From Theory to Application (Optimization and Optimal Control in Space Applications)

Transcription

1 From Theory to Application (Optimization and Optimal Control in Space Applications) Christof Büskens Optimierung & Optimale Steuerung

2 The paradox of mathematics If mathematics refer to reality, it is not save. Albert Einstein (ca. 1950)

3 The paradox of mathematics If mathematics refer to reality, it is not save. If mathematics is save, it does not refer to reality. Albert Einstein (ca. 1950)

4 From Theory to Application

5 From Theory to Application

6 Topics London: a short review first step into space applications second step into space applications further space applications

7 Nonlinear Optimization

8 Sparse NLP Solver WORHP We Optimize Really Huge Problems > variables > constraints

9 History of sparse NLP solvers: WORHP, just another NLP solver? <1997 SNOPT 2000 IPFILTER 2001 KNITRO 2002 IPOPT Can WORHP be competative? 2010 WORHP

10 User, Market & Scientific Requirements In contrast to most established and grown NLP solvers, WORHP has undergone extensive design on the drawing board before its implementation was started, making use of -user requirements, -current architectures, -computational standards and compilers to construct a modern NLP solver for largescale nonlinear optimisation. Programming Language C / C++ 3 Fortran / C 9 Fortran GR 8 Algorithms IP SQP 12 Sequential Quadratic Programming Primal-Dual Interior Point Generalized Reduced Gradient Successive Quadratic Programming Benders Decomposition Levenberg Marquardt Coordinate Search Direct; 14 Optimization Classes Solvers 28 for nonlinear constraints 19 dense & sparse Commercialization / License Type Company; 9 GPL type; RU; 3 1 BR; 1 AU; 1 US 13 Distribution by Country EU/ESA 16 LP-problem, mixed integer, stochastic QP-problem, mixed integer Semidefinite and 2nd-order cone prog. Geometric programming Non Linear programming Minimization of nonsmooth functions Semi-infinite programming Mixed integer nonlinear programming Network constraints Special/constraint solvers UK 5 DE 5 GR; 1 AT; 2 FI; 1 DK; 1 CH; 1 PT; 1 BE; 1

11 CUTEr(920) SNOPT IPFILTER 0.3 KNITRO IPOPT WORHP 2.0 Problems solved Optimal level Acceptable level Not solved Percentage 89,89% 89,67% 96,41% 95,33% 99,79% Time s s s s 5060 s

12 COPS 3.0(68) SNOPT KNITRO IPOPT WORHP 2.0 Problems solved Optimal level Acceptable level Not solved Percentage 94,12% 94,12% 100% 100% Time 8858 s 6352 s 5682 s 1463 s

13 Interfaces Fortran Traditional Basic-Feature Full-Feature C/C++ Traditional Basic-Feature Full-Feature AMPL MATLAB Plattforms: Linux Unix Windows Mac OS

14 Optimal Control Problems with Perturbations: ODE

15 Methods for solving Optimal Control Problems First Optimize then Discretize! First Discretize then Optimize! Not real-time capable

16 Direct approaches for OCP I: ODE

17 Optimal Control Solver TRANSWORHP TRANScription method for WORHP > states and controls > constraints > dicretization points in time

18 Optimization of Coverage

19 Satellite Mission Analysis (Coverage) Find a satellite constellation s.t. any target on Earth within a fixed time frame can be monitored at least once with as few satellites as possible. Requested result: Number of satellites and optimal orbits Earth: Sphere WGS-84 Swath: left and right looking sensors

20 Details of Satellite Constellation Periodic solution (~14d, repeating forever ) No orbital manoeuvres allowed Control unit is only available to compensate errors in model, avoid collision with space debris. Commercial software package STK analysis of constellations no interface for optimization

21 Point Coverage Area of coverage Monitor coverage over discrete grid (STK) Disadvantages No information if region between grid points was (not) covered Area not differentiable w.r.t. orbit parameters

22 Point Coverage Area of coverage Monitor coverage over discrete grid (STK) Disadvantages No information if region between grid points was (not) covered Area not differentiable w.r.t. orbit parameters

23 Point Coverage Area of coverage Monitor coverage over discrete grid (STK) Disadvantages No information if region between grid points was (not) covered Area not differentiable w.r.t. orbit parameters

24 Point Coverage Area of coverage Monitor coverage over discrete grid (STK) Disadvantages No information if region between grid points was (not) covered Area not differentiable w.r.t. orbit parameters

25 Orbital Elements for Satellite Trajectory Elliptical orbit with Earth at one focus Longitude of ascending node W Inclination i Argument of perigee w Semi major axis a Eccentricity e Mean anomaly M W M w a i Reference point: Vernal Equinox

