High Warehouse Racks: Optimal Feedback Control and High Warehouse - - PowerPoint PPT Presentation

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High Warehouse Racks: Optimal Feedback Control and High Warehouse - - PowerPoint PPT Presentation

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High Warehouse Racks: Optimal Feedback Control and High Warehouse Racks: Optimal Feedback Control and Adaptive Control Improvement Adaptive Control Improvement Christof Bskens skens Christof B INRIA, Optimal Control: Algorithms and

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High Warehouse Racks: Optimal Feedback Control and High Warehouse Racks: Optimal Feedback Control and Adaptive Control Improvement Adaptive Control Improvement

Christof Bü Christof Büskens skens

INRIA, Optimal Control: Algorithms and Applications (May/June 2007)

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2 Classification of Optimal Control Solutions

  • pen-loop:

closed-loop:

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3 Open-Loop vs. Closed-Loop

nonlinear linear Model Method

  • pen-loop

closed-loop knowledge Part I Part II

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4 Part II linear closed-loop  nonlinear closed-loop

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5 Topic Slide: Optimal Control of Cranes

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6

Optimal Control Problem for High-Rack Trolley

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7

  • Part I: (Motivation)

– Example: Warehouse Cranes

  • Part II: (Linear Quadratic Regulator (LQR))

– Perturbed Optimal Control Problems – Linear Quadratic Regulator Problems (LQR)

  • Discussion
  • Solution
  • LQR vs. NLP

– Example: Warehouse Crane

  • Part III: (Perturbed Linear Quadratic Regulator)

– Perturbed Linear Quadratic Regulator (Riccati) – Real-Time Optimal Adaption of Optimal Controllers – Example: Inverse Pendulum

Overview

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8

Methods for solving Optimal Control Problems

  • The result of any performance process is limited by the most

scarcely available ressource: the Time. (Drucker)

  • We have enough Time, if only we use it wisely. (Goethe)
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9

Optimal Control Problems with Perturbations: ODE

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10 Linear Quadratic Regulator (LQR)

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11

  • Part I: (Motivation)

– Example: Warehouse Cranes

  • Part II: (Linear Quadratic Regulator (LQR))

– Perturbed Optimal Control Problems – Linear Quadratic Regulator Problems (LQR)

  • Discussion
  • Solution
  • LQR vs. NLP

– Example: Warehouse Crane

  • Part III: (Perturbed Linear Quadratic Regulator)

– Perturbed Linear Quadratic Regulator (Riccati) – Real-Time Optimal Adaption of Optimal Controllers – Example: Inverse Pendulum

Overview

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12 Explanations: The Model

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13 Explanations: The Quadratic Objective

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14 Explanations: The Quadratic Objective

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15 Explanations: The Quadratic Objective

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16 Summing up the Explanations

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17

  • Part I: (Motivation)

– Example: Warehouse Cranes

  • Part II: (Linear Quadratic Regulator (LQR))

– Perturbed Optimal Control Problems – Linear Quadratic Regulator Problems (LQR)

  • Discussion
  • Solution
  • LQR vs. NLP

– Example: Warehouse Crane

  • Part III: (Perturbed Linear Quadratic Regulator)

– Perturbed Linear Quadratic Regulator (Riccati) – Real-Time Optimal Adaption of Optimal Controllers – Example: Inverse Pendulum

Overview

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18 Approach using the Hamiltonian

Methods:

  • Pontryagin‘s Minimum Principle
  • Calculus of variations
  • Hamilton-Jacobi-Bellmann equation
  • Karush-Kuhn-Tucker conditions
  • Dynamical Optimization
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19 Approach using the Hamiltonian

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20 Relationship: Hamiltonian  LQR-Problem

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21 Strictly Local Minimum

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22 The Suspicion

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23 Resolution (with )

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24 Algebraic Matrix-Riccati-Equation

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25 Summary

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26 Extension

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27 Extension

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28 Solution of LQR

click me

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29 Alternative Formulations and Equivalences (LQR) (NLP)

