Fast Nonlinear Model Predictive Control Algorithms and Applications in Process Engineering

Transcription

1 Fast Nonlinear Model Predictive Control Algorithms and Applications in Process Engineering Moritz Diehl, Optimization in Engineering Center (OPTEC) & Electrical Engineering Department (ESAT) K.U. Leuven, Belgium INRIA-Rocquencourt, May 30, 2007

2 Outline of the Talk K.U.Leuven s Optimization in Engineering Center OPTEC Nonlinear Model Predictive Control (NMPC) How to solve dynamic optimization problems? Four crucial features for fast NMPC algorithms Application to a Distillation Column

3 OPTEC Aim: Connect Optimization Methods & Applications Applications: Smart problem formulations allow efficient solution (e.g. convexity) Methods: New developments are inspired and driven by application needs

4 Optimization in Engineering Center OPTEC Five year project, from 2005 to 2010, Euro per year, about 20 professors, 10 postdocs, and 60 PhD students involved in OPTEC research Promoted by four departments: Electrical Engineering Mechanical Engineering Chemical Engineering Computer Science Many real world applications at OPTEC...

5 Quarterly Stevin Lecture: Everyone Invited! Quarterly Simon Stevin Lecture on Optimization in Engineering : Dec 6: Larry Biegler, CMU Pittsburgh Apr 18: Stephen Boyd, Stanford July 9: Steve Wright, Madison, Wisconsin Oct 24: Manfred Morari, ETH Zurich Dec X: David Mayne, Imperial, London K.U. Leuven Seminar on Optimization in Engineering : Jan. 31: Mario Milanese (Torino): MPC of semi-active damping Feb. 8: Philippe Toint (Namur): large scale optimization methods Feb. 22: Peter Kuehl (Heidelberg): Robust optimal feedback control March 1: Yurii Nesterov (UCL)/ Florian Jarre (Duesseldorf): new optimization algorithms Simon Stevin, ) Lecture and following Reception in Arenberg Castle, Leuven

6 Outline of the Talk K.U.Leuven s Optimization in Engineering Center OPTEC Nonlinear Model Predictive Control (NMPC) How to solve dynamic optimization problems? Four crucial features for fast NMPC algorithms Online MPC of a combustion engine

7 First Principle Dynamic System Models E.g. some equations modelling a distillation column (in Stuttgart) Nonlinear differential algebraic equations (DAE) often in modeling languages like gproms, SIMULINK, Modelica typical order of magnitude: some hundreds to thousands variables difficulties: stiffness, discontinuities, high index Can we use these models directly for optimization and feedback control?

8 Principle of Optimal Feedback Control / Nonlinear MPC: x 0 u 0 x 0 u 0

9 Nonlinear Model Predictive Control When We Drive a Car Always look a bit into the future! Brain predicts and optimizes: e.g. slow down before curve Main challenge for NMPC: fast and reliable real-time optimization!

10 NMPC applications, with decreasing timescales Distillation column (with Univ. Stuttgart) Chromatographic Separation (with Univ. Dortmund) Looping kites for power generation, with TU Delft, Politecnico di Torino Polymerisation reactor (with BASF) Combined Cycle Power Plant (with Univ. Pavia) PET plant: Plant wide control project with Politecnico di Milano Robot arms (with Columbia Univ. & INRIA Grenoble) Car Engines: EU Project with Univ. Linz, Stuttgart, Politecnico di Milano

11 Outline of the Talk K.U.Leuven s Optimization in Engineering Center OPTEC Nonlinear Model Predictive Control (NMPC) How to solve dynamic optimization problems? Four crucial features for fast NMPC algorithms Application to a Distillation Column

12 Optimal Control Family Tree

13 Optimal Control Family Tree

14 Optimal Control Family Tree

15 Optimal Control Problem in Simplest Form

16 Simplest Approach: Single Shooting

17 Nonlinear Program (NLP) in Single Shooting After control discretization, obtain NLP: Solve with NLP solver, e.g. Sequential Quadratic Programming (SQP)

