IEEE CDC 2013chemori/Temp/Afef/Presentation_Conf/ACROV_CDC1… · IEEE CDC 2013 (Florence, Italy) Speaker: A. CHEMORI (LIRMM / CNRS, France) 2 Outline of the presentation o L1 adaptive

Divine Maalouf , Ahmed Chemori , Vincent Creuze

Laboratory of Informatics, Robotics and Microelectronics of Montpellier

LIRMM, University of Montpellier 2 - CNRS

161, rue Ada 34095

Montpellier, France

Florence, December, 13th, 2013

IEEE CDC 2013 Regular session : Maritime Control

Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 2

Outline of the presentation

o L1 adaptive control and its extended version Background on L1 adaptive control

Time lag limitation : A simple example

Proposed solution : Basic idea

First validation : Back to the simple example

o Stability analysis of the extended version Basic idea


o Application in underwater robotics Our demonstrator (experimental setup)

Its dynamic modeling

Application of the proposed solution for depth control

o Real-time experimental results Scenario 1 : Control in nominal case

Scenario 2 : External disturbance rejection

o Conclusion & future work


L1 adaptive Stability analysis Application Experiments Conclusion

Background on L1 adaptive control

Time lags limitation : A simple example

Proposed extension : Basic idea




Main features

Recently developed controller [Hovakimyan 2010]

Inspired from MRAC controller (+ low pass filter)

Decoupling robustness from adaptation

Fast adaptation can be guaranteed

Validated on various systems (mainly in aerospace)




Inspired by direct MRAC (Model Reference Adaptive Control)

µ : is a vector of unknown constant parameters

µ̂ : is the estimate of µ

r : is a piecewise-continuous bounded reference signal





With a State predictor instead of the reference model

The tracking error is replaced by the prediction error






With low pass filter

C(s) : is a stable and strictly proper transfer function

C(s) = 1 Direct MRAC






With a low pass filter

With a projection operator to bound the estimated parameters




u= ua+um

um =¡kTmx(t)!(t) : is an unknown constant representing

uncertainty on the input gain

¾(t) : is a parameter modeling input disturbances

µ(t) : is a vector of unknown constant parameters

General case of MIMO systems

µ(t)!(t) ¾(t)


Am = A¡ bkTm


Time lag limitation : A simple example

0 10 20 30 40 50-150

-100

-50

0

50

100

150

Time (s)

Ou

tpu

t y(t

)

Consider the following system [Hovakimyan 2010]

For a bounded reference trajectory to be tracked :

r(t) = 100cos(0:2t)

A =

·0 1

¡1 ¡1:4

¸; B =

·0

1

¸; C =

£1 0

¤; µ =

·4

¡4:5

¸

_x(t) = Ax(t) +B³u(t) + µ(t)Tx(t)

´; x(0) = x0

y(t) = Cx(t)

C(s) =!kD(s)

1+!kD(s)= 160

s+160, ¡ = 10000 , km = 0

The proposed design parameters are the following:

A time lag in the tracking is noticed

Due to the presence of the filter in the control loop



Proposed Solution : Basic idea

The new control law in this case :

The adaptive term

The state-feedback term

The proposed extension term

u= ua+um+uPID


Proposed extension



C(s) = 160s+160

, ¡ = 10000 , km = 0

The proposed design parameters are the same:

The same reference trajectory to be tracked :

r = 100cos(0:2t)

The obtained tracking for both controllers :

The time lag is very reduced



Illustration example

Effects of the PID gains on stability

L1 adaptive Application Experiments Conclusion Stability analysis



Illustrative example

The estimated parameters are away from bounds

The parameters of the PID extension are :

The adaptation gain and the low pass filter :

KP = 3 ; KI = 0:5 ; KD = 0:2

¡ = 100000 ; C(s) = 1s+1




The controlled system :

State predictor :

Adaptation stage :

Control input :

Let’s now compute the loop transfer function for both cases :

L1 adaptive control :

Proposed extended version :

_x(t) =¡x(t) + µ(t) +u(t)

_̂x(t) =¡x̂(t) + µ̂(t) + u(t)

_̂µ(t) = ¡¡~x(t)

u(t) =¡C(s)(µ̂¡ r(t)) + uPID

Gextended(s) =¡(s+ ¡

s+1)uPID+¡C(s)

s(s+1)+¡(1¡C(s))

Gnominal(s) =¡C(s)

s(s+1)+¡(1¡C(s))

Nyquist plot to evaluate stability and its margins




Gnominal(s)

Gextended(s)

Both systems are stable

Stability margins are slightly increased

What about the affects of the gains ?



