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Proceedings of the American Control Conferenceâ¨Chicago, Illinois â¢ June 2000 Maceration Control of a Sugar Cane Crushing Mill i Minyue Fu1 Graham C. Goodwin1 Turker Ozkocak JUICE CIRCUIT AT MACKNADE - as or 1998 â fifed wÂ«ur Abstract eft,0 -- & jz T1 sees J I 4 filth ^mitl T4 SOCS ; (uvnb mill; T3 SOCO fifed BfeUj Raw sugar is produced from juice in sugar cane crushedâ¨by a series of mills. To improve the extraction a liqÂ¬â¨uid bath is applied to the cane between mills. Thisâ¨liquid bath, commonly called maceration, consists ofâ¨water and some of the produced juice. Although theâ¨extraction is improved with higher water content inâ¨the maceration, the total juice output is restricted byâ¨the storage capacity of the plant. The aim of macerÂ¬â¨ation control is to manipulate the added water withinâ¨the process limits placed by the storage capacity whilstâ¨optimising sugar extraction. In this paper, mathematÂ¬â¨ical models of the processes pertaining to macerationâ¨are derived from first principles. A cascaded modelâ¨predictive controller is then designed using the derivedâ¨models. fixed ifcijy / VKiahk 2ndMJ delay Heater ESi overflow 2ndMJ -Heater ESi u - Clarifier m Figure 1: Process Schematics in a boot is controlled by a boot valve which splits theâ¨juice from the following mill between that boot andâ¨the preceeding boot. The part of the second mill juiceâ¨which is not returned back to boot no. 1 and the firstâ¨mill juice are then sent to a screen which filters tinyâ¨fibres of bagasse. The screened juice is next sent toâ¨downstream processes, e.g. the evaporators. The totalâ¨capacity of juice storage before the evaporators exertsâ¨an upper bound on the nett juice output of the millingâ¨train. See Fig. 1 for a schematic of the process. Key words: Model predictive control, modelling,â¨sugar mills, maceration 1 Introduction The function of a raw sugar factory is to produce crysÂ¬â¨tal sugar from the juice in sugar cane delivered toâ¨the factory [1]. The extraction process in Australiaâ¨is mostly done by crushing mills. The prepared cane isâ¨passed through a series of mills called the milling train,â¨see Fig. 1. The mills crush the cane to separate theâ¨juice which contains the sugar from its fibrous part.â¨The fibrous material left after the juice is removed isâ¨called bagasse. To help the extraction of juice, someâ¨of the produced juice is returned to the bagasse beÂ¬â¨tween the mills. Water is also added before the lastâ¨mill to wash out any remaining sugar. Measurementsâ¨of bagasse mass and fedback juice flow between millsâ¨are often not available, making it difficult to estimateâ¨mass balance on-line. Bagasse is carried by a fixedâ¨speed inter-carrier between any two mills. From theâ¨carrier, the bagasse is fed into the next mill. At theâ¨exit of a mill the bagasse dives into a boot where itâ¨absorbs the feedback juice or water. The liquid level 2 Integration of three subprocesses Regarding the control and constraints the problem canâ¨be considered as an interaction of three subprocesses: 2.1 Juice subsystem After dry crushing in the first mill, the crushing proÂ¬â¨cess can be seen as the replacement of cane juice withâ¨maceration, see Fig. 1. The juice subsystem formuÂ¬â¨lates the juice circuit of the milling train where theâ¨juice and maceration flows axe defined in terms of theâ¨added water flow, transport lags and boot valves: Ji+i(t) â ~ Ti)(l â Vi(t â Ti)) + u(t â Tj) Ji+2(t)Vi+i(t) for i = 1,2,3â¨addw(t) for i = 4 { u(t) = (1) where Ji and V; are the ith mill juice and ith boot valveâ¨opening respectively, Tj is the transport lag from the ithâ¨boot to the following mill, mill no (i + 1), and addw is department of Electrical and Computer Eng., The UniverÂ¬â¨sity of Newcastle, Callaghan 2308, Australia 2255 0-7803-5519-9/00 $10.00 Â© 2000 AACC The total juice sent to downstream tanks, Ja can beâ¨written as follows: the volumetric flow of the water added to the last boot.