18 Sequential Quadratic Programming (SQP)

19 Toy Problem with One ODE for Illustration Mildly nonlinear and unstable system.

20 Single Shooting

21 Single Shooting: First Iteration

22 Single Shooting: Second Iteration

23 Single Shooting: Third Iteration

24 Single Shooting: 4th Iteration

25 Single Shooting: 5th Iteration

26 Single Shooting: 6th Iteration

27 Single Shooting: 7th Iteration (Solution)

28 Single Shooting: Pros and Cons

29 Alternative: Direct Multiple Shooting [Bock, Plitt 1981]

30 Nonlinear Program in Multiple Shooting

31 SQP for Multiple Shooting Summarize NLP:

32 Toy Example: Multiple Shooting Initialization

33 Multiple Shooting: First Iteration

34 Multiple Shooting: Second Iteration

35 Multiple Shooting: 3 rd Iteration (already solution!)

36 Multiple Shooting: 3 rd Iteration (already solution!) Single shooting converged much slower!

37

38 The MUSCOD-II Developer Team [Heidelberg, Leuven, Madrid]

39 Outline of the Talk K.U.Leuven s Optimization in Engineering Center OPTEC Nonlinear Model Predictive Control (NMPC) How to solve dynamic optimization problems? Four crucial features for fast NMPC algorithms Online MPC of a combustion engine

40 NMPC Computation from 1998 to : 5th order distillation model allows sampling times of only 5 minutes [Allgower, Findeisen, 1998] 2001: 206th order distillation model, sampling times of 20 seconds [D. et al. 01] 2006: 5th order engine model, sampling times of milliseconds [Ferreau et al. 06] 5*60*1000 / 20 = times faster, due to Moore s law + Algorithm Development

41 NMPC Computation from 1998 to : 5th order distillation model allows sampling times of only 5 minutes [Allgower, Findeisen, 1998] 2001: 206th order distillation model, sampling times of 20 seconds [D. et al. 01] cf. [Biegler] 2006: 5th order engine model, sampling times of milliseconds [Ferreau et al. 06], [Albersmeyer, Findeisen `06] 5*60*1000 / 20 = times faster, due to Moore s law + Algorithm Development

42 Four Crucial Features for Fast NMPC Direct, simultaneous optimal control: Multiple Shooting Efficient derivative generation for ODE/DAE solvers Initialization by Initial Value Embedding Real-Time Iterations for fast tracking of optimal solutions

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44 x 0

45 x 0

46 x 0

47 x 0

48 x 0

49 x 0

50 x 0

51 Never simulate a nonlinear system open-loop! Conventional: Initial Value Embedding:

52 x 0 x 0

53 x 0 x 0

54 x 0 x 0

55 x 0 u 0

56 Real-Time Iterations minimize feedback delay

57 x 0 u 0

58 Outline of the Talk K.U.Leuven s Optimization in Engineering Center OPTEC Nonlinear Model Predictive Control (NMPC) How to solve dynamic optimization problems? Four crucial features for fast NMPC algorithms Application to a Distillation Column

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61 Transient in 15 minutes instead of 2 hours!

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64 Conclusions Recent progress makes Nonlinear MPC with first principles models in millisecond range possible (now x faster than 1998) Emerging consensus for NMPC algorithms: employ direct, simultaneous methods use Initial Value Embedding (first order predictor) perform Real-Time Iterations to trace NMPC problem solution while data change Use SQP type method to track active set changes

65 An Invitation 13th Czech-French-German Conference on Optimization (CFG07), Heidelberg, Germany, September 17-21, (inv. speakers: Fletcher, Scherer, Trelat, Waechter,...) Traditionally strong in optimal control.

66 4 PhD Positions in Numerical Optimization: Sequential Convex Programming Algorithms for Nonlinear SDP Large Scale & PDE Constrained Real-Time Optimization Algorithms Fast Model Predictive Control Applications in Mechatronic Systems Shape Optimization of Mechanical Parts under Inertia Loading (deadline: June 21, 2007)