model-based predictive control - project mbpc•mbpcis a control algorithm based on solving an...

Model-based Predictive Control -

project

MbPC

S.l. dr. ing. Constantin Florin Caruntu

www.ac.tuiasi.ro/~caruntuc

[email protected]

Predictive control based on

Diophantine equationsOutline

1. Introduction

2. Prediction model

3. Objective functions

4. Predictive control algorithm

13.01.2014

15:22


Diophantine equations

Lecture 1 - Introduction

• MbPC is a control algorithm based on solving an on-line optimal control

problem

• All the MBPC algorithms posses common elements:

- explicit use of a model to predict the process output at future time instants

(horizon)

- computation of a control sequence minimizing a cost function

- receding strategy: at each instant the horizon is displaced forwards the future

⇒ the first control signal is applied to the process.

Model

Optimizer

Past Inputs

and OutputsPredictor

Outputs

Reference

Trajectory

Future

InputsFuture

Errors

Cost

FunctionConstraints

_

+

1.1. Model based Predictive Control elements1. Introduction13.01.2014

15:22

Fig. 1. Basic structure of MPC

Fig. 2. The predictive control concept

k : current time hi : minimum prediction horizon

hc: control horizon hp: prediction horizon

future control, computed at time k

predicted values of the output

set-point

reference trajectory

dead time

k-1 k k+1 k+2 ……….. time

( )r k i+

ˆ( )y k i k+ ˆ( ), [ , ]y k i k i hi hp+ ∈

( 1 ), [1, ]u k i k i hc+ − =( )u k

( )y k

hc

futurepast

( 1 ), 1, :u k i k i hc+ − =

ˆ( ), , :y k i k i hi hp+ =

( ), , :r k i i hi hp+ =

( )w k i k+

hi hp

( ), , :w k i k i hi hp+ =

1.2. Predictive control concept1. Introduction13.01.2014

15:22

• at each current time k: the process output is predicted over the horizon [hi,hp]

- the forecast depends on past inputs, past outputs and future control scenario

• a reference trajectory , starting at , is

defined to describe how to drive the process output to the set-point

• the control sequence is computed in order to minimize

a cost function depending on the predicted control errors

• the first element is applied to the process

- at the next sampling instant all time-sequences are shifted ⇒ the whole

procedure is repeated ⇒ receding horizon strategy

{ }( ), ,w k i k i hi hp+ = ( ) ( )w k k y k=( )r k i+

{ }( 1 ), 1,u k i k i hc+ − =

{ }ˆ( ) ( ) , ,y k i k w k i k i hi hp + − + =

( )u k k

13.01.2014

15:22 1.2. Predictive control concept1. Prediction model

Model-based Predictive Control -

project

MbPC

S.l. dr. ing. Constantin Florin Caruntu

www.ac.tuiasi.ro/~caruntuc

[email protected]


Diophantine equationsOutline

1. Introduction

2. Prediction model

3. Objective functions

4. Predictive control algorithm

13.01.2014

15:22



Lecture 2 – Prediction model

• Real system model→ two additive sub-models:

– a process model

→ relationship between the outputs and measurable inputs of the physical

plant

– a disturbance model

→ describes a part of the measurable outputs of the physical plant which

where not included in the process model

→ gathers the effects of

» non-measurable inputs

» external disturbances

» measuring noise

» modeling errors

13.01.2014

15:22 2.1. Models2. Prediction model

• Model (“corner-stone” of MPC) able to:

- capture the process dynamics

- allow the prediction

- permit theoretical analysis{

• System model = process model + disturbance model

)(kn

)(ku )(kx )(ky

Process +

+

))(),..,1(),(),..,1(()( BA ndkudkunkxkxfkx −−−−−−=

1 1

1 1 1

process model disturbance model

( ) ( )( ) ( 1) ( ) ( ) ( )

( ) ( ) ( )

dq B q C qy k u k e k x k n k

A q D q A q

− − −

− − −= − + = +144424443 144424443

B

B

A

A

nn

nn

qbqbbqB

qaqaqA

−−−

−−−

+++=

+++=

...)(

...1)(

110

1

11

1

D

D

C

C

nn

nn

qaqdqD

qcqcqC

−−−

−−−

+++=

+++=

...1)(

...1)(

11

1

11

1

(2.1.1)

(2.1.2)

13.01.2014

15:22 2.1. Models – general form2. Prediction model

• ARX model (Auto-Regressive eXogenous)

10;0,, ==⇒==≤= DCnnnnnn DCBA

)()(

1)1(

)(

)()(

11

1

keqA

kuqA

qBqky

d

−−

−−+−=

)()1()()()( 11 kekuqBqkyqA d +−= −−−

(2.1.3)

• ARIX model (Auto-Regressive Integrated eXogenous)

11 1)(,11;0,, −− −=∆==⇒==≤= qqDCnnnnnn DCBA

(2.1.4)

,)()(

1)1(

)(

)()(

11

1

keqA

kuqA

qBqky

d

∆+−=

−−

−−

)()1()()()( 11 kekuqBqkyqA d +−∆=∆ −−−

13.01.2014

15:22 2.1. Linear models2. Prediction model

• Modelul ARMAX (Auto-Regressive Moving-Average eXogenous)

(2.1.5)

10;0,, =⇒=>≤= Dnnnnnn DCBA

)()(

)()1(

)(

)()(

1

1

1

1

keqA

qCku

qA

qBqky

d

−

−

−

−−

+−=

)()()1()()()( 111 keqCkuqBqkyqA d −−−− +−=

• Modelul ARIMAX (Auto-Regressive Integrated Moving-Average eXogenous)

(2.1.6)

∆=⇒=>≤= − )(1;0,, 1qDnnnnnn DCBA

)()(

)()1(

)(

)()(

1

1

1

1

keqA

qCku

qA

qBqky

d

∆+−=

−

−

−

−−

)()()1()()()( 111 keqCkuqBqkyqA d −−−− +−∆=∆

13.01.2014


• Modelul FIR (Finite Impulse Response - Raspuns la impuls de durata finita)

• Modelul FSR (Finite Step Response)

(2.1.7)

(2.1.8)

DCnnnnn DCHBA ,,,;1;0 −== : does not exist

1

0

( ) ( 1) with for 0Hn

i i i

i

y k hu k i b h d−

=

= − − = =∑

DCnnnnn DCSBA ,,,;1;0 −== : does not exist

10 0

0 1

( ) ( 1); , 1, for 0Sn

i

i i i i

b sy k s u k i i d

b s s

−

= −

== ∆ − + ∀ ≥ =

= −∑

13.01.2014


{ }ˆ( | ), 1,y k i k i p+ =i-step ahead predictor of the system output:

Data { available measurementsfuture (postulated) control input

}..),1(..),1(),({ −− kukyky

},1),|1({ pikiku =−+

a)Diophantine equations (GPC)

b)Filtering techniques (EPSAC)

13.01.2014

15:22 2.2. i-step ahead predictor2. Prediction model

A

FqE

A

iii

−+=1

(2.2.4)

−=++=

−=++=

−

−

1;

1;

0

0

AFn

ni

En

ni

nnqffF

inqeeE

iiF

iF

iiE

iE

K

K(2.2.5)

ii

i FqAE −+=1 ii

i FqAE −−=1( , )

k→ k + i

� (ARX models)

Processes without dead time with C=D=1


0>An

For C = D = 1 (2.1.1) becomes: )()1()()()( 11 kekuqBkyqA +−= −−(2.2.1)

(2.2.1) )()1()()()( 11 ikeikuqBikyqA ++−+=+ −−(2.2.2)

)(1

)1()( ikeA

ikuA

Biky ++−+=+ (2.2.3)

⇒+ )4.2.2(]*)2.2.2[( iE )()()1()( ikeEkyFikBuEiky iii +++−+=+ (2.2.6)

