Inernaional Journal of Informaion Technology and nowledge Managemen July-December 0, Volume 5, No., pp. 433-438 INVERSE RESPONSE COMPENSATION BY ESTIMATING PARAMETERS OF A PROCESS COMPRISING OF TWO FIRST ORDER SYSTEMS Mandeep Singh and Abhay Sharma Absrac: An inverse response in a sysem occurs when he iniial response of an oupu variable occurs in he opposie direcion o ha of he seady sae value. The classical example of his behaviour can be seen in a boiler wih wo differen firs order sysems. This paper helps o esimae he ransfer funcion parameers of he process from is uni sep response. Classical mehod for his esimaion involves simulaneous soluion of hree non-linear equaions conaining exponenial erms. Mos of he available ools fail o achieve his soluion. The paper proposes a novel ieraive mehod o overcome his limiaion. eywords: inverse response, ransfer funcion, non-linear equaions, parameer esimaion. INTRODUCTION Whenever a conrolled variable encouners wo (or more) compeing dynamic effecs wih differen ime consans from he same manipulaed variable, he resuling composie dynamic behaviour of his process can exhibi a roublesome inverse response or a large overshoo response. In process conrol, inverse response (IR) processes can be commonly encounered and classical examples are illusraed in exbooks (Tyner and May(968) [], Vidyasagar (985)[]). The imporan characerisic of processes wih inverse response is ha he process ransfer funcion has odd number of zeroes on he righ hand side of origin [3]. IR processes are frequenly encounered in process conrol like drum boiler, disillaion column and boos converer [4], [5]. Scali e al [6] and Zhang e al [3] proposed an analyical design for IR processes wihou dead ime. This inverse response problem affecs he achievable closed-loop performance because he conroller operaes on wrong sign informaion in he iniial ime of he ransien [7]. Le us ake wo firs order sysems wih opposing gains. The problemaic siuaion is he resul of wo opposing dynamics. Wih he given opposie firs order sysems le us sudy he combined process conrol of a process. Figure : Process Block Diagram for IR Process, Deparmen of Elecrical and Insrumenaion Engineering, Thapar Universiy, Paiala, India, E-mail: mandy_ie@yahoo.com, s.abhay4@gmail.com G( s) y( s) u( s) () G(s) refers o he overall ransfer funcion of he process G(s) = G (s) + G (s) G (s) and G (s) are parallely coupled opposing firs order sysems as shown in figure. G( S) G ( s) and G ( s) 0s s ( ) s s s ( s )( s ) refers o he proporionaliy gain consan of iniial firs order sysem. refers o he proporionaliy gain consan of oher firs order sysem. and are he ime consans of he wo firs order sysems respecively. y(s) and u(s) are he final oupu and sep inpu respecively as in figure. We can ge he zeroes of he ransfer funcion by equaing he numeraor of he ransfer funcion o zero. We ge he following equaion ( ) s 0, S ( ) This value of s gives he value of zero on he righ hand side of he origin. We have inverse response when, i.e., iniially process, which reacs faser han process, dominaes he response of he overall sysem, bu ulimaely process reaches a higher seady-sae value han process and forces he response of he overall sysems in he opposie direcion as shown in Figure. () (3)
434 INTERNATIONAL JOURNAL OF INFORMATION TECHNOLOGY AND NOWLEDGE MANAGEMENT The derivaive of he plo in figure 3 is given by he equaion dy d * e * e The value a he funcion a is minima is given as (5) Y ( e ) ( e ) (6) / / min Derivaive of he funcion holds zero value a is minima Figure : Inverse Response Curve In his conex, inverse response appears due o compeing effecs of slow and fas dynamics. In concree erms, i appears when he slower process has higher gain, he open-loop response o a sep inpu presens undershoo, he more pronounced he more i increases he value of zero.. PROBLEM DEFINITION The problem of inverse response has been a challenging ask for he process engineers for a long ime. Basically, if process parameers are esimaed hen a compensaor may be defined for i. We esimae he parameers such ha he process follows firs order slowly and hen reaches a seady value [8]. The main problem is ha Classical mehod for his esimaion involves simulaneous soluion of hree non-linear equaions conaining exponenial erms. Mos of he available ools fail o achieve his soluion. The paper proposes a novel ieraive mehod o overcome his limiaion. 3. METHOD This secion would be helpful in evaluaing he process parameers by mahemaically sudying he graph. For solving he equaions we consider no dead ime delay. From he graph shown in figure 3, we can see ha is minima occurs a ime =. The funcion holds a zero value a ime = and from observaion i is clearly undersood ha he plo has is derivaive as zero a ime =. The funcion reaches is seady sae value afer some ime. The derivaive of he funcion remains zero a seady sae. The following equaion (4) expresses he oupu funcion in ime domain which was earlier expressed in frequency domain. Y( ) ( e ) ( e ) (4) / / Figure 3: Graphical Represenaion of Oupu Funcion Showing Required Parameers dy d * e * e 0 The value of he funcion is zero a ime ( =. So his can be wrien as / / (7) Y( ) ( e ) ( e ) 0 (8) The final equaion can be obained a he seady sae value where he funcion holds a consan value. This can be done by puing a limi on Y() o ge a desired value a infinie ime. I is saed as below lim ( ) Y (9) This is because he exponenial erm vanishes as ime approaches infiniy. So we are jus lef wih he difference of he proporionaliy gain consans of he wo funcions i.e.. There can also be a case ha he final oupu follows iniial firs order sysem raher han oher, in ha case equaions would become Y ( e ) ( e ) (0a) / / max dy d * e * e 0 / / (0b) Y( ) ( e ) ( e ) 0 (0c) lim ( ) Y (0d) The above saed four equaions (0a-0d) are sufficien o solve four variables (,,, ). The remaining variables which are,, Y max or Y min, dy can be esimaed d graphically from he figure 3 o ge an approximae value of he unknown variables. The dynamic characerisics of he model as shown by above equaions (0a-0d) are evaluaed in MATLAB sofware o ge he value of he unknown variables. The soluion canno be obained by using he solve command because he unmodified form of he hree exponenial equaions does no reurn any soluion using his command. This kind of problem canno be ackled in MATLAB and so we use an alernaive approach o i.
