Modelling, Simulation and Control of Quadruple Tank Process

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1 Modelling, Simlation and Control of Qadrple Tan Process Seran Özan, Tolgay Kara and Mehmet rıcı,, Electrical and electronics Engineering Department, Gaziantep Uniersity, Gaziantep, Trey bstract Simple processes with only one otpt that may controlled by one inpt (ariable) are nown as single inpt single otpt process. Bt many processes are not sch simple. They hae more than one inpt (ariable) and one otpt, which are called Mlti Inpt-Mlti Otpt (MIMO) processes. Common MIMO systems hae some difficlties, sch that they are large and complex. In addition, they hae nonlinearities and also loop interactions which are between inpts and otpts. On the prpose of stdying mltiariable systems and designing controllers, the qadrple tan process (QTP) is chosen as a benchmar. This system is sitable for stdying linear and nonlinear controllers and exhibits minimm and non-minimm system behaiors by simply changing configration ale positions. We linearize nonlinear process model, than apply arios control methods and compare the controlled system performance reslts.. Introdction Mltiariable system inoles more than one control loop, These loops interact with each other, in sch a manner that single inpt not only affects its own otpt bt also affects other process otpts.[,] QTP has for tans and two pmps, this benchmar is simply a water leel control problem. The aim of the process is to eep the liqid leel in the lower tans at the desired ales. Related laboratory process introdced by Johansson in Figre.. [], the process is sed to show mltiariable interactions which are also nown as copling and it limits performance in mltiariable control systems. Mltiariable interactions in a QTP are each otpt (water leels) of the system has affected by two pmps. De to these reasons, it can be regarded as a prototype for many MIMO control applications in indstry sch as paper prodction processes, chemical processes, metallrgy and biotechnological areas, medical indstries. The strctre of paper is as follows. The mathematical modeling of QTP benchmar system is described in Section, Linearization detail is gien in Section. Centralized and Decentralized control design steps are gien in Section, Simlation reslts are depicted in Section 5 and the Conclsions of the article are in Section 6.. Mathematical Model The process inpts are pmps oltages and the process otpts are lower tans water leels in Figre.. For each tan the mathematical model is obtained by sing Bernolli s law yields and mass balance law. Tan nmbers are represented by i, which may be,,,. Figre.. Qadrple-Tan Process.. The Nonlinear Model The mathematical eqialent of the process is gien by Bernolli s law and mass balance eqation as follows: Rate of accmlation (Rate of in-flow)-(rate of ot-flow) d( ρv) ρq in - ρqot (sin ce ρ ρ ρ as same liqid) () i i q in - q ot icross sectional area of the tan hi the tan water leel qin_i in-flow of the tan qot_i ot-flow of the tan The inflow of the tan (qin_i) only depends on the inpt pmp oltage and ot-flow of the tan (qot_i) depends on the graity and acceleration de to head of the water in the tan. Based on Bernolli s eqation qot_i can be determined as follows q in V q i in V γ () qin V qin V ( γ ) where, are the pmp constants; γ, γ ale ratio of ale positions qot a i gh i i () ai, cross sectional area of the otlet pipes; g, acceleration de to graity

2 .. Linearized Model The eqations in [] hae sqare root terms which case to nonlinearity. Becase of that to design a controller becomes more difficlt. The eqation [5] is soled sing by Taylor series expansion at the operating points and Jacobian matrix transformation to get a state space form of QTP. R(t) System Y(t) Figre.. Single Tan Diagram Using the law of conseration of mass i i q in qot q ot γ V a gh a gh () The non-linear eqations of the QTP are gien as follow: gh a gh gh a gh gh ( γ ) -a gh (-γ ).. Relatie Gain rray If decentralized control strctre is chosen as a mlti inpt mlti otpt controller, an appropriate pairing of inpt and otpts is needed. In this case of an m x m plant transfer fnction, there is m! different pairings. By the way, physical interpretation of system gies idea abot which pairing is sefl or which one is not. Relatie Gain rray (RG) is a method that can be sed to sggest pairings throgh a nown qantity. RG is defined as a matrix Λ []: (5) Λ G(0)* G T (0) (6) If diagonal entries of Λ is negatie than controlling the system is particlarly difficlt. pairing with 0.67 < λ <.50 in main diagonal elements sally gies good performance []. The RG of the qadrple system is gien as: λ λ Λ λ λ, γγ λ γ (7) R(t): inpt as oltage Y(t): otpt as water leel dx f( h, h, h... hn,,,... n) dx n f n( h, h, h... h n,,,... n) (8) The general ector form x f ( x, ) (x represents states) Let He heδ h Ue eδ Using Taylor series to yield the linear approximation dx x f ( He, Ue) f ( heδ h, eδ) df df f ( x, ) f ( he, e) ( he, e) ( he, e) higher order terms d 0 For simplification the higher order terms are neglected. a gh a gh γv, ( V; V) 0 a g a g γ h h0 h h0 V V 0 h0 h0 Let x : h h i i io and : i i io ; so the system can be represented in state space form as follows: a g a g γ x x x ho ho and a g a g γ x x x ho ho a g ( γ ) x x ho a g ( γ) x x (9) ho

