Revista INGENIERÍA UC ISSN: Universidad de Carabobo Venezuela

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1 Revisa INGENIERÍA UC ISSN: Universidad de Carabobo Venezuela Cornieles, Erneso; Saad, Maarouf; Areaga, Francisco; Obediene, Luis Sliding mode conrol for mulivariable processes Revisa INGENIERÍA UC, vol., núm., abril, 24, pp Universidad de Carabobo Valencia, Venezuela Available in: hp:// How o cie Complee issue More informaion abou his aricle Journal's homepage in redalyc.org Scienific Informaion Sysem Nework of Scienific Journals from Lain America, he Caribbean, Spain and Porugal Non-profi academic projec, developed under he open access iniiaive

2 . INTRODUCTION The sliding mode conrol SMC is a simple procedure o synhesize conrollers for linear and nonlinear processes. The design of a SMC depends on he process model, and paricularly, he number of uning parameers relaed o he model order []. Since mos processes modeled using firs principles end o be of higher order and complexiy, he radiional sliding mode conrol procedures presen disadvanages in heir applicaion (he sliding surface divides he phase plane ino regions where he swiching funcion has differen signs). Even hough sliding mode conrol has been widely invesigaed for a variey of sysem ypes, few papers has presened a general approach for process conrol. An efficien 62 Rev. INGENIERÍA UC. Vol., N o, Abril 24 REVISTA INGENIERÍA UC. Vol., N o, 62-68, 24 Conrol en modo deslizane para procesos mulivariable Erneso Cornieles (), Maarouf Saad (2), Francisco Areaga (3), Luis Obediene (3) () Escuela de Ingeniería Elécrica, Faculad de Ingeniería, Universidad del Zulia, Maracaibo, Venezuela (2) Groupe de Recherche en Élecronique de Puissance e Commande Indusrielle, Déparemen de Génie Élecrique, Monréal, Canada (3) Unidad de Invesigación en Auomaización Indusrial, Escuela de Ingeniería Elécrica, Faculad de Ingeniería, Universidad de Carabobo, Valencia, Venezuela ecornieles@luz.edu.ve, msaad@ele.esml.ca, lobedien@uc.edu.ve Resumen Ese arículo presena el enfoque de conrol por modo deslizane, el cual puede ser usado para una gran variedad de procesos mulivariables. El proceso esudiado en ese rabajo se modela como un sisema de primer orden con zona muera. El conrol diseñado se evalúa ano en el conrol de emperaura como en el de nivel de un sisema de anque de agua. El proceso consise de un anque con dos servo válvulas y dos sensores para la medición de nivel y de emperaura en el sisema de nivel de líquido. El conrolador es una implemenación en compuador del conrol por modos deslizanes. La implemenación del conrolador y el sisema insrumenado ha mosrado mejor funcionamieno, para las diferenes condiciones experimenales (perurbaciones, y punos de ajuse), en comparación con el conrolador PI. Palabras Claves: Conrol por modo deslizane, sisema mulivariable, zona muera. Sliding mode conrol for mulivariable processes Absrac This paper inroduces he approach of sliding mode conrol SMC which can be used for a large class of mulivariable processes. The process sudied in his work is modeled as a firs order sysem wih a dead zone. The designed conroller is evaluaed in boh, he level and emperaure conrol of a waer ank sysem. The process consiss of a ank wih wo servo valves and wo sensors for he level and emperaure measuremen in he liquid level sysem. The conroller is a compuer implemenaion of SMC. The conroller implemenaion and he insrumened sysem has shown beer performance, for he differen experimenal condiions (i.e., disurbance and se poins), compared wih a PI conroller. Keywords: Sliding mode conrol, mulivariable sysem, dead zone. alernaive could be he use of empirical modeling mehods. Empirical mehods use low order linear models. Mos of he ime firs-order-plus dead ime models are adequae for process conrol analysis and design. In spie of reduced order, hese models presen uncerainies arising from imperfec knowledge of he sysem, and he process nonlinear effecs conribue o performance degradaion of conrollers, a sliding model conrol can be designed wih he assumpion ha he robusness of he conroller will compensae for modeling error arising from he linearizaion of he nonlinear process model. The purpose of he presen approach is o design a general sliding model conrol using reduced order models [2]. The obained conroller resuls have

