Advanced Model Predictive Control Part 12 potx

Model Predictive Control and Optimization for Papermaking Processes 319 that the highest costing MV’s are driven to their minimum operating points, and the lowest costing MV’s are driven to their maximum operating points. The TAD1 dry end differential pressure is left as the MV that is within limits and actively controlling the paper moistures. Figure 10 shows that throughout this trial, the MV’s are optimized without causing any disturbance to the CV’s. Fig. 5. The MPC model matrix for the tissue machine control and optimization example Fig. 6. Natural gas costs and electricity costs during the trial Stock Flow TAD1 Supply Temp TAD1 DE DP TAD1 Gap Pres TAD2 Exh Temp TAD2 DE DP TAD2 Gap Pres Yankee Hood Temp Yankee Supply Fan S p eed Machine Speed Stock Flow TAD1 Gap Pressure Tickler Refiner Dry Weight Reel Moisture TAD Moisture TAD1 Exhaust Pressure Advanced Model Predictive Control 320 Fig. 7. Total costs during the trial Table 2. The MV cost rankings. MV eng unit Low Limit High Limit Linear Obj Coef (Cost/eng unit) Process Gain (%Moi/eng unit) Cost (Cost / % Moi) Rank Optimization Behavior TAD1 Supply Temp deg F 300.0 450.0 0.68 -0.12 5.48 4 450 (max) TAD1 DE DP inch H2O 1.0 3.9 47.30 -5.12 9.24 3 controlling Moi TAD1 Gap Prs inch H2O 0.4 1.5 -0.03 1.95 0.02 6 0.4 (max) TAD2 Exh Temp deg F 175.0 250.0 5.86 -0.45 13.02 1 175 (min) TAD2 DE DP inch H2O 1.0 3.5 40.26 -3.14 12.82 2 1 (min) TAD2 Gap Prs inch H2O 0.2 1.5 -16.40 4.25 3.86 5 0.2 (max) Model Predictive Control and Optimization for Papermaking Processes 321 Fig. 8. Manipulated variables during the trial Fig. 9. Manipulated variables during the trial Advanced Model Predictive Control 322 Fig. 10. Controlled variables during the trial 3.4 Grade change strategies Grade change is a terminology in MD control. It refers to the process of transitioning a paper machine from producing one grade of paper product to another. One can achieve a grade change by gradually ramping up a set of MVs to drive the setpoints of CVs from one operating point to another. During a grade change, the paper product is often off- specification and not sellable. It is important to develop an automatic control scheme to coordinate the MV trajectories and minimize the grade change transition times and the off- spec product. An offline model predictive controller can be designed to produce CV and MV trajectories to meet these grade change criteria. MPC is well-suited to this problem because it explicitly considers MV and CV trajectories over a finite horizon. By coordinating the offline grade change controller (linear or nonlinear) and an online MD-MPC, one can derive a fast grade change that minimizes off-spec production. This section discusses the design of MPC controllers for linear and nonlinear grade changes. Figure 11 gives a block diagram of the grade change controller incorporated into an MD control system. The grade change controller calculates the MV and CV trajectories to meet the grade change criteria. This occurs as a separate MPC calculation performed offline so that grade change specific process models can be used, and so that the MPC weightings can be adjusted until the MV and CV trajectories meet the design criteria. The MV trajectories are sent to the regulatory loop as a series of MV setpoint changes. The CV trajectories are sent as setpoint changes to the MD controller. If the grade change is performed with the MD controller in closed-loop, additional corrections to the MV setpoints are made to eliminate any deviation of the CV from its target trajectory. Controlled Variables 11.6 11.7 11.8 11.9 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 8:34:07 8:40:34 8:47:01 8:53:28 8:59:55 9:06:22 9:12:49 9:19:16 9:25:43 9:32:10 9:38:37 9:45:04 9:51:31 9:57:58 10:04:25 10:10:52 10:17:19 10:23:46 10:30:13 10:36:40 10:43:07 10:49:34 10:56:01 11:02:28 11:08:55 11:15:22 11:21:49 11:28:16 11:34:43 11:41:10 11:47:37 11:54:04 