Adaptive Control of Level in Water Tank: Simulation Study

Transcription

1 Adaptive Control of Level in Water Tank: Simulation Study Jiri Vojteek and Petr Dotal Abtract An adaptive control i a popular, o called modern control method which could be ued for variou type of ytem with negative propertie from the control point of view. Thi method i baed on the feature from the nature where plant, animal or even human being adopt it behavior to the actual environment. The goal of thi paper i to how how adaptive control could be ued for controlling of the level in the tank a a repreentation of the water dam, reervoir etc. Propoed control trategy wa teted on the real model of the cylindrical water tank a a part of the Proce Control Teaching ytem PCT40. The mathematical model for imulation wa derived with the ue of material balance inide and the reulting nonlinear ordinary differential equation i olved numerically with the ue of the mathematical oftware Matlab. Thi mathematical model wa alo verified by the meaurement on the real model. The adaptive approach here ue polynomial approach, recurive identification and pole-placement method with pectral factorization. The adaptive controller here could be tuned by the choice of the poition of the root inide the cloed loop. Keyword Modelling, Simulation, Water Tank, Adaptive Control, Mathematical Model, Pole-placement Method, Recurive Identification. I. INTRODUCTION HE modelling and imulation i of the firt tep before the Tchoice of the optimal working point and control trategy. Thee day, when computation power and peed of peronal or indutrial computer are very high and the price i low the role of the imulation grow. The ytem could be decribed either mathematically or practically [1], []. The mathematical decription for example ue material, heat etc. balance [3] depending on the type of the ytem, whether it i chemical reactor [4] and [5], heat exchanger, electric motor or other indutrial proce [6]. On the other hand, real model i uually mall repreentation of the originally nonlinear ytem and we expect that reult of experiment on thi model are alo valid or comparable to thoe on the real ytem. The big advantage of the mathematical modelling i in it afety experiment on ome real ytem could be ometime hazardou. Neverthele, J. Vojteek i with the Department of Proce Control, Toma Bata Univerity in Zlin, nam. TGM 5555, Zlin, Czech Republic (correponding author to provide phone: ; fax: ; vojteek@fai.utb.cz). P. Dotal i with the Department of Proce Control, Toma Bata Univerity in Zlin, nam. TGM 5555, Zlin, Czech Republic ( dotalp@fai.utb.cz). experiment on the real or abtract model are uually much cheaper than thoe on the original ytem which i ometime big and component are expenive. Thi goal of thi contribution i to decribe how the imulation could help u with the deigning of the controller for real model of the water. Thi real model i repreented here by the Armfield' Proce Control Teaching Sytem PCT40 [4] which ha everal proce control model and one of them i the water tank. The mathematical model of thi water tank ytem i mathematically decribed by the firt order nonlinear Ordinary Differential Equation (ODE) [1]. Thi mathematical model i then ubtracted to tatic and dynamic analye. The tatic analyi mean olving of thi ODE in the teadytate, i.e. the derivative with the repect to time are equal to zero [3]. The nonlinear ODE i then reduced to the nonlinear algebraic equation which can be olved for example with the ue of imple iteration method [8]. The reult of the tatic analyi could be optimal operating point or the range where the input variable could vary from the practical point of view. On the other hand, the dynamic analyi oberve the behavior of the ytem after the tep change of the input quality, in thi cae the change of the feed volumetric flow rate inide the water tank. The dynamic analyi mean mathematically the ue one of numerical method for olving of the ODE. The main group of numerical method are onetep method for example Euler method, Runge-Kutta method, or multi-tep method Predictor-Corrector etc. [9]. The advantage of thee method i that they are eaily programmable even more they are build-in function in the mathematical oftware like Matlab [10], Mathematica etc. [11]. The control of nonlinear procee could be a problem with conventional control method. Fortunately, there are new control method baed on adaption [1], prediction or chao theory [13]. There wa ued the adaptive control approach baed on the choice of the External Linear Model (ELM) of the originally nonlinear ytem [14], parameter of which are etimated recurively and the control deign employ polynomial approach with pole-placement method and pectral factorization. Thee method atify baic control requirement uch a tability, diturbance attenuation and reference ignal tracking. The ELM here ue o called delta-model [15] which are pecial type of dicrete-time model parameter of which are related to the ampling period which mean that thee ISSN:

