, PI tuning rules using two on H∞ optimization and IMC principles, tuning rules for PI/PID controllers are reported in Gu et al. with and methods without time delay, PI tuningin rules two Based (2006). A PID controller is designed in Pai et al. (2010) different have been reported Sree using and Chi(2006). A PID controller is designed in Pai et al. (2010) using direct synthesis approach for disturbance rejection. been proposed. For first-order inverse response processes with and methods without PI tuning two rules different methods have been reported in rules Sree using and ChiforA PI/PID controllers are reported et al. (2006). controller is designed in Pai in et Gu al. (2010) different have been reported in Sree and Chisynthesis approach for rejection. dambaram (2003). time In thedelay, first method, controller parameusing direct directPID synthesis approach for disturbance disturbance rejection. with and methods without time delay, PI tuning rules using two using different have been reported in Sree and Chidambaram (2003). In the first method, controller parame(2006). A PID controller is designed in Pai et al. (2010) Minimization of integral performance criteria generally using direct synthesis approach for disturbance rejection. dambaram (2003). In the first method, controller parameters are obtained byhave matching the coefficients of respective different methods reported in Sree and Chi- Minimization of performance criteria generally dambaram (2003). thebeen first the method, controller parameters are are of obtained byIn matching the coefficients of respective respective direct synthesis approach for disturbance rejection. Minimization of integral integral performance criteria generally results in small undershoot and shorter settling time ters obtained by matching coefficients of powers ‘s’ in the numerator and denominator of closed using dambaram (2003). In the first the method, controller parameMinimization of integral performance criteria generally results in small undershoot and shorter settling time ters are obtained by matching coefficients of respective powers of ‘s’ in the numerator and denominator of closed results inand small undershoot and shorter settling time Atherton (1991); Kaya et al. (2007); Padula powers of ‘s’ in the numerator and denominator ofmethod closed (Zhuang loop transfer function for servo response. Second Minimization of integral performance criteria generally ters are obtained by numerator matching the coefficients of respective results in small undershoot and shorter settling time (Zhuang and Atherton (1991); Kaya et al. (2007); Padula powers of ‘s’ in the and denominator of closed loop transfer function for servo response. Second method (Zhuang and Atherton (1991); Kaya et al. (2007); Padula Visioli (2011)). PI and PID controller settings have loop transfer function forprinciples servo response. Second method and which is based on IMC yields a PI controller results in small undershoot and shorter settling time powers of ‘s’ in the numerator and denominator of closed (Zhuang and Atherton (1991); Kaya et al. (2007); Padula and Visioli (2011)). PI and PID controller settings have loop transfer function for servo response. Second method whichfirst-order is based based on on IMCPIprinciples principles yields PI tocontroller controller and Visioli (2011)). PI and PID settings have reported in Majhi (2005) andcontroller Zhuang and Atherton which is IMC yields aa PI with filter. controller is tuned achieve been (Zhuang and(2011)). Atherton (1991); Kaya et al. (2007); Padula loop transfer function for servo response. Second method and Visioli PI and PID controller settings have been reported in Majhi (2005) and Zhuang and Atherton which is based on IMC principles yields a PI controller with first-order filter. PI controller is tuned to achieve been reported in Majhi (2005) and Zhuang and Atherton for FOPTD process models using integral perforwith first-order filter.log PImodulus controller is tuned to achieve (1993) maximum closed-loop ofyields +2 dB first-order Visioli (2011)). PI and PID settings have which is based on IMC principles a for PI to controller been in Majhi (2005) and Zhuang and Atherton (1993) for process models using integral perforwith first-order filter. PIresponse controller is tuned achieve and 2 controller maximum closed-loop log modulus of +2 +2 dB for first-order (1993)reported for FOPTD FOPTD process models using integral performance criteria (ISE, ISTE, IST E). For integrating promaximum closed-loop log modulus of dB for first-order plus time delay inverse processes in Marchetti 2 been reported in Majhi (2005) and Zhuang and Atherton with first-order filter. PI controller is tuned to achieve (1993) for FOPTD process models using integral perfor2 mance criteria (ISE, ISTE, ISTrules E). have For integrating integrating promaximum closed-loop logresponse modulus processes of +2 dB for plus time (2000). delay inverse inverse response processes in first-order Marchetti mance criteria (ISE, ISTE, IST E). For processes, optimal PI/PID tuning been proposed plus time delay in Marchetti and Scali 2 for FOPTD process models integral performaximum closed-loop logresponse modulus processes of +2 dB for mance criteria (ISE, ISTE, IST E).using For integrating processes, optimal PI/PID tuning rules have been proposed plus time (2000). delay inverse in first-order Marchetti (1993) and Scali Scali (2000). cesses, optimal PI/PID tuning rules have been proposed in Ali and Majhi (2011). The controller parameters are and 2 criteria (ISE, ISTE, ISTcontroller E). have For integrating proplus time (2000). delayNygardas inverse response in Marchetti cesses, optimal PI/PID tuning rules been proposed in Ali and Majhi (2011). The parameters are In