3 0 γ 0 0 T T 0 0 γ 0 dx T T x ( γ ) T ( γ) T c y x 0 c 0 0 where the time constants are i hio Ti ai g, i,..,, Δh Δ Δh Δy x y h Δ Δ Δy Δ h (0) Table. Operating points Parameters Minimm-Phase Nonminimm-Phase (ho, ho) cm (.6,.78) (.,.7) (ho, ho) cm (.6,.) (.7,.99) (, ) V (.00,.00) (.5,.5) (, ) cm /Vs (.,.5) (.,.9) (ɣ, ɣ) (0.70, 0.60) (0., 0.) By sing the operating point parameters, physical modeling gies minimm-phase and non-minimm-phase transfer matrices in Eqation and Eqation respectiely: s (.8 s)( 6.) G (s) G G..8 G G ( 0 s)( 90.6 s) 90.6s s (8.7)(6.8) s G(s)..6 (56.6)(9) s s 9 s () () Transfer fnction matrix of the linearized system can be written as: γ c ( γ) c T ( Ts)( st) G(s) ( γ) c γc ( st)( st) st γ c ( γ) c T ( Ts)( st) ( γ) c γc ( st)( st) st () where c T c / and c T c /. Here the ratio / is approximately eqal to. The parameters γ, γ ( 0, ) are determined from how the ales are set prior to an experiment []. De to ales process act minimm or nonminimm phase which is shown on Table.. Centralized and Decentralized control In this part, arios control methods are applied to the nonlinear system simlation to alidate modeling and obsere the system performance. Firstly, a state feedbac controller is designed for minimm-phase system. The system is controllable and obserable. The goal is to obtain a leel tracing controller for lower two tans. By Eqation, it s clear to see the plant is a Type 0 system which has no integrator. The basic idea to design Type sero system is adding an integrator in the feedforward path between the error comparator and the plant as shown in Figre.. Table. Vale Setting. Vale ales Process Zero Location minimm phase Zero is in left half < γ < plan nonminimm Zero is in right half 0 < γ < phase plan Zero is located at the γ origin Figre.. gmented error state feedbac controlled system bloc strctre fter mathematical calclations and deriations, linearized system state space model is obtained at minimm phase operating point as gien in Eqation. Table.Process parameters Parameter Vale Height of tans, hmax 0 cm Bottom area, Tan, Tan,, 8 cm Bottom area, Tan, Tan,, cm Ot pipe cross-sections, a, a 0.07 cm Ot pipe cross-sections, a, a Leel measrement deice constant, c V/cm Graity g 98 cm/s dx x y x The system and controller eqations are as follows [5]: ()

