COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN

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1 Int. J. Chem. Sc.: (4), 04, ISSN X COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN R. GOVINDARASU a, R. PARTHIBAN a and P. K. BHABA b* a Department of Chemcal Engneerng, Sr Venkateswara College of Engneerng, SRIPERUMBUDUR 607 (T.N.) INDIA b Department of Chemcal Engneerng, FEAT, Annamala Unversty, ANNAMALAI NAGAR (T.N.) INDIA ABSTRACT Desgn of controllers usng the coeffcent dagram method s a novel approach and can be mplemented to all knd of processes, whch can be approxmated to FOPTD. In ths paper, the varous approaches used n tme delay approxmatons are dscussed. Coeffcent dagram (CD) are drawn for FOPTD processes and second order process. From the CD, tme response, stablty ndces and robustness of the system are analyzed. Polynomal controller has been desgned and ts control acton on the FOPTD process for a servo problem s dscussed n detal. The result ndcates that polynomal based controller s most successful n the operaton of closed loop system. Key words: Polynomal controller, CD, FOPTD, Stablty, Robustness. INTRODUCTION Due to the extensve use of control mechansm n varous applcatons, t s essental to desgn a consstent control system. The conventonal control desgn technques are used for smple but not for complex systems. Modern control had been developed but there are many dffcultes assocated wth t. The coeffcent dagram overcomes these dffcultes. Manabe ntroduced the Coeffcent Dagram n 99. CD s an algebrac approach that s appled to a polynomal loop, n whch a coeffcent dagram s used as a crteron for good desgn. Plant transfer functon G(S) = N(/D/( s specfed before the desgn of the controller. The performance of the control system s determned by drawng the Coeffcent Dagram and observng the tme response, stablty and robustness propertes. In CD (Fg. ), the coeffcent a s read on the left sde scale, and the stablty ndex γ, equvalent tme constant τ, and the stablty lmt γ * are read by the rght hand scale. If the curvature of a * Author for correspondence; Emal: govnd.ntt@gmal.com

2 646 R. Govndarasu et al.: Coeffcent Dagram: A Novel Tool n. becomes larger, the system becomes more stable correspondng to larger stablty ndex. If the a curve s leftend down, the equvalent tme constant s small and response s fast. The equvalent tme constant (τ eq ) s calculated to the desred settlng tme (t s ) usng τ = t s /(.5 ~ 3) and the stablty ndces are selected as γ = (.5,,,, ) for = to n accordng to standard Manabe form 4. Coeffcent dagram stablty ndces was studed n 953 by Graham, who proposed the ITAE. Ths was followed by Kessler n 960 to decrease the oscllatons and overshoot. On comparng the ITAE and Kessler model, t was found that Kessler model was more stable and has 8% overshoot. In Kessler model, all stablty ndces were chosen as whereas n the CD model, stablty ndces are selected as [,,..., and.5]. In the case of CD (Manabe s standard form), the responses are obtaned wthout overshoot and wth smallest settlng tme compared to other methods 5. Table.: Stablty ndces of the standard Forms n γ 4 γ 3 γ γ Bnomal ITAE Kessler 4 CD. 4.5 In the block dagram (Fg. ), N(S) s the numerator polynomal of degree m and D(S) s the denomnator polynomal of degree n (m n) 6. In case of lag n the system, e θs s represented by frst order Pade approxmaton 7. N( and D( are numerator and denomnator polynomals of the transfer functon of the plant. A( s the forward denomnator polynomal whle F( and B( are the reference numerator and the feedback numerator polynomals of the controller transfer functon, respectvely. Snce, the transfer functon of the controller has two numerators, t resembles to a two degree of freedom (DOF) system structure. A( and B( are desgned n such a way to satsfy the desred transent behavor. F( s determned as zero order polynomal and used to provde the steadystate gan 8.

