![Page 1: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/1.jpg)
Real-Time Optimal Control and Trajectory Planning with Applications to Industrial Robots
Christof Büskens
INRIA, Optimal Control: Algorithms and Applications (May/June 2007)
![Page 2: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/2.jpg)
2
C. BüskensC. Büskens
Optimal Control: Industrial Robots
Foces:
-centrifugal
-Coriolis
-gravity
-frictional (dry)
-restoring
![Page 3: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/3.jpg)
3
C. BüskensC. Büskens
Parcel Robots and minimal Robot Trajectory Planning
Unloading of ContainersParameter Identification/
Trajectory PlanningRobot Dynamics- kinematics- (minimal) dynamic- collision prevention
with: Deutsche Post, DHL Express, BIBA, EADS
with: SEF–Robotics, ADC, Lüneburg
![Page 4: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/4.jpg)
4
C. BüskensC. Büskens
Classification of Optimal Control Solutions
open-loop:
closed-loop:
![Page 5: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/5.jpg)
5
C. BüskensC. Büskens
Mixed Strategy
![Page 6: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/6.jpg)
6
C. BüskensC. Büskens
Open-Loop vs. Closed-Loop
nonlinearlinearModel
Method
open-loop
closed-loop
knowledge
Part I
Part II
![Page 7: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/7.jpg)
7
C. BüskensC. Büskens
Part I
nonlinear open-loop nonlinear closed-loop
![Page 8: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/8.jpg)
8
C. BüskensC. Büskens
• Part I: (Motivation)– Example: Industrial Robot
• Part II: (Optimal Control)- Perturbed Optimal Control Problems- Perturbed NLP Problems- Numerical Solution of NLP-Problems
• Part III: (Real-Time Optimal Control)- Parametric Sensitivity Analysis / Solution Differentiability- Real–Time Solution
• Open–Loop Strategy• Mixed Strategies (Constraints)
- Examples: - Industrial Robot ABB IRB 6400 - Emergency Landing
Overview
![Page 9: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/9.jpg)
9
C. BüskensC. Büskens
Example: ABB IRB 6400 (Optimal Control Problem)
![Page 10: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/10.jpg)
10
C. BüskensC. Büskens
Optimal Control Problems with Perturbations: ODE
![Page 11: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/11.jpg)
11
C. BüskensC. Büskens
Methods for solving Optimal Control Problems
![Page 12: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/12.jpg)
12
C. BüskensC. Büskens
• Part I: (Motivation)– Example: Industrial Robot
• Part II: (Optimal Control)- Perturbed Optimal Control Problems- Perturbed NLP Problems- Numerical Solution of NLP-Problems
• Part III: (Real-Time Optimal Control)- Parametric Sensitivity Analysis / Solution Differentiability- Real–Time Solution
• Open–Loop Strategy• Mixed Strategies (Constraints)
- Examples: - Industrial Robot ABB IRB 6400 - Emergency Landing
Overview
![Page 13: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/13.jpg)
13
C. BüskensC. Büskens
Direct approaches for OCP I: ODE
![Page 14: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/14.jpg)
14
C. BüskensC. Büskens
Direct approaches for OCP II: ODE
MACH1: [B./Gerdts]WORHP: [B./Gerdts]
NUDOCCCS: [B.]
![Page 15: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/15.jpg)
15
C. BüskensC. Büskens
Sparse NLP Solver
WORHP
We Optimize Really Huge Problems
![Page 16: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/16.jpg)
16
C. BüskensC. Büskens
WORHP Born for Space Applications
![Page 17: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/17.jpg)
17
C. BüskensC. Büskens
Principle Method
QP
SQP
Optimization Process
Objective function
Constraints
Sparsity
User
![Page 18: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/18.jpg)
18
C. BüskensC. Büskens
SQP Algorithm
Evaluation ofInitial Estimate
Main Iteration
2nd order Information(Hessian)
Compute new iteration point
Preparation of QPinput data
Check Optimality Criteria
QP Solver
Solution of QuadraticSubproblem
Provides search direction
Hessian Update
Find a suitable stepsise
![Page 19: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/19.jpg)
19
C. BüskensC. Büskens
Requirements
What is a good NLP Solver? • fast• robust• precise• large-scale• user-friendly• flexible• …
fast
robust
precise
large-scale
![Page 20: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/20.jpg)
20
C. BüskensC. Büskens
Main Features of WORHP
• Sparse Structures– efficient storage and computation
• Automatic Hessian structure– simpler usability for large-scale sparse problems
• Reverse Communication– flexible interface for demanding problem formulations
• Modularity– modules for different intentions and purposes
![Page 21: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/21.jpg)
21
C. BüskensC. Büskens
Sparse NLP Solver: Sparse Matrices
Sparse Matrix(here: )
Higher efficiency by storing only nonzero entries in column-
compressed format
Lower storage
requirement(here: -87.5%)
Reduced computational time
![Page 22: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/22.jpg)
22
C. BüskensC. Büskens
Reverse Communication
Advantages
Simple implementationNo worries after start
Caller
Evaluation
ObjectiveGradient
Evaluation
ConstraintsJacobian
Disadvantages
Inflexible user-interfaceNo influence after start
Traditional Calling Sequence:
SQP
![Page 23: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/23.jpg)
23
C. BüskensC. Büskens
Reverse Communication
Caller
New Calling Sequence:
Evaluation
ObjectiveGradient
Evaluation
ConstraintsJacobian
Advantages
Flexible problem evaluationFull control on optimization
Disadvantages
Harder to implementWORHP
![Page 24: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/24.jpg)
24
C. BüskensC. Büskens
Features of WORHP: Sentinel
User Solution WORHP
Sentinel
![Page 25: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/25.jpg)
25
C. BüskensC. Büskens
Group Strategy for Gradient Computation
Challenges of large-scale problems:– How to determine the minimum number of groups?– Efficient algorithm?– Suboptimal groups acceptable?– Extendable to Hessian computation?