12:00:31 12:06:58 12:13:25 12:19:52 Time DW (lb/ream) 0 5 10 15 20 25 Moisture (%) ReelDwt PV ReelMoi PV ExpressMoi PV Model Predictive Control and Optimization for Papermaking Processes 323 Fig. 11. Block diagram of MD-MPC control enhanced with grade change capability The MV and CV trajectories are generated in a two step procedure. First there is a target calculation step that generates the MV setpoints required to bring the CV’s to their target values for the new grade. Once the MV setpoints are generated, then there is a trajectory generation step where the MV and CV trajectories are designed to meet the specifications of the grade change. The MV targets are generated from solving a set of nonlinear equations: () () () () 11 dw dw 1 2 3 1 2 3 22 dw dw 1 2 3 1 2 3 11 moi moi 1 2 3 1 2 3 22 moi moi 1 2 3 1 2 3 y f u ,u ,u , ,C ,C ,C , 0, y f u ,u ,u , ,C ,C ,C , 0, y f u ,u ,u , ,C ,C ,C , 0, y f u ,u ,u , ,C ,C ,C , 0, −……= −……= −……= −……=   (11) Here y dw /y moi represents the CV target for the new grade. The functions f ( ∙ ) are the models of dry weight and moisture. The process MV’s are denoted u i and model constants are denoted C i . The superscripts indicate the same paper properties measured by different scanners. Since the number of MV’s and the number of CV’s is not necessarily equal, these equations may have one, multiple or no solutions. To allow for all of these cases, the problem is recast as: min   F ( u  ,u  ,… ) , (12) Subject to: G ( u  ,u  ,… ) ≤0, H ( u  ,u  ,… ) =0, Grade Change Controller Process Scanner R 1 R 2 MD - MPC Operator Operator u 1,OP u 2,OP y 2 y 1 u C1,SP u 1,SP Δu 1,GC u 2,SP u C2,SP y 1,SP y 1,SP y 2,SP y 2,SP User Input Advanced Model Predictive Control 324 Where F ( ∙ ) is a quadratic objective function formulated to find the minimum travel solution. H ( ∙ ) represents the equality constraints given above, and G ( ∙ ) represents the physical limitations of the CVs and MVs (high, low, and rate of change limits). Once the MV targets have been generated, the MV and CV trajectories are then designed. Figure 12 gives a schematic representation of the trajectory generation algorithm. The process models are linearized (if necessary) and then scaled and normalized for application in an MPC controller. Process constraints such as the MV and CV targets, and the MV high and low limits are also given to the MPC controller. Internal controller tuning parameters are then used to adjust the MV and CV trajectories to meet the grade change requirements. Fig. 12. Diagram of MPC-based grade change trajectory generation. 3.4.1 Linear grade change In a linear grade change, the MD process models that are used in the MD-MPC controller are also used as the models for determining the MD targets, and for designing the MD grade change trajectories. 3.4.2 Nonlinear grade change In a nonlinear grade change, a first principles model may be used for the target and trajectory generation. For example, a simple dry weight model is: m  =K    , (13) MPC Module Linearization Nonlinear Model MPC Controller Process Model Scaling & Normalization Grade Change Constraints • MV physical limits • MV and CV targets • Customized Weights CV Target CV Trajectories MV Trajectories Model Predictive Control and Optimization for Papermaking Processes 325 Where m  is the paper dry weight, q  is the thick stock flow, and v is machine speed. K is the expression of a number of process constants and values including fibre retention, consistency, and fibre density. (Chu et al. 2008) gives a more detailed treatment of this dry weight model. (Persson 1998, Slätteke 2006, and Wilhelmsson 1995) are examples of first principles moisture models that may be used. 3.4.3 Mill implementation results In this section, some results of MPC-based grade changes for a fine paper machine are given. The grade change is from a paper with a dry weight of 53 lb/3000ft 2 (86 g/m 2 ) to a paper with a dry weight of 44 lb/3000ft 2 (72 g/m 2 ). Both paper