3 Int. J. Chem. Sc.: (4), Fg. : Coeffcent dagram Fg. : Closed loop block dagram wth polynomal Control System Therefore, the output of the steady state system s gven by y = N( F( A( F( r + d () P( P( where P( s consdered as the characterstc polynomal of the closedloop system and s defned by n P ( s ) = A ( s ) D ( s ) + B ( s ) N ( s ) = a s () = 0 At a > 0 n whch A p ( = l s and ( s ) = = 0 q B = 0 k s The polynomal CD controller desgn conssts of equvalent tme constant (τ eq = t s /.5) and stablty ndces (γ). Accordng to Manabe s standard form, γ values are

4 648 R. Govndarasu et al.: Coeffcent Dagram: A Novel Tool n. selected as {.5,,...}. Usng the desgn parameters (τ eq, γ * ), a target characterstc polynomal s determned as P ( τ + τs + n target ( = a 0 eq (3) j = j= λ j Equatng the polynomals represented n the above equatons, a Dophantne equaton of A(D( + B(N( = P t arget ( s obtaned. It s then transformed nto Sylvester matrx. Solvng the algebrac equatons, the controller parameters (k and l ) are computed. The CD controller polynomals A(, B( and closed loop characterstc polynomal P( are determned usng k, and l. 0 The reference numerator, F( s obtaned from F( = (P(/N() s = 0. Development of coeffcent dagram Coeffcent dagrams are drawn and llustrated wth two frst order systems wth tme delay and one second order system wthout tme delay 7. Example : Process transfer functon, G( = e 0.θ 0.5 s + Approxmated G( = 0. s 0.s +. s +...(4)...(5) From the transfer functon, N( = 0.s and D( = 0.s +.s + are obtaned. It s assumed as A( = l s + l s, B( = k s + k s + k 0. Accordng to the standard Manabe form γ = [.5]. τ = for t s =.5 are selected. Target polynomal s found to be P( = 0.008s s s + s +. Equatng the target polynomal to the rght sde of the Dophantne equaton and solvng, A( = 0.08s s, B( = 0.s s and γ * = [ ] are obtaned. From the coeffcent dagram (Fg. 3) t s notced that a > (A+B), and γ >.5γ * ; hence, t s reported as the gven system s stable and robust. Example : Process transfer functon, G( = 0.e s 8 s + (6)

5 Int. J. Chem. Sc.: (4), Approxmated G( = 8 s 0. 0.s + 9 s + (7) 0 0 a a γ τ γ τ γ * a γ* Fg. 3 (a): CD (Example ) Fg. 3 (b): Effect of γ (Example ) 0. a Fg. 3 (c): Effect of τ (Example ) From the transfer functon, N( = 0. 0.s and D( = 8s + 9s + are obtaned. It s assumed as A( = l s + l s and B( = k s + k s + k 0. Accordng to the standard Manabe form γ = [.5] and τ = for t s =.5 are selected. Target polynomal s found to be P( = 0.008s s s + s +. Equatng the target polynomal to the rght sde of the Dophantne equaton and solvng, A( = 4.4*0 4 s s, B( = 0.99s s + 0 and γ * = [ ] are obtaned. From the coeffcent dagram (Fg. 4) t s notced that a > (A+B), and γ >.5γ * ; hence, t s reported as the gven system s stable and robust. Example 3: Process transfer functon, G( = s 3 8s + 6s + 8s s + 4 (8)

6 650 R. Govndarasu et al.: Coeffcent Dagram: A Novel Tool n. From the transfer functon, N( = 8s + 8s + 3 and D( = s s + 4s + 4 are obtaned. It s assumed as A( = l 3 s 3 + l s + l s and B( = k 3 s 3 + k s + k s + k 0. Accordng to the standard Manabe form, γ = [.5] and τ = for t s =.5 are selected. Target polynomal s found to be P( = 0 5 s s s s s + s +. Equatng the target polynomal to the rght sde of the Dophantne equaton and solvng, A( = 0 5 s s s, B( = s s s and γ * = [ ] are obtaned. From the coeffcent dagram (Fg. 4), t s notced that a > (A+B), and γ >.5γ * ; hence, t s reported as the gven system s stable and robust. 0 γ τ a 0. γ * Fg. 4 Coeffcent dagram of example: 0 0. γ τ γ * a Fg. 5: Coeffcent dagram of example The effect γ and τ are studed n Fg. 3b and Fg. 3c respectvely. In Fg. 3, the curvature of a s larger for γ =.5 than others and hence, the system s more stable. Even