Example:Gradient computation for the Rayleigh optimal control problem with
100.000 variables
Evaluations
Standard: 100.000 Groups: 7
![Page 26: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/26.jpg)
26
C. BüskensC. Büskens
Modularity
Modules to adapt to the User‘s requirements:
Speed, Precision, Robustness, Simplicity, …
GradientFixed Stepsize
GradientAdaptive Stepsize
GradientGroup Strategy
Advanced ErrorReporting
Matrix structureCreation/Analysis
Pre-Iteration-Analysis
WORHP
QPSOL
QP-Matrixdiagonal-BFGS
QP-MatrixHessian
QP-MatrixIdentity
QP-Matrixdiag.-Hessian
QP-MethodNonsmooth Newton
QP-MethodInterior Point
direct SolversMA47, SuperLU
iterative SolversCGN, CGS, BiCG
InterfaceTranscriptor
InterfaceReverse Comm.
InterfaceTraditional
InterfaceAMPL
![Page 27: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/27.jpg)
27
C. BüskensC. Büskens
Example: ABB IRB 6400 (optimal control)
![Page 28: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/28.jpg)
28
C. BüskensC. Büskens
Hierarchical and Pareto Optimization
![Page 29: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/29.jpg)
29
C. BüskensC. Büskens
Offline
Real-Time Capable?
UnperturbedProblem
![Page 30: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/30.jpg)
30
C. BüskensC. Büskens
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)
![Page 31: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/31.jpg)
31
C. BüskensC. Büskens
• Part I: (Motivation)– Example: Industrial Robot
• Part II: (Optimal Control)- Perturbed Optimal Control Problems- Perturbed NLP Problems- Numerical Solution of NLP-Problems
• Part III: (Real-Time Optimal Control)- Parametric Sensitivity Analysis / Solution Differentiability- Real–Time Solution
• Open–Loop Strategy• Mixed Strategies (Constraints)
- Examples: - Industrial Robot ABB IRB 6400 - Emergency Landing
Overview
![Page 32: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/32.jpg)
32
C. BüskensC. Büskens
Perturbed NLP problems
![Page 33: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/33.jpg)
33
C. BüskensC. Büskens
Parametric Sensitivity Analysis / Solution Differentiability
![Page 34: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/34.jpg)
34
C. BüskensC. Büskens
Sensitivity Analysis of OCP
![Page 35: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/35.jpg)
35
C. BüskensC. Büskens
Example: ABB IRB 6400 (parametric sensitivity analysis)
![Page 36: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/36.jpg)
36
C. BüskensC. Büskens
• Part I: (Motivation)– Example: Industrial Robot
• Part II: (Optimal Control)- Perturbed Optimal Control Problems- Perturbed NLP Problems- Numerical Solution of NLP-Problems
• Part III: (Real-Time Optimal Control)- Parametric Sensitivity Analysis / Solution Differentiability- Real–Time Solution
• Open–Loop Strategy• Mixed Strategies (Constraints)
- Examples: - Industrial Robot ABB IRB 6400 - Emergency Landing
Overview
![Page 37: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/37.jpg)
37
C. BüskensC. Büskens
offline
Real-Time Optimization (General Idea)
UnperturbedProblem
SensitivityAnalysis
onlineReal-Time Optimization
Real-Time Optimal Control
SensitivitiesSolution
Solution
![Page 38: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/38.jpg)
38
C. BüskensC. Büskens
Real-Time Optimization
![Page 39: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/39.jpg)
39
C. BüskensC. Büskens
Small Perturbation?
