Control of a string of identical pools using non-identical feedback controllers

Transcription

1 Delft Univerit of Technolog Delft Center for Stem and Control Technical report -5 Control of a tring of identical pool uing non-identical feedback controller Y. Li and B. De Schutter If ou want to cite thi report, pleae ue the following reference intead: Y. Li and B. De Schutter, Control of a tring of identical pool uing non-identical feedback controller, Proceeding of the 9th IEEE Conference on Deciion and Control, Atlanta, Georgia, pp. 5, Dec.. Delft Center for Stem and Control Delft Univerit of Technolog Mekelweg, 68 CD Delft The Netherland phone: ecretar fax: URL: Thi report can alo be downloaded viahttp://pub.dechutter.info/ab/_5.html

2 Control of a String of Identical Pool Uing Non-Identical Feedback Controller Yuping Li and Bart De Schutter Abtract In the ditant-downtream control of irrigation channel, the interaction between pool and the internal timedela for water to travel from uptream to downtream, impoe limitation on global performance, i.e. there exit propagation of water level error and amplification of flow over gate in the uptream direction. Thi paper anale thee coupling propertie for a tring of identical pool, both with identical feedback controller and with non-identical feedback controller. The definition of tring tabilit in term of bounded water level error and bounded flow i given. It i hown that for a tring of infinite number of pool, tring tabilit cannot be achieved b decentralied ditant-downtream feedback control. However, for a tring of finite number of pool, a better global performance can be achieved b non-identical feedback controller uch that the cloed-loop bandwidth of the ubtem increae from downtream to uptream. I. INTRODUCTION When deigning decentralied feedback controller for irrigation network, one uuall onl take local performance into account, i.e. regulating the water-level in a pool at it etpoint while rejecting offtake diturbance. Such a deign might preent ver bad global performance, e.g. in repone to offtake diturbance in the downtream pool, the gate in the uptream pool ma go beond aturation or the water-level in the uptream pool ma drop too low to atif the water demand. Therefore, in thi paper, deign of decentralied feedback controller i dicued baed on global performance conideration. In large-cale irrigation network, water i often ditributed via open water channel under the power of gravit i.e. there i no pumping. The flow of water through the network i then regulated b automated gate poitioned along the channel [], [6], []. The tretch of a channel between two gate i commonl called a pool. Water offtake point to farm and econdar channel are ditributed along the pool. A uch, an important control objective i etpoint regulation of the water-level immediatel uptream of each gate, which enable flow demand at the often gravit-powered offtake point to be met without over-uppling. When the number of pool to be controlled i large and the gate are widel dipered, it i natural to emplo a decentralied control tructure. In practice, a ditant-downtream control tructure i.e. uing the uptream gate to control the downtream Y. Li and B. De Schutter are with the Delft Center for Stem and Control, Delft Univerit of Technolog, Mekelweg, 68 CD Delft, the Netherland. {uping.li,b.dechutter}@tudelft.nl Thi reearch ha been upported b the European 7th framework STREP project Hierarchical and ditributed model predictive control HD-MPC, contract number INFSO-ICT-85. Tpicall at the downtream end of pool. water-level of a pool i implemented for management of water ervice and water ditribution efficienc [8]. Fig. how a ide view of a channel under decentralied ditantdowntream control. For uch a control tructure, when offtake diturbance occur in the downtream pool, the interaction between pool, due to the fact that the flow into one pool equal to the flow upplied b it uptream pool, and the internal time-dela for the tranportation of water from uptream to downtream put requirement on managing the water-level error propagation and attenuating the amplification of flow over gate in the uptream direction, ee [], [] for anali of coupling between pool with ditantdowntream control. Thi paper