13.01.2014

15:22 2.2. i-step ahead predictor2. Prediction model

ii

ii HqGBE −+= (2.2.7)

−=++=

−=++=

−

−

1;

1;

0

0

BHn

ni

Gn

ni

nnqhhH

inqggG

iiH

iH

iiG

iG

K

K

(2.2.8)

(2.2.7) =>

past future

( ) ( ) ( 1) ( 1) ( )i i i iy k i F y k H u k G u k i E e k i+ = + − + + − + +144424443 14444244443

(2.2.9)

13.01.2014

15:22


2.2. i-step ahead predictor2. Prediction model


[ ]{ }2)(ˆ)( ikyikyJ +−+=E

{ }{ } { }{ }

2

2 2

ˆ( ) ( 1) ( 1) ( ) ( )

ˆ( ) ( ) ( 1) ( 1) ( )

ˆ2 ( ) ( ) ( 1) ( 1) ( )

i i i i

i i i i

i i i i

J F y k H u k G u k i E e k i y k i

E e k i F y k H u k G u k i y k i

E e k i F y k H u k G u k i y k i

= + − + + − + + − + =

= + + + − + + − − + +

+ + + − + + − − +

E

E E

E

(2.2.11)

(2.2.10)

Minimizing the cost function (2.2.11) yields:

)1()()1()(ˆ −++−+=+ kuHkyFikuGiky iii(2.2.12)

13.01.2014

15:22



Minimum variance prediction:

Procedure to obtain the i-step ahead predictor:

1. Compute the polynomials and of the form (2.2.5) by solving the

diophantine equation (2.2.4);

2. Knowing compute and of the form (2.2.8) by solving the

diophantine equation (2.2.7);

3. With , and known one can obtain the i-step ahead predictor using

the relation (2.2.12).

iE iF

iE iHiG

iG iHiF

Observations:

• The polynomial coefficients are the first i coefficients of the process

impulse response.

(2.2.4)*B => (2.2.13)

• The solution for becomes simple: and .

A

BFqBE

A

B iii

−+=

Substituting (2.2.10) in (2.2.13) => (2.2.14)

++= −

iii

i HA

BFqG

A

B

BEG ii = 0=iH0=Bn

iG

13.01.2014

15:22



[ ][ ][ ]

ˆ ˆ ˆ( 1), , ( )

( ), , ( 1)

( ), , ( 1), ( 1), , ( )

T

T

T

A B

y k y k p

u k u k p

y k y k n u k u k n

= + +

= + −

= − + − −

y

u

s

K

K

K K

Matriceal form for the i-step ahead predictor, for :pi ,1=

[ ]ˆ, 1p= ×y

ΨsPuy +=ˆ

[ ], ( ) 1A Bn n= + ×s

[ ], 1p= ×u

=

−− 021

01

0

0

00

ggg

gg

g

pp K

K

K

P

=

−−

−−

1.0.1.0.

1.10.11.10.1

BA

BA

nppnpp

nn

hhff

hhff

KK

KKKKKKKK

KK

Ψ

(2.2.18)

(2.2.17)

(2.2.16)

(2.2.15)

(2.2.19)

(2.2.20))(][, BA nnp +×=Ψ

pp×=][, P

13.01.2014

15:22



� (for FIR and FSR models)

Changes:

– diophantine equation (2.2.4) =>

–

– degree depends on i => if

degree[ ] 0 min( , ) 1, 1i ii G H H HE n i n n n i= => = − = − −

0,1 == ii FE

0=An

iH : 0H ii n H≥ =

(2.2.21)

(2.2.22)

FIR: i-step ahead minimum variance predictor

)1()(ˆ −+=+ ikBuiky

future past

ˆ( ) ( 1) ( 1)i iy k i G u k i H u k+ = + − + −1442443 14243

(2.2.7) =>

13.01.2014

15:22



)()()(

)()1(

)(

)()(

11

1

1

1

keqDqA

qCku

qA

qBky

−−

−

−

−+−= (2.3.1)