2 parameter approache to the continuou one if the ampling period i adequately mall. The on-line identification wa realized by the Recurive Leat-Square (RLS) Method which i imple, eaily programmable method and if we combine thi method with ome kind of forgetting, for example exponential or directional, it provide atifactory reult. There exit everal mathematical program which can be ued for imulation like Matlab, Mathematica, Witne [16] etc. All method are verified by imulation in the mathematical oftware Matlab, verion h q in II. MODEL OF THE WATER TANK The equipment under the conideration i the real model of the water tank which i one part of the Multifunctional proce control teaching ytem PCT40 from Armfield [4] ee Fig. 1. The PCT40 include alo other model of procee uch a Continuou Stirred Tank Reactor (CSTR) or heat exchanger. h v q r r 1 Fig. Schematic repreentation of the water tank CSTR Water tank Heat exchanger PSV Fig. 1 Multifunctional proce control teaching ytem PCT40 Thi ytem combine both modelling technique it i mall repreentation of the water tank with the volume of 4- liter original of which i uually much bigger with huge volume. The mathematical model of thi ytem could be alo eaily derived. The chematic repreentation of the water tank can be found in Fig.. The model conit of platic tranparent cylinder with inner radiu r 1 = m. There i another platic tranparent cylinder inide due to quicker dynamic repone of the ytem lower uage of feeding water. The outer radiu of thi maller cylinder i r = m and the maximal water level in the tank i h max = 0.3 m. In the Fig., q denote the volumetric flow rate, h i ued for the water level and r are radiue of inner and outer cylinder. The input variable i the volumetric flow rate of the feeding water q in and tate variable are water level h in the tank and output volumetric flow rate of the water which come from the tank, q. The mathematical model come from the material balance inide the tank which i in the word form [3]: Flow Flow rate rate into into the the ytem Flow Flow rate rate Rate Rate of of = out out of of the the ytem + accumulation and mathematically: dv = q+ (1) dt where V i a volume of the water inide the tank and t i ued for the time. The volume of the tank i generally V = F h () for F a a area of the bae due to cylindrical hape of the tank. It mean, that balance (1) could be rewritten to the form dh = q+ F (3) dt where F i in thi cae F = π r1 π r = m (4) It i alo known, that volumetric flow rate through the water valve i nonlinear function of the water level, i.e. q = k h (5) where k i a valve contant which i pecific for each valve and depend on the geometry and type of the valve. If we put equation (5) inide (3) the reulting mathematical model i: dh k h = (6) dt F There hould be introduced one implification the height of the dicharging valve, h v in Fig., i neglected. ISSN:

3 The unknown contant k could be computed for example from the teady tate (variable with upercript ( ) ), where q in = q and equation (5) i q q = k h k = (7) h The water tank i fed via Proportioning Solenoid Valve (PSV) which could be operated in the range 0 100%. Thi range i from the practical point of view limited to the range m The equation (7) have till one unknown the contant of the outlet valve, k. Thi contant could be computed from the reference meaurement where we know the input volumetric flow rate, q in, and after ome time alo the teady-tate value of the water level, h. There were made everal meaurement and the reulted teady-tate value of the water level, h, together with the valve contant, k, for variou input volumetric flow rate, q in, are hown in Table I. TABLE I MEASURED STEADY-STATE VALUES OF WATER LEVEL AND COMPUTED VALUES OF VALVE CONSTANT FOR DIFFERENT VOLUMETRIC FLOW RATE Flow rate q in [m ] Steady-tate