Tracking Control of Nanosatellites with Uncertain Time Varying Parameters
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Transcript of Tracking Control of Nanosatellites with Uncertain Time Varying Parameters
Problem StatementControl Formulation
Conclusions
Tracking Control of Nanosatelliteswith Uncertain Time Varying Parameters
Divya Thakur1 and Belinda G. Marchand2
Department of Aerospace Engineering and Engineering MechanicsThe University of Texas at Austin
AAS/AIAA Astrodynamics Specialist ConferenceJuly 31 - August 4, 2011 Girdwood, Alaska
1Graduate Student, Department of Aerospace Engineering2Assistant Professor, Department of Aerospace Engineering, AIAA Associate Fellow
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 1/ 18
Problem StatementControl Formulation
Conclusions
MotivationError DynamicsModeling a Time-Varying Inertia Matrix
Motivation
◮ Spacecraft tracking problem is widely studied.
◮ Many adaptive control solutions for systems with constant uncertain inertiaparameters.
◮ Limited research in adaptive control of time-varying inertia matrix.
◮ Focus of study: Adaptation mechanism that maintains consistent trackingperformance in the face of uncertain time-varying inertia matrix.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 2/ 18
Problem StatementControl Formulation
Conclusions
MotivationError DynamicsModeling a Time-Varying Inertia Matrix
Attitude-Tracking Error Dynamics
◮ Attitude-error dynamics:
qe0 = −12
qTevωe
qev=
12
(qe0I +
[qev
×])
ωe
Angular-velocity tracking error dynamics:
ωe = J−1(
−Jω − [ω×]Jω + u)
+ [ωe×]BCR(qe)ωr −BCR(qe)ωr
◮ Control objective: Find u(t) s.t. limt→∞
[qe, ωe
]= 0 for any
[qr(t),ωr(t)] for all [q(0),ω(0)], assuming full feedback[q(t),ω(t)] anduncertainty inJ(t).
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18
Problem StatementControl Formulation
Conclusions
MotivationError DynamicsModeling a Time-Varying Inertia Matrix
Attitude-Tracking Error Dynamics
◮ Attitude-error dynamics:
qe0 = −12
qTevωe
qev=
12
(qe0I +
[qev
×])
ωe
Angular-velocity tracking error dynamics:
ωe = J−1(
−Jω − [ω×]Jω + u)
+ [ωe×]BCR(qe)ωr −BCR(qe)ωr
◮ Control objective: Find u(t) s.t. limt→∞
[qe, ωe
]= 0 for any
[qr(t),ωr(t)] for all [q(0),ω(0)], assuming full feedback[q(t),ω(t)] anduncertainty inJ(t).
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18
Problem StatementControl Formulation
Conclusions
MotivationError DynamicsModeling a Time-Varying Inertia Matrix
Attitude-Tracking Error Dynamics
◮ Attitude-error dynamics:
qe0 = −12
qTevωe
qev=
12
(qe0I +
[qev
×])
ωe
Angular-velocity tracking error dynamics:
ωe = J−1(
−Jω − [ω×]Jω + u)
+ [ωe×]BCR(qe)ωr −BCR(qe)ωr
◮ Control objective: Find u(t) s.t. limt→∞
[qe, ωe
]= 0 for any
[qr(t),ωr(t)] for all [q(0),ω(0)], assuming full feedback[q(t),ω(t)] anduncertainty inJ(t).
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18
Problem StatementControl Formulation
Conclusions
MotivationError DynamicsModeling a Time-Varying Inertia Matrix
Attitude-Tracking Error Dynamics
◮ Attitude-error dynamics:
qe0 = −12
qTevωe
qev=
12
(qe0I +
[qev
×])
ωe
Angular-velocity tracking error dynamics:
ωe = J−1(
−Jω − [ω×]Jω + u)
+ [ωe×]BCR(qe)ωr −BCR(qe)ωr
◮ Control objective: Find u(t) s.t. limt→∞
[qe, ωe
]= 0 for any
[qr(t),ωr(t)] for all [q(0),ω(0)], assuming full feedback[q(t),ω(t)] anduncertainty inJ(t).
