Model Predictive Control Alberto Bemporad imt.lu/ab Course page: http://cse.lab.imtlucca.it/~bemporad/mpc_course.html Course structure • Basic concepts of model predictive control (MPC) and linear MPC • Linear time-varying and nonlinear MPC • Quadratic programming (QP) and explicit MPC • Hybrid MPC, stochastic MPC, data-driven MPC • Numerical examples: – MPC Toolbox for MATLAB (linear/explicit/parameter-varying MPC) – Hybrid Toolbox for MATLAB (explicit MPC, hybrid systems) For additional background on linear systems and numerical optimization: http://cse.lab.imtlucca.it/~bemporad/intro_control_course.html http://cse.lab.imtlucca.it/~bemporad/optimization_course.html "Model Predictive Control" - © A. Bemporad. All rights reserved. 2/132 Model Predictive Control: Basic Concepts Model Predictive Control (MPC) optimization prediction model algorithm model-based optimizer process set-points inputs outputs r(t) u(t) y(t) measurements simplified likely Use a dynamical model of the process to predict its future evolution and choose-------------------- the “best” control action a good "Model Predictive Control" - © A. Bemporad. All rights reserved. 3/132 Model Predictive Control (MPC) • Goal: find the best control sequence over a future horizon of N steps past future r(t) N−1 y X k predicted outputs k y − k2 k u − k2 min W (yk r(t)) 2 + W (uk ur(t)) 2 uk k=0 manipulated inputs s:t: xk+1 = f(xk; uk) prediction model yk = g(xk) t t+k t+N umin ≤ uk ≤ umax constraints ymin ≤ yk ≤ ymax x0 = x(t) state feedback numerical optimization problem t+1 t+1+k t+N+1 • At each time t: – get new measurements to update the estimate of the current state x(t) – solve the optimization problem with respect to fu0; : : : ; uN−1g ∗ – apply only the first optimal move u(t) = u0, discard the remaining samples "Model Predictive Control" - © A. Bemporad. All rights reserved. 4/132 Daily-life examples of MPC • MPC is like playing chess ! • On-line (event-based) re-planning used in GPS navigation • You use MPC too when you drive ! "Model Predictive Control" - © A. Bemporad. All rights reserved. 5/132 MPC in industry • The MPC concept dates back to the 60’s (Rafal, Stevens, AiChE Journal, 1968) (Propoi, 1963) • MPC used in the process industries since the 80’s (Qin, Badgewell, 2003) (Bauer, Craig, 2008) Today APC (advanced process control) = MPC ©SimulateLive.com "Model Predictive Control" - © A. Bemporad. All rights reserved. 6/132 3 MPC in industry (Qin, Badgewell, 2003) S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764 745 Table 6 • IndustrialSummary of linear survey MPC applications of MPC by areas applications (estimates based on vendor conducted survey; estimates in do mid not include 1999 applications by companies who have licensed vendor technology)a Area Aspen Honeywell Adersab Invensys SGSc Total Technology Hi-Spec Refining 1200 480 280 25 1985 Petrochemicals 450 80 — 20 550 Chemicals 100 20 3 21 144 Pulp and paper 18 50 — — 68 Air & Gas — 10 — — 10 Utility — 10 — 4 14 Mining/Metallurgy 8 6 7 16 37 Food Processing — — 41 10 51 Polymer 17 — — — 17 Furnaces — — 42 3 45 Aerospace/Defense — — 13 — 13 Automotive — — 7 — 7 Unclassified 40 40 1045 26 450 1601 Total 1833 696 1438 125 450 4542 First App. DMC:1985 PCT:1984 IDCOM:1973 IDCOM-M:1987 RMPCT:1991 HIECON:1986 1984 1985 OPC:1987 Largest App. 603 Â 283 225 Â 85 — 31 Â 12 — a The numbers reflect a snapshot survey conducted in mid-1999 and should not be read as static. A recent update by one vendor showed 80% increase in the number of applications. b Adersa applications through January 1, 1996 are reported here. Since there are many embedded Adersa applications, it is difficult to accurately Estimatesreport their number based or distribution. on vendor Adersa’s survey product literature indicates over 1000 applications of PFC alone by January 1, 1996. c The number of applications of SMOC includes in-house applications by Shell, which are unclassified. Therefore, only a total number is estimated here. "Model Predictive Control" - © A. Bemporad. All rights reserved. 7/132 Nonlinear The PFC algorithm can be used with several different model types. The most general of these is a discrete-time Aspen Target first-principles model that can be derived from 8 by PFC MVC NOVA-NLC integrating across the sample time: Process Perfecter xkþ1 ¼ fðxk; u k; vk; wkÞ; ð9aÞ First Empirical Principles y ¼ g ðxk; u kÞþn ; ð9bÞ DMCplus k k HIECON RMPCT although special care should be taken for stiff systems. PFC Connoisseur SMOC 3.2.2. Linear empirical models Linear Linear empirical models have been used in the majority of MPC applications to date, so it is no Fig. 4. Classification of model types used in industrial MPC surprise that most of the current MPC products are algorithms. based on this model type. A wide variety of model forms are used, but they can all be derived from 9 by linearizing about an operating point to get: x þ1 ¼ Ax þ B u þ B v þ B w ; ð10aÞ 3.2.1. Nonlinear first-principles models k k u k v k w k Nonlinear first-principles models used by the NOVA- y ¼ Cxk þ Du k þ n : ð10bÞ NLC algorithm are derived from mass and energy k k balances, and take exactly the form shown above in 8. The SMOC and PFC algorithms can use this model Unknown model parameters such as heat transfer form. An equivalent discrete-time transfer function coefficients and reaction kinetic constants are either model can be written in the form of a matrix fraction estimated off-line from test data or on-line using an description (Kailath, 1980): extended Kalman filter (EKF). In a typical application À 1 À 1 À 1 À 1 y ¼½I À Uy ðq Þ ½Uuðq Þu k þ Uvðq Þvk the process model has between 10 and 100 differential k À 1 algebraic equations. þ Uwðq Þwkþnk; ð11Þ MPC in industry M. Bauer, I.K. Craig / Journal of Process Control 18 (2008) 2–18 5 (Bauer, Craig, 2008) • EconomicM. assessment Bauer, I.K. Craig / Journal of ofAdvanced Process Control 18 (2008) Process 2–18 Control (APC) 5 Petrochemical North America Petrochemical North America Chemicals Africa Chemicals Africa Petroleum refining Europe Petroleum refining Europe Mineralsprocessing Mineralsprocessing Asia Pacific Asia Pacific Oil & Gas Oil & Gas South America South America Power & utilities Power & utilities 0% 10% 20% 30% Pulp & paper 0% 10% 20% 30% Pulp & paper M. Bauer,M. Bauer, I.K. Craig I.K. Craig / Journal / Journal of Process of Process Control Control 18 18 (2008) (2008) 2–18 2–18 7 7 Industrial gasesIndustrial gases Coal products Model predictive control Coal products Suppliers Suppliers Model predictive control StandardStandard Water & wastewaterWater & wastewater Users Users ConstraintConstraint control control FrequentlyFrequently Other Other Split-rangeSplit-range control control RarelyRarely Linear programming (LP) NeverNever 0% 20%0% 40% 20% 60% 40% 80% 60% 100% 80% 100% Linear programming (LP) Don't know Nonlinear control algorithms or models Don't know Fig. 2. Participants of APC survey by industry and by continent (total: 66 participants).Nonlinear Several answers control werealgorithms allowed or formodels industry sectors. Fig. 2. Participants of APC survey by industry and by continent (total: 66 participants). Several answers were allowedDead-time for industry compensation sectors. participants of APC survey by industry (worldwide) Dead-time compensation sections identify the respondent and capture the size of the the estimation of qualitative,Statistical non-monetary process control benefits sections identify the respondent and capture the size of the the estimation of qualitative,Statistical non-monetary process control benefits plant as well as the use of control technology within the (15%) as well as rules of thumb [47], such as that a new con- plant as well as the use of control technology within the (15%) as well as rules of thumb [47]Neural, suchnetworks as based that control a new con- organization. The third section queries the use of APC trol scheme will increaseNeural the networks throughput based control by a fixed percent- organization. The third section queries the use of APC trol scheme will increase the throughputExpert system by based a fixed control percent- technology and tools. Sections four to six are key to the age (7%). For the waterExpert and system waste-water based control industries, Lant technologyeconomic and tools. assessment Sections four process. to six While are Section key tofour the coversage (7%).and For Steff theens water[48] provide and waste-water a qualitativeFuzzy self-assessmentindustries,logic control Lant form Fuzzy logic control economic assessmentthe managerial process. process While by asking Section about four reporting, covers decisionand Stetoffens establish[48] provide how good a qualitative the processInternal model self-assessment control control is. (IMC) The disadvan- form Internal model control (IMC) the managerialmaking process and by budgeting, asking about Section reporting, five is concerned decision withto the establishtage ofhow such good a table the processis thatAdaptive it control is industry / selftuning is. The related control disadvan- and has to making andengineering budgeting, process Section and five the is estimation concerned of withcost and the benefits.tage ofbe such updated a table with is that the developmentAdaptive it is industry / selftuningDirectsynthesis of related newcontrol technologies. and(DS) has to engineeringSection process sixand requests the estimation the respondents of cost and opinion benefits. aboutbe the updatedThe with questionnaire the development resultsDirectsynthesis ofsupport new technologies. the (DS) industry–univer- Section sixaccuracy, requests importance the respondents and satisfaction opinion with about the current the esti-Thesity questionnaire study by Marlin results and support co-workers the industry–univer-[49], which0% high- 20% 40% 60% 80% 100% accuracy, importancemation process and satisfaction in place as well with as the about current future esti- trends.sity studylighted by theMarlin importance and co-workers of determiningFig.[49] 4., Industrial which the0% base high-use 20% of case APC 40% methods: 60% survey 80% results.