Development of the Frequency PID Tuning Method with Filter

Transcription

1 139 A publiation o CHEMICAL ENGINEERING TRANSACTIONS VOL Guest Editors: Petar Sabev Varbanov Peng-Yen Liew Jun-Yow Yong Jiří Jaromír Klemeš Hon Loong Lam Copyright 2016 AIDIC Servizi S.r.l. ISBN ; ISSN The Italian Assoiation o Chemial Engineering Online at DOI: /CET Development o the Frequeny PID Tuning Method with Filter Kozin A. Kozin* a Anna M. Egorova a Sergei S. Mikhalevih a Flavio Manenti b Franeso Rossi b a National Researh Tomsk Polytehni University Department o Eletronis and Automation o Nulear Plants Lenin Street Tomsk Russian Federation b Politenio di Milano Dipartimento Chimia Materiali e Ingegneria Chimia Giulio Natta Piazza Leonardo da Vini Milano Italy kozin@tpu.ru In the previous researh authors developed tunable method or PID ontrollers to ahieve the desired phase margin. Crossover requeny is used as a parameter that allows orreting time response o losed-loop system. The aim o the researh is to ind proportional integral derivative and ilter oeiients o PID ontroller that will provide desired harateristis (phase margin and rossover requeny) in the system. The analytial approah o proposed PID tuning method is presented. The omparison proposed method with onventional was done. Pratial approah ontains developing the ontrol system o extration proess. This system is a part o Russian Federation large projet losed-loop nulear uel yle. The plant was linearized and digital ontrol system was developed. 1. Introdution One o the atual researh trends o ontrol theory is a development o PID tuning methods. It is due to wide spread PID ontrollers in engineering pratie. As is well known PID ontroller is appropriate or the most linear or quasi-linear proesses (Moradi and Vossoughi 2015). At the present time there are a lot o dierent methods that are based on a desired time-domain or requeny harateristis. But most o them are developed or ideal PID ontroller despite o all ontrollers ontain ilter or derivative part o PID ontroller in modern engineering pratie. Ideal PID ontroller ontains ideal dierentiation that ould not be hardware realized. Thereore the irst order ilter is used or pratial reasons. Nowadays there are a lot o dierent PID tuning rules and methods whih ignore ilter onstant in tuning proedure and then reommend hoosing it additionally (Kristiansson and Lennartson 2006). Any deviation o ilter onstant hanges desired stability or time-response harateristis. There are several PID tuning methods whih allow to determinate our unknown (proportional gain integral derivative and ilter time onstants). For instane method (Sanhis et al. 2010) used as desired data: phase margin gain margin and additional parameter input data is requeny plant harateristis. The method (Sanhis et al. 2010) also uses an optimization algorithm to provide desired harateristis o the system. Obvious disadvantage o this method is neessary to provide a lot o initial inormation about proess and desired ontrol system. In the artile (Vadutov 2014) the author suggests to use onstrained optimization to provide minimum o IAE or ISE. In addition the author imposes restritions to the poles o losed-loop transer untion. This method an t be used or plant transer untion without delay. There is also researh (Aström et al. 1998) where authors plae ilter in strutural sheme beore PID ontroller. This way dereases transient rate o ontrol system. In the previous researh (Mikhalevih et al. 2015) authors developed tuneable method or PID ontrollers to ahieve the desired phase margin. Crossover requeny is used as a parameter that is allowed to orret time response o losed-loop system. The aim o researh is to ind proportional integral derivative and ilter oeiients o PID ontroller that will provide desired harateristis (phase margin and rossover requeny) in the system. Please ite this artile as: Kozin K. A. Egorova A. M. Mikhalevih S. S. Manenti F. Rossi F Development o the requeny pid tuning method with ilter Chemial Engineering Transations DOI: /CET