4 x x B ; y Cx ; Kx Ie ξ r y r Cx (5) where ξ is the otpt of the integrator, r is the reference control signal. By agmenting the states ξ with states x, we can get the integral action in the controller for better tracing. The agmented system eqations are as follows: where x e () t 0 x () t B e ξ e () t C 0 ξe ( t) 0 x e () t x() t x( ), ξe () t ξ() t ξ( ), e() t Kxe() t I ξ e() t e () t (6) to nonminmm phase model. We can also see that by looing at diagonal elements of RG matrix. From Table, ale positions ɣ and ɣ are 0.7 and 0.6 respectiely. RG matrix in this configration is:. 0. Λ (7) 0.. RG, in this case, sggests ( y, ) and ( y, ) pairings. PI controllers transfer fnction: G ( s) K ( ), i, (8) ci i Ts i Controller parameters are tned so that they gie acceptable performance sch as less than %0 and 50s settling time. The controller settings (K, T) (.0, 0) and (K, T) (.7, 0) gie the response shown in Figre.. 6 x Here new states become xt () [ x () t () t] R e ξe. Closed loop poles of system are placed: [ ] P ± i ± i Figre.. PI-controlled system bloc strctre The controlled system performance is obsered ia simlations. System simlation time is 50s; system initially starts with operating point parameters gien in Table.. fter 50s a step change of cm in the reference signal is applied for Tan leel. For Tan a constant reference is chosen to trac. The response of the system can be seen from Figre.. The leel of Tan is tracing the reference signal with zero steady state error and a settling time of approximately 60s. Tan leel deiates from its reference a little between 60s and 0s as a reslt of interaction of tans. Figre.. PI-controlled system simlation (, reference signal).. Decopler Figre.. State feedbac controlled system responses. Secondly, a decentralized PI controller is designed by sing system transfer matrix. Tan and Tan leels are the otpts to be controlled. Since we se minimm phase system configration, the interaction between tans are small in contrast MIMO problems can be conerted to SISO problems by seeral methods. One of these methods is non interacting or decopling control schemes. This ind of control aoids the effects of loop interactions totally. The decopler diides a MIMO process into a few independent single-loop sbsystems. [6]. Figre.5 shows the decopling control plot. ccording to ideal decopling procedre in [7] as gien in Eqation (.6). T T T s (9) T T

5 where the diagonal elements, T T (ideal decopler case) and off diagonal elements, G T nd G T. G G 0.57 T.7s, 0.5 T 0.07s (0) Comparison of PI controlled system and Decentralized system is shown on Figre.6 Figre.5. Bloc diagram of system with decopler the distrbance will hae more effect on the other tan leel. For sch a case, a dynamical decopler is designed and integrated to decentralized PI controller. Performance of controllers is compared ia simlations. cnowledgement This wor is spported by The Scientific and Technological Research Concil of Trey (TUBİTK), throgh project 6E References [] K. H. Johansson. Relay feedbac and mltiariable control. PhD thesis, Department of tomatic Control, Lnd Institte of Technology, Sweden, Noember 997 [] D.. Vijla, K. n, P. M. Honey, P. S. Poorna, Mathematical Modelling of Qadrple Tan System International Jornal of Emerging Technology and danced Engineering, ol., Isse, December 0. [] K. H. Johansson, The qadrple-tan process: a mltiariable laboratory process with an adjstable zero, IEEE Trans. Control Syst.Technol., ol.8, no., pp.56-65, May [] G. C., Goodwin, S.F., Grabe Control system design, Pearson [5] K., Ogata Modern control engineering, Pearson [6] P. Nordfel and T. Hagglnd, Decopler and PID controller design of TITO systems, Jornal of Process Control, ol.6, no.9, pp.9-96, 006 [7] W. L. Lyben, Distillation decopling, IChE Jornal, ol.6, no., pp.98-0, 970. [8] M. rıcı, T. Kara, daptie Falt Tolerant Control for Liqid Tan Process, International Jornal of pplied Mathematics, Electronics and Compters, ol., Special Isse, pp. -7, 06. [9] D. E. Seborg, D.. Mellichamp, Process Dynamics and Control, Wiley, 00. [0] M. Rezaie, B. Rahmani. Fzzy predictie control of threetan system based on a modeling framewor of hybrid systems, Jornal of Systems and Control Engineering, 8(6), 69-8, 0. Figre.6. Comparison of controller 6. Conclsions In this paper, a mltiple interacting copled tans system is chosen as a case stdy. The linear system model is obtained and the conditions which the system behaes as minimm or nonminimm phase are gien. Minimm phase system strctre is chosen for this stdy than a centralized agmented state feedbac controller is designed. For this controller, any frther performance improement is not considered. We only show that sch a controller design is possible. Since interactions between leels of tans are in acceptable range a decentralized control system design can be another option. In this case, interactions between tans may be considered as distrbance. Howeer, for cases sch as high changes in the reference signal for one tan,