grades have the same reel moisture setpoint of 4.8%. For the grade change, stock flow, 6 th section steam pressure, and machine speed are manipulated. Figures 13 and 14 show a grade change performed on the paper machine using linear process models, and keeping the regular MPC in closed-loop during the grade change. The grade change was completed in 10 minutes, which is a significant improvement over the 22 minutes required by the grade change package of the plant’s previous control system. In Figure 13, the CV trajectories are shown. Here it can be seen that although there is initially a small gap between the actual dry weight and the planned trajectory, the regular MPC takes action with the thick stock valve (as shown in Figure 14) to quickly bring dry weight back on target. The deviation in the reel moisture is more obvious. This might be expected as the moisture dynamics of the paper machine display more nonlinear behaviour for this range of operations. The steam trajectory in Figure 14 is ramping up at its maximum rate and yet the paper still becomes too wet during the initial part of the grade change. This indicates that the grade change package is aggressively pushing the system to achieve short grade change times. Fig. 13. CV trajectories under closed-loop GC with linear models Advanced Model Predictive Control 326 Fig. 14. MV trajectories under closed-loop GC with linear models Figures 15 and 16 show a grade change performed on a high fidelity simulation of the fine paper machine. This grade change uses a nonlinear process model, and the regular MPC is kept in closed-loop during the grade change. Here it can be seen that the duration of the grade change is reduced to 8 minutes. Part of the improvement comes from using stock flow setpoint instead of stock valve position, allowing improved dry weight control. Another improvement is that the planned trajectories allow for some deviation in the reel moisture that cannot be eliminated. Both dry weight and reel moisture follow their trajectories more closely. At the end of the grade change, the nonlinear grade change package is able to anticipate the need to reduce steam preventing the sheet from becoming dry. Fig. 15. CV trajectories under closed-loop GC with nonlinear models Model Predictive Control and Optimization for Papermaking Processes 327 Fig. 16. MV trajectories under closed-loop GC with nonlinear models 4. Modelling, control and optimization of papermaking CD processes To produce quality paper it is not enough that the average value of paper weight, moisture, caliper, etc across the width of the sheet remains on target. Paper properties must be uniform across the sheet. This is the purpose of CD control. 4.1 Modelling of papermaking CD processes The papermaking CD process is a large scaled two-dimensional process. It involves multiple actuator arrays and multiple quality measurement arrays. The process shows very strong input-output off-diagonal coupling properties. An accurate CD model is the prerequisite for an effective CD-MPC controller. We begin by discussing a model structure for the CD process and the details of the model identification. 4.1.1 A two-dimensional linear system The CD process can be modelled as a linear multiple actuator arrays and multiple measurement arrays system, Y(s)=G(s)U(s)+D(s), (14) and Y(s)= y  (s) ⋮ y   (s) ,G(s)= G  (s) … G   (s) ⋮⋱⋮ G    (s) … G     (s) , U(s)= u  (s) ⋮ u   (s) , (15) Advanced Model Predictive Control 328 where Y(s)∈ℂ (  ⋅) is the Laplace transformation of the augmented CD measurement array. The element y  (s)∈ℂ  (i=1,…,N  ) is the Laplace transformation of the i th individual CD measurement profile, such as dry weight, moisture, or caliper. N  is the total number of the quality measurements, and m is the number of elements of individual measurement arrays. U(s)∈ℂ ( ∑      ) is the Laplace transformation of the augmented actuator setpoint array. The element u  (s)∈ℂ   (j=1…N  ) is the Laplace