7 Int. J. Chem. Sc.: (4), though, the a curvature for τ = 0.5 and τ = are left end down, the curvature of a for τ = s larger than τ = 0.5 comparatvely. Hence, the system wth τ = has gven better response 8. CD polynomal controller Polynomal CD controller s desgned (Fg. ) for a FOPTD process (Example ). Approxmaton of dead tme n the gven process (Table 3) s done usng Pade approxmaton, numerator approxmaton and denomnator approxmaton. Approxmated transfer functon models are smulated n closed loop wth step nput and the responses are recorded (Fg. 9). The step responses ndcates that Pade approxmaton 9 s most sutable method for the gven system. Performance of the PN CD controller s analyzed usng step response analyss (Fg. 6). The effect of stablty ndces and equvalent tme constant on the system performance (Y(t)) are nvestgated (Fg. 7, Fg. 8 and Table ). It s observed that the stablty ndex of.5 and equvalent tme constant of performed better than others 3. y (t) Step up and down Step down and up Tme (sec) Fg. 6: Step up and step down response of example wth Pn controller. 0.8 y (t) y = y =.5 y = Tme (sec) Fg. 7: Effect of stablty ndces of example wth Pn controller

8 65 R. Govndarasu et al.: Coeffcent Dagram: A Novel Tool n y (t) τ = 0.5 τ =.0 τ = Tme (sec) Fg. 8: Effect of equvalent tme constant of example wth Pn controller.4. y (t) Tayler num Tayler Den Pade appx Tme (sec) Fg. 9: Effect of dead tme approxmaton of example wth Pn controller Table : Effect of equvalent tme constant and stablty ndex of example Equvalent tme constant and stablty ndex A( B( Stablty ndex (γ ) =.5 Tme constant (τ) = τ = s + 0.7s 0.305s s τ = s s 0.s s τ = s s 0.0s + 0.s γ = s 0.35s 0.435s s γ = s s 0.s s γ = s 0.53s 0.05s s + 0.5

9 Int. J. Chem. Sc.: (4), Table 3: Tme delay approxmaton of example Transfer functon Pade approxmaton Tayler s numerator approxmaton Tayler s denomnator approxmaton G( 0.S/(0.S +.S + ) 0.S/(0.5S + ) /(0.5s + 0.7s + ) CONCLUSION In ths paper, coeffcent dagram (a novel tool) s drawn and dscussed wth three examples. The approaches used n the approxmatons of the tme delay process are dscussed and observed that Pade approxmaton s most sutable. The most mportant characterstc propertes of the system namely tme response, stablty ndces and robustness are recorded n a sngle and smple coeffcent dagram. The effect of stablty ndces and equvalent tme constant on the system performance are analyzed usng coeffcent dagram as well as polynomal CD controller based closed loop response. It s notced that the desgned polynomal controller exhbts better performance for a servo problem. ACKNOWLEDGEMENT Fnancal support from AICTE, New Delh under Research Promoton Scheme (0/AICTE/RIFD/RPS (POLICYIII) 38/03) s gratefully acknowledged. REFERENCES. S. Manabe, 4 th IFAC Symposum on Control n Aerospace, Seoul, Korea, 99 (998).. S. E. Hamamc, M. Koksal and S. Manabe, 4th Asan Control Con, Sngapore, 6 (00). 3. S. Skogestad, J. Proc. Control, 3(4), 9 (003). 4. S. Manabe, 4 ed IEEE Conference on Decson and Controls Hawa, USA, 3489 (003) 5. S. E. Hamamc, J. Electrcal Engg., 87(3), 63 (005). 6. K. H. Ang, G. Chong and Y. L, IEEE Trans. Control Sys. Tech., 3(4), 559 (005).

10 654 R. Govndarasu et al.: Coeffcent Dagram: A Novel Tool n. 7. M. S. Tavazoe and M. Haer, Trans. on Engg. Comp. Technol., 30 (005). 8. P. K. Bhaba and S. Somasundaram, Modern Appl. Sc., 3(5), 38 (009). 9. P. K. Bhaba and S. Somasundaram, Sensors Transducers J., 33(0), 53 (0). 0. R. R. Rnu Raj and L. D. Vjay Anand, J. Theo. App. Res. Mech. Engg.,, 49 (03). Accepted :