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18
Problem StatementControl Formulation
Conclusions
MotivationError DynamicsModeling a Time-Varying Inertia Matrix
Type of Inertia Matrix Considered
◮ Time-varying inertia matrix of the form
J(t) = JoΨ(t)
◮ Jo:(
Jo > 0, JTo = Jo
)
constant, unknown or uncertain
◮ Ψ(t):(
Ψ > 0, ΨT = Ψ)
time-varying, known
◮ Uncertainty itself is constant, multiplicative
◮ May be used to model spacecraft undergoing1. Thermal Variations2. Fuel slosh3. Appendage deployment (sensor booms, solar sails, antennas, etc.)
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 4/ 18
Problem StatementControl Formulation
Conclusions
MotivationError DynamicsModeling a Time-Varying Inertia Matrix
Example of a Time-Varying Inertia Matrix (1/2)◮ Consider a spacecraft undergoing boom deployment (e.g., GOES-R spacecraft):
◮ Boom extension rate controlled by miniature DC-torque motors◮ Initial mass of prism:m0◮ Mass of fully extended boom:αm0, 0 < α < 1
DEPLOYED BOOM
12l
SENSOR
SATELLITE
MAIN BODY
1l
13l
1l
STOWED
COLLABPSIBLE
BOOM
SATELLITE
MAIN BODYSTOWED
BOOM
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 5/ 18
Problem StatementControl Formulation
Conclusions
MotivationError DynamicsModeling a Time-Varying Inertia Matrix
Example of a Time-Varying Inertia Matrix (2/2)◮ Rod length, rod mass, and prism mass are (respectively)
r(t) =2l1τ
, mp(t) =αm0
τt, mc(t) = m0 − 2mp(t)
◮ Inertia matrix given by
Jo =
56m0l21 0
0 56m0l21 0
0 0 16m0l21
For 0≤ t ≤ τ ,
Ψ(t) =
1− 2α
τt 0 0
0 1− 75α
τt + 12
5α
τ2 t2 + 16
5α
τ3 t3 0
0 0 1+ α
τt + 12α
τ2 t2 + 16α
τ3 t3
,
for t > τ ,
Ψ(t) =
1− 2α 0 00 1− 7
5α+ 125 α+ 16
5 α 00 0 1+ α+ 12α+ 16α
.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 6/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Control Formulation
◮ Control method based on the non-certainty equivalence (non-CE) adaptivecontrol results of Seo and Akella (2008)3.
◮ Provides superior performance over traditional CE based methods whenreference trajectory does not satisfy certain persistence of excitation (PE)conditions.
◮ Original result treats constant inertia matrix.
◮ Present investigation modifies original result to handle time-varying inertiamatrix of the specific form
J(t) = JoΨ(t).
3Seo, D. and Akella, M. R., High-Performance Spacecraft Adaptive Attitude-Tracking Control ThroughAttracting-Manifold Design,Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, 2008, pp. 884–891
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 7/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-CE Adaptive Controller◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ
(
−W(
θ + δ)
+ WfΓWTf
(kp(qev
− ωef ) + ωe
))
˙θ = ΓWT
f
[(β + kv)ωef + kpqev
]− ΓWT
ωef
δ = ΓWTf ωef ,
◮ Regressor matrix
Wθ∗ = −Ψ
−1JoΨω −Ψ−1[ω×]JoΨω + Jo
(
[ω×]BCR(qe)ωr −BCR(qe)ωr
)
+ Jo
(kpβqev
+ kpqev+ kvωe
),
◮ Parameters:θ∗ = [Jo11, Jo12, Jo13, Jo22, Jo23, Jo33]T.
◮ Filter variables
ωef = −βωef + ωe
Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-CE Adaptive Controller◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ
(
−W(
θ + δ)
+ WfΓWTf
(kp(qev
− ωef ) + ωe
))
˙θ = ΓWT
f
[(β + kv)ωef + kpqev
]− ΓWT
ωef
δ = ΓWTf ωef ,
◮ Regressor matrix
Wθ∗ = −Ψ
−1JoΨω −Ψ−1[ω×]JoΨω + Jo
(
[ω×]BCR(qe)ωr −BCR(qe)ωr
)
+ Jo
(kpβqev
+ kpqev+ kvωe
),
◮ Parameters:θ∗ = [Jo11, Jo12, Jo13, Jo22, Jo23, Jo33]T.
