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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
First validation : Back to the simple example
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
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 3
L1 adaptive Stability analysis Application Experiments Conclusion
Background on L1 adaptive control
Time lags limitation : A simple example
Proposed extension : Basic idea
First validation : Back to the simple example
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 4
Background on L1 adaptive control
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)
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 5
Background on L1 adaptive control
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
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 6
Background on L1 adaptive control
Inspired by direct MRAC (Model Reference Adaptive Control)
With a State predictor instead of the reference model
The tracking error is replaced by the prediction error
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 7
Background on L1 adaptive control
Inspired by direct MRAC (Model Reference Adaptive Control)
With a State predictor instead of the reference model
With low pass filter
C(s) : is a stable and strictly proper transfer function
C(s) = 1 Direct MRAC
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 8
Background on L1 adaptive control
Inspired by direct MRAC (Model Reference Adaptive Control)
With a State predictor instead of the reference model
With a low pass filter
With a projection operator to bound the estimated parameters
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 9
Background on L1 adaptive control
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)
L1 adaptive Stability analysis Application Experiments Conclusion
Am = A¡ bkTm
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 10
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
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 11
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
L1 adaptive Stability analysis Application Experiments Conclusion
Proposed extension
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 12
First validation : Back to the simple example
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
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 13
Illustration example
Effects of the PID gains on stability
L1 adaptive Application Experiments Conclusion Stability analysis
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 14
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
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 15
L1 adaptive Application Experiments Conclusion Stability analysis
Illustrative example
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
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 16
L1 adaptive Application Experiments Conclusion Stability analysis
Illustrative example
Gnominal(s)
Gextended(s)
Both systems are stable
Stability margins are slightly increased
What about the affects of the gains ?
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 17
L1 adaptive Application Experiments Conclusion Stability analysis
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
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 18
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
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 19
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
L1 adaptive Experiments Conclusion Stability analysis Application
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 20
Our demonstrator : Experimental setup of the AC-ROV Ha
rdw
are
Con
figur
atio
n
L1 adaptive Experiments Conclusion Stability analysis Application
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 21
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
L1 adaptive Experiments Conclusion Stability analysis Application
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 22
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
L1 adaptive Experiments Conclusion Stability analysis Application
5
6
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 23
L1 adaptive Conclusion Stability analysis Application Experiments
2 experimental scenarios
External disturbances Nominal case
Scenario 1 Scenario 2
Depth
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 24
Scenario 1: Control in nominal case
Time history of depth tracking Time history of the estimated parameters
The closed-loop behavior is improved
L1 adaptive Conclusion Stability analysis Application Experiments
PID Augmentation
PID Augmentation
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 25
Scenario 2: External disturbance rejection
L1 adaptive Conclusion Stability analysis Application Experiments
Time history of depth tracking Time history of the control inputs
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 26
Scenario 2: External disturbance rejection
Time history of the estimated parameters
L1 adaptive Conclusion Stability analysis Application Experiments
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 27
Real-time experiments : An illustration movie
L1 adaptive Conclusion Stability analysis Application Experiments
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 28
Conclusion
Future work
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 29
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
L1 adaptive Stability analysis Application Experiments Conclusion
Speaker: A. CHEMORI (LIRMM / CNRS, France) IEEE CDC 2013 (Florence, Italy) 30
Conclusion & future work
L1 adaptive Stability analysis Application Experiments Conclusion