Multidisciplinary System Optimization of Spacecraft Interferometer Testbed

Transcription

1 Multidisciplinary System Optimization of Spacecraft Interferometer Testbed Final Presentation 7 May 2003 Deborah Howell Space Systems Laboratory Chart: 1

2 SIM: Space Interferometry Mission Mission: to determine the positions and distances of stars and to search for Earth-like planets SIM is several hundred times more accurate than any previous program in finding positions and distances of stars (4 µ arcseconds) Improved accuracy due to use of interferometry Being developed by JPL under contract from NASA Launch Date: 2009 in Earth-trailing solar orbit Artist s concept of SIM* Modeling and simulation are key to understanding and evaluating the design space Current models are extremely complex and utilize powerful simulation techniques SIM layout* *Source: planetquest.jpl.nasa.gov Chart: 2

3 Motivation Multidisciplinary System: Structures, Disturbance, Optics, Thermal, Control Expensive to build and then test must rely on simulation to evaluate designs Trade-offs between control and passive systems to counteract disturbance sources Many options here: material, configuration, placement of the reaction wheels Testbed: structurally represents a S/C interferometer, provides useful, real-world benchmark for simulation Chart: 3

4 OPTSIM testbed OPTSIM = Optical Performance Testbed for Space Interferometry Models Shaker (V S ) m p m p Spring Suspension Light collected at each end of long truss Truss positioned above the bus Disturbance source located below optics Possible problem: gravity effects Chart: 4

5 Flow chart and mathematical problem statement Design Vector M1, M2 M1, M2 K Legend M1, M2: Mirror positions PL: Pathlength PE: Pointing Error θ: commanded FSM angle φ: Resulting FSM angle K: Control gain values Structures Optics φ PL θ Control PE, PL, θ Problem Statement Min PE = F(K, M1,M2, Mp, Vs) Max OPL = F(M1,M2) Subject to: OPLD < 20 nm (2e-8m) OPL>3m Out Parameters (determine configuration) Mp: Bus mass Vs: Disturbance value Chart: 5

6 Model of System Siderostat Mirror #1 Fast steering mirrors (2) Siderostat Mirror #2 Performance Finite Element model and optical trace created using IMOS (JPL) Possible to do all analysis in Matlab (one computing environment) Structural State Space model: x = Ax + Bu y = Cx Performance: PE = FCx where F is the optical sensitivity matrix Chart: 6 Pointing Error Optical Pathlength Design Variables: K P,K I, K D Control gains for PID Control of FSM s L1, L2 : Placement of siderostat mirrors

7 Optical Layout RHS breadboard Sidero ODL TOP VIEW CCD plane FSM CCD Laser (Source) Pointing Error: distance btwn centroids Beam shear: area intersect of beams, interferometric fringes occur here Optics LHS breadboard Sidero ODL (optical delay line) CCD images the incident beams Pointing Error FSM s add control Can get PE, OPL, OPLD by modeling the optical trace Mirrors will move according to the disturbance, and will effect optical metrics Optical Trace in Simulation Siderostat mirror Source CCD Siderostat mirror Chart: 7

8 Initial Tradespace Exploration Appoint PE as single objective: Min PE Keep OPL and OPLD as constraints: OPL > 3 m OPLD < 0.2 m Note: net value for OPLD LHS points Initial Tradespace Exploration No Control, examine passive effects Orthogonal array of encircled points Two optical breadboards RHS, LHS, have discrete points on where to put optics 4 variables: x1, y1, x2,y2; each with 3 levels Variables Levels x1 y1 x2 y RHS points RHS LHS Chart: 8

9 Tradespace exploration Results Orthogonal Array PE (m) OPL (m) OPLD(m) 1 x1_1 y1_1 x2_1 y2_ E-06 2 x1_1 y1_2 x2_2 y2_ E-07 3 x1_1 y1_3 x2_3 y2_ E-08 4 x1_2 y1_1 x2_2 y2_ E-06 5 x1_2 y1_2 x2_3 y2_ E-06 6 x1_2 y1_3 x2_1 y2_ E-07 7 x1_3 y1_1 x2_3 y2_ E-07 8 x1_3 y1_2 x2_1 y2_ E-06 9 x1_3 y1_3 x2_2 y2_ E-07 overall means Pointing error affected most by y values OPL and OPLD mostly affected by x values OPLD constraint not violated, maybe I should lower the constraint Mp = 200 lbs, Vs = 1.0 V PE = 5e-4 PE mean of experiments effect on PE x1_ E E-04 y1_ E E-04 x2_ E E-04 y2_ E E-04 x1_ E E-05 y1_ E E-02 x2_ E E-04 y2_ E E-02 x1_ E E-05 y1_ E E-02 x2_ E E-04 y2_ E E-02 OPL mean of experiments effect on OPL x1_ E E-02 y1_ E E+00 x2_ E E-02 y2_ E E+00 x1_ E E+00 y1_ E E-04 x2_ E E+00 y2_ E E-04 x1_ E E-02 y1_ E E-04 x2_ E E-02 y2_ E E-04 OPLD mean of experiments effect on OPLD x1_ E E-02 y1_ e e-02 x2_ e e-01 y2_ e e-02 x1_ e e-02 y1_ e e-02 x2_ e e-02 y2_ e e-02 x1_ e e-01 y1_ e e-02 x2_ E E-02 y2_ e e-02 Chart: 9