water level h [m] Valve contant k [m 5/. -1 ] The coure of the imulated tep repone of the water level for the tep change of the volumetric flow rate 60% (i.e. q in = m ) and 100% (i.e. q in = m ) are hown in Fig. 3 and Fig. 4 olid line meaured data imulated data 0 Time (t /) Fig. 3 Meaured and imulated data for q in = m computation without the valve meaured data imulated data Time (t /) Fig. 4 Meaured and imulated data for q in = m computation without the valve The contant of the valve, k, i then computed from equation (7) and the value of thi contant for thee tep change in Fig. 3 and Fig. 4 are: / 1 k = = = m h (8) / 1 k = = = m h 0.85 The value of thi contant and teadied water level, h, for the other meaurement are hown in Table I. The imulation i very often connected to the verification part becaue it i good to know if the derived mathematical model i accurate enough. We have done the imulation of the dynamic behavior of the propoed mathematical model (6) with computed value of the valve contant, k, and reult are hown in Fig. 3 and Fig. 4 a a dahed line. It i clear, that although imulated and meaured output reache the ame final value, the dynamic i much different the mathematical model ha quicker output repone. Thi tatement mean that the mathematical model (6) i not accurate and we mut neglect ome implification. In thi cae we did not take into the account the height of the valve, h v = m, diplayed in Fig. which ha alo impact to the mathematical model of the ytem. Inaccuracy of the mathematical model i alo indicated by very different value of the valve contant in the lat column of Table I. If we count with thi height, new contant of the valve for the ame tep change a in previou cae are / 1 k = = = m h ( ) (9) / 1 k = = = m h ( ) and value for the next volumetric flow rate are again hown in Table II. The comparion of the meaured and imulated data for thi new contant of the valve i hown in Fig. 5 and Fig. 6. ISSN:

4 meaured data imulated data 0 Time (t /) Fig. 5 Meaured and imulated data for q in = m computation with the valve Both graph in Fig. 5 and Fig. 6 how uitability of the new mathematical model which i alo indicated by very imilar value of the new valve contant, k n, in the lat column of the Table II. The new mathematical model which will be ued for later imulation i then: dh kn h+ hv = (10) dt F With the valve contant k n = m 5/. -1 a a mean value of the lat column in Table II meaured data imulated data Time (t /) Fig. 6 Meaured and imulated data for q in = m computation with the valve TABLE II MEASURED STEADY-STATE VALUES OF WATER LEVEL AND COMPUTED VALUES OF VALVE CONSTANT FOR DIFFERENT VOLUMETRIC FLOW RATE RECOMPUTED VALUES OF VALVE CONSTANT Flow rate q in [m ] Steady-tate water level h [m] New valve contant k n [m 5/. -1 ] A. Steady-tate Analyi. The teady-tate analyi mean that we olve the mathematical model with the condition d( )/dt = 0, i.e. ODE (6) i tranferred to the nonlinear algebraic equation: h ( q ) in = (11) k where the optional variable i the input volumetric flow rate, q in. There were done imulation analyi for the range q in = <0; > m and reult are hown in the Fig h max h min imulated data meaured data q in,min q in,max -0.1 x Input volumetric flow rate (q in /m ) Fig. 7 The teady-tate analyi of the mathematical model Thi analyi how nonlinear behavior of the ytem and alo we can chooe the volumetric flow rate in the range q in = < ; > m becaue lower value of q in mean that we did not get enough water in the tank and vice vera the flow rate bigger than q in = m reult in bigger water level than it maximal value h max. Dot in the Fig. 7 diplay reult of meaured teady-tate value. B. Dynamic Analyi. The dynamic analyi olve the ODE with the ue of ome numerical method. In thi cae, the Runge-Kutta tandard method wa ued becaue it i eaily programmable and even more it i build-in function in ued mathematical oftware Matlab. The working point wa characterized by the input volumetric flow rate = m which i in the middle of the operating interval defined after the tatic analyi in the Fig. 7. The input variable, u(t), i the change of the initial q in in % and the output variable i the water level in the tank. The input and the output variable are then generally: ( t) u() t = 100 [%]; y () t = h()[ t m] (1) The imulation time wa 3000, ix tep change of the input variable u(t) were done and reult are hown in Fig. 8. ISSN:

