MODEL PREDICTIVE CONTROL SC4060 Ton J.J. van den Boom Delft Center for Systems and Control Delft University of Technology Email: [email protected]

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Model Predictive Control (February-March 2011)

Preliminaries - slide 1

COURSE SCHEDULE 2008-2009 Tue 01 Feb : Introduction/Basics (C1/C2) Thu 03 Mar : Stability (C6) Thu 03 Feb : Basics (C2)

Tue 08 Mar : cancelled

Tue 08 Feb : Prediction (C3)

Thu 10 Mar : Tuning (C8)

Thu 10 Feb : Standard formulation (C4)

Tue 15 Mar : cancelled

Tue 15 Feb : Solution of the SPCP (C5)

Thu 17 Mar : cancelled

Thu 17 Feb : cancelled

Tue 22 Mar : reserve

Tue 01 Mar : Solution of the SPCP (C5)

Thu 24 Mar : Question hour

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Tuesday

10:45-12:30

3mE Room-J (Coenraad Storkzaal)

Thursday

08:45-10:30

3mE Room-J (Coenraad Storkzaal)

Model Predictive Control (February-March 2011)

Preliminaries - slide 2

EXAMINATION

• Written exam on Tuesday April 4, 9:00-11:00 hrs • Homework assignment set (MATLAB) Available in week 9, Hand-in on Monday April 18

QUESTIONS Minh Dang Doan Room: 8C-3-10, tel: 015-278 1362, email: [email protected]

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Model Predictive Control (February-March 2011)

Preliminaries - slide 3

Some properties of discrete time systems CONTINUOUS TIME

DISCRETE TIME state space model

d x(t) = A x(t) + B u(t) dt y(t) = C x(t) + D u(t)

x(k + 1) = A x(k) + B u(k) y(k) = C x(k) + D u(k) operators

d x(t) S x(t) = dt

q x(k) = x(k + 1) transfer function

H(s) = C (s I − A)−1 B + D

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H(z) = C (z I − A)−1 B + D

Model Predictive Control (February-March 2011)

Preliminaries - slide 4

Consider two systems with common A and C matrices

H1

H2

x1 (k + 1) = A x1 (k) + B1 u1 (k)

x2 (k + 1) = A x2 (k) + B2 u2 (k)

y1 (k) = C x1 (k) + D1 u1 (k)

y2 (k) = C x2 (k) + D2 u2 (k)

Connect the systems

u2 (k)

?

y(k) = y1 (k) + y2 (k)

H2 y2 (k)

New system becomes:

x(k + 1) = A x(k) + B1 u1 (k) + B2 u2 (k)

u1 (k)-

H1

y1 (k) -? d

-

y(k)

y(k) = C x(k) + D1 u1 (k) + D2 u2 (k)

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Model Predictive Control (February-March 2011)

Preliminaries - slide 5

CHAPTER 1 INTRODUCTION

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Model Predictive Control (February-March 2011)

Notes chapter 1 - slide 6

Model predictive control Enormous challenge for industry:

• Tighter product quality specifications • Increasing productivity demands • Fast changes in the economical market Model Predictive Control (MPC) is able to respond in an effective way to these demands. MPC is more a methodology, not a single technique. Differences in translation into a mathematical formulation.

• • • • SC4060

Model Predictive Control (MPC) Model Based Predictive Control (MBPC) Receding Horizon Control (RHC) Moving Horizon Control (MHC) Model Predictive Control (February-March 2011)

Notes chapter 1 - slide 7

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Model Predictive Control (February-March 2011)

Notes chapter 1 - slide 8

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Model Predictive Control (February-March 2011)

Notes chapter 1 - slide 9

Ingredients of MPC Five ingredients are recognizable:

• • • • •

Process + disturbance model Performance index Constraint handling Optimization Receding horizon principle

Theory:

• analysis is difficult Practice:

• accurate tuning for stability & robustness SC4060

Model Predictive Control (February-March 2011)

Notes chapter 1 - slide 10

History of MPC

• pioneered by Richalet (1979) and Cutler & Ramaker (1979) • well accepted by industry: - MPC can handle multivariable processes with large time-delays, non-minimum-phase, unstable poles - easy concept, easy tuning - MPC can handle constraints - MPC can handle structure changes and actuator failures

• academic interest - Self-tuning control & Minimum Variance control - Generalized Predictive Control (GPC)

• many successful applications have been reported

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Model Predictive Control (February-March 2011)

Notes chapter 1 - slide 11

COURSE SETUP 2. Basics of MPC

? 3. Prediction -

? 4. Standard formulation 5. Solving the MPC problem

? 6. Stability

-

 ? 8. Tuning

? 7. LMI-based MPC

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Model Predictive Control (February-March 2011)

Notes chapter 1 - slide 12

COURSE SETUP 2. Basics of MPC

? 3. Prediction -

? 4. Standard formulation 5. Solving the MPC problem

? 6. Stability

-

 ? 8. Tuning

? 7. LMI-based MPC

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Model Predictive Control (February-March 2011)

Notes chapter 1 - slide 13

CHAPTER 2 THE BASICS OF MODEL PREDICTIVE CONTROL

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 14

BASICS OF MODEL PREDICTIVE CONTROL Discuss the five ingredients of MPC:

• • • • •

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1. Process + disturbance model 2. Performance index 3. Constraint handling 4. Optimization 5. Receding horizon principle

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 15

BASICS OF MODEL PREDICTIVE CONTROL Discuss the five ingredients of MPC:

• • • • •

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1. Process + disturbance model 2. Performance index 3. Constraint handling 4. Optimization 5. Receding horizon principle

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 16

1. The process & disturbance model The models applied in MPC serve two purposes:

• Prediction of the behavior of the future output of the process on the basis of inputs and known disturbances applied to the process in the past

• Calculation of the input signal to the process that minimizes the given objective function These models are not necessarily the same.