and (1975), itprocesses is demostrated that a mance andWaller Scali in Ali and Majhi (2011). The controller parameters are obtained for integrating and unstable processes in Visioli In Waller Waller and Nygardas Nygardas (1975), it it is is tuning demostrated that optimal PI/PID tuning rules have been in proposed and Scali (2000). in Ali and Majhi (2011). The controller parameters are In and (1975), demostrated that aa cesses, obtained for integrating and unstable processes Visioli PID controller with Ziegler-Nichols (Ziegler and obtained for integrating and unstable processes in Visioli (2001) and Kaya (2003) by minimizing ISTE criterion. In Waller and Nygardas (1975), it is demostrated that a PID controller controller with Ziegler-Nichols tuning (Ziegler and and in Ali and Majhi (2011).and The controller parameters are obtained for integrating unstable processes in Visioli PID with Ziegler-Nichols tuning (Ziegler (2001) and Kaya (2003) by minimizing ISTE criterion. Nichols (1942)) gives acceptable control performance for (2001) and Kaya (2003) by minimizing ISTE criterion. similar PI/PID controller settings forin aVisioli class In Waller and Nygardas (1975), itcontrol is tuning demostrated that PID controller with Ziegler-Nichols (Ziegler and Nichols (1942)) gives response acceptable performance fora On obtained forlines, integrating and unstable processes (2001) and Kaya (2003) by minimizing ISTE criterion. Nichols (1942)) gives acceptable control performance for On similar lines, PI/PID controller settings for a class second-order inverse processes (without delay). Oninverse similarresponse lines, PI/PID controller settingsin for a work. class processes are proposed this PID controller with Ziegler-Nichols tuning (Ziegler and Nichols (1942)) gives model, acceptable control performance for of second-order inverse response processes (without delay). (2001) andresponse Kaya PI/PID (2003) by minimizing ISTE criterion. On similar lines, controller settings for a Eberclass second-order inverse response (without of inverse processes are proposed in this work. For the same process anprocesses analytical design ofdelay). a PID of inverse response processes are proposed in this work. Particle Swarm Optimization (PSO) (Kennedy and Nichols (1942)) gives acceptable control performance for second-order inverse response (without For the same same process model, anprocesses analytical design ofdelay). PID On similar lines, PI/PID controller settingsin for a Eberclass of inverse response processes are proposed this work. For the process model, an analytical design of aa PID Particle Swarm Optimization (PSO) (Kennedy and controller based on IMC theory is proposed in Scali and Particle Swarm Optimization (PSO)(1995); (Kennedy and Eberhart (1995); Eberhart and Kennedy Shi second-order inverse response processes (without delay). For the same process model, an analytical design of a PID controller based on IMC theory is proposed in Scali and of inverse response processes are proposed in this work. Particle Swarm Optimization (PSO) (Kennedy and Ebercontroller based on IMC theory is proposed in Scali and hart (1995); (1995); Eberhart and Kennedy (1995); Shi Rachid (1998). Iinoyamodel, and Altpeter (Iinoya and of Altpeter Eberhart and (1995); and Eber(1998); Eberhart andKennedy Shi(PSO) (2001); Poli Shi et al. For the same process an analytical design a PID controller based on IMC is response proposed in Scali and hart Rachid (1998). Iinoya and Altpeter (Iinoya and Altpeter Particle Swarm Optimization (Kennedy and(2007)) Eberhart (1995); Eberhart and Kennedy (1995); Shi Rachid (1998). Iinoya and Altpeter (Iinoya and Altpeter (1998); Eberhart and Shi (2001); Poli et al. (2007)) (1962)) have reported antheory inverse compensator hart (1998); Eberhart and Shi (2001); Poli et al. (2007)) is used to minimize the objective functions. The three controller based on IMC theory is proposed in Scali and Rachid (1998). Iinoyasaid and Altpeter (Iinoya and Altpeter (1962))for have reported an process inverse response compensator hart (1995); Eberhart and Kennedy (1995); Shi and(2007)) Eber(1998); Eberhart and Shi (2001); Poli et al. (1962)) have reported an inverse response compensator is used to minimize the objective functions. The three design the above model. Iinoya and Altis used to minimize the objective functions. The three integral performance criteria viz. integral of the squared Rachid (1998). Iinoya and Altpeter (Iinoya and Altpeter (1962)) have an in inverse response compensator design for for the reported above said process model. Iinoya and AltAlthart (1998); Eberhartcriteria andobjective Shiviz. (2001); Poliofetthe al. (2007)) is used to minimize the functions. The three design the above said process model. Iinoya and integral performance integral squared peter’s scheme is modified Zhang et al. (2000) using H ∞ integral performance criteria viz. integral of the squared (1962)) have reported an inverse response compensator design for the above said process model. Iinoya and Altpeter’s scheme scheme is is modified modified in in Zhang Zhang et et al. al. (2000) (2000) using using H H∞ is used to minimize criteria the objective functions. The three integral performance viz. integral of the squared peter’s ∞ design the above said in process Iinoyausing and Altpeter’s for scheme is modified Zhangmodel. et al. (2000) H∞ integral performance criteria viz. integral of the squared Copyright © 2018, 2018is IFAC peter’s scheme modified in Zhang et al. (2000) usingControl) H∞429Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Copyright © 2018 IFAC 429 Copyright 2018 responsibility IFAC 429Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 429 10.1016/j.ifacol.2018.05.063 Copyright © 2018 IFAC 429