5 Water level (h(t)/m) h max h min 40% 0% 10% -10% -5% -50% Time (t/) Fig. 8 The dynamic analyi for variou tep change of the input volumetric flow rate q in Output repone how that thi output ha aymmetric repone the final value i different in ign and alo in order for poitive and negative tep change. Even more, for it i inappropriate to chooe the input tep change of the volumetric flow rate lower than approximately -40% and bigger than +30% becaue the reulted water level i lower or higher than phyical propertie of the water tank. III. ADAPTIVE CONTROL The adaptive control wa ued for control of thi ytem. There are everal adaptive approache which can be ued. The method here ue External Linear Model (ELM) a a linear decription of the originally nonlinear ytem. Parameter of and the tructure of the controller are derived from thi ELM and it parameter are identified recurively during the control. Parameter of the controller are recomputed in each time period too which mean that thi controller adopt it parameter according to the actual tate and behavior of the controlled ytem. In thi cae, all output repone in Fig. 8 could be expreed by the firt or the econd order tranfer function (TF), for example in the continuou-time b( ) b0 G1 ( ) = = a( ) + a0 (13) b( ) b 1 + b0 G ( ) = = a + a+ a ( ) 1 0 A. Control Sytem Synthei The controller i contructed with the ue of polynomial ynthei and the control tructure with 1DOF i hown in Fig. 9. w - e Q v u G Fig. 9 1DOF control configuration The block G denote tranfer function (13) of controlled y plant, w i the reference ignal (wanted value), v i diturbance, e i ued for control error, u i control variable and y i a controlled output. The tranfer function of the feedforward part Q() of the controller i deigned with the ue of polynomial ynthei: q( ) Q ( ) = (14) p ( ) where degree of polynomial p () and q() are computed from: deg q( ) = deg a( ) + deg f ( ) 1 (15) deg p ( ) deg a( ) 1 and parameter of thee polynomial are computed by the Method of uncertain coefficient which compare coefficient of individual -power from the Diophantine equation, e.g. [16]: a( ) p ( ) + b( ) q( ) = d( ) (16) and the polynomial d() on the right ide of (16) i an optional table polynomial. It i obviou, that the degree of thi polynomial i: deg d( ) = deg a( ) + deg p ( ) + 1 (17) Root of thi polynomial are called pole of the cloed-loop and their poition affect quality of the control. Thi polynomial could be deigned for example with the ue of Pole-placement method. A choice of root need ome a priory information about the ytem behavior. It i good to connect pole with the parameter of the ytem via pectral factorization. The polynomial d() can be then rewritten to the form deg deg ( ) ( ) ( ) d d = n + α n i (18) where > 0 i an optional coefficient reflecting cloedloop pole and table polynomial n() i obtained from the pectral factorization of the polynomial a() * * n ( ) n( ) = a ( ) a( ) (19) The Diophantine equation (16), a it i, i valid for tep change of the reference and diturbance ignal which mean that deg f() = 1 in (15). The feedback controller Q() enure tability, load diturbance attenuation and aymptotic tracking of the reference ignal. B. External Linear Model (ELM) The ELM here come from the dynamic analyi a it i written above. The TF in (13) belong to the cla of continuou-time (CT) model. The identification of uch procee i not very eay. One way, how we can overcome thi problem i the ue of o called model. Thi model belong to the cla of dicrete model but it parameter are cloe to the continuou one for very mall ampling period a it proofed in [18]. The model introduce a new complex variable γ, for example z 1 γ = (0) T v ISSN:

6 If we chooe for implification firt order TF G 1 in (13), the differential equation will be y ( k) = b u ( k 1) a y ( k 1) (1) 0 0 where b 0 and a 0 are delta parameter different from the parameter b 0 and a 0 in (13) and the individual part in Equation (1) can be written a y( k) y( k 1 ) y ( k) = ; Tv u ( k 1) = u( k 1) () y ( k 1) = y( k 1 ); The regreion vector ϕ i then ϕ ( k 1) = [ y ( k 1), u ( k 1) ] T (3) and the vector of parameter θ i generally ( k) a1, a0, b1, b0 T θ = (4) which i computed from the differential equation T y k = θ k ϕ k 1 + e k (5) ( ) ( ) ( ) ( ) where e(k) i a general random immeaurable component. A it i written in the previou part, control ytem ynthei i done in continuou time, but recurive