L1 Adaptive Control for Autonomous...
Transcript of L1 Adaptive Control for Autonomous...
IntroductionHelicopter Model
L1 Adaptive ControllerResults
L1 Adaptive Control for Autonomous Rotorcraft
B. J. Guerreiro∗ C. Silvestre∗ R. Cunha∗
C. Cao† N. Hovakimyan‡
∗bguerreiro,cjs,[email protected]
Instituto Superior Tecnico, Portugal
University of Connecticut
University of Illinois at Urbana-Champaign
11 June 2009 – American Control Conference
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 1/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Outline
1 IntroductionAutonomous RotorcraftL1 Adaptive Control Theory
2 Helicopter ModelState Space EquationL1 Model Formulation
3 L1 Adaptive ControllerSystem and PredictorAdaptive and Control LawsStability and Performance
4 ResultsSimulation ResultsSummary
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 2/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Autonomous RotorcraftL1 Adaptive Control Theory
Outline
1 IntroductionAutonomous RotorcraftL1 Adaptive Control Theory
2 Helicopter ModelState Space EquationL1 Model Formulation
3 L1 Adaptive ControllerSystem and PredictorAdaptive and Control LawsStability and Performance
4 ResultsSimulation ResultsSummary
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 3/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Autonomous RotorcraftL1 Adaptive Control Theory
IntroductionAutonomous Rotorcraft
Applications:
Low altitude aerial surveillance;Automatic infrastructure inspection;3D mapping of unknown environments.
The platform:
High precision 3D maneuvers;Hover and VTOL capabilities;Carry multiple sensors.
Challenging Control problem:
Highly nonlinear and coupled modelWide parameter variations over the flightenvelope
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 4/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Autonomous RotorcraftL1 Adaptive Control Theory
IntroductionL1 Adaptive Control Theory
Some features:Separation (decoupling) between adaptation and robustness;Guaranteed fast adaptation;Guaranteed transient performance – inputs and outputs;Guaranteed (bounded away from zero) time-delay margin;
Formulation used:Nonlinear dynamics approximated by time-varying linearsystem for specific region of operation;Multi-input multi-output;State feedback;
Other L1 adaptive control formulation:Output feedback for systems with unknown dimension;Non-affine, nonlinear;Time-varying reference system;etc.
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 5/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Autonomous RotorcraftL1 Adaptive Control Theory
IntroductionFurther reading on L1 Adaptive Control
Further reading:
C. Cao and N. Hovakimyan.L1 Adaptive Controller for Systems with Unknown
Time-varying Parameters and Disturbances in the
Presence of Non-zero Trajectory Initialization Error.Int. Journal of Control, 81(7), 2008.
C. Cao and N. Hovakimyan.L1 Adaptive Controller for MIMO Systems in the
Presence of Unmatched Disturbances.Proceedings of the American Control Conference, 2008.
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 6/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
State Space EquationL1 Model Formulation
Outline
1 IntroductionAutonomous RotorcraftL1 Adaptive Control Theory
2 Helicopter ModelState Space EquationL1 Model Formulation
3 L1 Adaptive ControllerSystem and PredictorAdaptive and Control LawsStability and Performance
4 ResultsSimulation ResultsSummary
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 7/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
State Space EquationL1 Model Formulation
Helicopter Model
Helicopter model:
6 DoF Rigid body dynamics;Parameterized for the Vario X-treme R/Chelicopter
Actuation u = [ θ0 , θ1c , θ1s , θ0t]T :
θ0 – main rotor collective inputθ1c , θ1s – main rotor cyclic inputsθ0t
– tail rotor collective input
State variables:
vB = [ u v w ]′ – Body-fixed linear velocityωB = [ p q r ]′ – Body-fixed angular velocity
ΠλB =
[
1 0 00 1 0
]
φθψ
– Euler angles
State Vector:
xB =
vB
ωB
Π λB
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 8/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
State Space EquationL1 Model Formulation
Helicopter ModelState Space Equation
State Equation (velocity and attitude dynamics):
xB = f(xB ,uB) =
−ωB × vB + 1
m[fe (vB , ωB ,uB ,vw) + fg (Π λB)]
−I−1
B (ωB × IB ωB) + I−1
B ne (vB , ωB ,uB)ΠQ (Π λB) ωB
where xB ∈ X and uB ∈ U .
Main Rotor
VerticalFin
Tail Rotor
HorizontalTailplane
Fuselage
KinematicsDynamics
Rigid Body
+
+
HelicopterComponents
Gravity
f
fu
n
ng
g
e
e
vB
wB
lB
pB
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 9/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
State Space EquationL1 Model Formulation
Helicopter ModelL1 Model Formulation
For a sufficiently small region of operation:
x(t) = A(t)x(t) +Bw w(t) +B ku u(t)y(t) = C x(t) , x(0) = x0
Let A(t) = An +Aδ(t) and Kx(t) = Kn +Kδ(t)
Assumption 1
There exists a control matrix Kn such that Am = An −BKn is
Hurwitz.