10 Expanded System Disturbance Source Structures Parameter (Vs) Design Vector Added Control - PID K(s) = K1 + 1/s*K2 + sk3 Expanded design space, 45 possibilities for mirrors PE, Φnom + - Optics Objective: Pointing Error Chart: 10 DΦ commanded FSM angle Control Design Vector Dφ = required FSM angle Runs at Interesting Design Points Vs (V) K K K sidero position inner outer outer outer outer outer outer PEe4 (m) ~22% reduction in PE by moving mirrors in K3 able to lower PE the most Not clear what control combo is the best Voltage value significantly lowers PE OPL constraint not violated at inner positions (3.1m) move this constraint?

11 Gradient Based Search Results Gradient search only on continuous control gains, fixed structure Fast because only one structural evaluation Using SQP with quasi-newton approximation to the Hessian Single objective: min PE scaled this by 10 4 Control gains bounded: 0<K<=100 Starting K's K K K iterations PE*10^4 (m) K3 seems to be the most important K1 and K2 do not change very much, but K3 shoots right to 100 active constraint (non-zero Lagrangian multiplier) As long as K3 is 100, and K1 and K2 are non-zero, PE is invariant up to the 7 th decimal place PE does not appreciably change even with K3 = 500 Algorithm did iterate for highlighted run, the initial PE was e-5 m Chart: 11

12 Sensitivity Analysis of Optimal Design Taking the highlighted solution as the optimal solution DESIGN VECTOR SENSITIVITIES J( x*) K1 J( x*) K2 J( x*) K3 NORM NORM NORM = 6.0 e-9 = 0.0 = e-6 From gradient vector in iteration PARAMETER SENSITIVITIES J( x*) Vs J( x*) m p NORM NORM Using finite differences = e-5 = K3 has the most negative effect on PE expected J( x*) is small in the absolute sense mirror positions probably have larger effect K3 NORM Both parameter sensitivities are positive expected PE is much more sensitive to mass than to shaker voltage not immediately intuitive Chart: 12

13 System Design Optimization using a Genetic Algorithm Easy to implement using Matlab GA toolbox Did not use mixed integers initially, just rounded to the nearest integer for the mirror positions Tried 3 different settings for GA Initial population: K3 = 50, M1 = 10, M2 = 10 pop. size mutation rate max # gen iterations K M M PE (m) 3.897E E E-05 K3 did not jump to the maximum allowed (100) Mirrors are not at their outer or innermost extremes Probably did not find global optimum Chart: 13

14 Multiobjective Design Optimization: GA 2 objectives: min PE and max OPL, Intuition: opposing objectives, simulation says otherwise Altered the encoding and decoding functions to only accept and return types of numbers set by flag (0=floating point, 1 = integer) Computation time: 7 hrs Used weighted sum approach to get pareto-front, coupled with GA JtotOpt = λ*(-pee+5) +(1-λ)*OPL Performance vs Lambda Performance lambda PEe5 OPL JtotOpt Chart: 14

15 Pareto Front Pareto-front is not in proper orientation Problem with simulation Problem with GA Have some really good designs M1 = 45, M2 = 37, K = M1 = 9, M2 = 28, K = 54 5 Good designs Only true pareto pts Utopia Point OPL Pareto Front PEe RHS y x y LHS x Chart: 15

16 Interesting Trends Expectation: That the control gain would increase with increasing λ since more emphasis on PE Reality: control gain varies widely Ctrl Gain Lambda No matter what the value for λ, the mirrors are almost always at extreme inner or outer positions RHS y x y LHS x Chart: 16

17 Sensitivity Analysis Sensitivity wrt scaling Scaled the PE up by 10 5 Used finite differencing at λ = 0.4, to get sensitivity wrt exponent OPL Scale PE Scale = = Says that the PE would have changed significantly if the scaling were changed But can only be believed if the GA returns the global optimum just don t know Sensitivity wrt weighting Using the results from the previous analysis, and finite differencing OPL λ PE λ = = PE is affected much more that OPL for the same increase in λ Signs make sense Chart: 17

18 Conclusions Conclusions and Recommended Future Work Control not too effective probably need to implement a different type or in a new way PE much more sensitive to bus mass than to shaker voltage No definite Pareto Front, but tendency toward extreme mirror positions PE more sensitive to MO weighting Scaling looks like a problem Future Work Get a proper measure of sensitivity on discrete mirror positions Refine model Open up trade space Different answers with different MO optimization routines GA global optimum check Fix scaling issues Chart: 18