0>An�

)0( =d

k→ k + i(2.3.1) )()1()( ikeDA

Ciku

A

Biky ++−+=+ (2.3.2)

DA

FqE

DA

C iii

−+= (2.3.3),( ii

i FqEDAC −+= )ii

i FqCEDA −−=

−+−=++=

−=++=

−

−

)1,max(;

1;

0

0

DACF

n

ni

E

n

ni

nninnqffF

inqeeE

i

iF

iF

i

iE

iE

K

K(2.3.4)

⇒+ )3.3.2(]*)2.3.2[( iEDA )()()1()( ikeEkyC

Fiku

C

BDEiky i

ii +++−+=+ (2.3.5)

13.01.2014

15:22

2.3. Predictors for ARMAX and

ARIMAX models2. Prediction model

(2.3.6)

(2.3.7)

(2.3.8)

(2.3.9)

[ ]{ }2)(ˆ)( ikyikyJ +−+=E

+−+++−+=

2

)(ˆ)()()1( ikyikeEkyC

Fiku

C

BDEJ i

iiE

+−−++++=

22 )(ˆ)1()()]([ ikyiku

C

BDEky

C

FikeEJ ii

iE

)()1()(ˆ kyC

Fiku

C

BDEiky ii +−+=+

(2.3.10)C

HqG

C

BDE ii

i

i −+=

1 , max( 1, 1)i i

B D CG Hn i n n n n= − = + − −

futurepast

ˆ( ) ( 1) ( ) ( 1)i ii

F Hy k i Gu k i y k u k

C C+ = + − + + −

1442443144424443

(2.3.11)

13.01.2014

15:22

Minimum variance prediction:



Algorithm to compute the minimum variance prediction:

1. Diophantine equation (2.3.3) =>

2. Diophantine equation (2.3.10) =>

3. => predictor (2.3.11)

,iE

,iG

iF

iH

iG,iH,iF

13.01.2014

15:22



(2.2.4)*C and (2.3.3)*D =>

)()1()( keAD

Cku

A

Bky +−=

• C = D = 1 (ARX)

)1()()1()(ˆ −++−+=+ kuHkyFikuGiky iii

)()1()(ˆ kyFikBuEiky ii +−+=+

• ARMAX, ARIMAX

)()1()1()(ˆ kyC

Fku

C

HikuGiky ii

i +−+−+=+

(2.3.12)

)()1()(ˆ kyC

Fiku

C

BDEiky ii +−+=+

A

FqED

A

CFqCE ii

iii

i−− +=+

13.01.2014

15:22

2.3.1. Generalized i-step

ahead predictor2. Prediction model

−−=

+−=+= −−

1

),min(1;

DCN

DCM

ini

iinnn

nninNqMCE

i

iD

• Coefficients until order :1−+ Dni

1; −=+= −DQ

ini

i nnA

RqQ

A

F

i

D

1; −=+= −DQ

ini

i nnA

RqQ

A

CFi

D

(2.3.16)

(2.3.13)

(2.3.14)

(2.3.15)

A

RqQqED

A

RNqQqM ini

ii

ii

ini

ii

iDD −−−−−− ++=

+++

• With (2.3.13) ÷ (2.3.15), (2.3.12) becomes:

iii

ii

iii

i

RRAN

QqEDQqM

=+

+=+ −−

(2.3.17)

13.01.2014

15:22



(2.3.18)

• From (2.3.13) and (2.3.17) =>

(2.3.21)

)( in

iii

ii NqQQqCEED D−− −−+=

1),max(];[ −=−−−= −DCKi

niii nnnNqQQK D

• Introducing polynomial defined by:iK

(2.3.19)

(2.3.20)ii

ii KqCEED −−=• (2.3.18) becomes:

• From (2.2.4), (2.3.3) and (2.3.20) => iii AKCFF +=

• Back to the predictor equation (2.3.9) and using (2.3.20) and (2.3.21) yields:

[ ]additional term

ˆ( ) ( 1) ( ) ( ) ( 1)ii i

Ky k i E Bu k i F y k Ay k Bu k

C+ = + − + + − −

144424443

(2.3.22)

• The additional term becomes null for the models: )1()( −= kBukAy (2.3.23)

13.01.2014

15:22



(2.3.24)

• A,B,C,D (obtained through identification), (2.3.22) becomes:

where:

(2.3.25)

(2.3.26)

• In applications:

• Using (2.2.7) the predictor equation becomes:

(2.3.27)

DCBA ˆ,ˆ,ˆ,ˆ

[ ])(ˆ)(ˆ

ˆ)()1(ˆ)(ˆ kyky

C

AKkyFikuBEiky i

ii −++−+=+

)1(ˆ

ˆ)(ˆ −= ku

A

Bky

11 1)(ˆ

)1(ˆ

−−

∆

−=∆=

==

qqD

TC

[ ]future past

additional term

ˆˆ ˆ( ) ( 1) ( ) ( 1) ( ) ( ) .i

i i i

K Ay k i G u k i F y k H u k y k y k

T+ = + − + + − + −

1442443 144424443144424443

ii

ii HqGBE −+=

13.01.2014

15:22



=> (FIR)

• Observations:

0An• =

(2.3.28)

(2.3.30)

(2.3.31)

(2.2.4) =>1

0

=

=

i

i

E

F11

1

0

++−−+−− +++=

=

DT

TDD

nnnninii

i

qtqttN

Q

K

innn DTNi−−= => if => if0=iN DT nni −> 1,0 ≥∀= iNi DT nn ≤

(2.3.19) becomes: in

ii NqQK D−+=

0Dn• >

0Dn• =

[ ])(ˆ)()1(ˆ)(ˆ kykyT

KikuBiky i −+−+=+

IF 0;

IF 0 0, 1

T i i

T i

i n K N

n K i

> ⇒ = =

= ⇒ = ∀ ≥)1(ˆ)(ˆ −+=+ ikuBiky

(2.3.29)

0, ,1 ,0,0)1,1max( ≥∀≥∀≠≠⇒−−= TDiiDTFnniKQnnn

i

: (2.3.14) => iii NKQ =⇒= 0

13.01.2014

15:22



2.3.2. computationiK

(2.3.33)

(2.3.34)

(2.3.32)

• ARX: and

• ARMAX:

• ARIMAX:

001ˆ ===⇒==⇒= iiiDT NQQnnD1=T

0=iK

001ˆ ==⇒=⇒= iiD QQnD

ii NK =(2.3.13) => is computed with the last coefficients of polynomialiK Tn

iTE1ˆ 1 ( 1) and :D i iD q n Q Q−= − = ⇒ scalars

iiii QNTEQTC

q+−=⇒

=

=)1()1(

ˆ

1(2.3.18) with

(2.3.32) and (2.3.13) iii QMQ +=⇒ )1(

(2.3.19) and (2.3.33) iii NqMK 1)1( −+=⇒

(2.3.13) => is computed with the last coefficients of polynomialiN 1−Tn

iTE

13.01.2014

15:22 2. Prediction model

(2.3.19): 1),max(];[ −=−−−= −DCKi

niii nnnNqQQK D

(2.3.19):

(2.3.36)

(2.3.38)

(2.3.35)

1),max(];[ −=−−−= −DCKi

niii nnnNqQQK D

• ARIX: and

• FIR: with and

• FSR: with and

:,1,01ˆ 1iiDT QQnnqD ⇒==⇒−= −1=T

0=⇒ iN⇒= 0Tn

scalars

(2.3.18) ][ 1iii

iii NqQQqEE −− −−+=∆⇒

(2.3.13)

(2.3.19) + (2.3.36) )1(ii EK =⇒

iii QEQ +=⇒ )1(

(2.3.37)

(2.3.35));0(1 =∆=q

1)1(1cu)20.3.2(

1,01ˆ 1

=⇒===

===−= −

ii

DTA

KTEq

nnnqD1=T

1=T 1ˆ 1 identical with FIRD q −= −

[ ])(ˆ)()1(ˆ)(ˆ kykyikuBiky −+−+=+

13.01.2014

15:22 2.3.2. computationiK2. Prediction model

� if and => the last rows of are equal to zero

because if ;

� if then and .