26 ODE System for Satellite Orbit Kepler's Second Law

27 ODE System for Satellite Orbit Kepler + Oblateness of Earth

28 ODE System for Satellite Orbit Kepler + Oblateness of Earth + Geopotential

29 Nonlinear Optimization Problem Vector of free orbit parameters of m satellites Area covered by constellation after time t Constraints (box constraints, coupling of satellites)

30 Store covered area as set of convex disjoint polygons Polygon Coverage Add new polygons Reuse existing corners Remove overlapping parts Simplify polygon set Remove unused corners Merge adjacent corners Remove inner edges

31 Store covered area as set of convex disjoint polygons Polygon Coverage Add new polygons Reuse existing corners Remove overlapping parts Simplify polygon set Remove unused corners Merge adjacent corners Remove inner edges

32 Store covered area as set of convex disjoint polygons Polygon Coverage Add new polygons Reuse existing corners Remove overlapping parts Simplify polygon set Remove unused corners Merge adjacent corners Remove inner edges

33 Store covered area as set of convex disjoint polygons Polygon Coverage Add new polygons Reuse existing corners Remove overlapping parts Simplify polygon set Remove unused corners Merge adjacent corners Remove inner edges

34 Store covered area as set of convex disjoint polygons Polygon Coverage Add new polygons Reuse existing corners Remove overlapping parts Simplify polygon set Remove unused corners Merge adjacent corners Remove inner edges

35 Store covered area as set of convex disjoint polygons Polygon Coverage Add new polygons Reuse existing corners Remove overlapping parts Simplify polygon set Remove unused corners Merge adjacent corners Remove inner edges

36 Store covered area as set of convex disjoint polygons Polygon Coverage Add new polygons Reuse existing corners Remove overlapping parts Simplify polygon set Remove unused corners Merge adjacent corners Remove inner edges

37 Store covered area as set of convex disjoint polygons Polygon Coverage Add new polygons Reuse existing corners Remove overlapping parts Simplify polygon set Remove unused corners Merge adjacent corners Remove inner edges

38 Polygon Coverage on Sphere Interpolate edges of polygons in cylindrical coordinates

39 Polygon Coverage on Sphere

40 Area of Polygon on Sphere Polygon area as sum of trapezoids Surface integral on sphere Efficient area calculation for closed n-gon:

41 Area of Polygon on Sphere Polygon area as sum of trapezoids Surface integral on sphere Efficient area calculation for closed n-gon:

42 Area of Polygon on Sphere Polygon area as sum of trapezoids Surface integral on sphere Efficient area calculation for closed n-gon:

43 Conclusion of Satellite Coverage Satellite constellation Integration Projection Swath data Polygon Coverage Differentiable area Integration Polygon set

44 Dynamical Coverage of Climes

45 Time Schedule Optimization of Satellites

46 Objectives Optimization variable : - Starting time of observation - Length of observation Objective Function : - Covered area of defined targets Task : Maximize coverage by arranging swath pieces

47 Constraints - No overlapping - Time frame restriction - Duty cycle restrictions - Limited data storage

48 Constraints - No overlapping - Time frame restriction - Duty cycle restrictions - Limited data storage

49 Constraints - No overlapping - Time frame restriction - Duty cycle restrictions - Limited data storage

50 Constraints - No overlapping - Time frame restriction - Duty cycle restrictions - Limited data storage

51 Mathematical Formulation Polynomial coverage calculation as before.

52 Maximum revisit time Is the maximum duration of a gap in coverage in the entire coverage interval. Sliding time frame as replacement formulation for maximum revisit time. Example: Day 0-2 Day 3-5 Day 6-8 Day 9-11

53 Hierarchical optimization Different sensors for different target areas. Sensors have different properties: - Swath angle - Dutycycle - uplink-rate Hierarchical optimization: - Constraints for previous solutions

54 More Requirements Fixed timeframes. Constraints are fulfilled according to the optimization process. Consideration of sun elevation. In the case of optical sensors. Sensor is only switched on if the midpoint of the swath is at a certain sun elevation angle.

55 Some other stuff

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57 Application: Emergency Landing click me

58 Multidisciplinary Design Optimization PRESTIGE = PRrogram in Education for Space, Technology, Innovation and knowledge University of Bremen + Politecnico di Milano Disciplines involved: Trajectory Propulsion Aerodynamics Thermal protection Propellant Tanks Cost and Risk of design

59 Four-Body-Problem Lagrange Points Halo Orbits Flight along Mannifolds Optimal Control

60 You can calculate on us!