Optimal control Outline

1.  Introduction
University of Strasbourg
2.  System modelling
Telecom Physique Strasbourg, ISAV option
Master IRIV, AR track 3.  Cost function
Part 2 – Predictive control 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. Introduction
1.1. Definition of MPC 1.2. Principle of MPC

—  Model Predictive Control (MPC) ⎡ r ( t +1) ⎤ ⎡ u (t ) ⎤

–  Use of a model to predict the behaviour of the ⎢ ⎥ N2 future ⎢ ⎥ Nu future control
⎢  ⎥ references ⎢  ⎥ signals
system.
⎣ ( 2 )⎦ ⎣ ( )⎦
⎢r t + N ⎥ ⎢u t + N − 1 ⎥
u (t )
u
–  Compute a sequence of future control inputs that
minimize the quadratic error over a receding
+
− Optimization System
horizon of time.
–  Only the first sample of the sequence is applied to y (t )
the system. The whole sequence is re-evaluated at Prediction
⎡ y ( t +1) ⎤
each sampling time. ⎢ ⎥
⎢  ⎥
⎢ y t + N ⎥ N2 predicted
⎣ ( )
2 ⎦ outputs

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 1. Introduction 1. Introduction
1.2. Principle of MPC 1.3.Various flavours of MPC
y —  DMC (Dynamic Matrix Control)
–  Uses the system’s step response.
r
–  The system must be stable and without integrator.
—  MAC (Model Algorithmic Control)
Receding
–  Uses the system’s impulse response.
Horizon
—  PFC (Predictive Functional Control)
–  Uses a state space representation of the system.
–  Can apply to nonlinear systems.
—  GPC (Generalized Predictive Control)
t + N1 t + N2 t –  Uses a CARMA model of the system.
–  The most commonly used.
Goal of the optimization : minimizing

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1. Introduction 2. Modelling
1.4. Advantages / drawbacks of MPC 2.1. Example of MAC

—  Advantages —  Input-output relationship :

∞
–  Simple principle, easy and quick tuning. y ( t ) = ∑ hiu ( t − i )
i=1
–  Applies to every kind of systems (non minimum
phase, instable, MIMO, nonlinear, variant). —  Truncation of the response :
N

–  If the reference or the disturbance is known in ŷ ( t + k |t ) = ∑ hiu ( t + k − i |t )

i=1
advance, it can drastically improve the reference
—  Drawbacks :
tracking accuracy.
–  Numerically stable. –  Model is not in its minimal form.
–  Computationally demanding.
—  Drawback
–  Good knowledge of the system model.

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2. Modelling 3. GPC cost function
2.2. The case of the GPC

—  CARMA modelling (Controller Auto- —  For the GPC :

Regressive Moving Average) :
C (q ) -1 N2 Nu
A( q ) y ( t ) = B ( q ) u ( t − 1) + e (t )
∑ ⎡⎣ ŷ (t + j | t ) − r (t + j )⎤⎦ + ∑ λ ⎡⎣ Δu (t + j − 1)⎤⎦
-1 -1 2 2
D (q ) -1 J=
j= N1 j=1

—  With : ( )
⎧ A q -1 = 1+ a q -1 + a q -2 +…+ a q -na
⎪ 1 2 na
⎪ -1
( ) -1 -2
⎨ B q = b0 + b1q + b2 q +…+ bnbq
-nb

⎪ -1
( ) -1 -2
⎪⎩ C q = 1+ c1q + c2 q +…+ cnc q
-nc

Quadratic error Energy of the control signal

—  Usually : D q ( ) = Δ ( q ) = 1− q
-1 -1 -1

—  Tuning parameters : N1 N2 Nu λ
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4. GPC prediction equations 4. GPC prediction equations

—  First Diophantine equation : —  Using the Diophantine equation :
C = E j ΔA+ q Fj -j
(1− q F ) y (t + j ) = E BΔu (t + j − 1) + E e(t + j )
-j
j j j

—  With C=1 : —  Which yields :

⎧deg E = j − 1
⎪
1= E j ΔA+ q Fj with ⎨
-j j ( ) y ( t + j ) = Fj y ( t ) + E j BΔu ( t + j − 1) + E j e ( t + j )

⎪⎩deg Fj = na ( ) —  Thus, the best prediction is :

—  Let : ŷ ( t + j |t ) = E j BΔu ( t + j − 1) + Fj y ( t )
⎡ e (t ) ⎤
⎢ Ay ( t ) = Bu ( t − 1) + ⎥ × ΔE j q
j