5th International Conference on Advances in Control and 414 Optimization of Dynamical Systems Mohammad Irshad et al. / IFAC PapersOnLine 51-1 (2018) 413–418 February 18-22, 2018. Hyderabad, India

time-weighted error (ISTE ), integral of the squared timesquared weighted error (IST 2 E) and integral of the squared time-cubed weighted error (IST 3 E) have been minimized for set-point tracking. For each of the process model, the steady-state gain and time-lag constant have been assumed as unity. The combined value of time delay and positive zero time constant or value of positive zero time constant is varied and for each value, PSO algorithm is run to obtain optimal PI/PID controller parameters. Finally, analytical expressions are obtained for the optimal controller parameter values using curve fitting toolbox. The paper has been organized as follows: Section 2 presents processes undertaken and integral functions used for minimization. PI/PID tuning formulas for the processes studied are presented in Section 3. Section 4 contains illustrative examples and simulation results. Lastly, conclusions are drawn in Section 5. 2. PROCESSES STUDIED

where α symbolizes controller parameters to be selected to minimize Jn (α). Here J1 (α), J2 (α) and J3 (α) denote ISTE, IST 2 E and IST 3 E criterion respectively 3. TUNING FORMULAS 3.1 FO-IR and FOPTD-IR process models Optimal tuning rules for FO-IR process models, relating coefficients of PI controller to the process parameters are given in Table 1. The above tuning rules are obtained by minimising performance indexes mentioned in (6) for set-point tracking. For a wide range of normalized ττz ratios, optimal PI values have been obtained. The optimal PI coefficients have been analytically interpolated using curve-fitting toolbox to derive the tuning rules. Tuning rules obtained for FO-IR process models are valid for FOPTD-IR process models if ττz is replaced by τz +D τ .

The inverse response processes considered in the present work are as follows:

3.2 IFOPTD-IR process model

First-order inverse response (FO-IR) process (1)

For the IFOPTD-IR process models, PI controllers fail to give stable output. Using optimal PID values obtained for ratios, tuning rules are a wide range of normalized τz +D τ derived. The obtained tuning rules are given in Table 2.