identification ue dicrete time tep. The reulted, o called hybrid, controller work in the continuou time but parameter of the polynomial a() and b() are identified recurively in the ampling period T v. Thi aumption reult in the condition, that the parameter of the -model are cloe the continuou one for the mall ampling period. IV. SIMULATION RESULTS Simulation analye do the control exercie on the mathematical model of the water tank (6) where the reference ignal (wanted value) i the level of the water in the tank, h, which i controlled by the change of the input volumetric flow rate q in. The ampling period wa T v = 1, the imulation time wa 5000 and 5 different tep change of the reference ignal wa done during thi time. The controller could be tuned with the choice of the parameter. The affect of thi parameter are hown in following figure Water level (w,y/m) w(t) 0.05 y(t), = y(t), = 5 y(t), = Time (t/) Fig. 10 The coure of the reference ignal, w(t), and the output variable, y(t), for different value of The Fig. 10 clearly how the effect of the tuning parameter increaing value of thi parameter reult in quicker output repone with the overhoot. The output repone for the lowet value, i.e. =, produce more moother coure of the output variable without the overhoot at the very beginning in the firt tep change. Input variable (u / %) u(t), = u(t), = 5 u(t), = Time (t/) Fig. 11 The coure of the input variable, u(t), for different value of The coure of the input variable on the other ide i very imilar for = 5 and 10. The third coure for = i different and moother on the contrary. If we want to check the quality of the control with ome kind of criteria, we can introduce the input and the output variable quality criteria S u and S y computed for a time interval a N Su = ( u() i u( i 1 )) [ ] ; i= Tf for N = (6) N T v Sy = ( w() i y() i ) m i= 1 where T f i time of imulation in thi cae T f = Computed value of thee criteria are hown in Table III. Identified parameter (a 0 /-) = =5 = Time (t / ) Fig. 1 The coure of the identified parameter a 0 for different value of Preented value have hown, that increaing value of reult in better reference ignal tracking, i.e. decreaing value of criterion S y but bigger change of input variable increaing value of criterion S u. ISSN:

7 TABLE III VALUES OF CRITERIA S U AND S Y STANDARD CONTROL Root poition S u [-] S y [m ] = = = Identified parameter (b 0 / -) x = =5 = Time (t / ) Fig. 13 The coure of the identified parameter b 0 for different value of Graph in Fig. 1 and Fig. 13 how value of etimated parameter a 0 and b 0. It i clear, that ued recurive identification ha not problem with the on-line identification except the very beginning of the control, which i caued by the uncertainty of the ytem which need ome time for etimation of the real parameter of the ytem. Thi i typical feature of thi type of adaptive control. A written in the Introduction, ued polynomial ynthei atifie baic control requirement including diturbance attenuation. The diturbance i imulated here by the change of the valve contant k. Thi contant characterize the output valve and it change could be caued for example if the valve i damaged or the crack in the tank occur. The imulation of thi diturbance wa performed for two tep change of valve contant k, e.g.: v1 ( t) =+ 5% of k, in t = 000 (7) v t = 5% of k, in t = 4000 () Water level (w,y/m) v 1 (t) v (t) 0.04 w(t) y(t), = 0.0 y(t), = 5 y(t), = Time (t/) Fig. 14 The coure of the reference ignal, w(t), and the output variable, y(t), for two diturbance and variou Reult preented in Fig. 14 and Fig. 15 how that propoed adaptive controller deal with both diturbance quite good. Input variable (u/%) 80 u(t), = u(t), = 5 u(t), = Time (t/) Fig. 15 The coure of the input variable, u(t), for two diturbance and variou The value of the criteria S u and S y for thi imulation with the diturbance i hown Table IV. The computation again for the whole imulation interval, i.e. T f = TABLE IV VALUES OF CRITERIA SU AND SY CONTROL WITH DISTURBANCES Root poition S u [-] S y [m ] = = = The lat imulation tudy how that thi controller could produce good control reult if the reference ignal i different from the tep change in thi cae exponential change, poitive and negative ramp. The reult for = 5 are again hown in Fig. 16 and Fig. 17 and value of S u and S y for all three and T f = are preented in Table V. Water level (w,y/m) w(t) y(t) Time (t/) Fig. 16 The coure of the reference ignal, w(t), and the output variable, y(t), for different change of the reference ignal and = 5 ISSN:

