Parametric Sensitivity Analysis of NLP Problems and its Applications to Real-Time Controller Design
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1 SADCO - Sensitivity Analysis for Deterministic Controller Design Parametric Sensitivity Analysis of NLP Problems and its Applications to Real-Time Controller Design Part 1: Nonlinear Real-Time Optimization Nonlinear Real-Time Christof Büskens Optimierung & Optimale Steuerung Paris,
2 Roadmap (London) parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) (Stuttgart) method IV NLP (WORHP) method V -M (WORHP) OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
3 Overview Part I: (nonlinear open-loop nonlinear closed-loop) (How to Calculate Optimal Trajectories (in Real-Time)) - Perturbed Problems - Perturbed NLP Problems - Parametric Sensitivity Analysis / Solution Differentiability - Real Time Solution - Example: Emergancy Landing Part II: (linear closed-loop nonlinear closed-loop) (How to follow Optimal Trajectories (in Real-Time)) - Riccati-Controller - Adaptive ler - Example: Inverse Pendulum
4 NLP post optimality
5 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
6 Perturbed NLP problems
7 Solution Differentiability
8 Advanced Sensitivity Analysis
9 post optimality real time approximation
10 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
11 Real-Time (General Idea) Unperturbed Problem Solution Sensitivity Analysis offline Solution Sensitivities Real-Time Optimization Real-Time online
12 Real-Time
13 Large Perturbations?
14 Primary Goals of Real-Time Optimization Hierarchical AAO-order: real-time ability admissibility (feasibility) optimality
15 real time approximation repeated correction
16 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
17 Real-Time Optimization (Extended Idea) Unperturbed Problem Solution Sensitivity Analysis offline Solution Sensitivities Real-Time Optimization Real-Time online Advanced Information Mathematical Model online
18 Real-Time Optimization (Mathematical Feedback) iterative process (Newton Type, no gradient calculations) self-correcting any-time property
19 Iterative Process Convergence? Order of convergence? Existence of a fixed point? Uniqueness of a fixed point? Order of optimality: worse, unchanged, improved?
20 Convergence of the Mathematical Feedback Strategy
21 Convergence of the Mathematical Feedback Strategy
22 theory example
23 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
24 NLP Example
25 NLP Example
26 NLP Example
27 NLP Example
28 NLP Example Warning: large values!
29 NLP Example
30 NLP Example
31 example theory
32 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
33 Advanced Sensitivity Analysis
34 Higher Order Sensitivities of the Objective
35 Higher Order Sensitivities
36 Improved Approximation (cheap) iterative refinement similar to q
37 2. NLP Example:
38 2. NLP Example:
39
40 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
41 Open-Loop vs. Closed-Loop open-loop: closed-loop:
42 Expectations on a Real-Time Optimization Algorithm
43 Open-Closed-Loop Technique (idea)
44 Expectations on a Real-Time Optimization Algorithm
45 Open-Loop vs. Closed-Loop Method closed-loop Part II Part I open-loop linear nonlinear Model
46 Problems with Perturbations: ODE
47 Methods for solving Problems First Optimize then Discretize! First Discretize then Optimize! Not real-time capable
48
49 Direct approaches for OCP I: ODE
50 Direct approaches for OCP II: ODE TransWORHP [B./Knauer] WORHP: [B./Gerdts]
51 Solver TransWORHP Transcription method for WORHP > states and controls > constraints > dicretization points in time
52 Sparse NLP Solver WORHP We Optimize Really Huge Problems > variables > constraints
53 Real-Time : (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 (Linear Approximations) [ B., Maurer,...] Real Time (Nonlinear Feedback Approx.) [ B. ]
54 NLP OCP
55 Perturbed NLP problems
56 Solution Differentiability
57 z x,u
58 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
59 Real-Time Optimization
60 Sensitivity Analysis of OCP
61 z x,u
62 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
63 Real-Time Optimization (Mathematical Feedback) iterative process (Newton Type, no gradient calculations) self-correcting any-time property
64 Convergence of the Mathematical Feedback Strategy
65 Convergence of the Mathematical Feedback Strategy
66 theory application
67 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
68
69 Example: Emergency Landing
70 Example: Emergency Landing
71 Example: Emergency Landing click me
72 Example: Emergency Landing
73 Example: Industrial Robot ABB IRB 6400 Forces: - centrifugal - Coriolis - gravity - frictional (dry) - restoring
74 Hierarchical and Pareto Optimization
75 Example: ABB IRB 6400 (parametric sensitivity analysis)
76 Example: ABB IRB 6400 ( Problem)
77 Example: ABB IRB 6400 (real-time optimal control) click me
78 Example: ABB IRB 6400 (real-time optimal control)
79 idea further methods
80 Roadmap parametr. sensitivity analysis realtime control prediction (NLP & OCP) repeated correction (NLP & OCP) method IV NLP (WORHP) method V -M OCP (TransWORHP) realtime trajectory prediction adaptive (Riccati-) controller (Riccati-) controller method III method II method I tools offline online feedback application
81 Sensitivity Analysis of OCP
82 Speedup Potential (Interpolation Error vs. Approximation Error) Trajectory error estimation:
83 Speedup Potential (Interpolation Error vs. Approximation Error) iterative process (Newton Type, no gradient calculation) self-correcting any-time property
84 Speedup Potential (Interpolation Error vs. Approximation Error) Potential for speedup: dynamically moving horizon iterative process abortable anytime
85 Advanced Sensitivity Analysis
86 Example: Burger s Equation
87 Numerical Results
88 Quality large foreseeable perturbations optimality robustness CPU time method V-M????????? method IV very good very good fast method III good very good very fast method II good good very fast method I OK good very fast optimality robustness CPU time method V-M????????? method IV good very good fast method III OK good very fast method II OK good very fast method I failing failing very fast large unforeseeable perturbations
89 Thank you for today!
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