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Notes chapter 2 - slide 17

Two types of IO models are applied:

• Direct Input-Output models (IO models) in which the input signal is directly applied to the model.

• Increment Input-Output models (IIO models) in which the increments of the input signal are applied to the model instead of the input directly. Model assumptions:

• • • • • • SC4060

Linear Time-invariant Discrete time Causal Finite order State space description

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 18

Input-output (IO) Model

y(k)= Go (q) u(k) where

Go (q)= strictly proper plant model ∆u(k)= ∆(q)u(k) = u(k) − u(k − 1) Increment-input-output (IIO) Model

y(k)= Gi (q) ∆u(k) where

∆u(k)= ∆(q)u(k) = u(k) − u(k − 1) ∆(q)= (1 − q−1 ) Gi (q)= strictly proper plant model SC4060

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 19

State space description: IO-model

IIO-model

xi (k+1) = Ai xi (k)+Bi ∆u(k)

xo (k+1) = Ao xo (k)+Bo u(k) y(k) = Co xo (k)

y(k) = Ci xi (k)

Relation IO Model and IIO model: Introduce new state

xi =

"

y(k − 1) ∆xo (k)

and define the system matrices

then

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Gi (q) = Go (q)∆(q)−1

# 

Ai =

I Co 0 Ao





 Bi =

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0 Bo



h

 Ci = I Co

i

Notes chapter 2 - slide 20

Advantage of using IIO models Example 1: Given an IO system with kGo (1)k < ∞. Minimize an IO-performance index N2

J(k) =

∑ ky(k + j) − r(k + j)k2 + λ2ku(k + j − 1)k2

j=1

Let r(k) = rss

6= 0 for k −→ ∞.

Jss = kyss − rss k2 + λ2 kuss k2 = kGo (1)uss − rss k2 + λ2 kuss k2   T = uTss GTo (1)Go (1) + λ2 I uss − 2uTss GTo (1)rss + rss rss SC4060

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 21

Minimizing Jss over uss means that



−1

uss = GTo (1)Go (1) + λ2 I GTo (1)rss  −1 yss = Go (1) GTo (1)Go (1) + λ2 I GTo (1)rss

It is clear that yss

6= rss for λ > 0.

Conclusion: There will be a steady-state error.

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Notes chapter 2 - slide 22

Example 2: Given an IIO system Gi (q) = ∆−1 (q)Go (q) with kGo (1)k < ∞. Minimize an IIO-performance index N2

J(k) =

2 2 2 ky(k + j) − r(k + j)k + λ k∆u(k + j − 1)k ∑

j=1

Let r(k) = rss

6= 0 for k −→ ∞. Jss = kyss − rss k2 + λ2 k∆uss k2

= 0 is reached for uss = G−1 o (1)rss which gives yss = rss and ∆uss = 0. It is clear that yss = rss for any λ > 0. Minimum Jss

Conclusion: There will be no steady-state error.

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 23

Standard model IO Model

IIO Model

xo (k + 1) = Ao xo (k) + Bo u(k)

xi (k + 1) = Ai xi (k) + Bi ∆u(k)

y(k) = Co xo (k)

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y(k) = Ci xi (k)

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 24

Standard model IO Model

IIO Model

xo (k + 1) = Ao xo (k) + Bo u(k)

xi (k + 1) = Ai xi (k) + Bi ∆u(k)

y(k) = Co xo (k)

y(k) = Ci xi (k)

Standard Model

x(k + 1) = A x(k) + B v(k) y(k) = C x(k)

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Notes chapter 2 - slide 25

Standard model State space model:

x(k + 1) = A x(k) + B v(k)+B2 w(k) + B3 v3 (k) y(k) = C x(k)+D11 e(k) + D12 w(k) where

x(k) v(k) y(k) e(k) w(k)

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is the state of the system is the input of the system (either u or ∆u) is the output of the system is zero-mean white noise (unmeasurable disturbances) denotes of all known signals (measurable disturbances)

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 26

Standard model State space model:

x(k + 1) = A x(k) + B1 e(k) + B2 w(k) + B3 v(k) y(k) = C x(k) + D11 e(k) + D12 w(k) where

x(k) v(k) y(k) e(k) w(k)

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is the state of the system is the input of the system (either u or ∆u) is the output of the system is zero-mean white noise (unmeasurable disturbances) denotes of all known signals (measurable disturbances)

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 27

Standard model Input-Output relation:

y(k)= H(q) e(k) + F(q) w(k) + G(q) v(k)

e(k)

? Transfer functions:

H

G(q)=C ( q I − A )−1 B3 −1

v(k) -

F(q)=C ( q I − A ) B2 + D11

G

-? d y(k) 6

H(q)=C ( q I − A )−1 B1 + D12 =C ( q I − A )−1 ( q I − A )−1 B F 6

Noise signal:

w(k)

e(k) is ZMWN IO-model: H(q) e(k) is zero-mean IIO-model: H(q) e(k) is NOT zero-mean, but gives a trend. SC4060

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 28

Standard model Input-Output relation:

y(k)= H(q) e(k) + F(q) w(k) + G(q) v(k)

e(k)

? Noise signal:

H

IO-model:

H e(k) →

3 2

v(k) -

1 0

−1

G

-? d y(k) 6

−2 −3

0

10

20

30

40

50

60

70

80

90

100

6

20

H e(k) →

F w(k)

15

IIO-model:

10 5 0

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0

10

20

30

40

50

k −→

60

70

80

90

100

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 29

Other model descriptions Impulse and step response models

• • • •

Process industry: no detailed models Reliable models: step/impulse response Simple experiments First MPC models were step/impulse response models (Richalet, Cutler)

Polynomial models

• • • •

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Model on basis of physical laws Model on basis of system identification Less parameters than in impulse or step response models Suitable for SISO systems

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 30

Choice of model description For all models: Resulting performance ≈ the same Differences:

• effort of modeling • model accuracy • computational power Step response models:

• • • • • • SC4060

more parameters more uncertainty intuitive less a priori information easy to understand cheap experiments Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 31

Polynomial model:

• • • • •

very compact gives good physical insight controller is compact and low order making predictions is cumbersome not tractable for MIMO systems

State space models:

• • • • •

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especially suited for MIMO systems compact model description compact and low order controller computations are well conditioned algorithms easy to implement

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 32

BASICS OF MODEL PREDICTIVE CONTROL Discuss the five ingredients of MPC:

• • • • •

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1. Process + disturbance model 2. Performance index 3. Constraint handling 4. Optimization 5. Receding horizon principle

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 33

2. Performance index: The MPC controller minimize the standard performance index at time k: N

J(v, k)=

∑

j=Nm

N

zˆT1 (k+ j−1|k)ˆz1 (k+ j−1|k) + ∑ zˆT2 (k+ j−1|k)ˆz2 (k+ j−1|k) j=1

where z1 reflects tracking error and z2 reflects the control action. and where zˆi (k+ j−1|k) is prediction of zi (k+ j−1) at time k. In MPC literature three performance indices often appear:

• Generalized Predictive Control (GPC) performance index • Linear Quadratic Predictive Control (LQPC) performance index • Zone performance index

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 34

GPC performance index: N

J(u, k) =

∑

j=Nm

r(k) y(k) ∆u(k) y p (k) Nm N Nc λ SC4060

Nc 2 yˆ p (k + j|k) − r(k + j|k) + λ2 ∑ | ∆u(k + j − 1) |2 j=1

is the reference trajectory is the process output signal is the process control increment signal

= P(q)y(k) is the minimum cost- horizon is the prediction horizon is the control horizon is the weighting on the control signal

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 35

LQPC performance index:

N

J(u, k) =

∑

N

j=Nm

xˆT (k + j|k)Qx(k ˆ + j|k) + ∑ uT (k + j − 1)Ru(k + j − 1)

x(k) u(k) Q R

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j=1

is the state signal vector is the process control signal is the state weighting matrix is the control weighting matrix

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 36

Zone-control performance index: N

J(u, k) =

∑

j=Nm

Nc

εˆ 2 (k+ j|k) + λ2 ∑ ∆u2 (k+ j−1) j=1

where

εˆ (k) =

    0   

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for

| y(k)−r(k) ˆ | ≤ δmax

y(k)−r(k)−δ ˆ max

for

y(k)−r(k) ˆ ≥ δmax

y(k)−r(k)+δ ˆ max

for

y(k)−r(k) ˆ ≤ −δmax

δmax

is the size of the zone

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 37



 Tracking error y(k) − r(k)

y(k) − r(k) →

3 2

+δmax

1 0

−δmax

−1 −2 −3

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

1.5

ε(k) →

1 0.5 0

−0.5 −1 −1.5

k −→

zone performance signal ε(k)

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 38

Performance index - GPC ↔ Standard Choose:



z(k) = 

Then:

z1 (k) z2 (k)





=

y p (k + 1) − r(k + 1) λ∆u(k)

 

N

J(k)=

T T z ˆ (k+ j−1|k)ˆ z (k+ j−1|k) + z ˆ z2 (k+ j−1|k) 1 ∑ 1 2 (k+ j−1|k)ˆ

j=1 N

2 = ∑ yˆ p (k + j|k) − r(k + j|k) + λ2 | ∆u(k + j − 1|k) |2 j=1

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 39

Performance index - LQPC ↔ Standard Choose:



z(k) = 

Then:

z1 (k) z2 (k)





=

Q1/2 x(k + 1) R1/2 u(k)

 

N

J(k)=

T T z ˆ (k+ j−1|k)ˆ z (k+ j−1|k) + z ˆ z2 (k+ j−1|k) 1 ∑ 1 2 (k+ j−1|k)ˆ

j=1 N

=

T T x ˆ (k + j|k)Q x(k ˆ + j|k) + u (k + j − 1|k)Ru(k + j − 1|k) ∑

j=1

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 40

Performance index (Minimum cost horizon) #  " Choose 0 0      for 0 ≤ j < Nm −1   0 I Γ1 ( j) 0 = " # Γ( j) =   0 Γ2 ( j)  I 0   for Nm −1 ≤ j ≤ N  0 I Then

