TWO-DEGREE-OF-FREEDOM CONTROL SCHEME FOR PROCESSES WITH LARGE TIME DELAY

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1 50 Aian Journal of Control, Vol 8, No, pp 50-55, March 006 -Brief aper- TWO-DEGREE-OF-FREEDOM CONTROL SCHEME FOR ROCESSES WITH LARGE TIME DELAY Bin Zhang and Weidong Zhang ABSTRACT In thi paper, an analtical two-degree-of-freedom-control cheme i propoed for controlling procee with large time dela The main contribution of thi paper are that a etpoint repone controller and an H ID load-loop controller are developed baed on optimal control theor, and control parameter are derived analticall Thi tructure can alo be ued to control integrating or untable procee eword: Smith predictor, optimal control, robut tabilit, untable and integral procee with large time dela I INTRODUCTION In general, the conventional Smith predictor [] i conidered to be an effective control tructure for procee with large time dela Since the cloed-loop tranfer function doe not include the time dela, and the tem can achieve good tracking performance when the model matche the proce perfectl However, there i alwa model mimatch, which normall reult in deterioration of the control tem performance, thu limiting application of the Smith predictor While the Smith predictor method can, theoreticall, be implemented, there are man problem that need to be olved, for example, the cloed-loop performance deterioration when the diturbance i not included in the Smith predictor Furthermore, due to the tabilit problem, it cannot be directl applied to untable procee [] Taking advantage of the merit of the Smith predictor, reearcher have propoed improved approache [,3-7] Manucript received April 0, 005; accepted November, 005 The author are with Department of Automation, Shanghai Jiaotong Univerit, Shanghai 0040, China ( zhangbin770@jtueducn) Thi paper wa financial upported b the National Natural Science Foundation of China (607403, ), State 973 project (00cb300) that modif the Smith predictor and extend it application Hu et al [8] preented a frequenc-domain method that ue an iterative learning control to improve the performance of the Smith predictor controller Atrom et al [9] propoed a new Smith predictor for the control of procee that involve an integrator and long dead-time, but it can onl be ued when the integrating time contant i one Another new modified Smith predictor for procee involving an integrator and long dead time wa propoed b Hang et al [0], baed on the ue of a rapid load etimator cheme, the load etimator doe not involve the olution of a cloed-loop equation Majhi and Atherton [] propoed a modified Smith predictor which can be ued for the control of integrating and untable procee Inpired b thi method, Lu et al [] preented a double twodegree-of-freedom control cheme for achieving improved control of untable dela procee; unfortunatel, four controller are needed Tan et al [3] propoed a modified internal model control (IMC) tructure for controlling untable procee with time dela Recentl, Tian and Gao [4] preented a ver efficient double-controller tructure (DCS) that i imilar to Atrom tructure which hare the idea of the Smith predictor Baed on thi tructure, Vrecko et al [5] preented a new modified Smith predictor (MS), in which the trade-off between diturbance rejection and robutne to variation in proce parameter can be tuned b mean of a ingle parameter However, neither Tian nor Vrecko dicued the application of their two method to untable or integrating procee