INVERSE RESPONSE COMPENSATION BY ESTIMATING PARAMETERS OF A PROCESS COMPRISING OF TWO FIRST ORDER SYSTEMS 435 Firsly we use subsiuion echnique in which we replace he value of by in all equaions using equaion (9) as lim Y( ) holds a given unique value. Now we are lef wih hree unknown parameers, and and hree simulaneous equaions. Since he MATLAB shows an error in solving hree nonlinear algebraic equaions, we use wo non-algebraic equaions whose resul is always unique. For he hird variable we use ieraive mehod. The hird variable is chosen o be because i is in exponenial form and any small variaion in is value will cause a large change in he value of oupu funcion hereby limiing he number of consideraions. The basic approach used is LEAST SQUARE ERROR (L.S.E.). For an insance, suppose we assume o be a variable and assume is value o be, hen we would compue he value of unknown variables and plo he corresponding curve. Since he iniial curve was given o us, we would now calculae he difference of ploed curve o he iniial curve a each poin and ake is square. This L.S.E. mehod is done for a large number of values of. The L.S.E. curve is now ploed and by number of experimens we observe a global minimum for differen given ieraions. The opimized value is acually he one occurring a he minima of he L.S.E. curve. User s concern is o idenify he minima boundary limis and hen using simple ieraive mehod we can approximae he posiion of minima hereby evaluaing he corresponding unknown variables. The error will reflec he accuracy of he soluion. The following Malab programme hrows some ligh on he soluion: s=solve ( (( +Y seady-sae )/n)*exp( /n) ( / )*exp( / ), ( +Y seady-sae )*( exp( /n)) *( exp( / )) ) n)) =s. =s. =[:00] y()= ( Y seady-sae )*( exp( / )) + *( exp( / h=(y() m()) l=ranspose(h) q=h*l 4. RESULTS AND DISCUSSIONS This mehod of parameric esimaion of Inverse Response is primarily dependen on he uni sep of he process. For his purpose, we have o simulae his response using MATLAB SIMULIN. The process defined by equaion (4) is simulaed for is response o he sep inpu. The values of,, and are chosen for six differen cases and he ransfer funcion so formed is shown in able. Care is aken while chosing hese values so as o follow he condiion. Afer geing he values of, and Y min or dy/dx and y max from he simulaed graph, he proposed mehod is applied for. These values are repored in able. The resuling value of esimaed,, and are also shown in able. I is clearly seen ha he errors are only on acoun of he manual measuremens done for of,, Y min and y max. The simulaed graphs are shown in figure 4(a-f). =s. =s. =[:00] y()=( + Y seady-sae )*( exp( /n)) *( exp( / )) h=(y() m()) l=ranspose(h) q=h*l In he above programme solve command is helpful in evaluaing he equaions wrien in i. n is rial value given o, m()gives he value of he given plo a inervals defined by (=[:00]) and q is he value of L.S.E. A differen programme is used when he given curve has is posiive maxima. The following Malab programme explains his: s=solve ( (( Y seady-sae )/ )*exp( / ) + ( / n)*exp( /n), ( Y seady-sae )*( exp( / )) + *( exp( /n)) ) Figure 4(a): Assumed Process Curve Figure 4(b): Assumed Process Curve
436 INTERNATIONAL JOURNAL OF INFORMATION TECHNOLOGY AND NOWLEDGE MANAGEMENT Figure 4(c): Assumed Process Curve Figure 4(d): Assumed Process Curve Figure 4(e): Assumed Process Curve Figure 4(f): Assumed Process Curve Table Resuls of Graphical Esimaion Sl no Original Transfer Minima of L.S.E Value of Value of Value of Y min Value of Y max Funcion lies beween in sec in sec.. 3. 4. 5. 6. 7s 5s 0 8 s 0 8 6s 3s 0 3 s 3s 5 7 5s 7s 3 35 8s [6, 8] 4.4 3.55 0.8376 [3, 5].6 6.4 3.088 [5, 7] 3.6 3.8 3.5547 [, 3] 0.8.83 3 0.5 [4, 5] 3.6 0.5 0.86 [3, 5] 4.8 8.94 3 6.57 Table Resuls of Parameric Esimaion Serial Original Transfer Value of Value of Value of Value of Value of L.S.E. Referred Figure number Funcion and %error and %error and %error and %error. 7s 5s.8803, 0.8803, 7., 5.096, 0.0076 4a.667%.0888%.48% 0.096% Table Con d