N

J=

∑

N

j=Nm N−1

=

∑

zˆT1 (k+ j−1|k)ˆz1 (k+ j−1|k) + ∑ zˆT2 (k+ j−1|k)ˆz2 (k+ j−1|k) j=1

zˆT1 (k+ j|k)Γ1 ( j)ˆz1 (k+ j|k) + zˆT2 (k+ j|k)Γ2 ( j)ˆz2 (k+ j|k)

j=0 N−1

=

∑

zˆT (k+ j|k)Γ( j)ˆz(k+ j|k)

j=0 SC4060

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 41

BASICS OF MODEL PREDICTIVE CONTROL Discuss the five ingredients of MPC:

• • • • •

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1. Process + disturbance model 2. Performance index 3. Constraint handling 4. Optimization 5. Receding horizon principle

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 42

3. Constraint handling Inequality constraints:

umin ≤ u(k) ≤ umax ymin ≤ y(k) ≤ ymax

∆umin ≤ ∆u(k) ≤ ∆umax xmin ≤ x(k) ≤ xmax

or

ψ(k) ≤ Ψ(k) ∀k For future values we use the predicted value of ψ:

ˆ + j|k) ≤ Ψ(k + j) ∀ j > 0 ψ(k

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 43

Equality constraints: Motivated by control algorithm itself:

Control horizon constraint:

∆u(k+ j|k) = 0 for j ≥ Nc

The state end-point constraint:

x(k ˆ + N|k) = xss

or

φ(k) = 0 k ∈ κ For future values we use the predicted value of φ:

ˆ + j|k) = 0 ∀ j > 0 A j φ(k

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 44

Multiple constraints can be combined by stacking Equality constraints:

 φ1 (k) = 0    .. .

φn (k) = 0

  

Inequality constraints:

 ψ1 (k) ≤ Ψ1 (k)    .. .

ψm (k) ≤ Ψm (k)

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  



 φ(k) =   

 ψ(k) =  

φ1 (k) .. .

φn (k)

ψ1 (k) .. .

ψm (k)

Model Predictive Control (February-March 2011)



 =0  



  ≤  

Ψ1 (k) .. .

Ψm (k)



  = Ψ(k) 

Notes chapter 2 - slide 45

Two-sided inequality constraint

Ψ1,min (k) ≤ ψ1 (k) ≤ Ψ1,max (k) can be translated into one-sided inequality constraint



ψ(k) = 

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ψ1 (k) −ψ1 (k)





≤

Ψ1,max (k) −Ψ1,min (k)

Model Predictive Control (February-March 2011)



 = Ψ(k)

Notes chapter 2 - slide 46

Structuring the input signal To obtain tractable optimization problem. Decrease the degrees of freedom in

{ u(k) , u(k + 1) , . . . , u(k + N − 1) } • Control horizon Nc . u(k+ j|k) = u(k+Nc −1|k) for j ≥ Nc • Blocking. u(k+ j|k) = u(k+mℓ −1|k) for mℓ ≤ j < mℓ+1 , ℓ = 1, . . . , nbl • Basis functions. u(k+ j|k) = ∑M i=0 Si ( j) αi (k) - Si ( j) are the basis functions - αi (k), i = 1, . . . , M are scalar parameters SC4060

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 47

Control horizon

SNc−1 SNc−2 SNc−3 SNc−4 SNc−5 S2 S1 S0 0

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Model Predictive Control (February-March 2011)

Nc −1

Notes chapter 2 - slide 48

Blocking

SM SM−1 SM−2 SM−3 SM−4 S2 S1 S0 0

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Model Predictive Control (February-March 2011)

N−1

Notes chapter 2 - slide 49

BASICS OF MODEL PREDICTIVE CONTROL Discuss the five ingredients of MPC:

• • • • •

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1. Process + disturbance model 2. Performance index 3. Constraint handling 4. Optimization 5. Receding horizon principle

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 50

Standard Predictive Control Problem: The MPC controller minimizes the following performance index at time k: N−1

J(v, k) =

∑ zˆT (k + j|k)Γ( j)ˆz(k + j|k)

j=0

subject to the constraints

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ˆ + j|k) = 0 A j φ(k

∀j ≥ 0

ˆ + j|k) ≤ Ψ(k + j) ψ(k

∀j ≥ 0

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 51

An optimization algorithm will be applied to compute a sequence of future control signals that minimizes the performance index subject to the given constraints. For

• linear models • linear constraints • quadratic (2-norm) performance index we can use quadratic programming algorithms. For 1-norm or ∞- norm performance index: linear programming algorithms. Both types of algorithms are convex and show fast convergence.