[ ]ˆpred. with 1

additional term

ˆˆ ˆ( ) ( 1) ( ) ( 1) ( ) ( )i

i i i

T D

K Ay k i G u k i F y k H u k y k y k

T= =

+ = + − + + − + −144444424444443

144424443

T

kuBkyAkc

)1(ˆ)(ˆ)(~

−−=

pi ,1=

0=Dn

(2.3.40)

(2.3.42)

(2.3.39)

(2.3.41)

• For

(2.3.27) :

{additional termˆpred. with 1

ˆ

T D= =

= + +y Pu Ψs K14243

cKK ~c=

becomes:

1)1(]~[;)](~,),(~[~ ×+=−= KK nnkckc cc Kwhere:

(2.3.43)[ ] )1(,

.0.

.10.1

+×=

= Kc

npp

n

c np

kk

kk

K

K

KK

K

KKK

K

Tnp −0=An cK

0=iK Tni >

0=== DTA nnn 0=cK 0=K

13.01.2014

15:22

2.3.3. Matriceal form of the

generalized predictor 2. Prediction model

(2.4.2)

(2.4.3)

(2.4.1)

(2.4.5)

(2.4.4)

• Minimum variance predictor: )()1(ˆˆ

)(ˆ kyT

Fiku

T

DBEiky ii +−+=+

• Process: )()1()( kkuA

Bky ξ+−=

DTBADCBA ˆ,,ˆ,ˆ,,, →

)ˆˆ( ii

i EADTqF −=(2.3.3) =>

(2.4.1) and (2.4.3) => )()ˆˆ(

)1(ˆˆ

)(ˆ ikyT

EADTiku

T

DBEiky ii +

−+−+=+

(2.4.2) and (2.4.4) => )(ˆˆ

)1(ˆ

ˆˆˆ)()(ˆ ik

T

ADEiku

A

B

A

B

T

ADEikyiky ii +−−+

−++=+ ξ

modeling error

ˆ ˆˆ ˆˆˆ( ) ( ) ( ) ( 1) ( )

ˆi iE DA E DAB B

k i y k i y k i u k i k iT A TA

ε ξ

+ = + − + = − + − + + 14243

• Prediction error:

(2.4.6)

13.01.2014

15:22

2.4. i-step ahead

predictor properties 2. Prediction model

T polynomial effect over predictions

The prediction error is influenced by the modeling errors and disturbances, through the

polynomials and :

� if then and the disturbances with the modeling errors are filtered

by

� if and

� is a factor of =>

=> the modeling errors

=> i-step ahead predictors (2.3.9) and (2.3.24) are minimum variance predictors

ˆ T D

1=iE )(kξ1=iTAD /ˆˆ

DDddBBAAkeDA

Ck ˆ,ˆ,ˆ,ˆ),()( =====ξ TCC == ˆ

;)()( ikeEik i +=+⇒ ε { } 22)( eik σε =+E

D̂∆∀= 0stε

13.01.2014

15:22 2. Prediction model2.4. i-step ahead

predictor properties

k k+1 k+2 k+pk+d+1

u(k+i)

y(k)

k+d

past future

)(ˆ iky +

r

w(k+i)

rpkwdkwpkydkydpkuku →+++→+++−−+ )}(),..,1({)}(ˆ),..,1(ˆ{:)}1(),..,({

• i-step ahead predictor:

1;,..,1));1(()( +≥+=−−+=+ dppdidikufiky (2.5.1)

13.01.2014

15:22 2. Prediction model2.5. i-step ahead predictor for

processes with dead time

1

1

−+=

−−=+=

−

−−

+−−

dnn

dinHqGBE

BH

G

didi

dii

di

di

)()1()()()( 11 kekuqBqkyqA d +−= −−−• Process:

� C=D=1

(2.5.2)

• 0>An

(2.2.6) 0)(; =→ − keBqB d1),()1()( +≥+−−+=+ dikyFdikBuEiky ii (2.5.3)

(2.5.4)

future past

ˆ( ) ( 1) ( ) ( 1) , 1i d i i dy k i G u k i d F y k H u k i d− −+ = + − − + + − ≥ +144424443 144424443

(2.5.5)

• The i-steap ahead minumum variance predictor becomes:

13.01.2014



diidii HHGG −− →→ ;

• Matriceal form:

(2.5.6)

(2.5.7)

(2.5.8)

ΨsPuy +=ˆ[ ] [ ][ ]TBA

TT

dnkukunkyky

dpkukupkydky

)(,),1(),1(,),(

)1(,),()(ˆ,),1(ˆˆ

−−−+−=

−−+=+++=

KK

KK

s

uy

=

−−−− 021

01

0

0

00

ggg

gg

g

dpdp K

KKKK

K

K

P

−−−−−−−−−−−−−−−−−−−−−−−=

−+−−−−−

−+−

1,0,1,0,

1,10,11,10,1

dndpdpndpdp

dnn

BA

BA

hhff

hhff

KK

KK

Ψ

future past

ˆ( ) ( 1) ( 1) , 1i d i dy k i G u k i d H u k i d− −+ = + − − + − ≥ +144424443 14243

(2.5.10)(2.2.9)

• 0=An

(2.5.9)

13.01.2014



(2.5.11)

(2.5.12)

(2.5.13)

:1+≥⇒ di

• (2.3.9) : )()1()(ˆ kyC

Fiku

C

BDEiky ii +−+=+

⇒→>≠ DTBADCBAnd Aˆ,,ˆ,ˆ,,,,0,0

1),()1(ˆˆ

)(ˆ +≥+−−+=+ dikyT

Fdiku

T

DBEiky ii

1),max(

1ˆˆ

ˆˆ −++=

−−=+=

−

−−+−−

TDBH

Gdididi

i

ndnnn

din

T

HqG

T

DBE

di

di

(2.5.11) and (2.5.12)

ˆ( ) ( 1) ( ) ( 1)i i di d

F Hy k i G u k i d y k u k

C C

−−+ = + − − + + −

13.01.2014

15:22 2. Prediction model2.5.1. i-step ahead generalized

predictor for processes with dead time

future pastadditional term

ˆˆ ˆ( ) ( 1) ( ) ( 1) [ ( ) ( )]i

i d i i d

K Ay k i G u k i d F y k H u k y k y k

T− −+ = + − − + + − + −

144424443 144424443144424443

(2.5.14)

(2.5.15)

(2.5.16)

⇒

• (2.3.24) :

(2.5.14) şi (2.5.4)

[ ]ˆ

ˆˆ ˆ0 ( ) ( 1) ( ) ( ) ( )ii i

K Ad y k i E Bu k i d F y k y k y k

T≠ ⇒ + = + − − + + −

)1(ˆ

ˆ)(ˆ −=

−ku

A

Bqky

d

iK

•Matriceal form:

cKK

ΨsPuKΨsPuy

c~

1ˆ:ˆ

=

==+++=

DT

[ ]T

dkuBkyAkcnkckc

Tk

)1(ˆ)(ˆ)(~;)(~,),(~~ −−−=−= Kc

(2.3.44) cK⇒

does not depend on d→ Section 2.3.1

13.01.2014

15:22 2. Prediction model2.5.1. i-step ahead generalized

predictor for processes with dead time

(2.6.1)