â¨The left side of Eqn. 1 formulates the maceration flowsâ¨subject to transportation delays. Next, we rewrite Eqn.â¨1 in discrete-time state space form. To this end, weâ¨choose a sampling time Ta and assume that (=71,)â¨is approximately an integer for all n = 1, â â¢ â ,4. Withâ¨this assumption, Eqn. 1 is approximated by: J.[k) = Ji[k] + J2[fc]Vi[fc] (6) Ji[k] is the measured juice output of the first mill.â¨J2[k]Vi[k] is the portion of the juice output of the secÂ¬â¨ond mill which is sent to downstream process. Vi[fc] isâ¨the first boot valve and J2[k] is estimated from Eqn.â¨1. Valve position, V) of each boot is available online.â¨Although the boot valve is a nonlinear element, its non-â¨linearity is well defined. Time varying transport lags,â¨Ti are found from the variable surface speeds of theâ¨rolls, fixed speeds of the carriers, and related distances.â¨They appear as parameters in our model, but their calÂ¬â¨culations will be omitted in this paper. Xi[k + 1] = Ai[k]Xi[k\ + Bi[k]Ji+2[k - m] (2) where 0 0 1 0 0 0 Ai[k\ = 0 1 0 _ 1- Vi[k-Jii] 0 0 2.2 Fibre subsystem The fibre subsystem defines the mass flow of fibre atâ¨the boots along the milling train. Those mass flowsâ¨define the target maceration for proper crushing. Theâ¨fibre rate is calculated from the first mill juice and theâ¨cane input which are the only flow measurements availÂ¬â¨able online. This is then delayed using the necessaryâ¨transport lags to calculate the fibre rate at each boot: mfi(t) = (mi(t - D0i) - - D02)) fbgi mfi(t) = - Di-1) fori = 2,3,4 where Sji and fbgi axe the assumed values of the densityâ¨of the first mill juice and the fibre ratio of the bagasse,â¨mi is the mass flow of input cane, Ji is the first millâ¨juice flow, Â£>oi is the transport lag from the point whereâ¨the mass flow of input cane is measured to the first bootâ¨and Â£>02 is the transport lag from the flow measurementâ¨of the first mill juice to the first boot. The other delay,â¨Â£>i, is the transport lag from the ith boot to the (i + 1)4'1â¨boot. The variable transport lags used by the fibreâ¨subsystem are calculated as in the juice subsystem. 0 - Ti) Xi[k\ â Bi[k] = 0 Ji+l(t~ 1) . _ Vi+1 [k-rii] _ addw(t â T4) can also be represented in discrete-timeâ¨state space form: Xw[k + 1] = Aw[k]Xw[k] + Bw[k]addw[k\ (3) (7) Expanding each row of Eqn. 1 and augmenting it withâ¨the state space representation for addw(tâT4), we have: " Ai Mi O A2 o â â â o o o o + [ O â¢â¢â¢ O Bw ]Taddw[k]â¨X[k + 1] = AjdX[k] + BjdaAdw[k] O ' *i[*] " ' Xi[k + 1] ' XA[k + 1]â¨_ Xw[k + 1] _ o o X4[k] Mi . Xw[k) _ Ayj (4) Outputs of the system are chosen as the macerationâ¨flows to all four boots, the juice output of the secondâ¨mill and immediate macerations. The flow of the secÂ¬â¨ond mill juice is chosen because it makes up for theâ¨portion of the total juice output which can be conÂ¬â¨trolled by added water. The outputs of the system canâ¨be written as follows: 2.3 Tank subsystem The dynamics of the downstream tanks; 1st and 2ndâ¨mixed juice and ESJ tanks, provides a mechanism toâ¨calculate the upper bound placed on the nett juiceâ¨output of the milling train, see Fig. 1. Because ofâ¨their closely coupled operation and the lack of meaÂ¬â¨surements of intermediary flows, they can be lumpedâ¨into a fictitious dynamic tank. The volume of this tank,â¨Voldtnk is the total volume of the tanks. If the radiusâ¨and height of this fictitious dynamic tank are tdtnkâ¨and hpTNK respectively, the nett juice flow into theâ¨tank subsystem, Js(t) â fESj(t), determines the levelâ¨of the juice in the dynamic tank: dhpTNKjt) y[k] = [ O