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 52

BASICS OF MODEL PREDICTIVE CONTROL Discuss the five ingredients of MPC:

• • • • •

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1. Process + disturbance model 2. Performance index 3. Constraint handling 4. Optimization 5. Receding horizon principle

Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 53

RECEDING HORIZON PRINCIPLE

• a time k: Performance index J(k) is minimized • first element of optimal control sequence v(k) is applied to the system • horizon shifted • optimization restarted for time k + 1

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Model Predictive Control (February-March 2011)

Notes chapter 2 - slide 54

RECEDING HORIZON PRINCIPLE FUTURE -

PAST 

b b b b b

b b

z(k)

b b b

b b b b b b b b b

v(k) Nm

Nc

N k+ j

k

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Model Predictive Control (February-March 2011)

-

Notes chapter 2 - slide 55

RECEDING HORIZON PRINCIPLE FUTURE -

PAST  b b b b

z(k + 1) zˆ(k + 1|k)

b

z(k)

b b

v(k) k+1

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Model Predictive Control (February-March 2011)

k+ j

-

Notes chapter 2 - slide 56

RECEDING HORIZON PRINCIPLE FUTURE -

PAST  b

b

b b

b b

b

z(k)

b b

b b b b b b b b b b b

v(k) Nm

Nc

k+1

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Model Predictive Control (February-March 2011)

N k+ j

-

Notes chapter 2 - slide 57

RECEDING HORIZON PRINCIPLE FUTURE -

PAST  b b

b b

b

zˆ(k + 1|k) z(k + 1)

b

z(k)

b b

v(k) k+2

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Model Predictive Control (February-March 2011)

k+ j

-

Notes chapter 2 - slide 58

RECEDING HORIZON PRINCIPLE FUTURE -

PAST  b b b

b b

b b

z(k)

b b b b b b b b b b b b b b

v(k) Nm

Nc

k+2

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Model Predictive Control (February-March 2011)

N k+ j

-

Notes chapter 2 - slide 59

CHAPTER 3 PREDICTION

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Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 60

PREDICTION MODEL Prediction on basis of model:

x(k+1) = Ax(k) + B1 e(k) + B2 w(k) + B3 v(k) y(k) = C1 x(k) + D11 e(k) + D12 w(k) p(k) = Cp x(k) + D p1 e(k) + D p2 w(k) + D p3 v(k)

x(k)

is the state

v(k)

is the control signal (u / ∆u)

e(k)

is ZMWN

w(k) consists of all known signals ( r / di / do ) p(k)

SC4060

is a signal, to be predicted.

Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 61

At time instant k: - make a set of j-step ahead predictions of Prediction, denoted as

p(k + j).

p(k ˆ + j|k)

- using information given at time k - future values of the control signal v(k +

SC4060

j)

Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 62

p(k) =C p x(k) + D p1 e(k) + D p2 w(k) + D p3 v(k) Define



p(k|k) ˆ



   p(k+1|k)  ˆ  p(k) ˜ = .   ..   p(k+N−1|k) ˆ



v(k)



   v(k+1)   v(k) ˜ = .   ..   v(k+N−1)

˜ p3 such that Goal: find p˜0 (k) and D



w(k)



   w(k+1)   w(k) ˜ = .   ..   w(k+N−1)

p(k) ˜ = p˜0 (k) + D˜ p3 v(k) ˜ with free-response signal:

p˜0 (k) = C˜ p x(k) + D˜ p1 e(k) + D˜ p2 w(k) ˜ SC4060

Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 63

Noiseless case The model:

x(k + 1) = Ax(k) + B2 w(k) + B3 v(k) p(k) = C p x(k) + D p2 w(k) + D p3 v(k) Successive substitution:

x(k+ ˆ j|k) = Ax(k + j − 1|k) + B2 w(k+ j−1) + B3 v(k+ j−1) = A2 x(k+ j−2|k) + AB2 w(k+ j−2) + AB3 v(k+ j−2) +B2 w(k + j − 1) + B3 v(k + j − 1) j

.. .

j

= A j x(k) + ∑ Ai−1 B2 w(k+ j−i) + ∑ Ai−1 B3 v(k+ j−i) i=1

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Model Predictive Control (February-March 2011)

i=1

Notes chapter 3 - slide 64

p(k+ ˆ j|k) = C p x(k+ ˆ j|k) + D p2 w(k+ j) + D p3 v(k+ j) j

= C p A j x(k) + ∑ C p Ai−1 B2 w(k + j − i) i=1

j

+ ∑ C p Ai−1 B3 v(k + j − i) + D p2 w(k+ j) + D p3 v(k + j) i=1

and so

p(k) ˜ = C˜ p x(k) + D˜ p2 w(k) ˜ + D˜ p3 v(k) ˜ = p˜0 (k) + D˜ p3 v(k) ˜ with free-response

p˜0 (k) = C˜ p x(k) + D˜ p2 w(k) ˜ SC4060

Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 65

where



C  p  CpA   ˜ C p =  C p A2 .  ..  C p AN−1

        



    D˜ p3 =     SC4060



    ˜ D p2 =     D p3 C p B3 C p AB3 .. .

C p AN−2 B3

D p2 C p B2

0 D p2

C p AB2

C p B2

.. .

C p B3

0 0

..

.

.. .

..

.

D p2 · · · C p B2

C p AN−2 B2 0 D p3

··· ···

··· ···

0 0

..

.

.. .

..

.

