Finite Set Control Transcription for Optimal Control Applications

Transcription

1 Finite Set Control Transcription for Optimal Control Stuart A. Stanton 1 Belinda G. Marchand 2 Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin 19th AAS/AIAA Space Flight Mechanics Meeting, February 8-12, 29 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

2 Outline Background System Description FSCT Method Overview Stanton, Marchand Finite Set Control Transcription 2

3 System Description FSCT Method Overview System Description Hybrid System Dynamics Continuous States Discrete Controls ẏ = f(t, y, u) y = ˆy 1 y ny T y i R u = [u 1 u nu ] T u i U i = {ũ i,1,..., ũ i,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

4 Background System Description FSCT Method Overview Solving an Optimal Control Problem Numerically Minimize J = φ(t, y, t f, y f ) + R t f t subject to ẏ = f(t, y, u), = ψ (t, y ), = ψ f (t f, y f ), = β(t, y, u) L(t, y, u) dt? Minimize J = F (x) c(x) = subject to h c Ṫ y (x) c T ψ (x) c T ψ f (x) c T β (x)i T = NLP Solver Stanton, Marchand Finite Set Control Transcription 4

5 Background System Description FSCT Method Overview FSCT Method Overview Parameter vector consists only of states and times x = [ y i,j,k t i,k t t f ] T Control history is completely defined by Pre-specified control sequence Control value time durations, t i,k, between switching points Key parameterization factors n y Number of States n u Number of Controls n n Number of Nodes n k Number of Knots n s Number of Segments (n s = n un k + 1) Stanton, Marchand Finite Set Control Transcription 5

6 System Description FSCT Method Overview FSCT Method Overview x = [ y i,j,k t i,k t t f ] T u 1 U 1 = {1, 2, 3}, u 2 U 2 = { 1, 1}.» u = y1 ny = 2 nu = 2 nn = 4 y2 nk = 5 Seg ns = 11 t1,1 t1,2 t1,3 t1,4 t1,5 t1,6 3 u1 2 1 t2,1 t2,2 t2,3 t2,4 t2,5 t2,6 1 u2-1 t tf t Stanton, Marchand Finite Set Control Transcription 6

7 Two Stable Linear Systems where» 1 1 A 1 = 1 1 ẏ = f(y, u) = A uy, u {1, 2},» 1 1, A 2 = [ ] 1 1 ẏ = y 1 1 [ ] ẏ = y y2 y y1 y1 (a) u = 1 (b) u = 2 Figure: Individually Stable Systems Stanton, Marchand Finite Set Control Transcription 7

8 Two Stable Linear Systems Several switching laws (a) Unstable u = j 1, y1 y 2 < 2, otherwise (b) Stable u = j 1, y1 > y 2 2, otherwise (c) Stable u = j 1, y T P 1 y < yp 2 y 2, otherwise where P ua u + A T u P u = I 1 ẏ = Auy ẏ = Auy { 1, y T P1y < y T P2y 1 { 1, u = y1y2 < 2, otherwise 2 u = 2, otherwise y2 3 y2 1 y ẏ = Auy 5 { 1, y1 > y u = 2, otherwise y1 (a) y1 (b) Figure: Three Switching Laws y1 (c) Stanton, Marchand Finite Set Control Transcription 8

9 Two Stable Linear Systems FSCT Optimization J = F (x) = t f t y T f y f = 1 u k = ( 1)k y2 1 y y1 y1 (a) (b) Figure: FSCT Locally Optimal Switching Trajectories Optimization implies the switching law j 1, 1 u = y 2 m y 1 m 2, otherwise Stanton, Marchand Finite Set Control Transcription 9

10 2-Dimensional Dynamics ẏ = ṙ 1 ṙ 2 v 1 v = v 1 v 2 u 1 g + u , Controls u 1 { 5,, 5} m/s 2, u 2 { 2,, 2} m/s 2 Initial and Final Conditions r = [2 15] T km v = [ 1.7 ] T km/s r f = v f = rf, vf, tf v r, t Stanton, Marchand Finite Set Control Transcription 1

11 2-Dimensional 2 r Minimum Time 2 r Minimum Fuel km km/s v u1 km km/s v u1 m/s 2 m/s u2 2 u2 m/s 2 m/s Time (s) (a) Time (s) (b) Figure: Optimal Solutions for the Minimum-Time (a) and Minimum-Fuel (b) Problem Stanton, Marchand Finite Set Control Transcription 11

12 : Fixed Thrust Fixed thrust cold gas propulsion for arbitrary attitude tracking Reference trajectory defined by r q i and r ω i (t) Minimize deviations between body frame and reference frame with minimum propellant mass consumption Thuster Pair r Thuster Pair l3 J = β 1p f β 2m pf l1 l2 ẏ = 6 4 p f p = 2 b q i b ω i ṁ p r q i ṗ 3 Z tf t ṗ dt = = f(t, y, u) 7 5 Z tf t r q v b T r q v b dt. u i U = { 1,, 1} where u i 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

13 : Fixed Thrust Actual Trajectory Desired Trajectory 1 Quaternions: Actual vs. Desired 1.6 Angular Velocity: Actual vs. Desired rad/s Time (s) (a) Figure: Fixed Thrust Attitude Control Time (s) (b) Stanton, Marchand Finite Set Control Transcription 13

14 : 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 States and Controls 2 3 y = 6 4 b q i b ω i m p d v r q i p 7 5 u =» w a Valve Core Motion 3 o 3 o 29.5o w i {, 1} a i { 1,, 1} Nozzle Throat Valve Core Rod w i indicates whether the i th thruster pair is on or off a i indicates the acceleration of the valve core rods of the i th thruster pair Stanton, Marchand Finite Set Control Transcription 14

15 : Variable Thrust Actual Trajectory Desired Trajectory 1 Quaternions: Actual vs. Desired 1.6 Angular Velocity: Actual vs. Desired rad/s Time (s) (a) Time (s) (b) Figure: Variable Thrust Attitude Control Stanton, Marchand Finite Set Control Transcription 15

16 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