2 B Zhang and WD Zhang: Two-Degree-of-Freedom Control Scheme for rocee with Large Time Dela 5 In thi paper, an analtical two-degree-of-freedom control tructure i propoed The main difference between thi reearch and DCS and MS i that our tructure can be applied to integrating or untable procee; furthermore, good etpoint tracking performance and load rejection can be achieved through optimal control theor A a reult, the deigner can concentrate on etpoint tracking and load rejection eparatel II TWO-DEGREE-OF-FREEDOM CONTROL SCHEME In order to decouple the etpoint repone from load diturbance rejection, a control cheme with two controller i propoed, a hown in Fig, () = G p () e θ i the proce tranfer function, G m i the dela-free proce model, and e θ m i the pure time dela model w G c G m u u u d d G c Then, () can be rewritten a θ m () = GCe m w () d() (7) G Note that if the cloed-loop control tem (7) i table and lim d( ) = 0, GC m =, then the tem can 0 Gc track the etpoint without tead-tate error However, GC m = cannot be implemented in practice Therefore, we need to introduce a low-pa filter, J(), which atifie the following equation: GC m = J () =, ( λ n ) n i a poitive integer III CONTROLLER DESIGN OF THE LOAD DISTURBANCE REJECTION LOO From Fig, we know that the load loop can be decribed b the unit feedback control tem hown in Fig c (8) e θ m Fig The tructure of the control tem r = 0 G c u d d From Fig, it can be een that the cloed-loop repone to the etpoint and the load input i given b () = () w () d(), () w d θm Gc( GcGme ) w() =, ( GcGm)( Gc) d () = G c If the model matche the proce exactl, then Eq () and (3) can be implified a follow G c w() =, GcGm d () = G c Thu, the etpoint repone and diturbance repone can be controlled b controller G c and G c, repectivel Define C = G (6) c : GG m c () (3) (4) (5) Fig Unit feedback control tructure of the load loop For the tructure hown in Fig, different method have been propoed, leading to a impler and eaier tuning controller, G c Recentl, Zhang [6,6] propoed the H ID controller and the H ID controller, both of which were developed on the bai of the optimal control theor, controller parameter are derived analticall In thi paper, thi method will be ued to deign the controller, G c Note from Fig that the tranfer function matrix from r = 0 and d to and u i Define G () c() () GG ( ) ( ) ( ) ( ) c GGc H() = C () G () c() GG () c() GG () c() Q G () c () = Gc( ) (9) (0)

3 5 Aian Journal of Control, Vol 8, No, March 006 We have Q () () ()( Q () ()) H() = Q () Q () () () If we can find the optimal Q() needed to guarantee that all the element of H() are table, then the optimal controller, G c, can be obtained b mean of (0): Q () Gc () = () Q ( ) ( ) It i known that the enitivit tranfer function of the load loop can be written a S () = Q () () (3) Let the tem nominal performance index be min W ( ) S( ), (4) W () i a weighting function However, in order to track the etpoint without teadtate error, another contraint for the tem containing m pole i needed; that i, Q () mut hold for Q ( ) ( ) lim = 0 0 k < m (5) 0 k Moreover, when (4) i olved, the obtained unique optimal Q im i uuall improper Thu, a low-pa filter, J (), need to be introduced to roll off at high frequencie Then, Q () = Qim J(), (6) m i β m β i β J() =, (7) n ( λ ) in which n i a poitive integer, λ i the performance degree, and β i (0 < i < m) i determined b (5) Then, G c () can be obtained through () IV CONTROLLER DESIGN FOR ROCESSES WITH LARGE TIME DELAY () Conider the firt-order table proce θ () = e T It i known from (8) that GC m = J () = n ( λ ) (8) Since it i a table firt-order proce, n = i the implet expreion for the above equation; then, C = G (9) λ m Subtituting (9) into (6), we have T Gc () = (0) λ Appling firt-order ade approximation to the time dela of the table proce, we obtain θ/ () = Gm () () θ/ It i known that () ha a zero at = /θ in the open right half plane; therefore, from (4), we have min W () S() = min W ()( () Q()) =θ / After the above equation i olved, b (5) and (6), the optimal Q() i ( T )( θ /) Q () = ( λ ) Subtituting the above equation into (), we can calculate G c (), a ID controller: Gc () = c TD, T I TF c λ θt =, TI =θ / T, TD =, and λ θ / TI TI = ( λ θ/ ) T F () Conider the integrating proce () θ () = e T We know from (8) that n = i the implet expreion Subtituting G m = /T into (8), we have T C = (3) λ And ubtituting (3) into (6), we have T Gc () = (4) λ To obtain an analtical controller, G c (), b the maximum modulu theorem, we have min W ( ) S( ) = min W ( )( ( ) Q( )) =θ / Solving the above equation, we obtain the optimal Q im () = T ( θ / ) According to (5), () Q() mut contain