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30 Sparse NLP Solver

WORHP We Optimize Really Huge Problems

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31 WORHP Born for Space Applications

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32 Reverse Communication Advantages

Simple implementation No worries after start

Caller

Evaluation Objective Gradient Evaluation Constraints Jacobian

Disadvantages

Inflexible user-interface No influence after start

Traditional Calling Sequence:

SQP

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33 Reverse Communication Caller

New Calling Sequence: Evaluation Objective Gradient Evaluation Constraints Jacobian

Advantages Flexible problem evaluation Full control on optimization Disadvantages Harder to implement

WORHP

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34 Features of WORHP: Sentinel

User Solution

WORHP Sentinel

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35 Modularity

Modules to adapt to the User‘s requirements: Speed, Precision, Robustness, Simplicity, …

Gradient Fixed Stepsize Gradient Adaptive Stepsize Gradient Group Strategy Advanced Error Reporting Matrix structure Creation/Analysis Pre-Iteration- Analysis

WORHP QPSOL

QP-Matrix diagonal-BFGS QP-Matrix Hessian QP-Matrix Identity QP-Matrix diag.-Hessian QP-Method Nonsmooth Newton QP-Method Interior Point direct Solvers MA47, SuperLU iterative Solvers CGN, CGS, BiCG Interface Transcriptor Interface Reverse Comm. Interface Traditional Interface AMPL

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36 Warehouse Crane

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37 Topic Slide: Optimization in High Rack Technology

Modeling (Real-Time) Optimization:

  • time
  • energy
  • jerk
  • scillatory behavior
  • non-oscillating at

Extended Constraints:

  • controlled stop
  • emergency stop
  • collision free

Extended Modeling

  • transfers from/into stock
  • robustness
  • intermediate stop

with: Westfalia Logistics

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38 Emergency Stop

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39

  • Part I: (Motivation)

– Example: Warehouse Cranes

  • Part II: (Linear Quadratic Regulator (LQR))

– Perturbed Optimal Control Problems – Linear Quadratic Regulator Problems (LQR)

  • Discussion
  • Solution
  • LQR vs. NLP

– Example: Warehouse Crane

  • Part III: (Perturbed Linear Quadratic Regulator)

– Perturbed Linear Quadratic Regulator (Riccati) – Real-Time Optimal Adaption of Optimal Controllers – Example: Inverse Pendulum

Overview

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40 Perturbed Linear Quadratic Regulator

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41 Perturbed NLP problems

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42

Parametric Sensitivity Analysis / Solution Differentiability

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43

  • ffline

Real-Time Optimization (General Idea)

Unperturbed Problem Sensitivity Analysis

  • nline

Real-Time Optimization Real-Time Optimal Control

Sensitivities Solution Solution

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44 Real-Time Optimization

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45

Existence und Uniqueness of perturbed Optimal Controllers

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46

Real-Time Optimal Adaption of the Optimal Controller

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47 Error Estimate

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48 Optimal Control of the Warehouse Crane Optimal Control of the Inverse Pendulum

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49 Application: The Inverse Pendulum

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50 Horizontally Acting Forces

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51 Further Forces

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52 Motivation: Inverse Pendulum

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53 Motivation: Inverse Pendulum (perturbed)

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54 Solution of the Inverse Pendulum

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55

Adaptive Optimal Control: Inverse Pendulum (perturbed)

click me

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56 Conclusion

  • Optimal controllers are not optimal

– Perturbations and Real-Time Trajectory Planning – Perturbed Nonlinear Optimization – Sensitivity Analyses / Solution Differentiability – A New Class of Higher Order Optimal Ricatti Controllers

  • Future Work

– Stabilizability – Controllability – Observability – Further assumption, e.g. PBH-Criteria – Extensions to the Ricatti Approach

  • Optimal Controllers
  • Tracking Type Controllers
  • Dynamic Observers
  • etc.
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57 Thank You!