H0K03a : Advanced Process Control Model-based Predictive Control 2 : - - PowerPoint PPT Presentation

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H0K03a : Advanced Process Control Model-based Predictive Control 2 : - - PowerPoint PPT Presentation

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H0K03a : Advanced Process Control Model-based Predictive Control 2 : Dynamic Optimization Bert Pluymers Prof. Bart De Moor Katholieke Universiteit Leuven, Belgium Faculty of Engineering Sciences Department of Electrical Engineering (ESAT)

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Bert Pluymers

  • Prof. Bart De Moor

Katholieke Universiteit Leuven, Belgium Faculty of Engineering Sciences Department of Electrical Engineering (ESAT) Research Group SCD-SISTA

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

H0K03a : Advanced Process Control

Model-based Predictive Control 2 : Dynamic Optimization

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1

Overview

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Lesson 2 : Dynamic Optimization

  • Optimization basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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2

General form : Legend :

  • : vector of optimization variables
  • : objective function / cost function
  • : equality constraints
  • : inequality constraints
  • : solution to optimization problem
  • : optimal function value

Notation

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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Gradient : (points in direction of steepest ascent) Hessian : (gives information about local curvature of )

Gradient & Hessian

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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Gradient & Hessian

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Example : Gradients for different Eigenvectors of hessian at the origin ( )

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Unconstrained Optimality Conditions

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Necessary condition for optimality of Sufficient conditions for minimum positive definite Classification of optima : positive definite minimum indefinite saddle point negative definite maximum

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Introduction of Lagrange multipliers leads to Lagrangian : with Lagrange multipliers of the ineq. constraints Lagrange multipliers of the eq. constriants Constrained optimum can be found as Minimization over but Maximization over !!!!

Lagrange Multipliers

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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Constrained optimum can be found as First-order optimality conditions in

Lagrange Multipliers

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Gradient of Gradient of ineq. Gradient of eq.

Interpretation ???

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Lagrange Multipliers

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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Lagrange Multipliers

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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Lagrange Multipliers

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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Lagrange Duality

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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Karush-Kuhn-Tucker Conditions

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

From previous considerations we can now state necessary conditions for constrained optimality : These are called the KKT conditions.

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Optimization Tree

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Optimization discrete continuous unconstrained constrained convex

  • ptimization

non-convex

  • ptimization

NLP LP QP SOCP SDP

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Convex Optimization

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

An optimization problem of the form is convex iff for any two feasible points :

  • is feasible
  • This is satisfied iff
  • `the cost function is a convex function
  • the equality constraints or linear or absent
  • the inequality constraints define a convex region
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Convex Optimization

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Importance of convexity :

  • no local minima, one global optimum
  • under certain conditions, primal and dual have

same solution

  • efficient solvers exist
  • polynomial worst-case execution time
  • guaranteed precision
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From LP to SDP

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

SDP SOCP QP LP

Semi-Definite Programming Second Order Cone Progr. Quadratic Programming Linear Programming

computational efficiency generality

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General form : Remarks :

  • always convex
  • optimal solution always at a corner of ineq. constraints
  • typically used in finance / economics / management

Linear Programming (LP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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Linear Programming (LP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Eliminating equality constraints : Reparametrize optimization vector : Leading to

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Quadratic Programming (QP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

General form : Remarks :

  • convex iff
  • LP is a special case of QP (imagine )
  • Used in all domains of engineering
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Second-Order Cone Programming (SOCP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

General form : Remarks :

  • Always convex
  • Second-Order, Ice-Cream, Lorentz cone :
  • Engineering applications with sum-of-squares,

robust LP, robust QP

SOC constraint

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Second-Order Cone Programming (SOCP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

QP as special case of SOCP : Rewrite this as which is equivalent to By introducing an additional variable we get the SOCP

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Semi-Definite Programming (SDP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

General form : with Remarks :

  • means that should be positive semi-definite
  • means that should be pos. semi-def.
  • always convex :
  • ineq. constraints called LMI’s : Linear Matrix Ineq.
  • LMI’s arise in many applications of

Systems & Control Theory

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Semi-Definite Programming (SDP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

convexity : Easily verified : hence and therefore which means that LMI’s are convex constraints.