a O c2 O c3 O a O ] X[fc] + [00010---0 ]T addw[k)â¨y[k] = CjdX[k] + Djdaddw[k] (5) where the nonzero columns, Cj are given as follows: a = [1-Vi[fc] 1 1 - Vi[fc] 0 0 0 ]T c2 = [ V2[k] 1 -V2[k] â¢â¢â¢ 1-V2[fc] 0 0 ]T c3 = [ 0 V3[k] 1 - V3[fc] â¢â¢â¢ 1-V3[k] 0 ]T c4 = [ 0 0 V4[k] 1 -V4[k] â â â l-V4[fc]]T 1 (Ja(t) - fESj(t)) (8) TtroTNK2 Above Ja(t) is the total juice entering the tank subsysÂ¬â¨tem as calculated from Eqn. 6 and fESj(t) is the flow dt 2256 out of the ESJ tank (i.e. the flow exiting the tankâ¨subsystem). This continuous time first order systemâ¨can be written in discrete time state space form with aâ¨sampling time Ts as: process to the target state in finite time. In reality onlyâ¨the very first calculated input is applied to the plant.â¨With that input and the new process state, another setâ¨of input values are calculated in the next cycle. Sinceâ¨an input profile over a finite time into the future is calÂ¬â¨culated, the dynamics of the process must be embeddedâ¨with a model of the process into the MPC algorithm.â¨If the model of the process is chosen to be linear withâ¨linear constraints then the algorithm reduces to linearâ¨model predictive control. Ts x[A; + 1] = x[&] + 2 <#] nroTNK 2/[&] = x[k] (9) In the discrete representation above, the state, x[A;] isâ¨taken as the deviation of the tank level off the setpoint. 4.2 Linear MPC algoritm The MPC algorithm used in this paper was proposed byâ¨Kenneth R. Muske and James Rawlings, [5]. We outÂ¬â¨line the method below for completeness. The systemâ¨subject to linear MPC is assumed to have the followingâ¨description: 3 Maceration control Extraction increases with increased amounts of macÂ¬â¨eration. However there is a desired ratio, kdesi, beÂ¬â¨tween the added maceration and the fibre content ofâ¨the bagasse for each boot i, beyond which the addiÂ¬â¨tional sugar extraction is negligible. This ratio mustâ¨be preserved at each boot. Hence the target maceraÂ¬â¨tion flow into each boot is determined by the fibre flow,â¨mfi[k], at that boot as calculated by the fibre subsysÂ¬â¨tem, see Eqn. 7. Whilst meeting the target maceraÂ¬â¨tion flows into the boots, the total juice flow, Ja, outâ¨of the milling train must be appropriate in order toâ¨maintain the tank levels within their safety limits. Asâ¨represented by the juice model, see Eqn. 1, other thanâ¨the first mill, the juice output of each mill and theâ¨maceration flow into each boot can be formulated asâ¨a function of added water flow, related transport lagsâ¨and boot valves. Accordingly the only component of Jsâ¨that can be controlled by the added water is the flowâ¨of the second mill juice, see Eqn. 6. The target flowâ¨of the second mill juice, J2d.es is calculated by a conÂ¬â¨troller based on the tank subsystem. The target valuesâ¨for the output of the discrete juice subsystem, yt, asâ¨represented by Eqns. 4 and 5, can be written as: Xk+1 = Axjfc + Bukâ¨yk = Cxk + Duk (11) The control is then based on the minimisation of theâ¨following infinite horizon quadratic objective functionâ¨at time k. 00 rmn ^ ((yk+j - yt)TQ(yk+j ~ Vt) j=0 +(uk+j - ut)TR(uk+j - ut) + Auk+jTSAuk+j) (12)â¨subject to constraints: umin ~ Ufc+j < Uâ¨Dmin ^ Ulc+j â Vâ¨AUmin < AUk+j < AU Here ji and j? provide a mechanism for relaxing theâ¨output constraints to circumvent possible temporaryâ¨infeasibilities. Since input constraints are defined byâ¨the limits of the actuators, relaxing the outputs is theâ¨only option. If infeasibility is unavoidable, the outputâ¨constraints are relaxed until the time, j\. The conÂ¬â¨straints are then applied between j\ and j2- 32 is theâ¨earliest possible time to guarantee the satisfaction ofâ¨the output constraints thereafter. In Eqn. 12, Q is aâ¨symmetric positive semidefinite penalty matrix on theâ¨outputs. R is a symmetric