D p3 · · · C p B3

Model Predictive Control (February-March 2011)





0  0   ..  .   0   D p2

0  0   ..  .   0   D p3 Notes chapter 3 - slide 66

Noisy case The model:

x(k+1) = Ax(k) + B1 e(k) + B2 w(k) + B3 v(k) y(k) = C1 x(k) + D11 e(k) + D12 w(k) p(k) = Cp x(k) + D p1 e(k) + D p2 w(k) + D p3 v(k) Successive substitution: j

j

i=1

i=1

x(k+ ˆ j|k) = A j x(k) + ∑ Ai−1 B3 v(k+ j−i) + ∑ Ai−1 B1 e(k+ ˆ j−i|k) j

+ ∑ Ai−1 B2 w(k+ j−i) i=1

e(k ˆ + j|k) = 0 for j > 0   −1 and e(k) = D11 y(k) −C1 x(k) − D12 w(k) Prediction of noise:

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Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 67

j

x(k+ ˆ j|k) = A j x(k) + ∑ Ai−1 B3 v(k+ j−i)+A j−1 B1 e(k) ˆ i=1

j

+

i−1 A ∑ B2w(k+ j−i)

i=1

and so:

p(k+ ˆ j|k) = C p x(k ˆ + j|k) + D p2 w(k + j) + D p3 v(k + j) j

= C p A j x(k) +C pA j−1 B1 e(k) + ∑ C p Ai−1 B p w(k + j − i) i=1

j

+ ∑ C p Ai−1 B3 v(k + j − i) + D p2 w(k + j) + D p3 v(k + j) i=1

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Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 68

p(k)= ˜ C˜ p x(k) + D˜ p1 e(k) + D˜ p2 w(k) ˜ + D˜ p3 v(k) ˜ = p˜0 (k) + D˜ p3 v(k) ˜ where



and free-response signal:

  D˜ p1 =  

D p1 Cp B1 C . p AB1 ..

Cp AN−2 B1

    

p˜0 (k)= C˜ p x(k) + D˜ p1 e(k) + D˜ p2 w(k) ˜ ˜ p2 w(k) = C˜ p x(k) + D˜ p1 D−1 (y(k) −C x(k) − D w(k)) + D ˜ 1 12 11 −1 −1 ˜ ˜ ˜ = (C˜ p − D˜ p1 D−1 C )x(k) + D D y(k) + ( D − D D ˜ p1 11 p2 p1 11 D12 Ew )w(k) 11 1

SC4060

Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 69

Summary On basis of model:

x(k+1) = Ax(k) + B1 e(k) + B2 w(k) + B3 v(k) y(k) = C1 x(k) + D11 e(k) + D12 w(k) p(k) = C p x(k) + D p1 e(k) + D p2 w(k) + D p3 v(k) we make prediction:

p(k) ˜ = C˜ p x(k) + D˜ p1 e(k) + D˜ p2 w(k) ˜ + D˜ p3 v(k) ˜ = p˜0 (k) + D˜ p3 v(k) ˜   −1 where e(k) = D11 y(k) −C1 x(k) − D12 w(k) and SC4060

Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 70

where



C  p  CpA   ˜ C p =  C p A2 .  ..  C p AN−1 

        

D p1 C p B1 C . p AB1

  D˜ p1 =    .. C p AN−2 B1 SC4060



    ˜ D p2 =          



    D˜ p3 =    

D p2 C p B2 C p AB2 .. .

0 D p2 C p B2

C p AB3 .. .

C p AN−2 B3

Model Predictive Control (February-March 2011)

0 0

..

.

.. .

..

.

D p2 · · · C p B2

C p AN−2 B2 D p3 C p B3

··· ···

0 D p3 C p B3

··· ···

0 0

..

.

.. .

..

.

D p3 · · · C p B3



0  0   ..  .   0   D p2



0  0   ..  .   0   D p3

Notes chapter 3 - slide 71

The use of a priori knowledge



We assume w(k +

   w(k) ˜ =  

w(k) w(k+1) ... w(k+N−1)

j|k) to be known at time k.

      

If not we choose



   w(k) ˜ =   SC4060

w(k|k) ˆ w(k+1|k) ˆ ... w(k+N−1|k) ˆ

Model Predictive Control (February-March 2011)

       Notes chapter 3 - slide 72

Zero-th order extrapolation: We assume w(k ˆ +

j) = w(k − 1) for j ≥ 0.

This results in



w(k − 1)

  w(k − 1)  w(k) ˜ =  ...  w(k − 1)

SC4060

Model Predictive Control (February-March 2011)

      

Notes chapter 3 - slide 73

Linear (first order) extrapolation:

ˆ + We assume w(k

j) = w(k − 1) + j (w(k − 1) − w(k − 2)) for j ≥ 0.

This results in



   w(k) ˜ =  

2w(k − 1) − w(k − 2) 3w(k − 1) − 2w(k − 2) ... (N + 1)w(k − 1) − Nw(k − 2)

      

Polynomial (higher order) extrapolation: For an nth order extrapolation we use n + 1 past values of w. The error estimate will grow with the degree of the polynomial extrapolation.