2 140 The paper onsists o introdution theoretial researh omparison study o proposed method with known desription o line o entriugal extrators and development o ontrol system. The riterions o desired ontrol system that are based on previous researh and Aström reommendations (Aström and Hägglund 2006) are shown in the theoretial part. Then the analytial approah o proposed PID tuning method is presented. Ater omparison proposed method with onventional was done. Pratial approah ontains developing the ontrol system o extration proess (Goryunov and Mikhaylov 2012). Blok o entriugal extrators as a plant is used. This blok ontains 12 entriugal extrators whih are onneted in serial line and used or extration uranium and plutonium. This equipment is a part o Russian Federation large projet losed-loop nulear uel yle. The plant was linearized and digital ontrol system was developed. 2. Theoretial researh Let us onsider the ontrol system that is presented on Figure 1. The system onsists o PID ontroller and plant. Initial inormation or PID tuning is transer untion o plant phase margin φm rossover requeny ω. Figure 1: Blok diagram o ontrol system where r is a set-point y is output d is a disturbane Transer untion o plant ould be presented as P s e N% s N% s L s N s D% s D% s M s D s (1) where L(s) and M(s) are numerator and denominator o lagging that is expanded into the Padé series. The ontroller assumes the ollowing orm: ki ks d C s k 1. p s T s (2) By inserting s=jω into Eq.(1) plant transer untion is stated as ollows: N s P j Re j Im Ds s j (3) where Re Im are real and imaginary parts o a plant. The same substitution to ontroller transer untion is ki kd j C( j) ( kp ). j T j1 (4) Aording to riterion desribed in the artile (Mikhalevih et al. 2015) and phase margin deinition we ould ormulate the system o equations: Re G( j ) os( ) Im G( j ) sin( ) d Re G( j ) 0. d m m where G( j ) P( j ) C( j ) is a transer untion o open-loop system. The system Eq.(5) allows to ind proportional integral and derivative parts o PID ontroller. Solutions o the system Eq.(5) are untions o phase margin rossover requeny and ilter onstant T. In ommon ase transer untion o open-loop system is (ReT Im ) k (ImT Re) k Re k (Re Im T ) k (ReT Im) k Im k G. p i d p i d j ( T j) ( T j) (5) (6)

3 141 So analytial solutions o the system o equations Eq(5) are T T a b T a k p ( 1)( 4os( )(Re Im)(Im Re ) ( )(Im Re ) 2(Re Im) 4(ReT Re)(ReTIm)(Im Re ) (7) ( T 1) a 2b ki 2(Im Re )(ReT Im) (8) 2 ( T 1) a kd 2(Im Re )(ReT Im) (9) where a ' ' 2 ((ReRe ImIm Im )os( ) (ReIm ' ReIm Re ' Im )sin( )) b (ReT Im)(Imos( ) Resin( )) dre dim Re Im d d m. In the book (Aström and Hägglund 2006) there is inormation i IE is minimal ki will be maximum. Aording to this suppose that ki has a maximum in stability area. One o the properties o an extrema is zero value o the irst derivation i.e. dk i 0. d (10) Eq.(10) ould be used or inding ilter onstant. Authors got an analytial solution o the Eq(10) with MATLAB sotware. Solution is not written in the artile beause it is too long. But rom Eq(8) and Eq(10) ould be seen that the maximum degree o Eq(10) by T is the third. It means that there are two types o solution: 2 omplex roots and 1 real root or 3 real roots. Complex roots are improper beause ilter onstant should be real. Additional requirement to T is bigger than zero. I all three root ulil onditions it have to be heked and most proper solution must be hosen. 3. Comparison study Let us onsider the ollowing ontrolled plants (see Table 1). Comparisons provided with onventional PID tuning methods whih are widespread or industrial purpose: AMIGO (Aström and Hägglund 2006) Ziegler- Nihols method (Ziegler and Nihols 1942). Filter onstant was hosen aording to reommendation in book (Visioli 2006): kd T (11) N where N ranges rom 2 to 20. Let us assume that N=10 or AMIGO and Ziegler-Nihols method. Table 1. PID ontroller parameters and IAE omparisons Plant PID tuning method kp ki kd T IAE 1 Proposed method s e ( 15s 1) AMIGO Ziegler-Nihols method s 2s1 Proposed method s e AMIGO s 4s 5s 1 Ziegler-Nihols method For the FOPTD example phase margin equals to 69 and rossover requeny equals 0.1 rad/s. For this data ilter onstant has one real positive root and two omplex roots (are ignored). Controller was speially tuned to provide smaller overshoot. Step and disturbane responses are shown on the Figure 2. The omparison

4 142 shows that the ontroller tuned by means o the proposed method is haraterized by a smaller overshoot at the step response. Unortunately disturbane attenuation has bigger overshoot and muh bigger settling time. Figure 2: Step and disturbane responses or the FOPTD plant For the seond example phase margin equals to 80 and rossover requeny equals 0.15 rad/s. For this data ilter onstant have three real roots but two o them are negative so we ould ignore it. Controller was also speially tuned to provide smaller overshoot. Step and disturbane responses are shown on the Figure 3. Figure 3: Step and disturbane responses or the third order plant Figure 4: Nyquist urves or the both examples (or the irst example at the let side the seond at the right side) The omparison shows that the ontroller tuned by means o the proposed method is haraterized by a smaller overshoot at the step response. The Nyquist urves o proposed AMIGO and Ziegler-Nihols methods or both examples are presented on Figure 4.