positive definite penalty maÂ¬â¨trix on the inputs. S is a symmetric positive semidefiÂ¬â¨nite penalty matrix on the rate of change of inputs inâ¨which Auk+j = uk+j - uk+j-i is the change of inputâ¨vector at time j. yt is the target output. Then ut andâ¨xt are the target input and state vectors, respectively,â¨which hold the output of the system in Eqn. 11 at theâ¨value yt with the minimum offset [5]. The solution ofâ¨the quadratic programme, uN contains N future controlâ¨moves as shown below: jv r u = [ uk uk+i â â â Ufc+V-l J j = 0,1, - - - , AT â 1â¨3 = 31,31 + 1, â¢ â¢ â¢ ,32â¨3 = 0,1,". ,N max i max > (13) max > 2It = [kdesimfi[k] â¢â¢â¢ kdesitnfilk] J2des[k]] (10) 4 Model predictive control Model predictive control, MPC, has been used in aâ¨large number of industrial applications and is freÂ¬â¨quently considered to be the first option for the controlâ¨of processes with hard constraints. Since our final aimâ¨is the control of a real plant by digital systems, theâ¨discussion here is restricted to the discrete time case. 4.1 Linear model predictive control A key feature of MPC is its ability to handle hard conÂ¬â¨straints on the outputs and inputs. Based on the curÂ¬â¨rent state of the process, previous input, and targetâ¨state and input values, the MPC algorithm calculatesâ¨a set of input values which are intended to bring the (14) 2257 After time k+Nâ1, the input is assumed to be constantâ¨at ut. Recall that only uk is applied to the plant. Withâ¨jik+j computed from Eqn. 11: juice MPC is given in Eqn. 10. Output of the juiceâ¨MPC is the optimal added water flow over the definedâ¨finite horizon. Only the first value in the profile is apÂ¬â¨plied to the plant. Tank MPC is configured with theâ¨tank model as calculated in Eqn. 9. Target outputâ¨for the tank MPC is zero, i.e., the tank level shouldâ¨be the steady state value achieved with zero nett flow.â¨Output of the tank MPC is the optimal profile of theâ¨nett flow into the tank subsystem over the related fiÂ¬â¨nite horizon. The target nett flow is the first value ofâ¨the profile. The target juice output of the mill, Jsdeaâ¨is found by the addition of the calculated target nettâ¨flow and the measured outflow, fesj. The target juiceâ¨output of the second mill, J2des is then found fromâ¨Eqn. 6. Both juice and tank subsystems are single inÂ¬â¨put. Accordingly the finite horizons for both systemsâ¨can be taken short (e.g. 2 samples) and the solution toâ¨all quadratic programs can be taken as unconstrainedâ¨minimums. The solutions are saturated at the relatedâ¨Umax and umin values as explained in Section 4. Theâ¨main difference between the Juice MPC and Tank MPCâ¨is the fact that the juice subsystem, (see Eqns. 4 andâ¨5) is stable where the dynamic tank subsystem, Eqn.â¨9 is unstable. Eventually the calculation of cost maÂ¬â¨trices, H and G (see Eqn. 18), are slightly differentâ¨for the tank subsystem [5j. The design has been simuÂ¬â¨lated for variable fibre rate where all the other processâ¨variables are kept constant. The variable fibre rate wasâ¨simulated by varying the first mill juice, see Fig. 2. j-1 Vk+j = C(A>xk + y Aj 1 1 Buk+i) + Duk+j (15) i=o and by redefining system states and inputs as: (16) Xk â Xk Xt, Uk â Uk Ut the objective function of Eqn. 12 can be rewritten inâ¨terms of the input, iiN only: min unTHun + 2unT (Gxk â Fuk-1) (17) UN The cost matrices H, G and F are functions of matricesâ¨of the system representation and MPC parameters: H = H(A,B,C,D,Q,R,S,N)â¨G = g(A,B,C,D,Q,N), F = F(S,N) (18) As the objective function, the constraints (see Eqn. 13)â¨must also be rewritten in input profile: auN < P where a and (3 are functions of: (19) P = P(A,B,C,D,N, ji,j2, constraints, Uk-i,ut) (20) a = a(A,B,C,D,N,j1,j2) However the constraints given in Eqn. 20, depend onâ¨the system representation (A,B,C,D). Hence theyâ¨are vulnerable to model errors. Online solution ofâ¨quadratic programs also poses a problem. For sysÂ¬â¨tems driven by single input, a quick remedy for theseâ¨problems is to define the finite horizon N rather short.