4 B Zhang and WD Zhang: Two-Degree-of-Freedom Control Scheme for rocee with Large Time Dela 53 at leat two zero, o we chooe the following filter which can meet thi requirement: (3 λ θ / ) J () = 3 ( λ ) Then we have [ ] T( θ/ ) (3 λ θ / ) Q () = 3 ( λ ) (5) (6) Subtituting (6) into (), we have the following controller: TD Gc () = c, T I TF (7) 3 λ θ θ T F =, T I = or 3 3 λ 3 λ θ / θ /4 λ, θ θ T D = 3λ or, TTI c =, and 3 λ 3 λ θ / θ /4 λ in the recommended range from 0 θ to 0 θ (3) Conider the firt-order untable proce θ () = e (8) T We know from (8) that n = i the implet expreion Subtituting G m = /(T ) into (8), we have T C = ( λ ) And ubtituting (9) into (6) lead to G c T () = λ T (9) (30) Note from () that the tranfer function of H() hould all be table, which i equivalent to the tabilit of Q(), and that the following equation hould hold: lim Q ( ) ( ) = 0, (3) / T lim Q ( ) ( ) = 0 (3) 0 Now, appling the firt-order Talor approximation to the time dela of the untable proce, we have ( θ) () = T Then, b the maximum modulu theorem, we have (33) min W() S() = min W()( () Q()) =θ (34) Solving (34), we find that the optimal Q() i T Qim () = (35) Moreover, in order to make (3) and (3) hold, the following filter hould be introduced: a J () = (36) λ ( ) Then ( T )( a ) Q () = ( λ ) Subtituting Q() into (3), we obtain λ λ T θt a =, T >θ T θ Subtituting (37) into (6) lead to Gc () = c, T I c λ λ T θt =, T I ( λ θ) λ λ T θt = T θ (37) (38) Unfortunatel, thi controller, G c (), i retricted to the condition T > θ Zhang [7] propoed an optimal H method, the deigned controller G c () can be ued in two cae, T > θ and T θ, and G c () can be expreed a Gc () = c TD, T I TF c a θ aθ =, TI = aθ, TD =, b a θ T F (39) λθ =, bt λθ λ λθ a = λ θ, b = aλ θ, and λ in T T the recommended range from θ to 5θ V ERFORMANCE ANALYSIS OF THE CLOSED-LOO CONTROL SYSTEM For our propoed control tem, we know from Eq () that the tranfer function of the cloed-loop tem i GC m θ m () = S () GC m ( S ()) e w () Gm S() d () (40) We know from Eq (8) that G m C/G m i table for the procee dicued in thi paper and from Section 3 that

5 54 Aian Journal of Control, Vol 8, No, March 006 the load rejection loop i internall table, o the whole tem i table B the Final Value Theorem, we know that lim t ( ) t GC m lim S () G m m C ( S ()) e θ = w () S () d () 0 Gm θ m = lim S( ) ( S ( )) e w Sd ( ) 0 Gm ( λ ) λ = w A a reult, the control tem can track the etpoint without tead-tate error in the cae of a tep input and a contant load diturbance Fig 3 Repone of the cloed-loop tem (propoed method, olid line; DCS, dotted line; H ID, dahed line) VI SIMULATIONS Example Conider the following table proce: θ () = e, T =, T =, and θ = 6 In the nominal cae, the controller parameter are λ = 06θ for the H ID [6] controller, and λ = 5 for the propoed method From the repone of the cloed-loop tem hown in Fig 3, we notice that the etpoint tracking with the propoed method i lower than that with DCS but fater than that with the ingle H ID method In order to tet the robutne to variation in parameter, imulation were conducted with θ = 7, T = 09, and = The cloed-loop repone hown in Fig 4 indicate that all three control tem can achieve a good performance in etpoint tracking and load diturbance rejection under tem parameter perturbation and that the method of Zhang achieve better robutne Example Conider the integrating proce θ () = e, T in which T = and θ = 5 The method of Majhi [] and Atrom [9] will be compared below The controller tuning parameter for our propoed method ued here were aigned to be λ = and λ = 073θ A tep load input d = 0 wa added at t = 00, and the cloed-loop repone are hown in Fig 5 The figure how that the propoed method produce an appropriate etpoint