Effects of the PID gains on the stability

KP = 3, KP = 15, KP = 30 KI = 0:5, KI = 2:5, KI = 5 KD = 0:1, KD = 0:2, KD = 0:3

Proportional gain Integral gain Derivative gain KP = 3 KI = 0:5 KD = 0:1


Our demonstrator : Experimental setup of the AC-ROV

Dynamic modeling of the AC-ROV

Application of the proposed solution for depth control

L1 adaptive Experiments Conclusion Stability analysis Application


Commercialized experimental setup

Control computer, Power Input,

Emergency stop,

Video input, Umbilical plug,

Ethernet plug, Video screen,

Ombilical, AC-ROV

Modified experimental setup

Our demonstrator : Experimental setup of the AC-ROV



Our demonstrator : Experimental setup of the AC-ROV Ha

rdw

are

Con

figur

atio

n



Frames definition

)(

)()()(

J

GDCM

Forces produced

by the thrusters

matrixsformation TranJ

putscontrol inVector of τ

cy forcesn / buoyangravitatioVector of G

, Damping), Coriolisices (MassModel matrM,C,D

eearth frams in the Coordinate

nd angularposition aVector of

ψθΦzyx

dy frame in the bovelocitiesVector of

rqpwvu

Τ

Τ

][ η

][ ν

Based on SNAME notation [SNAME1950]

[Fossen2002]

Pitch Yaw

Roll

SNAME : Society of Naval Architects and Marine Engineers

Dynamic model of the AC-ROV



Mz _w+Dzw¡ cos(')cos(#)(W ¡B) = ¿z +wdz

¿z = TKu

M¤z (´)Ä́+D¤z(º; ´) _́ + g¤z(´) = ¿¤z +w¤zd

·_́1_́2

¸=

"0 1

0¡D¤zM¤z

#·´1´2

¸¡"

0g¤zM¤z¡ w¤dz

M¤z

#+

·01M¤z

¸!¿z

¤

¾(t)

µ(t)

Dynamic model of the AC-ROV

The depth dynamics of he system writes :

The model can be expressed in earth-fixed-frame as :

In a state space representation :

: is a parameter regrouping the gravity, buoyancy and external disturbances

: represents the uncertainties on damping

Two controllers are implemented : L1 adaptive controller

Extended L1 adaptive controller

u=K¡1T¡1JT (ua +um+uPID) 2 R2

·_́1_́2

¸= Am

·´1´2

¸+

·01M¤z

¸(ua + µ(t)jj´(t)jjL1 + ¾(t) ) ; y = ´1


5

6


L1 adaptive Conclusion Stability analysis Application Experiments

2 experimental scenarios

External disturbances Nominal case

Scenario 1 Scenario 2

Depth


Scenario 1: Control in nominal case

Time history of depth tracking Time history of the estimated parameters

The closed-loop behavior is improved


PID Augmentation

PID Augmentation


Scenario 2: External disturbance rejection


Time history of depth tracking Time history of the control inputs


Scenario 2: External disturbance rejection

Time history of the estimated parameters



Real-time experiments : An illustration movie



Conclusion

Future work



Problem: Control of underwater vehicles

Difficulties inherent to that systems:

• High nonlinear dynamics

• Unknown/variable model parameters

• Non measurable states

Proposed Solution: An extension of L1 adaptive based on a PID controller

Validation: In real-time through experiments on the AC-ROV

Advantages of the proposed solution :

• Invariant fast adaptation

• No a priori knowledge of the parameters is needed

• Robustness towards uncertainties / disturbance rejection

• Time lag cancelation

Future work : Multivariable case

Implementation on the vehicle L2ROV

Control using vision

Conclusion & future work



Conclusion & future work


Documents

IEEE CDC 2013chemori/Temp/Afef/Presentation_Conf/ACROV_CDC1… · IEEE CDC 2013 (Florence, Italy) Speaker: A. CHEMORI (LIRMM / CNRS, France) 2 Outline of the presentation o L1 adaptive