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Semi-Definite Programming (SDP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Schur Complement : is equivalent with More general : Remarks :

  • Originally developed in a statistical framework
  • Today widely used in S&C in order to reformulate

problems involving eigenvalues as an LMI.

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Semi-Definite Programming (SDP)

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

SOCP as special case of SDP : is equivalent with (by using Schur complement) : (exercise : apply Schur complement to LMI and reconstruct SOC constraint)

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Convexity = SDP ?

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Formulated as SDP Convex optimization Convex optimization formulatable as SDP Example :

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Convexity ≠ SDP !

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

SDP SOCP QP LP convex optimization

structure easily exploitable (many toolboxes available) → significant efficiency gains ! convexity difficult to exploit (computationally)

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28

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

MPC Paradigm

  • At every discrete time instant , given information about

the current system state , calculate an ‘optimal’ input sequence over a finite time horizon :

  • Apply the first input to the real system
  • Repeat at the next time instant , using new state

measurements / estimates.

N N

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29

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Dynamic Programming

  • Finding the optimal input sequence is done by means
  • f Dynamic Programming
  • Definition* :

“DP is a class of solution methods for solving sequential decision problems with a compositional cost structure”

  • Invented by Richard Bellman (1920-1984) in 1953
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30

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Example 1 : The darts problem* : “Obtain a score of 301 as fast as possible while beginning and ending in a double.”

  • Decision : next area towards which to throw the dart
  • Cost : time

http://plus.maths.org/issue3/dynamic/

* D. Kohler, Journal of the Operational Research Society, 1982.

Dynamic Programming

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31

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Example 2 : DNA sequence allignment : G A A T T C A G T T A (sequence #1) G G A T C G A (sequence #2)

  • Decisions : which nucleotides to match
  • Cost : e.g. based on substitution / insertion prob.
  • Algorithms : Baum/Welch, Waterman/Smith, …

Dynamic Programming

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32

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

“Series of sequential decisions” :

Dynamic Programming in MPC

Typical optimization problem :

measured Optimization variables

→ standard QP formulation ?

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33

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Optimization vector : Cost function : For convexity hence .

Linear MPC as standard QP

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  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Equality constraints with Sparsity : many entries in equal to 0

Linear MPC as standard QP

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35

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Inequality constraints with Sparsity : many entries in equal to 0

Linear MPC as standard QP

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36

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Alternatives

  • slightly faster to solve
  • ‘chattering’ : optimal solution jumps around
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  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Alternatives

Non-convex optimization in general : to be avoided !!!

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38

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

MPC and optimization

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39

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

MPC and optimization

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40

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Active Set Methods

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41

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Active Set Methods

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42

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Interior Point Methods

  • 1. Choose initial point
  • 2. Linearize KKT conditions around current point
  • 3. Solve Linearized KKT system to obtain search

direction

  • 4. Calculate step length such that
  • 5. Repeat from step 2, until convergence
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43

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Interior Point Methods

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44

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

  • ASM allow hot start
  • reuse of active set, factorizations, …
  • ASM has feasible intermediate results
  • by construction
  • IPM can exploit sparsity
  • solution of KKT system by sparse solver

In industry currently mostly ASM due to first two advantages, but IPM under consideration…

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45

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

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46

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

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47

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

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48

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

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49

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Conclusion

  • convex optimization powerful tool !!!
  • different optimization algorithm have pro’s and con’s
  • try to avoid NLP’s !!!!
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50

  • Overview
  • Optimization Basics
  • Convex Optimization
  • Dynamic Optimization
  • Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

References