tudie the global control performance problem b analing decentralied feedback control of a tring of identical pool, for which we ugget a control trateg of uing non-identical feedback controller. A definition of tring tabilit in term of bounded water level error and bounded flow i given. It i hown that tring tabilit cannot be achieved for infinite number of pool with decentralied ditant-downtream feedback control. However, for finite number of pool which i true in practice, b deigning the non-identical feedback controller uch that the cloed-loop bandwidth of the ubtem increae from downtream to uptream, a much better global performance than that with identical feedback controller can be achieved. Furthermore, we extend the anali reult to a tring of heterogeneou pool and give guideline for deigning feedback controller baed on global performance. The paper i organied a follow. Section II give the definition of tring tabilit in term of bounded water level error and bounded flow. Both the cae of a tring of identical pool with identical feedback controller and with non-identical feedback controller are dicued. The global performance anali i extended to a tring of heterogeneou pool in Section III. Section IV how imulation reult. A brief ummar i finall given in Section V. II. BOUNDED WATER LEVEL ERRORS AND BOUNDED FLOWS Conider n + pool. Denote the firt downtream pool G, the econd downtream pool G, and o on, till the mot uptream pool, G n. The ideview of the interconnected cloed-loop tem i hown in Fig., where i i the water level in pool i and h i i the head over gate i. Baed on ma balance, a imple model of the water-level in pool i that capture the dnamic at low frequencie i

3 i+ DATUM h i gate i Fig.. obtained ee []: u i K i i K i pool i i r i h i gate i u i r i i h i pool i i Decentralied control of an open water channel G i : i = cie i u i ci u i +d i, where c i and c i are dicharge coefficient, function of the pool urface area and the width of uptream and downtream gate repectivel, and i i the internal time-dela that the water take to travel from the uptream end to the downtream end of the pool, u i := h / i i proportional to the flow over gate i, and d i i the water offtake diturbance. Denote the water level error a e i := r i i, where r i i the water level etpoint. Eentiall, the decentralied controller K i i a PI compenator: K i : u i = κ i + φi e i, with κ i > and φ i > ; the integrator i included for zero tead-tate water-level error in rejection to tep load diturbance d i, the phae-lead term help for cloed-loop tabilit. A previoul mentioned, the interaction between pool i.e. the flow out of pool i equal to the flow into pool i influence the global performance of the cloed-loop tem. Thi i repreented b the propagation of water level error and the amplification of control action in the uptream direction. To anale the above coupling propertie between pool, we tud a tring of identical pool with decentralied feedback control. In uch cae, G i : i = ce u c i ui +di, with c := c i and := i for i =,...,n. u i d i r i e i u i i K i e c d i r i e i u i i K i e c Fig.. Identical pool with decentralied feedback control Fig. how the configuration of a tring of identical pool with decentralied feedback control. The coupling tranfer function from one cloed-loop ubtem to the next one can be obtained a follow from the above pool model and We conider here open water channel with overhot gate, for which the flow over gate i can be approximated b c i h / i t []. Note that the dicuion in thi paper alo work for channel with underhot gate. the feedback controller: T ee,i := ei e i = cκ i +φ i + cκ i+φ i e T uu,i := ui cκi+φi u = 5 i + cκ i+φ i e Denote the coupling tranfer function from the firt downtream pool to the n th pool a E n := en e and F n := u n u. Definition.: Given a tring of n + pool under centralied or decentralied control, if lim n E n j < and lim n F n j < for all, the tem i aid to be tring table in term of bounded water level error and bounded flow. For a tring of pool with decentralied control one ha cκ i+φ i E n = T ee,i = +cκ i= i= i+φ ie 6 cκ i+φ i F n = T uu,i =. +cκ i+φ ie 7 i= i= A. Coupling of pool with identical feedback controller If one deign the decentralied controller baed on local