INVERSE RESPONSE COMPENSATION BY ESTIMATING PARAMETERS OF A PROCESS COMPRISING OF TWO FIRST ORDER SYSTEMS 437 Table Con d. 3. 4. 5. 6. 0 8 s 0 8 6s 3s 0 3 s 3s 5 7 5s 7s 3 35 8s 0.00583, 8.00583, 3.96,.069, 0.063 4b 0.05% 0.07% % 6.9% 0.4974, 8.4974, 6.0, 3.6, 0.4 4c.487%.7663% 0.67% 5.367% 9.956,.956,.08,.98, 0.07 4d 0.439% 0.33769% 0.64% 0.667% 4.6468, 6.6468, 4.95, 6.9967, 0.00077 4e.35% % % 0.33% 3.6764, 34.6764, 3.94, 8.0, 0.079 4f.0% 0.947%.5% 0.5% The above shown resuls are for model processes whose oupu funcions are shown by figures (4a-4f). Delay in each plo is assumed zero and graphical esimaion has played an imporan role in he esimaion of process parameers. The resul has been possible solely due o he presence of single global minima oherwise ieraive mehod would have possibly failed. The resuls show he error occurred in each case limied o he esimaion echnique of each user. 5. CONCLUSION This paper represens a novel mehod for he esimaion of inverse response parameers of a process model. Two opposing cases of firs order processes are considered and he given Malab sofware helps in esimaing he process parameers of he curve for he uni sep applied. I should be noiced ha process is ideal wihou any process delay. This paper shows ha hree exponenial equaions and a linear equaion has been solved which was no possible wih he classical mehod and he earlier publicaion relaed o parameric esimaion of inverse response process comprising of firs order sysem and a pure capaciive sysem [9] has been crucial in undersanding of process model and is dynamics. The resul is obained by leas square error and ieraive mehod and he limiaion due o non-linear equaions has been eliminaed. 6. FUTURE SCOPE The procedure described above gives a highly reliable mehod which is easy o use and implemen. The esimaing procedure is suiable for applicaion in indusrial environmen because of is simpliciy. I has is applicaion in dead ime compensaion and inverse response compensaion [0]. The resuls show ha i can be applied in he esimaion of large number of processes giving beer esimaion in less ime. This mehod is helpful in designing a suiable compensaor, by which he fine uning of he PID (Proporional Inegral and Derivaive) conroller []. This is possible only if he process parameers are known o us. REFERENCES [] Tyner, M.; May. F.P., Process Engineering Conrol, Ronald Press: New York, 968. [] Vidyasagar M., On Undershoo and Non-minimum Phase Zeros, IEEE Trans. Auom. Conrol 986, 3, 440-44. [3] Zhang W.; Xu. X.; Sun. Y., Quaniaive Performance Design for Inverse-response Processes, Ind. Eng. Chem. Res, 39, 056-06, 000. [4] Shinskey F.G., Process Conrol Sysem, nd ed.; McGraw-Hill Book Company: New York, 979. [5] Arulselvi S., Uma G., Chidambaram M., Design of PID Conroller for Boos Converer wih RHS Zero, Conf. Proc. IPEMC In. Power Elecron. and Moion Conrol Conf, pp. 53-537, 004. [6] Scali C.; Rachid A., Analyical Design of Proporional- Inegral-Derivaive Conroller for Inverse Response Processes, Ind. Eng. Chem.Res., 37, pp. 37-379, 998. [7] Skogesad S.; Poslehwaie I., Mulivariable Feedback Conrol, Wiley, 997. [8] Iinoia.; Alperer R.J., Inverse Response in Process Conrol, Ind.Eng. Chem., 54, pp. 39-43, 96. [9] Singh Mandeep; Saksena Dhruv, Parameric Esimaion of Inverse Response Process Comprising of Firs Order Sysem and a Pure Capaciive Sysem, Inernaional Journal of Informaion Technology and nowledge Managemen, 5(), June 0 (in press).
438 INTERNATIONAL JOURNAL OF INFORMATION TECHNOLOGY AND NOWLEDGE MANAGEMENT [0] Sephanopoulos.G., Chemical Process Conrol: An Inroducion o Theory and Pracice, Prenice-Hall, 984. [] Waller.T.V.; Nygårdas C.G., On Inverse Response in Process Conrol, Ind. Eng. Chem. Fundam, 4, pp. - 3, 975.