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Optimal control University of Strasbourg Telecom Physique Strasbourg, ISAV option Master IRIV, AR track Part 2 – Predictive control
Outline 1. Introduction 2. System modelling 3. Cost function 4. Prediction equation 5. Optimal control 6. Examples 7. Tuning of the GPC 8. Nonlinear predictive control 9. References
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1. Introduction 1.1. Definition of MPC
� Model Predictive Control (MPC) � Use of a model to predict the behaviour of the
system. � Compute a sequence of future control inputs that
minimize the quadratic error over a receding horizon of time.
� Only the first sample of the sequence is applied to the system. The whole sequence is re-evaluated at each sampling time.
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1. Introduction 1.2. Principle of MPC
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r t +1( )!
r t + N2( )
!
"
###
$
%
&&&
+!
Prediction
y t +1( )!
y t + N2( )
!
"
###
$
%
&&&
Optimization
u t( )!
u t + Nu !1( )
"
#
$$$
%
&
'''
u t( )System
y t( )
N2 future references
N2 predicted outputs
Nu future control signals
1. Introduction 1.2. Principle of MPC
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t t + N1 t + N2
Receding
Horizon
r
y
Goal of the optimization : minimizing
1. Introduction 1.3. Various flavours of MPC
� DMC (Dynamic Matrix Control) � Uses the system’s step response. � The system must be stable and without integrator.
� MAC (Model Algorithmic Control) � Uses the system’s impulse response.
� PFC (Predictive Functional Control) � Uses a state space representation of the system. � Can apply to nonlinear systems.
� GPC (Generalized Predictive Control) � Uses a CARMA model of the system. � The most commonly used.
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1. Introduction 1.4. Advantages / drawbacks of MPC
� Advantages � Simple principle, easy and quick tuning. � Applies to every kind of systems (non minimum
phase, instable, MIMO, nonlinear, variant). � If the reference of the disturbance is known in
advance, it can drastically improve the reference tracking accuracy.
� Numerically stable.
� Drawback � Good knowledge of the system model.
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2. Modelling 2.1. Example of MAC
� Input-output relationship :
� Truncation of the response :
� Drawbacks : � Model is not in its minimal form. � Computationally demanding.
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y t( )= hiu t ! i( )
i=1
"
#
y t + k | t( )= hiu t + k ! i | t( )
i=1
N
"
2. Modelling 2.2. The case of the GPC
� CARMA modelling (Controller Auto-Regressive Moving Average) :
� With :
� Usually :
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A q-1( ) y t( )= q-d B q-1( )u t !1( )+ C q-1( )
D q-1( ) e t( )
A q-1( )=1+ a1q-1 + a2q
-2 +…+ anaq-na
B q-1( )= b0 + b1q-1 + b2q
-2 +…+ bnbq-nb
C q-1( )=1+ c1q-1 + c2q
-2 +…+ cncq-nc
!
"##
$##
D q-1( )= ! q-1( )=1" q-1
3. GPC cost function � For the GPC :
� Tuning parameters : N1 N2 Nu λ
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J = y t + j | t( )! r t + j( )"# $%
2
j!N1
N2
& + ' (u t + j !1( )"# $%2
j=1
Nu
&
Quadratic error Energy of the control signal
4. GPC prediction equations
� First Diophantine equation :
� With C=1 :
� Let :
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C = E j!A+ q-j Fj
1= E j!A+ q-j Fj with deg E j( )= j "1
deg Fj( )= na
#$%
&%
Ay t( )= Bq-du t !1( )+ e t( )"
#
$%%
&
'(() "E jq
j
*"AE j y t + j( )= E j B"u t + j ! d !1( )+ E je t + j( )
4. GPC prediction equations
� Using the Diophantine equation :
� Which yields :
� Thus, the best prediction is :
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1! q-j Fj( ) y t + j( )= E j B"u t + j ! d !1( )+ E je t + j( )
y t + j( )= Fj y t( )+ E j B!u t + j " d "1( )+ E je t + j( )
y t + j | t( )= E j B!u t + j " d "1( )+ Fj y t( )
4. GPC prediction equations
� Second Diophantine equation :
� Separation of control inputs :
� Prediction equation : � With :
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E j B = Gj + q-j! j
y t + j | t( )= Gj!u t + j " d "1( )Forced response
! "### $###+# j!u t " d "1( )+ Fj y t( )
Free response! "#### $####
y = G !u+ f
y = y t +1+ d |t( )… y t + N2 + d |t( )!" #$T
!u = %u t | t( )…%u t + Nu &1| t( )!" #$T
f = f t +1| t( )… f t + N2 | t( )!" #$T
'
(
))
*
))
4. GPC prediction equations
� And :
� With g0 … gN2-1 the samples of the system’s step response.
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GN2!Nu=
g0 0 ! 0
g1 g0 ! 0
" " # "gN2"1 gN2"2 ! g0
" " " "gN2"1 gN2"2 ! gN2"Nu
#
$
%%%%%%%%%
&
'
(((((((((
5. Optimal control
� Cost function : � Let :
� With :
� Only the first optimal control sample is applied to the system.
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J = y ! r( )Ty ! r( )+" !uT !u
!uopt s.t. dJd !u
= 0
! !uopt = GTG +" I( )-1GT r # f( )
r = r t +1( )…r t + N2( )!" #$T
Future references
6. Examples 6.1. First order system
� A system in the CARMA form has the following parameters :
� Compute the system’s prediction equations 3 steps ahead.
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A = 1! 0.7q-1
B = 0.9! 0.6q-1
C = 1
"
#$
%$
6. Examples 6.1. First order system
� Using three times the CARMA model :
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6. Examples 6.1. First order system
� Putting everything in matrix form :
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6. Examples 6.1. First order system
� Optimal control (differential) :
� Optimal control (absolute) :
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7. Tuning the GPC � Parameter λ :
� Increase : response slow down. � Decrease : more energy in the control signal, thus
faster response.
� Parameter N2 : � At least the size of the step response of the system.
� Parameter N1 : � Greater than the system’s delay.
� Parameter Nu : � Tends toward dead-beat control when Nu tends
toward zero.
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8. Nonlinear predictive control � The system can be nonlinear. � The optimal solution is computed using
an iterative optimization algorithm. � The optimization is performed at each
sampling time. � Additional constraints can be added. � The cost function can be more complex. � Main drawback : very computationally
intensive.
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9. References � R. Bitmead, M. Gevers et V. Wertz,
« Adaptive Optimal control – The thinking man's GPC », Prentice Hall International, 1990.
� E. F. Camacho et C. Bordons, « Model Predictive Control », Springer Verlag, 1999.
� J.-M. Dion et D. Popescu, « Commande optimale, conception optimisée des systèmes », Diderot, 1996.
� P. Boucher et D. Dumur, « La commande prédictive », Technip, 1996.
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