First-order plus time delay inverse response (FOPTD-IR) process

A set-point filter with the following transfer function is considered for reducing the overshoot for integrating process models during simulations.

Gp (s) = Kp

Gp (s) = Kp

(−τz s + 1) (τ s + 1)

(−τz s + 1) −Ds e (τ s + 1)

Integrating first-order plus time delay inverse response (IFOPTD-IR) process Gp (s) = Kp

(−τz s + 1) −Ds e s(τ s + 1)

PI/PID controller having following transfer function is considered in this work :

Gc (s) = Kc (1 +

1 ) τi s

(4)

1 + sτd ) sτi

(5)

where Kc is proportional gain , τi is integral time constant and τd is derivative time constant. In order to obtain optimal controller parameters, an integral performance criterion has been minimized. Mathematically, the integral performance indexes are represented by Jn (α) =



∞

[tn e(α, t)]2 dt, n = 1, 2, 3

as2

(6)

0

430

1 + bs + 1

(7)

where a = τi τd and b = τi . Table 1. Tuning rules for optimal PI controllers for FO-IR process models

(3)

where ( τ , D, τz > 0 ); Kp is the steady-state gain; τ is time-lag constant; D is the time delay and τz is the positive zero time constant.

Gc (s) = Kc (1 +

Fsp =

(2)

ISTE Kc Kp =

a1 ( ττz )b1

+ c1

τi /τ = a2 ( ττz )b2 + c2

IST 2 E

IST 3 E

a1

0.483

0.366

0.318

b1

-1.069

-1.157

-1.191

c1

0.319

0.269

0.226

a2

0.344

0.211

0.144

b2

1.099

1.242

1.364

c2

1.019

0.981

0.969

Table 2. Tuning rules for optimal PID controllers for IFOPTD-IR process models Kc Kp τ = a1 ( τz τ+D )b1 + c1

τi /τ = a2 ( τz τ+D )b2 + c2

τd /τ =

τ +D τ +D a3 ( z τ )2 +b3 ( z τ )+c3 τ +D ( z τ +d3 )

ISTE

IST 2 E

a1

0.601

0.630

IST 3 E 0.670

b1

-1.295

-1.353

-1.242

c1

0.140

0.160

0.129

a2

5.192

3.439

3.033

b2

0.947

0.912

0.958

c2

2.167

1.118

1.496

a3

0.379

0.351

0.318

b3

0.683

0.619

0.588

c3

-0.043

-0.037

0.002

d3

-0.028

-0.016

0.032

5th International Conference on Advances in Control and Optimization of Dynamical Systems Mohammad Irshad et al. / IFAC PapersOnLine 51-1 (2018) 413–418 February 18-22, 2018. Hyderabad, India

Table 3. PI controller parameter values for Example 1. IST 3 E

ISTE

Kc

0.2537

0.802

0.635

0.544

τi

1.0428

1.363

1.192

1.113

1.5 1

Process variable

Sree and Chidambaram

IST 2 E

415

4. SIMULATIONS AND RESULTS

0.5

Sree and Chidambaram

0

IST3E 2

−0.5

IST E

−1 −1.5

4.1 Example 1. Consider a first-order inverse response process (Sree and Chidambaram (2003)) having transfer function:

−2 0

4.2 Example 2. An isothermal CSTR exhibiting multiple steady state solutions is considered (Sree and Chidambaram (2003)). The series-parallel reactions occuring in following manner: k

1 2 A −→ B −→ C

k

3 2A −→ D

15

(9)

For the species A and B, mass balance equations are given by: 431

20

Time(sec)

25

30

35

40

2

1.6 1.4

Sree and Chidambaram

1.2

IST3E

1

IST2E

0.8 0.6 0.4 5

10

15

(8)

20

Time(sec)

25

30

35

40

(b)

Fig. 1. (a) Output responses for Example 1 (nominal) (b) Controller outputs for Example 1 (nominal). 2

Process variable

1.5 1 0.5 0 −0.5

Sree and Chidambaram+20%

−1

IST3E+20% Sree and Chidambaram−20% 3

IST E−20%

−1.5 −2 0

5

10

15

20

Time(sec)

