Robust Adaptive Attitude Control of a Spacecraft

Transcription

1 Robust Adaptive Attitude Control of a Spacecraft AER1503 Spacecraft Dynamics and Controls II April 24, 2015 Christopher Au

2 Agenda Introduction Model Formulation Controller Designs Simulation Results 2

3 Introduction Several Ways to control spacecraft attitude Difficult because: MIMO nonlinear system Parametric uncertainties Non-parametric uncertainties Will present a controller with the following components: PD feedback Feedforward tracking response Adaptive robustness to parametric Sliding Mode robustness to non-parametric 3

4 Model Formulation Motion Equations: I ω + ω Iω = u ε = 1 2 ω ε ηω η = 1 2 ωt ε Euler Identities: ε T ε + η 2 = 1 ε = a sin 2, η = cos 2 Quaternion Altitude Error: ε e = η d ε ηε d ε d ε η e = ηη d + ε T η d Spacecraft plant implemented in Simulink 4

5 Controller Design: Adaptive Control Using feedback and feedforward control, the equil. is GAS Must know inertia matrix exactly With adaptive control, the inertia matrix can be estimated Goal: have the system be have as if the unknown parameters are known Adaptation law: a = ΓY T s Control law: u = Y a K D s Γ is the adaptation gain a = I 11 I 22 I 33 I 12 I 13 I 23 T is the unknown elements a is the estimated elements s is the smoothed error 5

6 Controller Design: Adaptive Control Y = Y is the regressor matrix corresponding to: I ω r + ω r Iω = Y a ω r1 ω 2 ω r3 ω 3 ω r2 ω 1 ω r3 ω r2 ω 3 ω r1 ω 1 ω r2 ω 2 ω r1 ω r3 ω r2 ω 1 ω r3 ω r3 + ω 1 ω r2 ω r1 + ω 2 ω r3 ω 1 ω r1 + ω 3 ω r3 ω 1 ω r1 ω 2 ω r2 ω r1 ω 3 ω r2 Parameter estimation error: a = a a Lyapunov Proof: V t = 1 2 st Is at Γ 1 a V = s T I s + a T Γ 1 a = s T I ω s T I ω r + a T Γ 1 a = s T u ω Iω s T (Ya ω r Iω) + a T Γ 1 a a = s T Y a K D s ω Iω s T (Ya ω r Iω) + a T Γ 1 a a = s T K D s + s T Y a a s T s Iω + s T YΓ Γ 1 a a = s T K D s ω 2 ω r2 ω 3 ω r3 ω r3 ω 2 ω r1 ω r2 + ω 3 ω r1 Barbalat s lemma: {V t lower bounded, V 0, V bounded}, then V 0. So s converges to zero and the system is GAS.(ε ε d,η η d ) 6

7 Controller Design: Adaptive Control Convergence is not exact in the estimated parameters Controller will generate values that allow the tracking error to converge to zero Trajectory must be sufficiently rich for convergence ( a a) Design parameters: λ, K D and Γ are limited in magnitude Due to high frequency unmodeled dynamics actuator dynamics structural resonant modes sampling limitations measurement noise 7

8 Controller Design: Adaptive Control Adaptive and feedforward block Closed loop system with adaptation 8

9 Controller Design: Robust Adaptive Non-parametric uncertainties can reduce performance of controller when placed online Drifting of estimated parameter terms in the adaptive controller Robustness in the adaptive controller can be achieved with sliding mode control This creates a dead zone where the system does not adapt New smoothed error: s Δ Such that: s s Δ = 0 s Δ = s sat s Illustration of saturator function with dead zone 9

10 Controller Design: Robust Adaptive Modified adaptation law: a = ΓY T s Δ Motion equations with disturbance: I ω + ω Iω = u + d Lyapunov Proof: V t = 1 2 Is Δ at Γ 1 a V = s T Δ I s + a T Γ 1 a = s T Δ Y a K D s ω Iω + d s T Δ (Ya ω r Iω) + a T Γ 1 a a = s Δ T K D s Δ + sat = s Δ T K D s Δ K d s Δ + s Δ d s Δ T K D s Δ Barbalat s lemma: V 0 s Δ 0 s + s Δ T Y a a s Δ T s Iω + s Δ T YΓ Γ 1 a a 10