repone and load rejection, a Atrom and Majhi method alo do Figure 6 how the repone to proce parameter perturbation, ie, θ = 47 It i ea to ee that the propoed method achieve better robutne than Atrom and Majhi method in etpoint tracking and i capable of rejecting load diturbance Fig 4 Repone of the cloed-loop tem with parameter perturbation (propoed method, olid line; DCS, dotted line; H ID, dahed line) Fig 5 Repone of the cloed-loop tem (propoed method, olid line; Majhi, dotted line; Atrom, dahed line) Fig 6 Repone of the cloed-loop tem with parameter perturbation (propoed method, olid line; Majhi, dotted line; Atrom, dahed line)

6 B Zhang and WD Zhang: Two-Degree-of-Freedom Control Scheme for rocee with Large Time Dela 55 Example 3 Conider the untable proce θ () = e, T in which =, θ = 5, and T = 0 The parameter of Majhi controller [8] were et to be k p =, T i = 0, k f = 3468 and k d = 44 The controller tuning parameter for our propoed H method were aigned here to be λ = 5 and λ = 5θ A tep load d = 0 wa added at t = 00, and the reulting cloed-loop repone are hown in Fig 7 The figure how that the propoed method achieve better reult than Majhi doe in etpoint tracking but wore reult in load rejection However, tuning in the propoed method i much eaier Fig 7 Output (upper) and input (lower) of the cloed-loop tep repone (propoed method, olid line; Majhi, dahed line) VII CONCLUSION An analtical two-freedom-of-degree control cheme for controlling procee with large time dela ha been propoed The major contribution of thi paper are a follow: firtl, the propoed method can deal with the control of the untable and integral procee; econdl, etpoint tracking repone and load diturbance rejection can be independentl tuned uing a ingle parameter In addition, the two controller are deigned on bai of optimal control theor, and the controller parameter can be eail tuned REFERENCES Smith, OJM, Cloer Control of Loop with Dead Time, Chem Eng rog, Vol 53, No 5, pp 7-9 (957) Wang, YC, redictor and Control of Time Dela rocee Mechanical Indutr re, China (986) 3 Hammartrom, LG and V Waller, On Optimal Control of Stem with Dela in the Control, IEEE Tran Ind Electron Contr Intrum, Vol IECI (7), pp (980) 4 Lee, D, MY Lee, SW Sung, and L Lee, Robut ID Tuning for Smith redictor in the reence of Model Uncertaint, J roce Contr, Vol 9, pp (999) 5 Gorecki, R and J Jekielek, Simplifing Controller for roce Control of Stem with Large Dead Time, ISA Tran, Vol 6, pp 37-4 (999) 6 Zhang, WD, YX Sun, and XM Xu, Two Degreeof-Freedom Smith redictor for rocee with Time Dela, Automatica, Vol 34, No 0, pp 79-8 (998) 7 Majhi, S and D Atherton, Obtaining Controller arameter for a New Smith redictor Uing Autotuning, Automatica, Vol 36, No, pp (000) 8 Hu, Q, JX Xu, and TH Lee, Iterative Learning Control Deign for Smith redictor, St Contr Lett, Vol 44, No 3, pp 0-0 (00) 9 Atrom, J, CC Hang, and BC Lim, A New Smith redictor for Controlling a roce with an Integrator and Long Dead-Time, IEEE Tran Automat Contr, Vol 39, No, pp (994) 0 Hang, CC, QG Wang, and X Yang, A Modified Smith redictor for a roce with an Integrator and Long Dead Time, Ind Eng Chem Re, Vol 4, No 3, pp (003) Majhi, S and D Atherton, Modified Smith redictor and Controller for rocee with Time Dela, IEE roc Contr Theor Appl, Vol 46, No 5, pp (999) Lu, X, YS Yang, QG Wang, and WX Zheng, A Double Two-Degree-of-Free Control Scheme for Improved Control of Untable Dela rocee, J roce Contr, Vol 5, pp (005) 3 Tan, W, JM Horacio, and TW Chen, IMC Deign for Untable rocee with Time Dela, J roce Contr, Vol 3, No 3, pp 03-3 (003) 4 Tian, YC and F Gao, Double-Controller Scheme for Control of rocee with Dominant Dela, IEE roc Contr Theor Appl, Vol 45, No 5, pp (998) 5 Vrecko, D, D Vrancic, C Juricic, and S Strmcnik, A New Modified Smith redictor: The Concept, Deign and Tuning, ISA Tran, Vol 40, No, pp - (00) 6 Zhang, WD, Analtical Deign Method for roce Control, ot-doctoral Reearch Report, Shanghai Jiaotong Univerit (998)