5 Obviously PID ontroller in both examples ould be tuned to another phase margin and rossover requeny whih will provide smaller settling time or other quality indiators. 4. Development o ontrol system or the blok o entriugal extrators Centriugal extrators are widely used in hemial and atom industries (Vedantam et al. 2012). Chain o extrators allows providing high extration level o the target omponents. Extrator desription is presented in details (Vedantam et al. 2012). The developing ontrol system is a small part o global Russian projet o nulear reproessing. Extratant is tributyl phosphate kerosene mixture. The mathematial model o extration olumn desribed in the artile (Goryunov and Mikhaylov 2012) ould be easily adapted or modelling o entriugal extrator by hanging the ell o olumn to annular mixing zone o entriugal extrator. Separator ould be approximate by an ideal mixing model. The model ontains 12 entriugal extrators that are onneted by baklow mode. Target omponent is uranium. Control provided by valve position. Sensor is uranium onentration meter. Sample time o sensor is equal to hal minute. The ontrol system is presented in Figure 5. Disturbane is hanging uranium onentration in inlet low. Inlet low 143 setpoint _ ontroller valve plant output sensor Figure 5: Struture o extrators ontrol system Plant identiiation (with valve and sensor) result near the working point (40-60 % o sensor sale) is transer untion: Gs ( ) s 1. In pratie ontrollers have digital orm so the disrete transer untion is kt kd z1 i s C(z) kp z 1 T z 1 T s where Ts is the sampling time. Beause o sample time o sensor is 30 s Ts is also equal 30 s. The ontrol system has to provide minimum overshoot or step response i.e. the system should be onservative. For this reason rossover requeny should be small. We propose to hoose ω = 0.01 rad/s. Phase margin is 80. PID ontroller parameters are kp= ki= kd = -1.2 T = Modelling results are presented in Figure 6. Conentration is shown in perentages o sensor sale. (12) (13) Figure 6: Step response and disturbane attenuation o the developed ontrol system or blok entriugal extrators

6 144 As seen o Figure 7 developed PID ontroller provides small overshoot or step response and disturbane o hanging uranium onentration in inlet low. In addition developed ontrol system allows to ontrol straight ater start-up despite o the plant in start-up mode is quasi-linear. 5. Conlusions The proposed method allows to ind PID ontroller with irst order ilter or derivative part and provides desired phase margin. Crossover requeny is used in the method as a tuning parameter. The artile presents lear mathematial desription o proposed method. Advantages are possibility to provide PID tuning or high order plants with pure delay provides desired time domain harateristis by vary o rossover requeny. Control system or blok o entriugal extrators is developed: presents plant identiiation and PID ontroller was tuned. Aording to modeling results developed system provides small overshoot and aeptable disturbane attenuation. Reerene Moradi H. Vossoughi G Robust ontrol o the variable speed wind turbines in the presene o unertainties: A omparison between H and PID ontrollers Energy Kristiansson B. Lennartson B Evaluation and simple tuning o PID ontrollers with high requeny robustness Journal o Proess Control 16(2) DOI: /j.jproont Sanhis R. Romero J.A. Balaguer P Tuning o PID ontrollers based on simpliied single parameter optimisation International Journal o Control 83(9) DOI: / Vadutov O.S Design o PID ontroller or delayed systems using optimization tehnique under pole assignment onstraints Bulletin o the Tomsk Polytehni University 325(5) (in Russian). Aström K.J. Panagopoulos H. Hägglund T Design o PI Controllers based on Non-Convex Optimization Automatia 34(5) DOI: /S (98) Mikhalevih S.S. Baydali S.A. Manenti F Development o a tunable method or PID ontrollers to ahieve the desired phase margin Journal o Proess Control Mikhalevih S.S. Rossi F. Manenti F. Baydali S.A Robust PI/PID ontroller design or the reliable ontrol o plug low reator Chemial Engineering Transations Aström K.J. Hägglund T Advaned PID Control in Researh Triangle Park NC ISA - Instrumentation Systems and Automation Soiety North Carolina USA.. Goryunov A.G. Mikhaylov V.S The automati ontrol system o a multi-omponent nonequilibrium extration proess in the pulse olumn Journal o Proess Control 22(6) DOI: /j.jproont Ziegler J.G. Nihols N.B Optimum Settings or Automati Controllers Transations o the A.S.M.E Vedantam S. Wardle K.E. Tamhane T.V. Ranade V.V. Joshi J.B CFD Simulation o Annular Centriugal Extrators International Journal o Chemial Engineering DOI: /2012/759397