11 Controller Design: Robust Adaptive Robust adaptive and feedforward block Closed loop system with adaptation and noise 11

12 Simulation Results Simulation Parameters Parameter Symbol Value Inertia Matrix I kg m 2 Spacecraft initial state x(0) T rad Proportional gain in PD controller K p 200 Derivative gain in PD controller K d 20 Smoothed error ε e weight λ 25 Initial estimate of inertia matrix parameters a(0) T kg m 2 Adaptation law gain Γ 15 Sliding mode dead zone ±

13 Simulation Results Smoothed error without sliding mode Smoothed error with sliding mode At steady state, s 0.1 so dead zone chosen to be = 0.15 to prevent parameter drift 13

14 Simulation Results Robust adaptive controller tracking results Robust adaptive controller tracking error Note how the tracking is almost perfect 14

15 Possible Controller Improvements Robust adaptive controller for s/c attitude tracking is not optimal Possible solutions: Nonlinear Quadratic Regulator Nonlinear Model Predictive Controller MPCs considers the system actuation restrictions Stability and robustness can be ensured through the proper choice of terminal constraints. 15

16 Conclusion Robust adaptive controller was designed for spacecraft attitude tracking Closed loop system was simulated entirely in software using Simulink Concepts from sliding mode control were used to add system robustness 16

17 Thank You Questions? 17

18 References [1] O. Egeland and J. -M. Godhavn, "Passivity-Based Adaptive Attitude," IEEE Transactions on Automatic Control, vol. 39, no. 4, pp , [2] J.-J. E. Slotine and D. D. Benedetto, "Hamiltonian Adaptive Control of Spacecraft," IEEE Transactions on Automatic Control, vol. 35, no. 7, pp , [3] J.-J. E. SLOTINE and W. LI, Applied Nonliner Control, Englewood Cliffs, New Jersey: Prentice Hall, [4] C. Dameren, "Spacecraft Dynamics and Control II," in AER 1503 Course Notes, Toronto, University of Toronto, [5] A. Bilton, "Adaptive Control," in MIE1068 Course Notes, Toronto, University of Toronto, [6] S. Boyd, "Model Predictive Control," in EE364b Course Notes, California, Stanford University, [7] B. T. Costic, D. M. Dawson, M. d. Queiroz and V. Kapila, "A Quaternion-Based Adaptive Attitude Tracking Controller Without Velocity Measurements," Decision and Control, vol. 3, no. 39, pp , [8] O. L. deweck, "Attitude Determination and Control," in Space Systems Product Development Course Notes, Massachusetts Institute of Technology,

19 / Supplementary Slides / 19

20 MPC 20

21 Controller Design: PD Feedback Quaternion Altitude Error: ε e = η d ε ηε d ε d ε η e = ηη d + ε T η d PD Control Law: u t = K d ω t kε e t, K d = K d T, k > 0 21

22 Controller Design: PD Feedback Lyapunov Function Candidate: V t = 1 2 ωt Iω + k ε e T ε e + η e 1 2 V = ω I ω + 2k εt e ε e + η e 1 = ω u ω Iω = ω K d ω kε e ω Iω = ω K d ω η e Figure 2: PD controller with plant 22

23 Controller Design: Feedforward Feedforward Torque: u d = I ω r + ω r Iω Controller Output: u = u d + u Smoothed Error:s = ω + λε e = ω ω r Reference Angular Velocity: ω r = ω d λε e Lyapunov Function Candidate: V t = 1 2 ωt I ω V = ω T I ω = ω T u ω Iω = ω T u Integrating both sides from t = 0 to T: 0 T ω T u dt = V T V 0 = V T 0 23

24 Controller Design: Feedforward Figure 3: Feedforward block Figure 4: PD and feedforward controller 24