◮ Filter variables
ωef = −βωef + ωe
Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-CE Adaptive Controller◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ
(
−W(
θ + δ)
+ WfΓWTf
(kp(qev
− ωef ) + ωe
))
˙θ = ΓWT
f
[(β + kv)ωef + kpqev
]− ΓWT
ωef
δ = ΓWTf ωef ,
◮ Regressor matrix
Wθ∗ = −Ψ
−1JoΨω −Ψ−1[ω×]JoΨω + Jo
(
[ω×]BCR(qe)ωr −BCR(qe)ωr
)
+ Jo
(kpβqev
+ kpqev+ kvωe
),
◮ Parameters:θ∗ = [Jo11, Jo12, Jo13, Jo22, Jo23, Jo33]T.
◮ Filter variables
ωef = −βωef + ωe
Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-CE Adaptive Controller◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ
(
−W(
θ + δ)
+ WfΓWTf
(kp(qev
− ωef ) + ωe
))
˙θ = ΓWT
f
[(β + kv)ωef + kpqev
]− ΓWT
ωef
δ = ΓWTf ωef ,
◮ Regressor matrix
Wθ∗ = −Ψ
−1JoΨω −Ψ−1[ω×]JoΨω + Jo
(
[ω×]BCR(qe)ωr −BCR(qe)ωr
)
+ Jo
(kpβqev
+ kpqev+ kvωe
),
◮ Parameters:θ∗ = [Jo11, Jo12, Jo13, Jo22, Jo23, Jo33]T.
◮ Filter variables
ωef = −βωef + ωe
Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-CE Adaptive Controller◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ
(
−W(
θ + δ)
+ WfΓWTf
(kp(qev
− ωef ) + ωe
))
˙θ = ΓWT
f
[(β + kv)ωef + kpqev
]− ΓWT
ωef
δ = ΓWTf ωef ,
◮ Regressor matrix
Wθ∗ = −Ψ
−1JoΨω −Ψ−1[ω×]JoΨω + Jo
(
[ω×]BCR(qe)ωr −BCR(qe)ωr
)
+ Jo
(kpβqev
+ kpqev+ kvωe
),
◮ Parameters:θ∗ = [Jo11, Jo12, Jo13, Jo22, Jo23, Jo33]T.
◮ Filter variables
ωef = −βωef + ωe
Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-CE Adaptive Controller◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ
(
−W(
θ + δ)
+ WfΓWTf
(kp(qev
− ωef ) + ωe
))
˙θ = ΓWT
f
[(β + kv)ωef + kpqev
]− ΓWT
ωef
δ = ΓWTf ωef ,
◮ Regressor matrix
Wθ∗ = −Ψ
−1JoΨω −Ψ−1[ω×]JoΨω + Jo
(
[ω×]BCR(qe)ωr −BCR(qe)ωr
)
+ Jo
(kpβqev
+ kpqev+ kvωe
),
◮ Parameters:θ∗ = [Jo11, Jo12, Jo13, Jo22, Jo23, Jo33]T.
◮ Filter variables
ωef = −βωef + ωe
Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Numerical Simulations
◮ Two sets of simulations:1. Non-PE reference trajectory2. PE reference trajectory
◮ Simulation features
◮ Quantities used to calculateJo andΨ
m0 = 30 kg, l = 0.2 m, α = 0.1, τ = 200 s
◮ Uncertain parameter
Jo =
0.2 00 0.2 00 0 1.0
−→ θ∗ = [0.2, 0, 0, 0.2, 0, 1.0]T
◮ Initial parameter estimate:θ(0) + δ(0) = 1.3 θ∗
◮ Simulation period isτ = 200 seconds.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 9/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-PE Reference Trajectory (1/3)◮ ωr =
(
0.1 cos(t)(1− e0.01t2) + (0.08π + 0.006 sin(t))te−0.01t2)
· [1, 1, 1]T
◮ Gain valueskp = 0.08,kv = 0.07,Γ = diag{100, 0.01, 0.01, 200, 0.01, 100}.