performance and if one take the interaction between pool a an unknown diturbance, then for identical pool, it i natural to elect K i in the ame for i =,...,n, i.e. u i = κ + φ e i, 8 where κ and φ are elected b jut conidering the local cloed-loop tem: regulating the water level in a pool to it etpoint while rejecting the offtake diturbance in the pool. Then the coupling between neighbouring pool are: T ee = T uu = cκ +φ + cκ +φ e 9 Similar a Lemma in [], we have the following reult. Lemma.: For a tring of identical pool with identical feedback controller, there exit an > uch that T ee j > and T uu j >. Proof. The proof follow the line of the proof for Lemma 9. of []. We firt prove ln T ee j d. Denote L := cκ +φ, then T ee = Le +Le e. Correpondingl, T eej = Lj expj +Ljexpj for all R. Appling Cauch Theorem to the integral of the function F := ln Lexp +L exp along the tandard Nquit contour C with infiniteimal indentation C ǫ around the origin, we have Fd = = Fd+ Fd+ Fd, C C ǫ C C i

4 u where C i i the imaginar axi minu the indentation C ǫ. Since L ha two pole at the origin, the integral along C ǫ i. B traightforward calculation, the integral along C i equal to jπ. Uing the conjugate mmetr of the integrand and rearranging term, ield ln Lj expj +Ljexpj d = π >. Indeed, L i trictl proper, hence ln T ee j < at high frequencie. It follow from that there exit an,, uch that T ee j >. From 9, T uu j >. Note that for the tring of pool with identical feedback controller, E n = T ee n. Hence there exit an > uch that lim n E n j i unbounded. Similarl, there exit an > uch that lim n F n j i unbounded. Following Definition., Theorem.: Conider a tring of infinite number of pool controlled b identical decentralied feedback controller 8, the cloed-loop tem i tring untable. Let u conider a numerical example for a tring of identical pool. The model of the pool i given in with the coefficient c =.68 and the tranportation time dela = min. For local performance, elect κ =. and φ = 8. for the feedback controller 8. The of the coupling tranfer functiont ee,i andt uu,i, for i =,...,, are hown in Fig.. It i oberved that Fig.. T ee ij and T Frequenc rad/min ij Cloed-loop coupling with identical feedback controller max T ee,i j.8. The maximum occur at the ame frequenc, around. rad/min for all i =,...,. Hence e max e =.8 u and max u =.8, which i intolerable in practice. B. Coupling of pool with non-identical feedback controller In fact, a tring of n + identical pool with identical feedback controller involve the tronget coupling between pool, e.g. max T ee,i j > occur at the ame for all i, which make bounded water level error impoible. To decouple the interaction and hence for global cloed-loop performance, we conider non-identical feedback controller a follow: K : u = κ + φ e, K i : u i = κ +αi+ φ =i,, e i for i =,...,n with α >. Subtituting - into 6-7 reult in E nj = T ee,ij i +A ei+b e = i +Ci+D i= i= F nj = T uu,ij i +A f i+b f = i +Ci+D 5 for >, where i= i= A e = αc κ α + 8αc κ α α c + α c κ α c + φ + κ α B e = α c + α c A f = καc + 8κ αc α c + α c κ c + φ + κ c + φ B f = c C = καc 8αc α c + α c + 8κ αc α c + α c D = 6 + κ c 8κ c + +φ c α c + α c κ c 8φ c +φ c + φ c + c + φ c α c + α c Note that for =, lim E nj = lim F nj =. n n The following ufficient condition for bounded water level error in Lemma. and bounded flow in Lemma.5 ue propertie a and b of the Gamma function defined in 6. a If the real part of the complex number x i poitive i.e. Re[x] >, then the integral Γx := e t t x dt 6 converge abolutel. b Uing integration b part, Γx+ = xγx. Lemma.: For a fixed >, lim E nj exit if n A e >, C >, D > and A e < C. Proof. For the cae of A e = C, one ha [ ] E nj B e D = +. 7 i i i= When n, expreion 7 correpond to equation of [5], which give lim n Enj = ΓΓ Γx. 8 Γx Note the convergence of the RHS of 8 i not enured. Indeed, even for the cae that A e > and equivalentl Here we ue a firt-order Padé approximation to repreent the tranportation time-dela. Such an approximation doe not change the anali reult in practice given that the offtake diturbance that induce e i ignificant at low frequenc range, while the high-frequenc reonance caued b timedela are dampened b the feedback controller with a low-pa filter, ee Section IV.