SC4060

Model Predictive Control (February-March 2011)

Notes chapter 3 - slide 74

CHAPTER 4 STANDARD FORMULATION

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Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 75

Standard formulation Many kinds of performance indices

• LQPC performance index • GPC performance index • zone performance index Many types of constraints

• equality constraints • inequality constraints Standard formulation:

• transform performance index • formalize constraints

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Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 76

Performance index Consider the system

x(k + 1) = Ax(k) + B1 e(k) + B2 w(k) + B3 v(k) y(k) = C1 x(k) + D11 e(k) + D12 w(k) Define performance index N−1

J(v, k) =

∑

zˆT (k + j|k)Γ( j)ˆz(k + j|k)

j=0

where

z(k) = C2 x(k) + D21 e(k) + D22 w(k) + D23 v(k)

SC4060

Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 77

LQPC: N

J(u, k) =

∑

j=Nm

N

xˆT (k + j|k)Qx(k ˆ + j|k) + ∑ uT (k + j − 1)Ru(k + j − 1) j=1



z(k) = 

GPC: N

J(u, k) =

∑

j=Nm

R1/2 u(k)



Nc 2 yˆ p (k + j|k) − r(k + j|k) + λ2 ∑ | ∆u(k + j − 1) |2 j=1



z(k) =  SC4060

Q1/2 x(k + 1)



yˆ p (k + 1) − r(k + 1) λ∆u(k)

Model Predictive Control (February-March 2011)

 

Notes chapter 4 - slide 78

LQPC performance index Consider IO-model

xo (k+1) = Ao xo (k)+Ko eo (k)+Lo do (k)+Bo u(k) y(k) = Co xo (k)+DH eo (k)+DF do (k) Choose

x(k) = xo (k) , v(k) = u(k) , w(k) = do (k) , e(k) = eo (k), then x(k+1) = Ao x(k) + Ko e(k) + Lo d(k) + Bo v(k) = A x(k) + B1 e(k) + B2 w(k) + B3 v(k) y(k) = C1 x(k) + D11 e(k) + D12 w(k)

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Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 79



z(k) =  

= 

Q1/2 x(k + 1) R1/2 v(k)

 

Q1/2 Ao x(k) + Q1/2 Ko e(k) + Q1/2 Lo w(k) + Q1/2 Bo v(k)

 

R1/2 v(k)         Q1/2 Ao Q1/2 Ko Q1/2 Lo Q1/2 Bo x(k)+  e(k)+  w(k)+  v(k) =  0 0 0 R1/2 = C2 x(k) + D21 e(k) + D22 w(k) + D23 v(k)

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Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 80

GPC performance index Consider the IIO-output model:

xi (k+1) = Ai xi (k)+Ki ei (k)+Li di (k)+Bi ∆u(k) y(k) =Ci xi (k) + DH ei (k) The realization y p (k) = P(q)y(k) described by:

x p (k + 1) = A p x p (k) + B p y(k) y p (k) = C p x p (k) + D p y(k) Choose e(k) = ei (k) , v(k) = ∆u(k),



w(k) =  SC4060

di (k) r(k + 1)





, and x(k) = 

x p (k) xi (k)

Model Predictive Control (February-March 2011)



, Notes chapter 4 - slide 81

then

 

x p (k+1) xi (k+1)





 = 

A p B pCi



x p (k)







+

B p DH



ei (k)+

xi (k) Ki     0 0 di (k) 0   + ∆u(k) Li 0 r(k + 1) Bi 0

Ai 

= Ax(k) + B1 e(k) + B2 w(k) + B3 v(k) y(k) = Ci x(k) + DH e(k) = C1 x(k) + D11 e(k) + D12 w(k)

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Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 82

For y p (k) we find:

y p (k) =

h

i

C p D pCi x(k) + D pDH e(k)

and we obtain the estimate y p (k+1):

  h i x p (k) + y p (k+1) = C p A p C p B pCi + D pCi Ai  xi (k) h i C p B p DH + D pCi Ki eˆi (k)+   h i h i di (k)  + D pCi Bi v(k) D pCi Li 0  r(k + 1)

where we used the fact that eˆi (k + 1|k) = 0. SC4060

Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 83

Now we find:



z(k) = 



= 

   

y p (k + 1) − r(k + 1)

 

λ∆u(k)

C p A p C p B pCi + D pCi Ai 0

0

C p B p DH + D pCi Ki 0 D pCi Li −I 0

0

 



 

 eˆi (k)+

di (k)

r(k + 1)



x p (k) xi (k)



+



+

D pCi Bi λI

= C2 x(k) + D21 e(k) + D22 w(k) + D23 v(k) SC4060

Model Predictive Control (February-March 2011)



 v(k)

Notes chapter 4 - slide 84

Standard performance index Define vector z˜(k) with predictions of zˆ(k +

j|k):





zˆ(k|k)    zˆ(k+1|k)   z˜(k) =  .   ..   zˆ(k+N−1|k)

Then using the formulas of chapter 3 we obtain:

z˜(k) = C˜2 x(k) + D˜ 21 e(k) + D˜ 22 w(k) ˜ + D˜ 23 v(k) ˜

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Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 85

where



C  2  C2 A   ˜ C2 =  C2 A2 .  ..  C2 AN−1 

        

D21 C2 B1 C . 2 AB1

  D˜ 21 =    .. C2 AN−2 B1 SC4060



    ˜ D22 =          



    D˜ 23 =    

D22 C2 B2 C2 AB2 .. .

0 D22 C2 B2

C2 AB3 .. .

C2 AN−2 B3

Model Predictive Control (February-March 2011)

0 0

..

.

.. .

..

.

D22 · · · C2 B2

C2 AN−2 B2 D23 C2 B3

··· ···

0 D23 C2 B3

··· ···

0 0

..

.

.. .

..

.

D23 · · · C2 B3



0  0   ..  .   0   D22



0  0   ..  .   0   D23

Notes chapter 4 - slide 86

Now define



then

Γ(0)

0

0

Γ(1)

   Γ¯ =   

.. .