3 Cornieles, Saad, Areaga y Obediene been esed and verified for a mulivariable sysem, wih level and emperaure as he process variables. 2. CONCEPTS ABOUT SLIDING MODE The SMC is a kind of Variable Srucure Conrol ha can modify is srucure. The design problem consiss of selecing he parameers of each srucure. The firs sep in SMC is o define he surface S() =, along which he process can slide o is desired final value. The sliding surface divides he phase plane ino regions where he swiching funcion S() has differen signs. The conroller srucure is inenionally alered as is sae crosses he surface in according o a prescribed conrol law. There are many opions for choosing he sliding surface. The surface S() seleced here is he inegral conrol presened by Sloine and Li [3], expressed by an inegral-differenial equaion acing on he racking-error expression: n d S () = + λ ed () () d where e() is he racking error beween he reference value or se poin, Ref(), and he oupu measuremen, Pv(), or e() = Ref() Pv(), n is he sysem order and λ is a uning parameer, which helps o define S(). This erm is designed in order o deermine he sysem performance on he sliding surface. The conrol objecive is o ensure ha he conrolled variable remains equal o is reference value over all he ime inerval. In oher words, he error e() and is derivaives mus be zero. Once he reference value is reached, Equaion () indicaes ha S() reaches a consan value. In order o mainain S() a his consan value (e() is zero a all imes), i is desired o ensure ha: ds() = (2) d Once he sliding surface has been seleced, he design of he conrol law ha akes he conrolled variable o is reference value and saisfies Equaion (2), he sliding conrol law, U(), consiss of wo addiive pars: a coninuous par, Uc(), and a disconinuous par, Ud() (), ha is: () () U () = Uc + Ud (3) The coninuous par is given by: where f [Pv(), Ref()] is a funcion of he oupu variable and he reference value. The disconinuous par, Ud(), generally incorporaes a nonlinear elemen ha includes he swiching elemen of he conrol law. In his case, a coninuous approximaion of he signum funcion was included o avoid he chaering problem. The sigmoid-like funcion given in (5) is used: Ud S () () = Kd S () + δ where Kd is he uning parameer responsible for he reaching mode, δ is a uning parameer used o reduce he chaering problem. 3. SMC FROM THE PROCESS MODEL This secion gives a summary of he general Sliding Mode Conrol explained in previous works [,3]. The developmen of his conroller significanly simplifies he applicaion of sliding mode approach o process conrol. The firs sep is o derive a way o handle he dead ime erm. I can be approximaed in wo differen ways: a firs-order Taylor series approximaed around, Equaion (7), or a firs-order Padé approximaion. Boh approximaions have shown o be good for chemical applicaion of he general form of Equaion (6). In he following equaions, he sliding model conrol synhesis using he Taylor approximaion approach is shown: s Ke Gs ( ) = (6) τ s + s e = (7) s e s+ Subsiuing Equaion (7) ino Equaion (6) yields Gs () = ( ) ( ) Uc() = f Pv, Ref (4) K ( τ s+ )( s+ ) In he ime domain, and wih zero iniial condiions, expression (8) becomes: (5) (8) Rev. INGENIERÍA UC. Vol., N o, Abril 24 63

4 2 d Pv() dpv() τ + ( + τ) + Pv( ) = KUc( ) (9) 2 d d And since his is a second-order differenial equaion (n = 2), he sliding mode surface S() is given as follows: de() S () = + λe () + λ ed () d () 2 where, λ = λ and λ = 2λ can be independen, resriced only by he sable characerisic of he sliding surface. From Equaion (2) and subsiuing he definiion of he error e(), ino he firs wo erms of he above equaion we obain: ds d R d Pv dr dpv d d d d d 2 2 () () () () () = + λ Solving for he highes derivaive from Equaion (9), subsiuing i ino he Equaion (), and solving for Uc() provides he coninuous par of he conroller (equivalen conrol procedure [4]): + τ dpv() Pv() λ + τ τ d τ Uc() = K 2 dr () dr () + λ e () + λ 2 + d d The derivaives of he reference value can be discarded wihou any effec on he conrol performance, as far as a regulaion is concerned, resuling in a simpler conroller. Thus, + τ dpv() λ τ τ d Uc() = K Pv() + + λe () τ λ Since is a designed parameer, one can fix is value in order o simplify i as follows: λ + τ 64 Rev. INGENIERÍA UC. Vol., N o, Abril 24 λe () () (2) (3) = (4) τ Sliding mode conrol for mulivariable processes This shows ha his choice for is he bes for he coninuous par of he conroller [2]. To assure ha he sliding surface behaves as a criical or over damped sysem, should be []: Wih: λ + τ λ = (5) 4 τ hen, he complee SMC can be represened as follows: τ Pv() S() Uc() = + λ e() + Kd (6) K τ S() + δ dpv () S () = signk ( ) + λe () + λ ed () d (7) Equaions (6) and (7) consiue he conroller equaions o be used, presening advanages from processes conrol poin of view. Firs, hey have a fixed srucure depending on he λ's parameers and he characerisic parameers of he model, and second, he acion of he conroller is considered in he sliding surface by including he erm sign(k) in Equaion (7). Noe ha sign(k) depends only on he saic gain of he process model, herefore i never swiches. To complee he SMC i is necessary o have a se of uning equaions. The parameers of he conroller s coninuous par and he sliding surface are Equaion (4) and Equaion (5). For he parameers of he conroller s disconinuous par, an opimizaion procedure o minimize he ISE performance index based on he Nelder-Mead and Camacho-Rojas searching algorihm [] is performed. The following uning relaions are obained:.5 τ Kd = K.76 The parameer values from Equaion (8) and Equaion (9) are considered as iniial esimaes [, 4, 5]. Equaions (8) and Eq. (9) are used when he signals from he ransmier and conroller are in fracions ( ). Someimes he conrol sysems work in percenages, ha is, he signals are given in percenages range ( o ). In hese cases he values of Kd and δ are muliplied by. λ (8) δ = ( KKdλ ) (9)