25

30

35

40

(a) 1.8

Control variable

The controller parameters obtained by ISTE, IST 2 E and IST 3 E criterion for the above process model are listed in the Table 3. The step responses and the control signals obtained by applying unit step set-point change at t = 0 and step disturbance change of magnitude 0.5 at t = 20 sec are shown in Fig. 1a and 1b respectively. For comparison, method reported in Sree and Chidambaram (2003) has been considered. ISTE criterion response is not shown as it results in very high undershoot which is undesirable. Except undershoot behavior which is least for IST 3 E criterion, IST 2 E and IST 3 E criterion result in similar responses. Both criteria lead to considerable reduction in rise time and settling time for servo response as compared to Sree and Chidambaram method. They also give superior regulatory performance. The various performance measures are shown in Table 4. For investigating model-mismatch condition, IST 3 E criterion is preferred because of least undershoot. All process parameters are perturbed by ±20% and the corresponding plots are shown in Fig. 2. It is observed that IST 3 E criterion results in lesser rise time and settling time as compared to Sree and Chidambaram method. Thus optimal controller obtained (IST 3 E criterion) is also giving robust control performance.

k

10

1.8

0.2 0

(1 − s) Gp (s) = (s + 1)

5

(a)

Control variable

To analyse the effectiveness of the proposed tuning rules, several examples of the processes mentioned above are considered. For physical implementation, a first-order filter is cascaded with the derivative term and filter time constant is taken to be 0.1 times the derivative time constant. Rise time (tr ) and settling time(ts ) in seconds and overshoot (Os ) in percentage are calculated to compare the closed loop performances of various tuning methods.

1.6

Sree and Chidambaram+20%

1.4

IST3E+20% Sree and Chidambaram−20%

1.2

IST3E−20%

1 0.8 0.6 0.4 0.2 0

5

10

15

20

Time(sec)

25

30

35

40

(b)

Fig. 2. (a) Output responses for Example 1 (perturbed) (b) Controller outputs for Example 1 (perturbed). dx1 F = −k1 x1 − k3 x1 2 + (CA0 − x1 ) dt V dx2 F = k1 x 1 − k 2 x 2 − x 2 dt V

(10)

where x1 and x2 are the concentrations of A and B respectively in the reactor,F is the flow rate(l/min), V is

5th International Conference on Advances in Control and 416 Optimization of Dynamical Systems Mohammad Irshad et al. / IFAC PapersOnLine 51-1 (2018) 413–418 February 18-22, 2018. Hyderabad, India

Table 4. Performance measures for various examples Servo

1.2

Regulatory

methods

tr

ts

Os (%)

ts

Sree and Chidambaram

6.88

13.25

0.00

13.10

IST 3 E

2.15

5.71

0.00

6.82

IST 2 E

1.55

5.62

0.00

7.00

Sree and Chidambaram

1.12

4.96

9.87

5.74

Example 1

Process variable

Tuning

1.4

Example 2 IST 3 E

0.76

3.16

14.35

3.70

0.71

2.84

15.84

3.43

ISTE

0.64

3.18

17.39

3.10

0.2

IST2E ISTE 2

4

6

14.16

6.23

10.43

IST 3 E

3.47

6.68

0.34

7.15

IST 2 E

3.31

6.18

1.18

6.96

ISTE

7.43

14.22

0.00

12.68

5.77

20.89

3.34

15.69

IST 3 E

5.60

15.08

3.82

15.26

10

Time(sec)

12

14

16

18

20

14

16

18

20

Sree and Chidambaram 3

Example 4 Jeng and Lin

8

1.4

Control variable

3.86

4.37

IST3E

1.2

Jeng and Lin

19.07

Sree and Chidambaram 0.4

(a)

Example 3

5.98

0.6

0 0

IST 2 E

IST 2 E

1 0.8

IST E

1

IST2E ISTE

0.8 0.6 0.4 0.2 0 −0.2 0

20.15

2

4

6

8

10

Time(sec)

12

(b)