(2.6.2)

pddd Mm ≤≤≤≤0

pdidikufky mm ,1)),1(()1( +=−−+=+

)()()(

)()1(

)(

)(~

)(11

1

1

1

keqAqD

qCku

qA

qBqky md

−−

−

−

−− +−=• Process model:

dndn

BB

qbqbbqB∆−−

∆+−− +++=

~~~)(

~ 110

1K (2.6.3)

(2.6.4)mM ddd −=∆

B~

0~b mm dddd bb −−−

~~1 1

~~+−+−+ mBmB ddnddn bb

mMB ddnb −+~

Bnb

. . . . . . . . .

B 0 . . . 0 b0

. . . 0 . . . 0

•

• the dead-time considered in (2.6.2) is and not

;∆)~(grad ~ dnnB BB

+== BnB =)(grad

md d

Observations:

13.01.2014


processes with unknown dead time

(2.6.5)

(2.6.6)

)1ˆ(, ==∆+→→ DTdnndd BBm• i-step ahead predictor:

1),()1(~

)(ˆ +≥+−−+=+ mimi dikyFdikuBEiky

1

1~

−+=

−−=+=

−−

+−−

−

MBdH

mG

didi

diidnn

dinHqGBE

mi

mdi

mm

m

• Matriceal form :

[ ] [ ]

[ ]T

dkuBkyAkcdnkukunkyky

dpkukupkydky

mTMBA

Tm

Tm

)-1-(ˆ-)(ˆ)(~;)--(,),1-(),1-(,),(

)1--(,),(;)(ˆ,),1(ˆˆ

=+=

+=+++=

KK

KK

s

uy

−−−−−−−−−−−−−−−−−−−−−−−=

−−−−−−−−−−−−=

−+−−−−

−+−

−−−− − 1,0,

1,10,11,10,1

021

01

0

1,0,

0

00

MBmmAnmm

MBA

mm

dndpdpdpdp

dnn

dpdphhff

hhff

ggg

gg

g

KK

KK

K

K

K

ΨP

ΨsPuy +=ˆ(2.6.7)1),1()()1()(ˆ +≥−++−−+=+ −− mdiimdi dikuHkyFdikuGiky

mm

13.01.2014


processes with unknown dead time

1−−dqR r(k) u(k) y(k+d+1) + + y(k)

ξ (k)

Controller Process

B

A +

- )()(1

1

krBRqA

BRqky

d

d

−−

−−

+=

(2.7.1)

1 − − d q R* r ( k ) u ( k ) y ( k + d +1) + y ( k )

ξ ( k )

B

A

B

A 1 − − d q

∧ + - y ( k )

y ( k + d +1)+c(k) ∧

∧

c(k) + +

+ +

-

y ( k + d +1) ∧

−−+

+=++

− )1(ˆ

ˆ)(

)(ˆ

ˆ)1(ˆ

kuA

Bqky

kuA

Bdky

d

� Smith predictor:

(2.7.2)

13.01.2014

15:22

2.7. Comparisons with

Smith predictor 2. Prediction model

(2.7.3)

�Minumum variance i-step ahead predictor with :

(2.7.4)

)()(ˆˆ

)1(ˆ11

kyT

Fku

T

DBEdky

dd ++ +=++

1+= di

AD

BFqBE

AD

TB ddd

ˆˆ

ˆˆ

ˆˆ

ˆ11

1+−−

+ +=(2.3.3)* B̂ 1+= di

AD

BFq

AD

TBBE dd

d ˆˆ

ˆ

ˆˆ

ˆˆ 111

+−−+ −=

1 − − d q R* r ( k ) u ( k ) y ( k + d +1) + y ( k )

ξ ( k )

B

A

B

A 1 − − d q

∧ + - y ( k )

y ( k + d +1) ∧

∧

+ +

+ +

-

T

Fd 1+

c(k)

(2.7.5)

(2.7.6)

−−+

+=++

−+ )1(ˆ

ˆ)(

)(ˆ

ˆ)1(ˆ

1 kuA

Bqky

T

F

kuA

Bdky

dd

13.01.2014

15:22 2. Prediction model2.7. Comparisons with

Smith predictor

model-based predictive control - project mbpc•mbpcis a control algorithm based on solving an...

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