transformation of the j th individual CD actuator setpoint profile, such as the headbox slice, water spray, steam box, or induction heater. N  is the total number of actuator beams available as MV’s, and n  is the number of individual zones of the jth actuator beam. In general a CD system has the same number of elements for all CD measurement profiles, but different numbers of actuator beam setpoints. D(s) ∈ℂ (  ⋅) is the Laplace transformation of the augmented process disturbance array. It represents process output disturbances. G  (s)∈ℂ ×  (i=1…N  andj=1…N  ) in (15) is the transfer matrix of the sub-system from the j th actuator beam u  to the i th CD quality measurement y  . The model of this sub- system can be represented by a spatial static matrix P  ∈ℝ ×  with a temporal dynamic transfer function h  (s). In practice, h  (s) is simplified as a first-order plus dead time system. Therefore, G  (s) is given by G  (s)=P  h  (s)=P      e    (16) where T  is the time constant and T  is the time delay. The static spatial matrix P  is a matrix with n  columns, i.e., P  = [ p  p  ⋯p   ] and its kth column p  represents the spatial response of the k th individual actuator zone of the j th actuator beam. As proposed in (Gorinevsky & Gheorghe 2003), p  can be formulated by, 22 kk 22 α((x x ) βω) α((x x ) βω) k ωω kk g ππ p{e cos(((xx)βω)) ecos(((xx)βω))} 2 ωω −− −+ −− =−−+−+ (17) where x is the coordinate of CD measurements (CD bins), g is the process gain, ω is the response width, α is the attenuation and β is divergence. x  is the CD alignment that indicates the spatial relationship between the centre of the k th individual CD actuator and the center of the corresponding measurement responses. A fuzzy function may be used to model the CD alignment. Refer to (Gorinevsky & Gheorghe 2003) for the technical details. Figure 17 illustrates the structure of the spatial response matrix P  . The colour map on the left shows the band-diagonal property of P  ; and the plot in the right shows the spatial response of the individual spatial actuator p  . It can be seen that each individual actuator affects not only its own spatial zone area, but also adjacent zone areas. 4.1.2 Model identification Model identification of the papermaking CD process is the procedure to determine the values of the parameters in (16, 17), i.e., the dynamic model parameters θ  ={T  ,T  }, the spatial model parameters θ  ={g,ω,α,β}, and the alignment x  . An iterative identification algorithm has been proposed in (Gorinevsky & Gheorghe 2003). As with MD model identification, this algorithm is an open-loop model identification approach. Identification experiment data are first collected by performing open-loop bump tests. [...]... algorithm for large-scale model predictive control, in Journal of Process Control, Vol 12, pp 775 – 795 Bemporad, A., & Morari, M (2004) Robust model predictive control: A survey In Proc of European Control Conference, pp 939–944, Porto, Portugal Chen, C (1999), Linear Systems Theory and Design Oxford University Press, 3rd Edition Chu, D (2006) Explicit Robust Model Predictive Control and Its Applications... overview of nonlinear model predictive control applications, in F Allgöwer and A Zheng (eds), Nonlinear Predictive Control, Birkhäuser, pp 369–393 Qin, S., & Badgwell, T (2003) A Survey of industrial model prediction control technology, in Control Engineering Practice, Vol 11, pp 733–764 Rawlings, J (1999) Tutorial: Model predictive control technology In Proc of the American Control Conference, pp... based on identified process model It can be seen from comparison to the model for Autoslice to caliper that the models for Autoslice to dry weight and to moisture have high model fit In general, the bump test with a larger bump magnitude and longer bump duration will lead to a more accurate process model (better model fit) However, the open- 338 Advanced Model Predictive Control loop bump tests degrade... uncertainties (such