5 Cornieles, Saad, Areaga y Obediene Level: Pv Ref α SMC + α + Gp(s) β Gp2(s) + α Gp2(s) Ref SMC2 The objecive of his work is o exend he scope of he designed sliding model conrol for SISO sysems o he mulivariable case. Process conrol is no an easy ask due o he nonlinear behavior, he ineracion among variables, and he presence of dead-ime. The ineracion arises from a close relaionship among manipulaed and conrolled variable (a manipulaed variable can considerably affec oher conrolled variables). The ineracion among he variables produces degradaion of he conrol sysem, since he effecs of a conroller on a loop considerably affecs he oher conrolled variables. The proposed approach is based on muliple SMC acing besides a decoupling sysem. Figure shows he proposed conrol scheme for a 2 inpus-oupus sysem. β 2 + characerisic using a 2x2 and he ineracion in he conroller uning equaions. The following equaions shows he developmen wihou loosing generaliy. Considering he equaion for he wo conrolled variables: Pv(s) where, and = Gp(s)M(s) (2) Pv ( s) Pv(s) =, Pv 2( s) Gp(s) Gp22(s) Decoupler Temperaure: Pv 2 Figure. Mulivariable conrol scheme. 4. MULTIVARIABLE SYSTEM M(s) Gp () s Gp () s 2 = Gp 2( s) Gp 22( s) M() s = M() 2 s In sysems wih srong ineracion, a common pracice echnique is o use decoupling ineracion loops [6, 7]. The basic characerisic of his echnique is ha he behavior of each loop will be independen of he oher loop. The purpose of he decoupling is o cancel he effec of he ineracion in such a way ha each conrolled variable is no affeced by changes in he manipulaed variables of he oher conrol loops. Once he sysem is decoupled, i is possible o use a Sliding Mode Conroller for a 2x2 sysem. The original Sliding Mode Conroller has a se of uning equaions for SISO sysems. The idea in his par is o consider he effec of he mulivariable Where Gp ij is dynamic funcion of he sysem and Mj() s is he oupu of he Sliding Mode Conroller (SMC and SMC2), he seady-sae gain relaing Pvi () s and M () j s is represened by: Pvi () s = K M j () s (2) Replacing he process ransfer funcions by heir respecive seady-sae gains, and making zero he second conrolled variable, he following equaions are obained [8] : P ( s) = ( K ) M ( s) + ( K ) M ( s) (22) v ij 2 2 Rev. INGENIERÍA UC. Vol., N o, Abril 24 65

