High Warehouse Racks: Optimal Feedback Control and High Warehouse Racks: Optimal Feedback Control and Adaptive Control Improvement Adaptive Control Improvement Christof Büskens skens Christof Bü INRIA, Optimal Control: Algorithms and Applications (May/June 2007)
2 Classification of Optimal Control Solutions open-loop: closed-loop:
3 Open-Loop vs. Closed-Loop Method Part II closed-loop Part I knowledge open-loop Model linear nonlinear
4 Part II linear closed-loop nonlinear closed-loop
5 Topic Slide: Optimal Control of Cranes
6 Optimal Control Problem for High-Rack Trolley
7 Overview • 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
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)
9 Optimal Control Problems with Perturbations: ODE
10 Linear Quadratic Regulator (LQR)
11 Overview • 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
12 Explanations: The Model
13 Explanations: The Quadratic Objective
14 Explanations: The Quadratic Objective
15 Explanations: The Quadratic Objective
16 Summing up the Explanations
17 Overview • 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
18 Approach using the Hamiltonian Methods: • Pontryagin‘s Minimum Principle • Calculus of variations • Hamilton-Jacobi-Bellmann equation • Karush-Kuhn-Tucker conditions • Dynamical Optimization
19 Approach using the Hamiltonian
20 Relationship: Hamiltonian LQR-Problem
21 Strictly Local Minimum
22 The Suspicion
23 Resolution (with )
24 Algebraic Matrix-Riccati-Equation
25 Summary
26 Extension
27 Extension
28 Solution of LQR click me
29 Alternative Formulations and Equivalences (LQR) (NLP)
30 Sparse NLP Solver WORHP We Optimize Really Huge Problems
31 WORHP Born for Space Applications
32 Reverse Communication Traditional Calling Sequence: Advantages Disadvantages Caller Simple implementation Inflexible user-interface No worries after start No influence after start Evaluation Evaluation SQP Constraints Objective Jacobian Gradient
33 Reverse Communication New Calling Sequence: Caller Disadvantages Advantages Evaluation Evaluation Harder to implement Flexible problem evaluation WORHP Constraints Objective Full control on optimization Jacobian Gradient
34 Features of WORHP: Sentinel WORHP User Solution Sentinel
35 Modularity Modules to adapt to the User‘s requirements: Speed, Precision, Robustness, Simplicity, … Interface Interface Interface Interface Transcriptor Traditional Reverse Comm. AMPL Gradient Matrix structure WORHP Fixed Stepsize Creation/Analysis Gradient Advanced Error QPSOL Adaptive Stepsize Reporting Gradient Pre-Iteration- Group Strategy Analysis QP-Matrix QP-Matrix QP-Method direct Solvers Identity diagonal-BFGS Nonsmooth Newton MA47, SuperLU QP-Matrix QP-Matrix QP-Method iterative Solvers Hessian diag.-Hessian Interior Point CGN, CGS, BiCG
36 Warehouse Crane
37 Topic Slide: Optimization in High Rack Technology Modeling (Real-Time) Optimization: - time - energy - jerk - oscillatory behavior - non-oscillating at Extended Constraints: - controlled stop - emergency stop - collision free Extended Modeling - transfers from/into stock - robustness - intermediate stop with: Westfalia Logistics
38 Emergency Stop
39 Overview • 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
40 Perturbed Linear Quadratic Regulator
41 Perturbed NLP problems
42 Parametric Sensitivity Analysis / Solution Differentiability
43 Real-Time Optimization (General Idea) Unperturbed Sensitivity Solution offline Problem Analysis Solution Sensitivities Real-Time Optimization online Real-Time Optimal Control
44 Real-Time Optimization
45 Existence und Uniqueness of perturbed Optimal Controllers
46 Real-Time Optimal Adaption of the Optimal Controller
47 Error Estimate
48 Optimal Control of the Warehouse Crane Optimal Control of the Inverse Pendulum
49 Application: The Inverse Pendulum
50 Horizontally Acting Forces
51 Further Forces
52 Motivation: Inverse Pendulum
53 Motivation: Inverse Pendulum (perturbed)
54 Solution of the Inverse Pendulum
55 Adaptive Optimal Control: Inverse Pendulum (perturbed) click me
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.
57 Thank You!
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