5 C > and D >, and hence Re[x ] <, Re[x ] <, Re[ ] <, Re[ ] < ; one onl ha that Γx Γ x and Γ Γ converge repectivel, baed on the previou propert a of the Gamma function. However, Γ Γ Γx Γx might till diverge. For A e C, one ha E nj = i= + x + i x i 9 Fig.. a b c u i ui t A ui t i t + i A ui r i Control action for zero tead-tate water-level error where x, x are the root of x +A e x+b e =, and, are the root of +C +D =. Note that expreion 9 correpond to equation of [5], i.e. E nj = ΓΓ Γx Γx Γn+xΓn+x Γn+ Γn+. Appling the previou propertie a and b of the Gamma function to the RHS of, it i direct that when < A e < C and D >, lim E nj = ; n while the limitation doe not exit for the cae of A e > C. The lemma i thu proved. Remark : a For all >, the condition A e > hold if and onl if κ > α. b For all >, the conditiona e < C hold if and onl if α c 8α c < 8αc, which i equivalent to αc > αc. Since α >, c > and >, hold if αc. c Note the denominator of D > for >. The numerator of D can be written a κc + φc + φ c +φ c + κ c κc. For all >, the condition D > hold if κ c. From the above point a, b and c, if κ > α c, then the condition < A e < C and D > hold for all >. Similarl, one ha the following reult for bounded flow. Lemma.5: For a fixed >, lim F nj exit if n < A f < C and D >. Proof. The proof follow the ame line a the proof of Lemma.. Remark : For >, the conditiona f < C hold if and onl if < 8αc, which i impoible given the aumption that α >. In fact, under ditant-downtream control, to compenate the influence of the internal time-dela, the amplification of control action in the uptream direction i unavoidable. Thi i hown in Fig.. Initiall, the tem i at tead-tate. At time t, the flow out of pool i increae, ee the change of u i the dahed line in Fig. a. To compenate for the influence of u i on i, the flow into the pool, u i, alo increae the olid line in Fig. a. However, the influence of u i on the downtream water-level i will be i min later than that of u i on i ee Fig. b. For zero tead-tate error of i from r i ee Fig. c, u i hould be greater than u i for ome time uch that the area of A ui i equivalent to the area of A ui. Hence, there exit > uch that lim n F n j i unbounded. Then to have bounded water level error for infinite number of identical pool with decentralied control, the energ of the control action goe to infinit, which i impoible in practice. Indeed, for robut tabilit of the cloed-loop, one ha the condition on the cloed-loop bandwidth uch that b < ee []. However, with the condition that α c, the bandwidth of the tring of pool increae from downtream to uptream. Hence, for a tring of infinite number of pool, there exit an N <, uch that the temporal tabilit condition for the ubtem i > N i not atified. From the above dicuion and Definition., the following concluion i obtained. Theorem.6: For a tring of infinite number of pool controlled b the decentralied feedback controller -, the cloed-loop tem i tring untable. Conider the numerical example given in Section II-A for a tring of identical pool. Select κ =., φ = 8. and α =.9 for the feedback controller in -. The of the coupling tranfer function T ee,i andt uu,i, fori =,...,, are hown in Fig. 5. The decoupling function of appling non-identical feedback controller i oberved. Indeed, for all i =,...,, T ee,i j for all. Hence, we can expect a decreaing propagation of the water-level error in the uptream direction, which i confirmed b the top graph of E n j in Fig. 6. Furthermore, an attenuation of the amplification of the control action i.e. flow over gate i alo achieved, ee in the bottom graph in Fig. 6 that max u u = 7., while a analed in Section II-A, max u u =.8 for the cae with identical feedback controller. Note Be > for all >.