6 Sliding mode conrol for mulivariable processes = ( K ) M ( s) + ( K ) M ( s) (23) M () s Then, he saic relaionship beween Pv( s) and can be obained by subsiuing M 2() s derived from Equaion (23) ino Equaion (22) and solving for Pv ()/ s M () s as follows: Pv ( s) K K = K (24) M s K 2 2 ( ) 22 Replacing Equaion (24) and Equaion (2) ino he ineracion index equaion [8], µ ij, he following general expression is obained: Pvm i Mm j K µ ij = = (25) Pvc K2K 2 i K Mc K 22 j The expression ( Pvm i/ Pvm j) denoes he parial derivaive ha is evaluaed wih all of manipulaed variables excep ha M j is held consan. Thus, his erm is he open-loop gain beween C and M i j. Similarly, ( Pvci / Pvc j) is evaluaed wih all he conrolled variables excep P v i held consan. Thus, his erm can be inerpreed as a closed-loop gain ha indicaes he effec of M j on Pv i when all of he oher feedback conrol loop are closed [7, 8]. So, he saic gain including he ineracion index is given by: Pv() s K M () s = µ (26) In general, he saic gain relaing he j-manipulaed wih he j-conrolled variables can be wrien as: K ' K = (27) µ In conclusion, he se of iniial uning parameers for mulivariable processes will be given by: λ + τ j = (28) τ 2 λ j λ j = (29) τ K D j = µ K j D j The parameers (, τ, K ) needed o complee he iniial uning of he conroller, are obained from he open-loop sep responses (for he j j manipulaed conrolled variable pairs afer decoupling) []. Compared o he original SMC uning values, only he K D uning parameer has changed o include he ineracion index for he necessary aggressiviy o reach he sliding surface for he mulivariable case. 5. EXPERIMENTAL VALIDATION (3) δ = ( K K λ ) (3) The slide mode conrol has been experimenally verified using he level and emperaure conrol. The process shown in Figure 2 consiss of a ank wih wo servo valves and wo sensors for he measuremen of he level and emperaure of he waer in he ank. The conroller is a compuer implemenaion of he sliding mode conroller. The sliding mode conrol is divided ino wo pars, he coninuous and disconinuous pars ha are algebraic funcions of he process variables and he se poin, and of S(). S() is calculaed from he sliding mode conroller [3,4]. To achieve he implemenaion, he overall algorihm mus have hree seps. Each ieraion, he sliding surface value S() is calculaed from he SMC. The uning parameers can be based on he sliding equaion [2] ( Level: K=.25, K2 =, K3 =, Temperaure: K =.27, K2 =, K3 = 35 ). Since he coninuous and disconinuous pars of he conroller are algebraic equaions, he LabView is used o compue heir values. This procedure is based on he PID algorihm ha exiss in mos of he indusrial conrollers. Figure 3 shows a ypical sep response of he level and emperaure conrol wih he sliding mode conroller. Pv is he level and Pv2 is he emperaure. U is he cold waer conrol and U2 is he ho waer conrol. The REF and he REF2 are he se poins for he level and emperaure. Figure 4 shows a ypical sep response of he level and emperaure conrol wih a PI conroller. I can be seen ha he sliding mode approach produces beer ime 66 Rev. INGENIERÍA UC. Vol., N o, Abril 24

7 Cornieles, Saad, Areaga y Obediene A A Valve 2 Valve Inerface 2 V/I Inerface V/I A m Level Sensor Temperaure Sensor A m2 Inerface 3 Impedance 3 W P v V Exi V Inerface 4 Impedance 3 W P v2 NATIONAL INST.CARD Inpu s Oupu s 68 (emp.) 22 (u ) IBM Slide Mode Conrol Figure 2. Experimenal es bench. 8. Value in Vols REF2 PV REF PV2-2. Number of poins 8 3. Value in Vols U2 U -3. Number of poins 8 Figure 3. A ypical sep response wih SMC. Rev. INGENIERÍA UC. Vol., N o, Abril 24 67

8 Sliding mode conrol for mulivariable processes REF2 PV2 Value in Vols REF PV.. Number of poins. Value in Vols 5.. U U Number of poins. Figure 4. A ypical sep response wih PI conroller. response. The conrol effor is also very small using he sliding mode approach. 6. CONCLUSION A sliding mode conrol approach has been presened for designing a conroller ha has fixed srucure and uning erms as a funcion of he characerisics parameers of he process. The sliding mode conrol can be implemened from a PID algorihm, where he expression of he PID represens he Sliding surface, and he res of he conroller is buil using algebraic block. The conroller presened in his paper for a mulivariable sysem has he simpliciy and he robusness characerisic of he Sliding Mode Conrollers. Briefly, he Sliding Mode Conrol designed for mulivariable sysems can be used for a wide class of nonlinear processes wih good resuls. The SMC designed for mulivariable sysems can be used for a wide class of nonlinear processes wih good resuls, and is robus for a process wih dead ime and disurbances. REFERENCES [2] Camacho, O. E. and Smih, C. A., 2, Sliding Mode Conrol; An Approach o Regulae Nonlinear Chemical Process, ISA Trans.39, pp [3] Sloine, J. J. and Li, W., 99, Applied Nonlinear Conrol, Prenice-Hall, New Jersey. [4] Ukin, V. I., 997, Variable Srucure Sysems Wih Sliding Modes, IEEE Trans. Auom. Conrol, AC-22, pp [5] Lee, C.K. and Kwok, N.M., 995, Chaering Reducion of a Digial Variable Srucure BLDC Moor Speed Conrol Sysem, pp , /95, IEEE, 995. [6] Cornieles, E. and Bougere, C.,997, Comparaison expérimenale de différenes echniques de réglage du régulaeur PID e PID Dual Loop, Rappor officiel R97. École Polyechnique de Monréal, Canada. [7] Seaborg, D. T. E. and Mellichamp, D., 989, Process Dynamics and Conrol, Wiley, New York. [8] Ogunnaike, B. and Ray, W. H. 994, Process Dynamics, Modeling and Conrol, Oxford Universiy. [] Camacho, O. E., Rojas, R.D., 2, A General Sliding Mode Conrol for Nonlinear Chemical Process ASME Trans. pp Rev. INGENIERÍA UC. Vol., N o, Abril 24