Assumption 2
There exist a time varying vector kw(t) and a time varying matrix
Kδ(t) such that B (Kδ(t)x(t) + kw(t)) = Bw w(t) +Aδ(t)x(t).
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 10/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
State Space EquationL1 Model Formulation
Helicopter ModelL1 Model Formulation
Equivalent system:
x(t) = Am x(t) +B (ku u(t) +Kx(t)x(t) + kw(t))y(t) = C x(t) , x(0) = x0
,
Bounded unknown parameters:
ku ∈ Ku , Kx(t) ∈ Kx and kw(t)
Control effectiveness: Ku = [ku, ku] ⊂ R+
0;
Uncertainty: 0 < ‖kw(t)‖ < ∆0, for all t ≥ 0;
Input gain matrix: Kx compact;
Derivatives: ‖Kx(t)‖2 ≤ dKx <∞ and ‖kw(t)‖2 ≤ dkw <∞;
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 11/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
System and PredictorAdaptive and Control LawsStability and Performance
Outline
1 IntroductionAutonomous RotorcraftL1 Adaptive Control Theory
2 Helicopter ModelState Space EquationL1 Model Formulation
3 L1 Adaptive ControllerSystem and PredictorAdaptive and Control LawsStability and Performance
4 ResultsSimulation ResultsSummary
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 12/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
System and PredictorAdaptive and Control LawsStability and Performance
L1 Adaptive ControllerSystem and Predictor
System:
x(t) = Am x(t) +B (ku u(t) +Kx(t)x(t) + kw(t))y(t) = C x(t) , x(0) = x0
,
Control objective
Ensure y(t) tracks a given bounded reference signal r(t), while all
other error signals remain bounded.
Predictor system:
˙x(t) = Am x(t) +B(
ku(t)u(t) + Kx(t)x(t) + kw(t))
y(t) = C x(t) , x(0) = x0
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 13/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
System and PredictorAdaptive and Control LawsStability and Performance
L1 Adaptive ControllerAdaptive laws
Adaptive Laws:
˙ku(t) = γ Proj(ku(t), B′ P x(t)u′(t))˙Kx(t) = γ Proj(Kx(t), B′ P x(t)x′(t))˙kw(t) = γ Proj(kw(t), B′ P x(t))
,
Prediction error: x = x − x;
Adaptation gain: γ > 0;
P solution of Lyapunov equation A′m P + P Am = −Q;
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 14/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
System and PredictorAdaptive and Control LawsStability and Performance
L1 Adaptive ControllerControl law
Control Law:
U(s) = −kdD(s) R(s)
r(t) = Kx(t)x(t) + ku(t)u(t) + kw(t) −Kg r(t)
kd ∈ R;
Kg ∈ Rnu×nu ;
D(s) is an nu × nu transfer function matrix;
Stable and strictly proper F (s):
F (s) = (I + ku kdD(s))−1 ku kdD(s) ,
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 15/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
System and PredictorAdaptive and Control LawsStability and Performance
L1 Adaptive Controller
Plant
AdaptiveLaws
Predictor
x( )t
+
u( )t
x( )t~
x( )t
u( )t
x( )t^
( )k t^u
K t( )^x
u( )t
+kd D( )s
r( )t-
Controller
( )tw
( )k t^w
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 16/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
System and PredictorAdaptive and Control LawsStability and Performance
L1 Adaptive ControllerReference System
Reference system:
xref (t) = Am xref (t) +B (ku uref (t) + r1(t))
yref (t) = C xref (t) , xref (0) = x0
with r1(t) = Kx(t)xref (t) + kw(t);
Reference control law:
Uref (s) = −kdD(s) Rref (s)
rref (t) = ku uref (t) + r1(t) −Kg r(t)
H(s) = (s I −Am)−1B;
H0(s) = C H(s).
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 17/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
System and PredictorAdaptive and Control LawsStability and Performance
L1 Adaptive ControllerStability
Consider G(s) = H(s) (I − F (s));
Upper bound for control parameters: L = maxKx∈Kx‖Kx‖L1
.
Result 1
The reference system is stable if kd and D(s) satisfy
(i) F (s) is strictly proper and stable and F (0) = I , (1)
(ii) F (s)H−1
0(s) is proper and stable , (2)
(iii) ‖G‖L1L < 1 . (3)
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 18/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
System and PredictorAdaptive and Control LawsStability and Performance
L1 Adaptive ControllerTransient Performance
Result 2
Given the system, the reference system and the L1 adaptive
controller defined above, subject to conditions of Result 1,
‖x‖L∞ ≤ γ0 (4)
‖x − xref‖L∞ ≤ γ1 (5)
‖y − yref‖L∞ ≤ ‖C‖L1γ1 (6)
‖u − uref‖L∞ ≤ γ2 (7)
Closed-loop system follows the reference system during thetransient for sufficiently large γ;
γi depend on γ, P , D(s), kd and on the parameter bounds;
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 19/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Simulation ResultsSummary
Outline
1 IntroductionAutonomous RotorcraftL1 Adaptive Control Theory
2 Helicopter ModelState Space EquationL1 Model Formulation
3 L1 Adaptive ControllerSystem and PredictorAdaptive and Control LawsStability and Performance
4 ResultsSimulation ResultsSummary
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 20/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Simulation ResultsSummary
Simulation ResultsImplementation
Region of operation:
Vc ∈ [0.75 , 1.25]m/sγc = ψc = 0 and ψct = 90o
Filter design: D(s) = 1
sI;
Iteratively find kd ≥ 273;
0.606 ≤ ‖G‖L1L ≤ 0.975;
Adaptive gain: γ = 10000;
Simulation velocity reference: (i)straight line moving sideways, (ii)Helix and (iii) hover;
0 5 10 15 20 25 30 35 40 45 50−1
0
1
2
u win
d [m/s
]
0 5 10 15 20 25 30 35 40 45 50−1
0
1
2
v win
d [m/s
]
0 5 10 15 20 25 30 35 40 45 50−1
0
1
2
ww
ind [m
/s]
Time[s]
Nominal stabilizing controller for the region of operation;
Von Karman turbulence models and wind gust at t = 22s;
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 21/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Simulation ResultsSummary
Simulation ResultsHelicopter Trajectory
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 22/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Simulation ResultsSummary
Simulation ResultsVelocity Errors – L1 controller versus nominal controller
0 5 10 15 20 25 30 35 40 45 50−2
−1
0
1
2
u [m
/s]
0 5 10 15 20 25 30 35 40 45 50−2
0
2
4
v [m
/s]
0 5 10 15 20 25 30 35 40 45 50−2
−1
0
1
w [m
/s]
Time[s]
(.)L1
(.)n
Linear Velocity Error
0 5 10 15 20 25 30 35 40 45 50−0.5
0
0.5
p [r
ad/s
]
0 5 10 15 20 25 30 35 40 45 50−0.2
−0.1
0
0.1
0.2
q [r
ad/s
]
0 5 10 15 20 25 30 35 40 45 50−2
0
2
4
r [r
ad/s
]
Time[s]
(.)L1
(.)n
Angular Velocity Error
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 23/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Simulation ResultsSummary
Simulation ResultsActuation and Parameters – L1 controller versus nominal controller
0 5 10 15 20 25 30 35 40 45 50−0.02
0
0.02
θ c 0 [rad
]
0 5 10 15 20 25 30 35 40 45 50−0.02
0
0.02
θ c 1c
[rad
]
0 5 10 15 20 25 30 35 40 45 50−0.05
0
0.05
θ c 1s
[rad
]
0 5 10 15 20 25 30 35 40 45 50−0.02
0
0.02
θ c 0t
[rad
]
Time[s]
(.)L1
(.)n
Actuation Error
0 5 10 15 20 25 30 35 40 45 501
1.05
1.1
1.15
hat k
u(t)
0 5 10 15 20 25 30 35 40 45 50−0.1
−0.05
0
0.05
0.1
hat K
x(t)
0 5 10 15 20 25 30 35 40 45 50−0.05
0
0.05
hat
k w(t
)
Time[s]
L1 Parameters
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 24/28
IntroductionHelicopter Model
L1 Adaptive ControllerResults
Simulation ResultsSummary
Summary
L1 Adaptive controller provide better performance than thecontroller used in the reference system;
Approach used:
Linear time-varying approximation of the nonlinear model;Velocity and attitude stabilization;L1 adaptive controller for time-varying parameters;Follow demanding reference signals;Wind disturbance rejection;
Further research on position control.
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 25/28
The end
Thank you for your time.
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 26/28
Short Bibliography
C. Cao and N. Hovakimyan.Design and analysis of a novel L1 adaptive control architecture with guaranteedtransient performance.IEEE Transactions on Automatic Control, 53(2):586–591, 2008.
C. Cao and N. Hovakimyan.L1 adaptive controller for multi-input multi-output systems in the presence ofunmatched disturbances.In American Control Conference, pages 4105–4110, Seattle, WA, June 2008.
C. Cao and N. Hovakimyan.L1 adaptive controller for systems with unknown time-varying parameters anddisturbances in the presence of non-zero trajectory initialization error.International Journal of Control, 81(7):1148–1162, 2008.
B. Guerreiro, C. Silvestre, R. Cunha, and D. Antunes.Trajectory tracking H2 controller for autonomous helicopters: and aplication toindustrial chimney inspection.In 17th IFAC Symposium on Automatic Control in Aerospace, Toulouse, France,June 2007.
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 27/28
L1 Adaptive Control for Autonomous Rotorcraft
B. J. Guerreiro∗ C. Silvestre∗ R. Cunha∗
C. Cao† N. Hovakimyan‡
∗bguerreiro,cjs,[email protected]
Instituto Superior Tecnico, Portugal
University of Connecticut
University of Illinois at Urbana-Champaign
11 June 2009 – American Control Conference
Guerreiro, Silvestre, Cunha, Cao and Hovakimyan L1 Adaptive Control for Autonomous Rotorcraft 28/28