⎢⎣ Δ ⎥⎦
⇒ ΔAE j y ( t + j ) = E j BΔu ( t + j − 1) + E j e ( t + j )
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4. GPC prediction equations 4. GPC prediction equations
—  Second Diophantine equation : —  And : ⎡ g 0  0 ⎤
⎢ 0
⎥
E j B = G j + q Γ j with deg(G j ) = j − 1
-j
⎢ g1 g0  0 ⎥
⎢ ⎥
—  Separation of control inputs :    
GN =⎢ ⎥
⎢ g N −1 ⎥
ŷ ( t + j |t ) = G j Δu ( t + j − 1) + Γ j Δu ( t − 1) + Fj y ( t )
2 × Nu gN  g0
2 −2
!##"## $ !###"### $ ⎢ 2
⎥
Forced response Free response ⎢     ⎥
⎢ g gN  gN −N ⎥
—  Prediction equation : ŷ = Gu + fˆ ⎢⎣ N 2 −1 2 −2 2 u ⎥⎦

—  With : ⎣ ( ) ( 2 )⎦
⎧ ŷ = ⎡ ŷ t +1|t … ŷ t + N |t ⎤T
⎪
—  With g0 … gN2-1 the samples of the
⎪
⎨u! = ⎡⎣ Δu ( t |t )…Δu ( t + N u − 1|t ) ⎤⎦
T
system’s step response.
⎪
⎪ fˆ = ⎡ fˆ ( t +1|t )… fˆ ( t + N 2 |t ) ⎤
T

⎩ ⎣ ⎦
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5. Optimal control 6. Examples

6.1. First order system

—  Cost function : J = ( ŷ − r ) ( ŷ − r ) + λ u T u —  A system in the CARMA form has the

T

dJ following parameters : ⎧ A = 1− 0.7q-1

—  Let : uopt s.t. =0
du ⎪
⎨ B = 0.9 − 0.6q
-1

( ) ( )
-1
⇒ uopt = G G + λ I T
G r − fˆ
T
⎪C = 1
⎩

r = ⎡⎣ r ( t +1)…r ( t + N 2 ) ⎤⎦ —  Compute the system’s prediction

T
—  With :
Future references equations 3 steps ahead.
—  Only the first optimal control sample is
applied to the system.

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6. Examples 6. Examples
6.1. First order system 6.1. First order system

—  Using three times the CARMA model : —  Putting everything in matrix form :

⎛ 0.8947 0.0929 0.0095 ⎞

(G G + λ I ) ⎜ ⎟
-1
T
3
G = ⎜ −0.8316 0.8091 0.0929 ⎟
T

⎝ −0.0766 −0.8316 0.8947 ⎠

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6. Examples 6. Examples
6.1. First order system 6.2. Simulation results

—  Optimal control (differential) :

Δu ( t ) = 0.644Δu ( t − 1) − 1.7483y ( t ) + 0.7513y ( t − 1)
+0.8947r ( t +1) + 0.0929r ( t + 2 ) + 0.0095r ( t + 3)

—  Optimal control (absolute) :

u ( t ) − u ( t − 1) = 0.644 ⎡⎣u ( t − 1) − u ( t − 2 ) ⎤⎦ − 1.7483y ( t )
+0.7513y ( t − 1) + 0.8947r ( t +1) + 0.0929r ( t + 2 ) + 0.0095r ( t + 3)
⇔
u ( t ) = 1.644u ( t − 1) − 0.644u ( t − 2 ) − 1.7483y ( t )
+0.7513y ( t − 1) + 0.8947r ( t +1) + 0.0929r ( t + 2 ) + 0.0095r ( t + 3)

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6. Examples 7. Tuning the GPC
6.2. Simulation results
—  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 9. References

—  The system can be nonlinear. —  R. Bitmead, M. Gevers et V. Wertz,
—  The optimal solution is computed using « Adaptive Optimal control – The thinking man's
an iterative optimization algorithm. GPC », Prentice Hall International, 1990.
—  E. F. Camacho et C. Bordons, « Model
—  The optimization is performed at each
Predictive Control », Springer Verlag, 1999.
sampling time.
—  J.-M. Dion et D. Popescu, « Commande
—  Additional constraints can be added.
optimale, conception optimisée des systèmes »,
—  The cost function can be more complex. Diderot, 1996.
—  Main drawback : very computationally —  P. Boucher et D. Dumur, « La commande
intensive. prédictive », Technip, 1996.

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