# Finite Set Control Transcription for Optimal Control Applications

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Information about Finite Set Control Transcription for Optimal Control Applications
Technology

Published on March 21, 2014

Author: BelindaMarchand

Source: slideshare.net

## Description

Talk charts from the 19th AAS/AIAA Space Flight Mechanics Meeting, February 8-12, 2009
Savannah, Georgia

university-log Background Applications Conclusions Finite Set Control Transcription for Optimal Control Applications Stuart A. Stanton1 Belinda G. Marchand2 Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin 19th AAS/AIAA Space Flight Mechanics Meeting, February 8-12, 2009 Savannah, Georgia 1 Capt, USAF; Ph.D. Candidate 2 Assistant Professor The views expressed here are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government. Stanton, Marchand Finite Set Control Transcription 1

university-log Background Applications Conclusions Outline Background System Description FSCT Method Overview Applications Linear Switched System Lunar Lander Small Spacecraft Attitude Control Conclusions Stanton, Marchand Finite Set Control Transcription 2

university-log Background Applications Conclusions System Description FSCT Method Overview System Description Hybrid System Dynamics ˙y = f(t, y, u) Continuous States y = ˆ y1 · · · yny ˜T yi ∈ R Discrete Controls u = [u1 · · · unu ]T ui ∈ Ui = {˜ui,1, . . . , ˜ui,mi } Examples Switched Systems Task Scheduling and Resource Allocation Models On-Oﬀ Control Systems Control Systems with Saturation Limits Stanton, Marchand Finite Set Control Transcription 3

university-log Background Applications Conclusions System Description FSCT Method Overview Solving an Optimal Control Problem Numerically Minimize J = φ(t0, y0, tf , yf ) + R tf t0 L(t, y, u) dt subject to ˙y = f(t, y, u), 0 = ψ0(t0, y0), 0 = ψf (tf , yf ), 0 = β(t, y, u) ? Minimize J = F(x) subject to c(x) = h cT ˙y (x) cT ψ0 (x) cT ψf (x) cT β (x) iT = 0 NLP Solver Stanton, Marchand Finite Set Control Transcription 4

university-log Background Applications Conclusions System Description FSCT Method Overview FSCT Method Overview Parameter vector consists only of states and times x = [· · · yi,j,k · · · · · · ∆ti,k · · · t0 tf ]T Control history is completely deﬁned by Pre-speciﬁed control sequence Control value time durations, ∆ti,k, between switching points Key parameterization factors ny Number of States nu Number of Controls nn Number of Nodes nk Number of Knots ns Number of Segments (ns = nunk + 1) Stanton, Marchand Finite Set Control Transcription 5

university-log Background Applications Conclusions System Description FSCT Method Overview FSCT Method Overview x = [· · · yi,j,k · · · · · · ∆ti,k · · · t0 tf ]T u1 ∈ U1 = {1, 2, 3}, u2 ∈ U2 = {−1, 1}. u∗ = » 1 2 3 1 2 3 −1 1 −1 1 −1 1 – 1 2 3 1 2 -1 1 -1 1 -1 y1 y2 Seg. u1 u2 t t0 tf ny = 2 nu = 2 nn = 4 nk = 5 ns = 11 ∆t1,1 ∆t1,2 ∆t1,3 ∆t1,4 ∆t1,5 ∆t1,6 ∆t2,1 ∆t2,2 ∆t2,3 ∆t2,4 ∆t2,5 ∆t2,6 1 2 3 4 5 6 7 8 9 10 11 Stanton, Marchand Finite Set Control Transcription 6

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control Two Stable Linear Systems ˙y = f(y, u) = Auy, u ∈ {1, 2} , where A1 = » −1 10 −100 −1 – , A2 = » −1 100 −10 −1 – −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 y1 y2 ˙y = −1 10 −100 −1 y (a) u = 1 −20 −10 0 10 20 30 −20 −15 −10 −5 0 5 10 15 20 y1 y2 ˙y = −1 100 −10 −1 y (b) u = 2 Figure: Individually Stable Systems Stanton, Marchand Finite Set Control Transcription 7

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control Two Stable Linear Systems Several switching laws (a) Unstable u =  1, y1y2 < 0 2, otherwise (b) Stable u =  1, y1 > y2 2, otherwise (c) Stable u =  1, yT P 1y < yP 2y 2, otherwise where P uAu + AT u P u = −I −8000 −6000 −4000 −2000 0 2000 −8000 −7000 −6000 −5000 −4000 −3000 −2000 −1000 0 1000 y1 y2 ˙y = Auy u = 1, y1y2 < 0 2, otherwise (a) −35 −30 −25 −20 −15 −10 −5 0 5 10 15 −30 −25 −20 −15 −10 −5 0 5 10 y1 y2 ˙y = Auy u = 1, y1 > y2 2, otherwise (b) −40 −30 −20 −10 0 10 20 30 40 −30 −20 −10 0 10 20 30 y1 y2 ˙y = Auy u = 1, yT P1y < yT P2y 2, otherwise (c) Figure: Three Switching Laws Stanton, Marchand Finite Set Control Transcription 8

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control Two Stable Linear Systems FSCT Optimization J = F(x) = tf − t0 yT f yf = 1 u∗ k = 3 2 + 1 2 (−1)k −20 −15 −10 −5 0 5 10 15 20 −30 −25 −20 −15 −10 −5 0 5 10 y1 y2 (a) −20 −15 −10 −5 0 5 10 15 20 −30 −25 −20 −15 −10 −5 0 5 10 y1 y2 (b) Figure: FSCT Locally Optimal Switching Trajectories Optimization implies the switching law u =  1, − 1 m ≤ y2 y1 ≤ m 2, otherwise Stanton, Marchand Finite Set Control Transcription 9