as, modeling errors, neglected 344 2 Advanced Model Predictive Control Will-be-set-by-IN-TECH nonlinearities, etc.) is not considered From a practical standpoint, it is very difficult to obtain the exact plant model If controllers are designed without the consideration of plant uncertainties then the controlled system might achieve very poor control performance, or even worse the controlled system... multivariable controller utilizing range control, U.S Patent 5,574,638 MacArthur, J.W (1996) RMPCT : A New Approach To Multivariable Predictive Control For The Process Industries, in Proc Control Systems 1996, Halifax, Canada, April 1996 Morari, M., & Lee, J (1999) Model predictive control: Past, present and future In Computer and Chemical Engineering, Vol 23, pp 667–682 Persson, H (1998) Dynamic modelling... provides a continuous-time transfer matrix model (defined in (14)) For convenience, the MPC controller design discussed in the next section will use the state space model representation Conversion of the continuous-time transfer matrix model into the discrete-time state space model is trivial (Chen 1999) and is omitted here 330 Advanced Model Predictive Control 4.2 CD-MPC design In this section, a... 342 Advanced Model Predictive Control Fan, J., Stewart, G., Dumont G., Backström J., & He P (2005) Approximate steady-state performance prediction of large-scale constrained model predictive control systems, in IEEE Trans on Control System Technology, Vol 13, pp.884 – 895 Fan, J., Stewart, G., & Dumont G (2004) Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional... strategy is a good candidate for CD control design Fig 21 The multiple CD actuator beams and quality measurement model display 4.3.3 Control performance of the CD-MPC controller Table 3 summarizes the performance comparison between the CD-MPC controller and the traditional single-input-single-output (SISO) CD controller (a Dahlin controller) Although traditional CD control is still quite common in paper... Multivariable Control and Energy Optimization of Tissue Machines, In Proc Control System 2010, Stockholm, Sweden, 2010 Duncan, S (1989) The Cross-Directional Control of Web Forming Process Ph.D Thesis, University of London, UK Fan, J (2003) Model Predictive Control For Multiple Cross-directional Processes : Analysis, Tuning, and Implementation, Ph.D Thesis, University of British Columbia, Canada 342 Advanced Model. .. flight controllers If disturbance data are supposed to be given a priori and the current state of plant is also available, then controllers using Model Predictive Control (MPC) scheme work well, as illustrated for active suspension control for automobiles (Mehra et al., 1997; Tomizuka, 1976) However, in those papers, it is supposed that the plant dynamics are exactly modeled; that is, robustness of controllers . CV from its target trajectory. Controlled Variables 11.6 11.7 11.8 11.9 12 12.1 12. 2 12. 3 12. 4 12. 5 12. 6 12. 7 8:34:07 8:40:34 8:47:01 8:53:28 8:59:55 9:06:22 9 :12: 49 9:19:16 9:25:43 9:32:10 9:38:37 9:45:04 9:51:31 9:57:58 10:04:25 10:10:52 10:17:19 10:23:46 10:30:13 10:36:40 10:43:07 10:49:34 10:56:01 11:02:28 11:08:55 11:15:22 11:21:49 11:28:16 11:34:43 11:41:10 11:47:37 11:54:04 12: 00:31 12: 06:58 12: 13:25 12: 19:52 Time DW. Variables 11.6 11.7 11.8 11.9 12 12.1 12. 2 12. 3 12. 4 12. 5 12. 6 12. 7 8:34:07 8:40:34 8:47:01 8:53:28 8:59:55 9:06:22 9 :12: 49 9:19:16 9:25:43 9:32:10 9:38:37 9:45:04 9:51:31 9:57:58 10:04:25 10:10:52 10:17:19 10:23:46 10:30:13 10:36:40 10:43:07 10:49:34 10:56:01 11:02:28 11:08:55 11:15:22 11:21:49 11:28:16 11:34:43 11:41:10 11:47:37 11:54:04 12: 00:31 12: 06:58 12: 13:25 12: 19:52 Time DW (lb/ream) 0 5 10 15 20 25 Moisture (%) ReelDwt PV ReelMoi PV ExpressMoi PV Model Predictive Control and Optimization for. with nonlinear models Model Predictive Control and Optimization for Papermaking Processes 327 Fig. 16. MV trajectories under closed-loop GC with nonlinear models 4. Modelling, control and