0

0

∑

0 0

..

N−1

J(v, k) =

...

.

.. .

. . . Γ(N − 1)

      

zˆT (k + j|k)Γ( j)ˆz(k + j|k)

j=0

¯ z(k) = z˜T (k)Γ˜

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Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 87

Handling constraints Equality constraints

ˆ + j|k) = 0 A j φ(k Stack all constraints at time k on predicted φ in one vector:





ˆ + 1|k) A1 φ(k    A φ(k  ˆ + 2|k)  2  ˜ φ(k) =  ..   .   ˆ + N|k) AN φ(k

In standard framework, we can express the equality constraints in the form:

˜ φ(k) = C˜3 x(k) + D˜ 31 e(k) + D˜ 32 w(k) ˜ + D˜ 33 v(k) ˜ =0 SC4060

Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 88

State end-point constraint:

h

i x(k + N|k) = AN x(k) + AN−1 B2 AN−2 B2 · · · B2 w(k)+ ˜ h i +AN−1 B1 e(k) + AN−1 B3 AN−2 B3 · · · B3 v(k) ˜ =0

by defining

C˜3 = AN

D˜ 31 = AN−1 B1 D˜ 33 =

we obtain:

h

D˜ 32 =

h

AN−1 B2 AN−2 B2 · · · B2

AN−1 B3 AN−2 B3 · · · B3

i

i

˜ = C˜3 x(k)+ D˜ 31 e(k)+ D˜ 32 w(k)+ φ(k) ˜ D˜ 33 v(k) ˜ =0

SC4060

Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 89

Inequality constraints

ψ(k) ≤ Ψ(k) where ψ reflect states, inputs, outputs and Ψ reflects limitations. Example:

x1,min ≤ x1 (k) ≤ x1,max v(k) ≤ vmax ymin ≤ y(k) result in :



  ψ(k) =    SC4060

x1 (k) −x1 (k) v(k) −y(k)

     



  Ψ(k) =   

Model Predictive Control (February-March 2011)

x1,max −x1,min vmax −ymin

      Notes chapter 4 - slide 90

So:















E1 0 0         −E1       0 0 ψ(k) =  x(k) +  e(k) +  w(k) +   0     0 0 −C1 −D11 −D12 = C4 x(k) + D41 e(k) + D42 w(k)+ D43 v(k)

where



0  0 v(k) I 0

E1 = [ 1 0 . . . 0 ] .

Now we obtain the inequality constraint:

ψ(k) = C4 x(k) + D41 e(k) + D42 w(k) + D43 v(k) ≤ Ψ(k)

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Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 91

Now consider the inequality constraint

ψ(k) = C4 x(k) + D41 e(k) + D42 w(k) + D43 v(k) ≤ Ψ(k) Stack all constraints at time k on predicted ψ in one vector:



ˆ + 1|k) ψ(k

  ψ(k  ˆ + 2|k) ˜ ψ(k) = ..  .  ˆ + N|k) ψ(k





Ψ(k + 1|k)

    Ψ(k + 2|k)   ≤ ..   .   Ψ(k + N|k)



   ˜  = Ψ(k)  

Using the formulas of chapter 3, the inequality constraints can be written in the form

˜ ˜ ψ(k) = C˜4 x(k) + D˜ 41 e(k) + D˜ 42 w(k) ˜ + D˜ 43 v(k) ˜ ≤ Ψ(k) where SC4060

Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 92



C  4  C4 A   C˜4 =  C4 A2 .  ..  C4 AN−1 

        

D41 C4 B1 C . 4 AB1

  D˜ 41 =    .. C4 AN−2 B1 SC4060



    D˜ 42 =          



    D˜ 43 =    

D42 C4 B2 C4 AB2 .. .

0 D42 C4 B2

C4 AB3 .. .

C4 AN−2 B3

Model Predictive Control (February-March 2011)

0 0

..

.

.. .

..

.

D42 · · · C4 B2

C4 AN−2 B2 D43 C4 B3

··· ···

0 D43 C4 B3

··· ···

0 0

..

.

.. .

..

.

D43 · · · C4 B3



0  0   ..  .   0   D42



0  0   ..  .   0   D43

Notes chapter 4 - slide 93

Standard formulation Consider the system

x(k + 1) = Ax(k) + B1 e(k) + B2 w(k) + B3 v(k) y(k) = C1 x(k) + D11 e(k) + D12 w(k) z˜(k) = C˜2 x(k) + D˜ 21 e(k) + D˜ 22 w(k) ˜ + D˜ 23 v(k) ˜ ˜ φ(k) = C˜3 x(k) + D˜ 31 e(k) + D˜ 32 w(k) ˜ + D˜ 33 v(k) ˜ ˜ ψ(k) = C˜4 x(k) + D˜ 41 e(k) + D˜ 42 w(k) ˜ + D˜ 43 v(k) ˜ Minimize performance index N−1

J(v, k) =

∑

zˆT (k + j|k)Γ( j)ˆz(k + j|k)

j=0

¯ z(k) = z˜T (k)Γ˜ subject to the constraints SC4060

˜ ˜ ˜ φ(k) = 0 and ψ(k) ≤ Ψ(k).

Model Predictive Control (February-March 2011)

Notes chapter 4 - slide 94