![Page 40: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/40.jpg)
40
C. BüskensC. Büskens
Primary Goals of Real-Time Optimization
Hierarchical AAO-order:
• real-time ability
• admissibility (feasibility)
• optimality
![Page 41: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/41.jpg)
41
C. BüskensC. Büskens
Idea 1: Adaptive optimal closed-loop control
Adaptive optimal control (closed-loop)
![Page 42: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/42.jpg)
42
C. BüskensC. Büskens
Idea 2: Real-Time model predictive control
Trajectory error estimation:
![Page 43: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/43.jpg)
43
C. BüskensC. Büskens
Real-Time model predictive control II
• iterative process (Newton Type, no gradient calculation)• self-correcting• any-time property
![Page 44: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/44.jpg)
44
C. BüskensC. Büskens
Real-Time model predictive control III
Potential for speedup:
• dynamically moving horizon
• iterative process abortable anytime
![Page 45: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/45.jpg)
45
C. BüskensC. Büskens
offline
Idea 3: Mixed Strategy
UnperturbedProblem
SensitivityAnalysis
onlineReal-Time Optimization
Real-Time Optimal Control
SensitivitiesSolution
Solution
onlineMathematical
Model
InformationAdvanced
![Page 46: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/46.jpg)
46
C. BüskensC. Büskens
Real-Time Optimization (Mixed Strategy)
• iterative process (Newton Type, no gradient calculation)• self-correcting• any-time property
![Page 47: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/47.jpg)
47
C. BüskensC. Büskens
Iterative Process
Convergence ?
• Order of convergence ?
• Existence of a fixed point ?
• Uniqueness of a fixed point ?
• Order of optimality:
worse, unchanged, improved?
![Page 48: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/48.jpg)
48
C. BüskensC. Büskens
Convergence of the Mixed Strategy
![Page 49: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/49.jpg)
49
C. BüskensC. Büskens
Convergence of the Mixed Strategy
![Page 50: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/50.jpg)
50
C. BüskensC. Büskens
Real-Time Optimal Control: (Sensitivity Derivatives)
Indirect Approaches (Minimumprinciple of Pontryagin)• Existence of Derivatives (Still Research)
[ Malanovski, Maurer, Pesch, ...]• Linear Perturbations (in the State) (Feedback Control Laws)
[ Kelley, Breakwell, Speyer, Bryson, Ho, Pesch, Bock, Krämer–Eis, ...]• General Perturbations (Linear Approximations)
[ Maurer, Augustin, Pesch, Kugelmann, ...]
Direct Approaches (NLP Problems)• Existence of Derivatives (NLP)
[ Fiacco, Robinson, ...]• Convergence (Discretized OCP OCP)
[ Alt, B., Dontchev, Felgenhauer, Malanowski, Maurer, ...]• Real–Time Optimal Control (Linear Approximations)
[ B., Maurer, ...]• Real–Time Optimal Control (Nonlinear Feedback Approx.)
[ B. ]
![Page 51: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/51.jpg)
51
C. BüskensC. Büskens
• Part I: (Motivation)– Example: Industrial Robot
• Part II: (Optimal Control)- Perturbed Optimal Control Problems- Perturbed NLP Problems- Numerical Solution of NLP-Problems
• Part III: (Real-Time Optimal Control)- Parametric Sensitivity Analysis / Solution Differentiability- Real–Time Solution
• Open–Loop Strategy• Mixed Strategies (Constraints)
- Examples: - Industrial Robot ABB IRB 6400 - Emergency Landing
Overview
![Page 52: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/52.jpg)
52
C. BüskensC. Büskens
Example: ABB IRB 6400 (real-time optimal control)
click me
![Page 53: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/53.jpg)
53
C. BüskensC. Büskens
Example: ABB IRB 6400 (real-time optimal control)
![Page 54: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/54.jpg)
54
C. BüskensC. Büskens
Example: Emergency Landing
![Page 55: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/55.jpg)
55
C. BüskensC. Büskens
Example: Emergency Landing
![Page 56: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/56.jpg)
56
C. BüskensC. Büskens
Example: Emergency Landing
![Page 57: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/57.jpg)
57
C. BüskensC. Büskens
Example: Emergency Landing
click me
![Page 58: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/58.jpg)
58
C. BüskensC. Büskens
Thank You!
![Page 59: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/59.jpg)
59
C. BüskensC. Büskens
Real-Time Optimization (Advanced Iteration Methods)
![Page 60: Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400 - Emergency Landing Overview. 37 C. Büskens offline Real-Time Optimization (General](https://reader036.vdocuments.net/reader036/viewer/2022071415/6111a33782acd703b1455442/html5/thumbnails/60.jpg)
60
C. BüskensC. Büskens
Aims and Cooperations