8 Water level (w,y/m) u(t) Time (t/) Fig. 17 The coure of the input variable, u(t), for different change of the reference ignal and = 5 TABLE V VALUES OF CRITERIA SU AND SY CONTROL WITH DIFFERENT CHANGES OF REFERENCE SIGNAL Root poition S u [-] S y [m ] = = = V. CONCLUSION The goal of thi contribution i to how one way how to deign the controller for the real ytem. At firt, the mathematical model with one ordinary differential equation i derived. Thi model wa then verified by the imulation of the teady-tate and dynamic and the reult are compared with the meaurement on the real ytem. Thi comparion how diproportion between imulated and meaured data which i caued by the inaccuracy of the model which doe not take into the account the height of the outlet valve. If we include thi height into the computation, the reult are much more accurate. The control approach here i baed on the choice of the external linear model of the originally nonlinear ytem, parameter of which are identified recurively. The imple controller with one degree-of-freedom i deigned with the ue of polynomial approach with pole-placement method and pectral factorization. The reulted controller i table and atifie baic control requirement. Moreover, it can be tuned by the choice of the parameter increaing value of thi parameter reult in quicker output repone but poible overhoot. Introduced hybrid adaptive controller produce good control reult although the ytem ha nonlinear behavior. There were done alo other imulation tudie which how that thi hybrid adaptive controller could be ued alo for different coure of reference ignal and for diturbance attenuation too. The future work i connected with the verification of the propoed control trategy on the real model of the water tank. REFERENCES [1] Luyben, W. L Proce Modelling, Simulation and Control for Chemical Engineer. McGraw-Hill, New York ISBN [] Maria A Introduction to modeling and imulation. In: Proceeding of the 1997 Winter Simulation Conference [3] Ingham, J.; I. J. Dunn; E. Heinzle; J. E. Prenoil. 000 Chemical Engineering Dynamic. An Introduction to Modeling and Computer Simulation. Second. Completely Revied Edition. VCH Verlaggeellhaft. Weinheim, 000. ISBN [4] Ruell, T.; M. M. Denn Introduction to chemical engineering analyi. New York: Wiley, 197, xviii, 50 p. ISBN [5] Macku, L., Samek, D. 013 Multilayer feed-forward neural network in prediction and predictive control of emi-batch reactor. International Journal of Mathematic and Computer in Simulation. Volume 7, Iue 1, pp [6] Matuu, R., Prokop, R. 01 Variou approache to olving an indutrially motivated control problem: Software implementation and imulation. International Journal of Mathematic and Computer in Simulation. Volume 6, Iue 1, pp [7] Armfield: Intruction manual PCT40, Iue 4, February 005 [8] Saad, Y Iterative Method for Spare Linear Sytem. Society for Indutrial & Applied. ISBN [9] Johnton, R. L Numerical Method. John Wiley & Son. ISBN [10] Mathew, J. H.; K. K. Fink Numerical Method Uing Matlab. Prentice-Hall. ISBN [11] Kaw, K.; C. Nguyen; L. Snyder Holitic Numerical Method. Available online < [1] Atrom, K.J., Wittenmark, B. 1989: Adaptive Control. Addion Weley. Reading. MA, 1989, ISBN [13] Senkerik, R., Pluhacek, M., Davendra, D., Zelinka, I., Kominkova Oplatkova, Z Chao driven evolutionary algorithm: A new approach for evolutionary optimization. International Journal of Mathematic and Computer in Simulation. Volume 7, Iue 4, pp [14] Bobal, V.; J. Böhm; J. Fel; J. Machacek. 005 Digital Self-tuning Controller: Algorithm. Implementation and Application. Advanced Textbook in Control and Signal Proceing. Springer-Verlag London Limited. 005, ISBN [15] Middleton, H., Goodwin, G. C. 004 Digital Control and Etimation - A Unified Approach. Prentice Hall. Englewood Cliff, 004, ISBN [16] Chramcov, B. 013 The optimization of production ytem uing imulation optimization tool in witne. International Journal of Mathematic and Computer in Simulation. Volume 7, Iue, pp [17] Kucera, V Diophantine equation in control A urvey. Automatica. 9, 1993, p [18] Stericker, D. L., Sinha, N. K. 1993: Identification of continuou-time ytem from ample of input-output data uing the -operator. Control- Theory and Advanced Technology. vol. 9, 1993, ISSN: