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

†[email protected]

University of Connecticut

‡[email protected]

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

bguerreiro,cjs,[email protected]

[email protected]

[email protected]



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




Autonomous RotorcraftL1 Adaptive Control Theory

Outline









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





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.





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.




State Space EquationL1 Model Formulation

Outline









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





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





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).





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 <∞;




System and PredictorAdaptive and Control LawsStability and Performance

Outline









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





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;





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) ,





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





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).





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)





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;




Simulation ResultsSummary

Outline









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;





Simulation ResultsHelicopter Trajectory





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





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





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.


The end

Thank you for your time.


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.


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

†[email protected]

University of Connecticut

‡[email protected]

University of Illinois at Urbana-Champaign

11 June 2009 – American Control Conference


bguerreiro,cjs,[email protected]

[email protected]

[email protected]

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