Table 5. PI parameter values for Example 2. ISTE

IST 2 E

IST 3 E

Kc

1.270

1.003

0.857

τi

0.890

0.773

0.718

Fig. 3. (a) Output responses for Example 2 (non-linear CSTR model) (b) Controller outputs for Example 2 (non-linear CSTR model). 4.3 Example 3. Consider an IFOPTD-IR process (Jeng and Lin (2012)) with transfer function:

the reactor volume (l) and CA0 is the feed concentration of A(mol/l)). The values of the parameters considered in Sree and Chidambaram (2003) are k1 = 0.8333 l/min, k2 = 1.6667 l/min, k3 = 0.16667 l/mol-min, CA0 = 10 mol/l. A steady-state value of x2 = 1.117 mol/l is obtained at F/V = 0.5714 min−1 . The transfer function obtained after linearization around above steady-state value is 0.5848(-0.3546s+1)/[(0.4149s+1)(0.4464s+1)]. With a measurement delay of 0.1 min, model obtained is reduced to FOPTD-IR (Sree and Chidambaram (2003)) having transfer function : −0.3567s  p (s) = 0.5848(−0.3546s + 1)e G (0.6302s + 1)

(11)

The controller parameters reported in Sree and Chidambaram (2003) are Kc = 0.3647 and Ti = 0.5065. Table 5 shows optimal PI parameter values for various integral criteria. For performance evaluation, non-linear CSTR model is considered. The closed loop performances are evaluated by applying unit step set-point change at t = 0 and step disturbance of magnitude 0.5 at t = 10 sec. Fig. 3 shows the output responses and corresponding control signals. Amongst the various integral performance criteria, servo and regulatory performances shown by IST 2 E criterion are optimum. From Table 4, it is observed that IST 2 E criterion has more overshoot but rise time and settling time are less as compared to Sree and Chidambaram method. Also, it has better load disturbance rejection. Hence, optimal controller obtained from proposed tuning rules is showing robust control performance. 432

Gp (s) =

0.547(−0.418s + 1) −0.1s e s(1.06s + 1)

(12)

The optimal controller parameter values are given in Table 6. Process variable plots obtained by applying unit step set-point change at t = 0 and step disturbance change of magnitude -1 at t = 25 sec, are shown in Fig. 4(a) whilst Fig. 4(b) shows corresponding control variable plots. The results are compared with the method reported in Jeng and Lin (2012). Table 4 shows various performance measures calculated for each of the tuning method. As is seen from the plots, the IST 2 E and IST 3 E criterion have similar responses but better than ISTE criterion. For setpoint tracking, settling time and percentage overshoot are considerably less for both the criteria as compared to Jeng and Lin method. They also show better load disturbance rejection. Now considering the case of model-mismatch in which all process parameters are assumed to have uncertainties of +20%. Figs. 5(a) and 5(b) show the related output and control signals respectively. It is observed that IST 3 E criterion yields best servo and regulatory responses amongst all performance criteria. It has less percentage overshoot and settling time for servo response and load disturbance rejection is much better compared to Jeng and Lin method. Thus optimal controller (IST 3 E criterion) shows more robust control performance compared to Jeng and Lin method. It is to be noted that Jeng and Lin already showed that their method has more satisfactory results compared to Luyben (2003), Gu et al. (2006) and Pai et al. (2010).

5th International Conference on Advances in Control and Optimization of Dynamical Systems Mohammad Irshad et al. / IFAC PapersOnLine 51-1 (2018) 413–418 February 18-22, 2018. Hyderabad, India

Table 6. PID parameter values for Example 3.

417

1.2

ISTE

IST 2 E

IST 3 E

Kc

1.608

2.852

3.128

3.025

τi

3.518

5.098

3.087

3.209

τd

1.06

0.878

0.784

0.744

τf

0.029

-

-

-

a

3.729

4.476

2.420

2.387

b

3.518

5.098

3.087

3.209

Process variable

1

Jeng and Lin

0.8 0.6 0.4

Jeng and Lin

0.2

IST3E

0

IST2E ISTE

−0.2 0

5

10

15

20

25

Time(sec)

30

35

40

45

50

30

35

40

45

50

(a)

1.2 1

Jeng and Lin

2.5

3

0.6

Control variable

Process variable

3 0.8

0.4

Jeng and Lin 0.2

IST3E

0

IST2E ISTE

−0.2 0

IST E

2

IST2E ISTE

1.5 1 0.5 0

5

10

15

20

25

Time(sec)

30

35

40

45

50 −0.5

(a)

−1 0

5

10

15

25

Time(sec)

(b)

2

Fig. 5. (a) Output responses for Example 3 (perturbed) (b) Controller outputs for Example 3 (perturbed).

