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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6. Examples 6.2. Simulation results

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6. Examples 6.2. Simulation results

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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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