Optimal control

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Optimal control

University of Strasbourg

Telecom Physique Strasbourg, ISAV option Master IRIV, AR track

Part 2 – Predictive control

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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

r t +1( )

! r t + N( ₂)

!

"

##

#

$

%

&

+!

Prediction

y t +1( )

! y t + N( ₂)

!

"

##

#

$

%

&

Optimization

u t( )

!

u t + N( _u !1)

"

#

$$

$

%

&

'' '

u t( )

System

y t( )

N₂ future references

N₂ predicted outputs

N_u future control signals

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1. Introduction

1.2. Principle of MPC

t + N¹ t + N² t

Receding Horizon

r

y

Goal of the optimization : minimizing

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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.

y t( )⁼ ^hⁱ^{u t}( )^!ⁱ

i=1

"

#

y t + k |tˆ( )⁼ ^hⁱ^{u t + k}( ^!^{i |t})

i=1

"N

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2. Modelling

2.2. The case of the GPC

  CARMA modelling (Controller Auto- Regressive Moving Average) :

  With :

  Usually :

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+ a}¹^q^-1^{+ a}²^q^-2^{+…+ a}^na^q^-na

B q( )^-1 ^{= b}⁰^{+ b}¹^q^-1^{+ b}²^q^-2^{+…+ b}^nb^q^-nb

C q( )^-1 ⁼^{1+ c}¹^q^-1^{+ c}²^q^-2⁺^…^{+ c}^nc^q^-nc

!

"

##

$

##

D q( )^-1 ⁼ ^!( )^q^-1 ⁼¹^"^q^-1

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3. GPC cost function

  For the GPC :

  Tuning parameters : N₁ N₂ N_u λ

J = _"#y tˆ

(

+ j |t

)

^! ^{r t}

( )

⁺ ^j _$%²

j!N₁ N₂

&

⁺ ^{' (}_"# ^{u t}

(

⁺ ^j ^!¹

)

_$%²

j=1 N_u

&

Quadratic error Energy of the control signal

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4. GPC prediction equations

  First Diophantine equation :

  With C=1 :

  Let :

C = E^j!A+ q^-jF_j

1= E_j!A+ q^-jF_j with deg

( )

E_j ^{= j} ^"¹ deg

( )

F_j ^{= na}

#

$%

&%

Ay t

( )

^{= Bq}^-d^{u t}

( )

^!¹ ⁺ ^{e t}

( )

"

#

$%

%

&

'(

( ) "E_jq ^j

( ) ( ) ( )

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4. GPC prediction equations

  Using the Diophantine equation :

  Which yields :

  Thus, the best prediction is :

1! q^-jF_j

( )

^{y t + j}

^{( )}

^{= E}^j^B^"^{u t + j}

⁽

^! ^d ^!¹

⁾

^{+ E}^j^{e t + j}

^{( )}

y t + j

( )

^{= F}^j^{y t}

( )

^{+ E}^j^B^!^{u t + j}

(

^" ^d ^"¹

)

^{+ E}^j^{e t + j}

( )

y t + j |tˆ

( )

^{= E}^j^B^!^{u t + j}

(

^" ^d ^"¹

)

^{+ F}^j ^{y t}

( )

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4. GPC prediction equations

  Second Diophantine equation :

  Separation of control inputs :

  Prediction equation :

  With :

E^jB = G_j + q^-j! _j

y t + j |tˆ

( )

^{= G}^j^!^{u t + j}

(

^" ^d ^"¹

)

Forced response

!###"###$+# _j!u t

(

" d "1

)

^{+ F}^j^{y t}

( )

Free response

!####"####$

y = Gˆ u +! fˆ

y =ˆ _!"y t +ˆ( 1+ d |t)^…^{y t + N}^ˆ( ²^{+ d |t})_#$^T

!

u = _!"%u t |t( )^…%^{u t + N}( ^u ^&^1|t)_#$^T

' ( )) )

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4. GPC prediction equations

  And :

  With g₀ … g_N2-1 the samples of the system’s step response.

G_N

2!N_u =

g₀ 0 ! 0

g₁ g₀ ! 0

" " # "

g_N

2"1 g_N

2"2 ! g₀

" " " "

g_N

2"1 g_N

2"2 ! g_N

2"N_u

#

$

%%

%

&

' (( (( (( (( (

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5. Optimal control

  Cost function :

  Let :

  With :

  Only the first optimal control sample is applied to the system.

J =

( )

yˆ ! r ^T

( )

^y^ˆ ^! ^r ⁺^"^u^!^T^u^!

!

u_opt s.t. dJ

du! = 0

! u!_opt = G

(

^TG +"I

)

^-1 ^G^T

( )

^r ^# ^f^ˆ

r = r t +1_!"

( )

^…^{r t + N}

(

²

)

_#$^T

Future references

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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.

A = 1! 0.7q^-1 B = 0.9 !0.6q^-1 C =1

"

#$

%$

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6. Examples

  Using three times the CARMA model :

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6. Examples

  Putting everything in matrix form :

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6. Examples

  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 N_u :

  Tends toward dead-beat control when N_u 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.