6 =i =i E F n n =i =i =i u =i =i =i Fig. 5. Waterlevelerorpropagationtranferfunction T Flowamplificationtranferfunction T Frequenc rad/min Frequenc rad/min ij ij ee Cloed-loop coupling with non-identical feedback controller Frequenc rad/min j j control, given that the temporal tabilit i enured for each ubtem, one can guarantee good global performance, i.e. management of the water level error propagation and attenuation of the amplification of flow over gate in the uptream direction, b enuring that the cloed-loop bandwidth increae from downtream to uptream. Remark : For a channel in which the pool length increae from uptream to downtream, the above condition that the cloed-loop bandwidth increae from downtream to uptream can be atified even b impl deigning the decentralied feedback controller jut baed on local performance. In realit, baed on the conideration of toring water to atif demand from farm, civil engineer deign irrigation network uch that the pool length, in general, tend to decreae from uptream to downtream. However, the previou guideline for decentralied feedback control deign hould till be kept in mind for a good tradeoff between local and global performance. IV. SIMULATION RESULTS In thi ection, imulation reult are hown for the cae of a tring of 5 identical pool with identical feedback controller and for the cae with non-identical feedback controller. In the imulation, a third-order model that capture the dominant wave-frequenc dnamic in the pool i ued. The parameter of the pool i given in Table I. 5 Saturation are et for gate poition and flow over gate. TABLE I Frequenc rad/min PARAMETERS OF THE POOL AND SATURATION VALUES SET Fig. 6. E nj and F nj for n =,..., with non-identical feedback controller III. FURTHER DISCUSSION In realit, the number of the pool in a channel i finite. When deigning decentralied feedback control for a tring of n+ < imilar pool in term of e.g. pool length, gate propertie, etc., one can firt elect κ, φ uch that the local performance in term of etpoint regulation i guaranteed. Then elect an α > uch that the tradeoff between local and global performance i managed, i.e. with the elected α, the bandwidth of the uptream cloed-loop b,n < n. Indeed, b including an α > in the non-identical feedback controller, the bandwidth of the cloed-loop ubtem increae in the uptream direction; hence, one can expect a fater repone of the interconnected tem to the offtake diturbance in the downtream pool than the cae with identical feedback controller. Note that in the ditributed control trateg dicued in [], [7], uch a peeding up of the cloed-loop repone i achieved b involving the known interaction between neighbouring pool in the input ignal to be rejected and b olving an optimiation problem to manage the tradeoff between local and global performance. One can extend the anali reult to channel with heterogeneou pool: For a channel with ditant-downtream Pool length Wave frequenc c 9 m. rad/min 6 min.9 Saturation of gate poition Saturation of flow max m.87 max Ml/da min m min Ml/da Correpondingl, the feedback controller involve an extra low-pa filter +.5 to guarantee no excitement of wave, i.e. a low gain at high frequencie. Hence, a the identical feedback controller are et a K i =.5+.8 for i =,,...,; +.5 b while the non-identical feedback controller are et a K =.5+.8, and for i =,..., +.5 K i =.5+.i Fig. 7 and 8 give the cloed-loop repone to an offtake diturbance in the downtream pool. Clearl, a much better decoupling performance i obtained b the trateg with the non-identical feedback controller. Fig. 7 how the water level error in the five pool when an offtake of 75 Ml/da at the downtream pool begin at time min. The water level etpoint for the pool are et 5 The parameter of the pool i the ame a that identified for pool of the Haughton Main Channel, ee [9].