1.5

Control variable

20

1

Table 7. PID parameter values for Example 4. 0.5

Jeng and Lin 3

IST E IST2E ISTE

0

−0.5 0

5

10

15

20

25

Time(sec)

30

35

40

45

50

Jeng and Lin

ISTE

IST 2 E

IST 3 E

Kc

0.95

1.115

1.186

1.204

τi

5.88

9.112

5.653

5.594

τd

1.161

1.263

1.144

1.060

(b)

τf

0.057

-

-

-

Fig. 4. (a) Output responses for Example 3 (nominal) (b) Controller outputs for Example 3 (nominal).

a

6.827

11.508

6.467

5.930

b

5.88

9.112

5.653

5.594

4.4 Example 4. Consider a higher order plant transfer function (Jeng and Lin (2012)):

Gp (s) =

0.5(−0.5s + 1) e−0.7s s(0.4s + 1)(0.1s + 1)(0.5s + 1)

(13)

5. CONCLUSION

The IFOPTD-IR process model (Jeng and Lin (2012)) obtained for controller design is: −0.81s  p (s) = 0.5183(−0.4699s + 1)e G s(1.1609s + 1)

and Lin method, IST 3 E criterion has less rise time and settling time for servo response. Load disturbance rejection is slightly better. Thus, more robust control performance is shown by optimal controller (IST 3 E criterion) with less controller parameters than Jeng and Lin method.

(14)

For the process model obtained, controller parameter values are calculated using proposed tuning rules and shown in Table 7. The method reported in Jeng and Lin (2012) has been considered for comparison. Performance evaluation is done on higher order plant model. The setpoint and load disturbance rejection responses obtained by applying unit step set-point change at t = 0 and step disturbance of magnitude -0.5 at t = 40 sec, are shown in Fig. 6. The step response of ISTE criterion is not shown as it has sluggish response which is unacceptable. The IST 3 E criterion shows better servo and regulatory responses than IST 2 E criterion. As compared to Jeng 433

In this paper, optimal tuning rules for stable and integrating first-order inverse response processes have been presented. Analytical expressions have been provided relating PI/PID controller parameters with the plant parameters. Proposed tuning rules give better servo and regulatory performances for nominal condition and robust control performance for model-mismatch condition. A comparative study of the closed-loop performances achieved by ISTE, IST 2 E and IST 3 E criterion has been done. Simulation results illustrate that ISTE criterion generally results in either large undershoot or sluggish response. IST 3 E criterion yields less undershoot and provides overall better set-point tracking and load disturbance rejection. REFERENCES Alc´antara, S., Pedret, C., Vilanova, R., and Zhang, W. (2009). Analytical H∞ design for a smith-type inverse-

5th International Conference on Advances in Control and 418 Optimization of Dynamical Systems Mohammad Irshad et al. / IFAC PapersOnLine 51-1 (2018) 413–418 February 18-22, 2018. Hyderabad, India

1.2

Process variable

1 0.8 0.6

Jeng and Lin

0.4

IST E

IST3E 2

0.2 0 −0.2 0

10

20

30

40

Time(sec)

50

60

70

80

(a) 1.2

Control variable

1 0.8 0.6 0.4 0.2

Jeng and Lin IST3E

0

2

IST E −0.2 0

10

20

30

40

Time(sec)

50

60

70

80

(b)