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control 2-Dimensional Lunar Lander Dynamics ˙y = 2 6 6 4 ˙r1 ˙r2 ˙v1 ˙v2 3 7 7 5 = 2 6 6 4 v1 v2 u1 −g + u2 3 7 7 5 , Controls u1 ∈ {−50, 0, 50} m/s2 , u2 ∈ {−20, 0, 20} m/s2 Initial and Final Conditions r0 = [200 15]T km v0 = [−1.7 0]T km/s rf = 0 vf = 0 v0 r0, t0 rf , vf , tf Stanton, Marchand Finite Set Control Transcription 10

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control 2-Dimensional Lunar Lander −20 0 20 40 60 80 100 120 0 100 200 Minimum Time r km rrrrrrrrrrrrrrrrrrrrrrrrrrrr −20 0 20 40 60 80 100 120 −4 −2 0 2 v km/s vvvvvvvvvvvvvvvvvvvvvvvvvvvv −20 0 20 40 60 80 100 120 −50 0 50 u1 m/s2 u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1 −20 0 20 40 60 80 100 120 −20 0 20 u2 Time (s) m/s2 u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2 (a) −20 0 20 40 60 80 100 120 140 160 0 100 200 Minimum Fuel r km rrrrrrrrrrrrrrrrrrrrrrrrrrrr −20 0 20 40 60 80 100 120 140 160 −2 −1 0 1 v km/s vvvvvvvvvvvvvvvvvvvvvvvvvvvv −20 0 20 40 60 80 100 120 140 160 −50 0 50 u1 m/s2 u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1 −20 0 20 40 60 80 100 120 140 160 −20 0 20 u2 Time (s) m/s2 u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2 (b) Figure: Optimal Solutions for the Minimum-Time (a) and Minimum-Fuel (b) Lunar Lander Problem Stanton, Marchand Finite Set Control Transcription 11

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control Small Spacecraft Attitude Control: Fixed Thrust Fixed thrust cold gas propulsion for arbitrary attitude tracking Reference trajectory deﬁned by rqi 0 and rωi(t) Minimize deviations between body frame and reference frame with minimum propellant mass consumption J = β1pf − β2mpf pf −p0 = Z tf t0 ˙p dt = Z tf t0 “ r qv b ”T “ r qv b ” dt. l3 Thuster Pair l1 r l2 Thuster Pair ˙y = 2 6 6 6 6 6 6 6 6 6 4 b ˙qi b ˙ωi ˙mp r ˙qi ˙p 3 7 7 7 7 7 7 7 7 7 5 = f(t, y, u) ui ∈ U = {−1, 0, 1} where ui indicates for each principal axis whether the positive-thrusting pair, the negative-thrusting pair, or neither is in the on position Stanton, Marchand Finite Set Control Transcription 12

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control Small Spacecraft Attitude Control: Fixed Thrust Actual Trajectory Desired Trajectory 0 2 4 6 8 10 12 14 16 18 20 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Quaternions: Actual vs. Desired (a) 0 2 4 6 8 10 12 14 16 18 20 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time (s) rad/s Angular Velocity: Actual vs. Desired (b) Figure: Fixed Thrust Attitude Control Stanton, Marchand Finite Set Control Transcription 13

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control Small Spacecraft Attitude Control: Variable Thrust Variable thrust cold gas propulsion Valve rod modiﬁes nozzle throat area Include additional states to model variable thrust Resulting dynamics are still hybrid Nozzle Throat 30o 30o 29.5o Valve Core Rod Valve Core Motion States and Controls y = 2 6 6 6 6 6 6 6 6 4 b qi b ωi mp d v r qi p 3 7 7 7 7 7 7 7 7 5 u = » w a – wi ∈ {0, 1} ai ∈ {−1, 0, 1} wi indicates whether the ith thruster pair is on or oﬀ ai indicates the acceleration of the valve core rods of the ith thruster pair Stanton, Marchand Finite Set Control Transcription 14

university-log Background Applications Conclusions Linear Switched System Lunar Lander Small Spacecraft Attitude Control Small Spacecraft Attitude Control: Variable Thrust Actual Trajectory Desired Trajectory 0 2 4 6 8 10 12 14 16 18 20 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Quaternions: Actual vs. Desired (a) 0 2 4 6 8 10 12 14 16 18 20 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time (s) rad/s Angular Velocity: Actual vs. Desired (b) Figure: Variable Thrust Attitude Control Stanton, Marchand Finite Set Control Transcription 15

university-log Background Applications Conclusions Conclusions This investigation explores the range of applications of the FSCT method The applicability of the method extends to all engineering disciplines FSCT vs. Multiple Lyapunov Functions Optimal control laws may be extracted whose performance exceeds those derived using a Lyapunov argument Multiple independent decision inputs managed simultaneously Solutions derived via the FSCT method are utilized in conjunction with a hybrid system model predictive control scheme Optimized control schedules can be realized in the context of potential perturbations or other unknowns Some continuous control input systems may be more accurately described as systems ultimately relying on discrete decision variables Continuous control variables may often be extended into a set of continuous state variables and discrete inputs Stanton, Marchand Finite Set Control Transcription 16

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