Fast Nonlinear Model Predictive Control Algorithms and Applications in

Process EngineeringMoritz Diehl,

Optimization in Engineering Center (OPTEC) &Electrical Engineering Department (ESAT)

K.U. Leuven, Belgium

INRIA-Rocquencourt, May 30, 2007

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

OPTEC Aim: Connect Optimization Methods & Applications

Methods: New developments are inspired and driven by application needs

Applications: Smart problem formulations allow efficient solution (e.g. convexity)

Optimization in Engineering Center OPTEC

Five year project, from 2005 to 2010, 500.000 Euro per year, about 20 professors, 10 pos tdocs, 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...

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, 1548-1620)

Lecture and following Reception in Arenberg Castle, Leuven

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

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?

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Principle of Optimal Feedback Control / Nonlinear MPC:

Brain predicts and optimizes:e.g. slow down before curve

Nonlinear Model Predictive Control When We Drive a Car

Always look a bit into the future!

Main challenge for NMPC: fast and reliable real-time optimization!

Distillationcolumn (withUniv. Stuttgart)

Polymeri-sation reactor (withBASF)

ChromatographicSeparation (withUniv. Dortmund)

Combined CyclePower Plant (withUniv. Pavia)

PET plant: Plant wide control project with Politecnico di Milano

Car Engines: EU Project with Univ. Linz, Stuttgart, Politecnico di Milano

NMPC applications, with decreasing timescales

Looping kites for power generation, with TU Delft, Politecnico di Torino

Robot arms (with Columbia Univ. & INRIA Grenoble)

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

Optimal Control Family Tree

Optimal Control Family Tree

Optimal Control Family Tree

Optimal Control Problem in Simplest Form

Simplest Approach: Single Shooting

Nonlinear Program (NLP) in Single Shooting

�After control discretization, obtain NLP:

�Solve with NLP solver, e.g. Sequential Quadratic

Programming (SQP)

Sequential Quadratic Programming (SQP)

Toy Problem with One ODE for Illustration

Mildly nonlinear and unstable system.

Single Shooting

Single Shooting: First Iteration

Single Shooting: Second Iteration

Single Shooting: Third Iteration

Single Shooting: 4th Iteration

Single Shooting: 5th Iteration

Single Shooting: 6th Iteration

Single Shooting: 7th Iteration (Solution)

Single Shooting: Pros and Cons

Alternative: Direct Multiple Shooting [Bock, Plitt 1981]

Nonlinear Program in Multiple Shooting

SQP for Multiple Shooting

Summarize NLP:

Toy Example: Multiple Shooting Initialization

Multiple Shooting: First Iteration

Multiple Shooting: Second Iteration

Multiple Shooting: 3rd Iteration (already solution!)

Multiple Shooting: 3rd Iteration (already solution!)

Single shooting converged much slower!

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

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

NMPC Computation from 1998 to 2006

�1998: 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 10-20 milliseconds [Ferreau et al. ‘06]

5*60*1000 / 20 = 15 000 times faster , due toMoore‘s law + Algorithm Development

NMPC Computation from 1998 to 2006

�1998: 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 10-20 milliseconds [Ferreau et al. ‘06], [Albersmeyer, Findeisen `06]

5*60*1000 / 20 = 15 000 times faster , due toMoore‘s law + Algorithm Development

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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ConventionalConventional::Initial Initial Value Value Embedding:Embedding:

Never simulate a nonlinear system open-loop!

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Real-Time Iterations minimize feedback delay

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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

Transient in 15 minutes instead of 2 hours!

Conclusions

�Recent progress makes Nonlinear MPC with first principles models in millisecond range possible (now 15 000 x faster than1998)

�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

An Invitation

�13th Czech-French-German Conference onOptimization (CFG07), Heidelberg, Germany, September 17-21, 2007. (inv. speakers: Fletcher, Scherer, Trelat, Waechter,...) Traditionally strong in optimal control.

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)