Fig. 6. (a) Output responses for Example 4 (real process) (b) Controller outputs for Example 4 (real process). response compensator. In American Control Conference., 1604–1609. IEEE. Ali, A. and Majhi, S. (2011). Integral criteria for optimal tuning of PI/PID controllers for integrating processes. Asian Journal of Control, 13(2), 328–337. Chen, P., Zhang, W., and Zhu, L. (2006). Design and tuning method of PID controller for a class of inverse response processes. In American Control Conference, 274–279. IEEE. Chien, I.L., Chung, Y.C., Chen, B.S., and Chuang, C.Y. (2003). Simple PID controller tuning method for processes with inverse response plus dead time or large overshoot response plus dead time. Industrial & Engineering Chemistry Research, 42(20), 4461–4477. Eberhart, R. and Kennedy, J. (1995). A new optimizer using particle swarm theory. In Proceedings of the Sixth International Symposium on Micro Machine and Human Science., 39–43. IEEE. Eberhart, R.C. and Shi, Y. (2001). Particle swarm optimization: developments, applications and resources. In Proceedings of the Congress on Evolutionary Computation, volume 1, 81–86. IEEE. Gu, D., Ou, L., Wang, P., and Zhang, W. (2006). Relay feedback autotuning method for integrating processes with inverse response and time delay. Industrial & Engineering Chemistry Research, 45(9), 3119–3132. Iinoya, K. and Altpeter, R.J. (1962). Inverse response in process control. Industrial & Engineering Chemistry, 54(7), 39–43. Jeng, J.C. and Lin, S.W. (2012). Robust proportional-integral-derivative controller design for stable/integrating processes with inverse response and time delay. Industrial & Engineering Chemistry Research, 51(6), 2652–2665. Kaya, I. (2003). A PI-PD controller design for control of unstable and integrating processes. ISA Transactions, 434

42(1), 111–121. Kaya, I., Tan, N., and Atherton, D.P. (2007). Improved cascade control structure for enhanced performance. Journal of Process Control, 17(1), 3–16. Kennedy, J. and Eberhart, R. (1995). Particle swarm optimization. In Proceeding of IEEE International Conference on Neural Network ,Perth, Australia, 1942– 1948. Luyben, W.L. (2000). Tuning proportional- integral controllers for processes with both inverse response and deadtime. Industrial & Engineering Chemistry Research, 39(4), 973–976. Luyben, W.L. (2003). Identification and tuning of integrating processes with deadtime and inverse response. Industrial & Engineering Chemistry Research, 42(13), 3030–3035. Majhi, S. (2005). On-line PI control of stable processes. Journal of Process Control, 15(8), 859–867. Marchetti, G. and Scali, C. (2000). Use of modified relay techniques for the design of model-based controllers for chemical processes. Industrial & Engineering Chemistry Research, 39(9), 3325–3334. O’Dwyer, A. (2009). Handbook of PI and PID Controller Tuning Rules. Imperial College Press. Padula, F. and Visioli, A. (2011). Tuning rules for optimal PID and fractional-order PID controllers. Journal of Process Control, 21(1), 69–81. Pai, N.S., Chang, S.C., and Huang, C.T. (2010). Tuning PI/PID controllers for integrating processes with deadtime and inverse response by simple calculations. Journal of Process Control, 20(6), 726–733. Poli, R., Kennedy, J., and Blackwell, T. (2007). Particle swarm optimization An overview. Swarm Intelligence, 1(1), 33–57. Scali, C. and Rachid, A. (1998). Analytical design of proportional- integral- derivative controllers for inverse response processes. Industrial & Engineering Chemistry Research, 37(4), 1372–1379. Shi, Y. and Eberhart, R.C. (1998). Parameter selection in particle swarm optimization. In International Conference on Evolutionary Programming, 591–600. Springer. Sree, R.P. and Chidambaram, M. (2003). Simple method of tuning PI controllers for stable inverse response systems. J. Indian Inst. Sci, 83, 73–85. Visioli, A. (2001). Optimal tuning of PID controllers for integral and unstable processes. IEE ProceedingsControl Theory and Applications, 148(2), 180–184. Waller, K.V. and Nygardas, C. (1975). On inverse repsonse in process control. Industrial & Engineering Chemistry Fundamentals, 14(3), 221–223. Zhang, W., Xu, X., and Sun, Y. (2000). Quantitative performance design for inverse-response processes. Industrial & Engineering Chemistry Research, 39(6), 2056– 2061. Zhuang, M. and Atherton, D. (1991). Tuning PID controllers with integral performance criteria. In International Conference on Control., 481–486. IET. Zhuang, M. and Atherton, D. (1993). Automatic tuning of optimum PID controllers. In IEE Proceedings D-Control Theory and Applications, volume 140, 216–224. IET. Ziegler, J.G. and Nichols, N.B. (1942). Optimum settings for automatic controllers. Trans. ASME, 64(11).