Transcript ADAPTIVE CONTROL SYSTEMS

ADAPTIVE CONTROL
SYSTEMS
MRAC
IV UNIT
BY
Y.BHANUSREE
ASST. PROFESSOR
MRAC
• MODEL REFERENCE ADAPTIVE CONTROL
SYSTEM
• It can be considered as an ADAPTIVE SERVO
SYSTEM.
• It consists of two loops inner loop &outer loop
• It consists of a reference model in outer loop.
• Ordinary feed back is called inner loop.
• Parameters are adjusted on basis of feedback
from the error.
• Error is the difference between produced output
& reference model value.
Adjustment of system parameters in
a MRAC can be obtained in two
ways.
GRADIENT METHOD (MIT RULE)
LYPNOV STABILITY THEORY
MODEL
CONTROLLER
YM
PARAMETERS
ADJUSTMENT
MECHANISM
UC
U
CONTROLLER
BLOCK
Y
PLANT
DIAGRAM OF MRAC
MRAC IS COMPOSED OF
• Plant containing unknown parameters
• Reference model
• Adjustable parameters containing control
law or loss law
• Ordinary feed back loop
MIT RULE
• It is developed by INSTRUMANTATION LABORATORY
at MIT.
• To consider MIT rule we use a closed loop response in
which controller has one adjustable parameter θ.
• The desired closed loop response is specified by a
model whose output is ym.
• Then error e= ym – y. (y is original output).
Error criteria selected here is
J(θ)=1/2 (e)2
And this loss function is to be minimized to make the system controlled.
To achieve this the parameters are changed in the direction of negative
gradient of J.
dθ = -γ ∂J = - γe ∂e
dt
dθ
∂θ
This is called GRADIENT or MIT rule.
γ
=adaptation gain
 ∂e/ ∂θ = sensitivity
derivative(informs how error is influenced by θ
The loss function can also be chosen as

J(θ)=|e|

dθ = - γ ∂e sign e [GRADIENT METHOD]
dt
∂θ
first MRAC is implemented by this function

dθ = - γ sign ∂e sign (e)
dt
∂θ
[ sign-sign algorithm]
used in telecommunication where simple
implementation & fast computing are required
For multivariable systems
θ is considered as vector
∂e/ ∂θ is considered as gradient of the
error with respect to the parameter
APPLICATIONS FOR MIT
RULE
ADAPTATION OF A FEED FORWARD GAIN
MRAS FOR A FIRST ORDER SYSTEM
ADAPTATION OF A FEED FORWARD
GAIN
PROBLEM :Adjustment of feed forward gain
ASSUMPTIONS:
• Process is linear with the transfer function KG(s)
• G(s) is known
• K is unknown parameter
DESIRED CONDITION:
Transfer function Gm(s) should be equal to KoG(s)
Ko is given constant
MODEL
Ym
KOG(S)
_
-γ/S
π
e
∑
+
UC
θ
U
π
PROCESS
KG(S)
Y
MRAC FOR ADJUSTMENT OF FEEDFORWARD GAIN BY MIT RULE
The constant k for this plant is unknown. However, a
reference model can be formed with a desired value of k,
and through adaptation of a feedforward gain, the
response of the plant can be made to match this model.
The reference model is therefore chosen as the plant
multiplied by a desired constant ko .
The cost function chosen here is,
The error is then restated in terms of the transfer functions
multiplied by their inputs.
As can be seen, this expression for the error
contains the parameter theta which is to be updated.
To determine the update rule, the sensitivity
derivative is calculated and restated in terms of the
model ouput:
Finally, the MIT rule is applied to give an expression for
updating theta. The constants k and ko are combined into
gamma.
To tune this system, the values of ko and gamma can be
varied.
NOTES ON DESIGN WITH MIT
It is important to note that the MIT rule by itself does not
guarantee convergence or stability.
 An MRAC designed using the MIT rule is very sensitive
to the amplitudes of the signals.
As a general rule, the value of gamma is kept small.
Tuning of gamma is crucial to the adaptation rate and
stability of the controller.
IF G(S)=1/S+1
UC is sinusoidal signal with frequency 1
rad/s
K=1
K0=2
Vary values of γ
 γ=0.5,1,2
We can observe the output approaches the
model output at γ=1
simulation results
OBSERVATIONS FROM GRAPH:
Convergence rate depends on adaptation gain γ.
Reasonable value of γ has to be selected .
Applications:
In robots with unknown load.
CD player where sensitivity of laser diode is not
known.
MRAC FOR A FIRST ORDER
SYSTEM
Process
dy = −ay+ bu
dt
Model
dym = −amym+bmuc
dt
Controller
U(t)=θ1Uc(t)- θ2 Y(t)
θ1 & θ2 are 2 adjustable parameters
LYAPUNOV THEOREM
ALGORITHM FOR LYAPUNOV THEORY
• Derive differential equation for the error e=y-ym.
• We attempt to find lyapunov function & an adaptation
mechanism such that the error will go to zero.
• We find dv/dt is usually only negative semi definite .so
finding error equation & lyapunov function with a
bounded second derivative .
• It is to show bounded ness & that error goes to zero.
• To show parameter convergence ,it is necessary to
impose further conditions ,such as persistently excitation
and uniform observability
MODEL
YM
KOG(S)
-
UC
e
∑
π
-γ/S
+
π
PROCESS
KG(S)
Y
B. DIAG FOR FEED FORWARD GAIN CONTROL USING LYAPUNOV RULE
θ
The error equations are
The value of dv/dt is <=0 [which is –ve semi
definite]
As time derivative of Lypanov function is negative
semi definite.
By using lemma of lypanov we can show error goes
to zero.
REFERENCES
http://www.control.hut.fi/Kurssit/AS-74.185/luennot/lu5ep.pdf
http://www.control.hut.fi/Kurssit/AS-74.185/luennot/lu6ep.pdf
http://www.control.lth.se/~FRT050/Exercises/ex4sol.pdf
http://www.control.lth.se/%7EFRT050/Exercises/ex5sol.pdf
http://www.igi.tugraz.at/helmut/Presentations/AdaptiveControl.html
http://www.irs.ctrl.titech.ac.jp/~dkura/ic2004/lec405.pdf
http://www.irs.ctrl.titech.ac.jp/~dkura/ic2004/lec406.pdf
http://mchlab.ee.nus.edu.sg/Experiment/Manuals/EE5140/adap1.pdf
http://www.rcf.usc.edu/~ioannou/RobustAdaptiveBook95pdf/Robust_Adaptive_Con
trol.pdf