0 50 100 150 2001e−007
1e−006
1e−005
0.0001
0.001
0.01
0.1
time (s)
Ang
ular
Vel
. Err
or V
ecto
r N
orm
Norm of angular velocity error vector‖ωe‖
0 50 100 150 2001e−007
1e−006
1e−005
0.0001
0.001
0.01
0.1
1
time (s)
Qua
tern
ion
Err
or V
ecto
r N
orm
Norm of quaternion error vector‖qev‖
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 10/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-PE Reference Trajectory (2/3)
10−2
10−1
100
101
102
103
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
time(s)
Con
trol
Tor
que
Nor
m (
N−
m)
Norm of control vector‖u‖
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 11/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
Non-PE Reference Trajectory (3/3)
10−1
100
101
102
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time(s)
J 0 Par
amet
ers
Jo(3,3)
Jo(1,1) = J
o(2,2)
Estimated
True
Parameter estimates converge to true values due to additionalpersistence of excitation introduced byΨ(t)
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 12/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
PE Reference Trajectory (1/3)◮ ωr =
[cos(t) + 2 5 cos(t) sin(t) + 2
]T
◮ Gain values:kp = 0.8, kv = 0.8,Γ = diag{1, 0.001, 0.001, 1, 0.001, 1}.
0 50 100 150 2000.0001
0.001
0.01
0.1
1
10
time (s)
Ang
ular
Vel
. Err
or V
ecto
r N
orm
Norm of angular velocity error vector‖ωe‖
0 50 100 150 2000.0001
0.001
0.01
0.1
1
time (s)
Qua
tern
ion
Err
or V
ecto
r N
orm
Norm of quaternion error vector‖qev‖
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 13/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
PE Reference Trajectory (2/3)
10−2
10−1
100
101
102
103
0
2
4
6
8
10
12
time(s)
Con
trol
Tor
que
Nor
m (
N−
m)
Norm of control vector‖u‖
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 14/ 18
Problem StatementControl Formulation
Conclusions
Adaptive ControlNumerical Simulations
PE Reference Trajectory (3/3)
10−1
100
101
102
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time(s)
J 0 Par
amet
ers
Jo(3,3)
Jo(1,1) = J
o(2,2)
Estimated
True
Parameter estimates converge to true values
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 15/ 18
Problem StatementControl Formulation
Conclusions
Conclusions
◮ A non-CE adaptive control law employed for spacecraft attitude tracking in thepresence of uncertain time-varying inertia matrix.
◮ Uncertainty has special multiplicative structure.
◮ Numerical simulations performed for PE and non-PE reference signals.
◮ Attitude and angular-velocity tracking errors converge to zero.
◮ Parameter estimates converge to true values even when reference signal isnon-PE.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 16/ 18
Problem StatementControl Formulation
Conclusions
Extra 1: Some Necessary Manipulations
◮ The following algebraic manipulations are necessary to enable the adaptivecontrol derivation
ωe = −kpβqev− kpqev
− kvωe︸ ︷︷ ︸
subtracted term
+ J−1o
Ψ
−1(
u − JoΨω − [ω×]JoΨω)
− Joφ+ Jo
(kpβqev
+ kpqev+ kvωe
)
︸ ︷︷ ︸
added term
,
whereφ =([ωe×]BCR(qe)ωr −
BCR(qe)ωr
)
◮ kp, kv > 0 andβ = kp + kv
◮ Note: Dynamics are unchanged
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 17/ 18
Problem StatementControl Formulation
Conclusions
Extra 2: Initial Conditions for Simulations
◮ Initial conditions
q(0) =[
0.9487, 0.1826, 0.1826, 0.18268]T
ω(0) =[
0, 0, 0]T
rad/s
qr(0) = [1, 0, 0, 0]T
Wf (0) = 0 , ωf (0) =ωe(0) + kpqve
(0)
kp
◮ Initial filter-states1 areWf (0) = 0 andωf (0) =ωe(0)+kpqve (0)
kp.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 18/ 18