7 / / / / / / / / / / r r Waterlevel m Waterlevel m Propagationofwaterleveleror withidenticalfeedbackcontroller Time min Propagationofwaterleveleror withnonidenticalfeedbackcontroller Time min Fig. 7. Water-level error propagation, with identical feedback controller top graph and with non-identical feedback controller bottom graph the ame: r =.5 m. Note that the local water level error in the downtream pool i.e. r i the ame for identical and non-identical feedback controller. With identical feedback controller the top graph, the water level error in the pool increae in the uptream direction. In the uptream pool, the maximum water level error caued b the offtake i.8 m. In contrat, with the non-identical feedback controller the bottom graph, the water level error in the pool decreae in the uptream direction. In the uptream pool, the maximum water level error caued b the offtake i.6 m. Flowovergate Ml/da Flowovergate Ml/da Amplificationofflow withidenticalfeedbackcontroller Time min Amplificationofflow withnonidenticalfeedbackcontroller Time min Fig. 8. Flow amplification, with identical feedback controller top graph and with non-identical feedback controller bottom graph Fig. 8 how the amplification of flow to compenate the influence of the offtake of 75 Ml/da at the downtream pool begin at time min. With identical feedback controller top graph, the amplification of flow i ignificant, e.g. the maximum flow over the mot uptream gate i Ml/da around 6 min; more erioul, the flow over the mot uptream gate goe beond aturation from 87 min to 7 min. While with non-identical feedback controller bottom graph, the amplification of flow over gate i well attenuated, e.g. the maximum flow over the mot uptream gate i Ml/da around 5 min. Note that, a expected, the control action in the uptream pool, i.e. ch / i t for i =,...,, in repone to the offtake diturbance are fater than the cae with identical controller. V. CONCLUSIONS Thi paper dicue the deigning of decentralied feedback controller for a tring of identical pool baed on the global performance of managing water-level error propagation and attenuating the amplification of flow over gate in the uptream direction. A definition of tring tabilit in term of bounded water level error and bounded flow i given. It i hown that for infinite number of pool with decentralied ditant-downtream feedback control, the cloedloop bandwidth limitation of each ubtem, impoed b the internal time-dela, make it impoible to achieve tring tabilit. However, for finite number of pool, b electing non-identical feedback controller uch that the cloed-loop bandwidth of the ubtem increae from downtream to uptream, a better global performance than that with identical feedback controller i achieved. Furthermore, the anali reult i extended to a tring of heterogeneou pool: In general, for ditant downtream control, the management of water-level error propagation and the attenuation of the amplification of flow over gate in the uptream direction require the cloed-loop bandwidth to increae from downtream to uptream. VI. ACKNOWLEDGEMENTS The author would like to thank Prof. Michael Cantoni of the Univerit of Melbourne for hi valuable comment regarding thi work. REFERENCES [] M. G. Bo, Dicharge meaurement tructure, Waageningen, The Netherland; 978. [] M. Cantoni, E. Weer, Y. Li, S.K. Ooi, I. Mareel and M. Ran, Control of large-cale irrigation network, Special Iue on the Technolog of Networked Control Stem, Proceeding of the IEEE, vol. 95, 7, pp [] A. J. Clemmen and J. Schuurman, Simple optimal downtream feedback canal controller: theor, Journal of Irrigation and Drainage Engineering, vol.,, pp. 6-. [] G. C. Goodwin, S.F. Graebe and M.E. Salgado, Control Stem Deign, Prentice Hall, Englewood Cliff, NJ;. [5] E. R. Hanen, A Table of Serie and Product, Prentice Hall, Englewood Cliff, NJ; 975. [6] I. Mareel, E. Weer, S. K. Ooi, M. Cantoni, Y. Li, and G. Nair, Stem engineering for irrigation tem: Succee and challenge, Annual Review in Control, vol. 9, pp. 9, Augut 5. [7] Y. Li and M. Cantoni, Ditributed controller deign for open water channel, Proceeding of the 7th IFAC World Congre, Seoul, South Korea, Jul 8, pp. 8. [8] X. Litrico and V. Fromion, Advanced control politic and optimal performance for an irrigation canal, Proceeding of the ECC, Cambridge, UK,. [9] S.K. Ooi and E. Weer, Cloed loop identification of an irrigation channel, Proceeding of the th IEEE CDC, Orlando, USA,, pp. 8-. [] S. Skogetad and I. Potlethwaite, Multivariable Feedback Control, John Wile and Son, Chicheter, UK; 996. [] E. Weer, Stem identification of an open water channel, Control Engineering Practice IFAC, vol. 9,, pp [] E. Weer, Control of irrigation channel, IEEE Tranaction on Control Stem Technolog, vol 6, Jul 8, pp