â¨Then the unconstrained global minimum can be takenâ¨as the solutions of the related quadratic programs. Theâ¨constraints are then applied by simply saturating theâ¨controller at the limits, umax and um,n, [2]. The unÂ¬â¨constrained global minimum of Eqn. 17 is given as: u* = -H^iGik -Fuk-1) 6 Discussion of results For the purpose of presentation, the maceration controlâ¨of the third boot together with the MPC controlledâ¨water flow is summarised in Fig. 2. While the changingâ¨fibre rate is closely tracked, the dynamic tank level isâ¨also kept at the target value, see Fig. 3. The nett flowsâ¨for the tank subsystem are shown in Fig. 4. (21) 7 Conclusions 5 MPC design for maceration We can now use the linear discrete time state spaceâ¨model for the maceration process as in Section 2 to apÂ¬â¨ply the MPC algorithm. The proposed design consistsâ¨of two cascaded linear MPCs: Juice MPC to regulateâ¨the added water and Tank MPC to estimate the targetâ¨second mill juice, J2des for the Juice MPC. Juice MPCâ¨uses the representation given for the juice subsystemâ¨in Eqns. 4 and 5. The target output vector for the Maceration control of a sugar cane crushing plant hasâ¨been examined. The maceration process involves threeâ¨subsystems: juice circuit, fibre flow and storage tanks.â¨All three subsystems have been modeled. Based onâ¨the models a cascaded linear model predictive controlâ¨scheme has been proposed, which is expected to provideâ¨a better integration of the related subsystems, henceâ¨improving sugar extraction. 2258 500 500 J1 fesj 450 â¢ mf3 480 - target Js Js mac3â¨addw . 400 â¢ 460 - i 350 " 440 - f r i | 300 i \ 420 - ! s 4'; I 400â¨E | 250 - 4 1 I i / 200 - 380 - f 360 340 - 100 - ~\ Y 50-: 320 - _ / 1 500 1500 isoo 300 0 2000 500 1000 1500 2000 2500â¨samples - 2 sees 3000 3500 4000 1000 2500 3000 4000 samples - 2 sees Figure 2: Maceration control of boot no 3 Figure 4: Flows for the dynamic tank of crushing sugar cane," Elsevier Publishing Company,â¨New York . 2.95 setpointâ¨level [4] Loughran, J.G. (1990), "Mathematical and exÂ¬â¨perimental modelling of the crushing of prepared sugarâ¨cane," PhD thesis, Mechanical Engineering, Universityâ¨of Queensland, Queensland, Australia. 2.9 - 2.85 â¢ 2.8 [5] Muske, R. and Rawlings, J. B. (1993) "Model Predictive Control with Linear Models," AIChE JourÂ¬ nal, 39(2), 262 2.75 â¢ [6] Hugot, E. (1972), "Handbook of cane sugar engiÂ¬â¨neering," Elsevier Publishing Company, New York, pp. 2.7 - 144-149. 2.65 â [7] West, M. (1997), "Modelling and control of aâ¨sugar cane crushing mill", MEECE Thesis, Electricalâ¨and Computer Engineering, The University of NewcasÂ¬â¨tle, New South Wales, Australia. [8] Partanen, A.G. (1995), "Controller refinementâ¨with application to a sugar cane crushing mill," PhDâ¨thesis, Systems Engineering, Research School of InforÂ¬â¨mation Sciences and Engineering, The Australian NaÂ¬â¨tional University. 2.6 - 500 1000 1500 2000â¨samples - 2 sees 2500 3000 3500 4000 Figure 3: Dynamic tank level 8 Acknowledgements The authors gratefully acknowledge invaluable assisÂ¬â¨tance from CSR especially Rob Peirce in the executionâ¨of this project. References [1] Ozkocak, T. and Fu, M. and Goodwin, G. C.â¨(1998), "A nonlinear modelling approach to the sugarâ¨cane crushing mill," Proc. CDC98. [2] De Dona, J. A. and Goodwin, G. C. and Seron,â¨M. M. (1998), "Connections between model predictiveâ¨control and anti-windup strategies for dealing with satÂ¬â¨urating actuators," Proc. ECC99. [3] Murry, C. and Holt, J. (1967) "The Mechanics 2259