DYNAMIC MATRIX CONTROL

Presented by

Chinta Manohar

D Surya Suvidha

DYNAMIC MATRIX CONTROL - INTRODUCTION

Developed at Shell in the mid 1970’s

Evolved from representing process dynamics with a set of numerical coefficients

Uses a least square formulation to minimize the integral of the error/time curve

DYNAMIC MATRIX CONTROL - INTRODUCTION

DMC algorithm incorporates feedforward and multivariable control

Incorporation of the process dynamics makes it possible to consider deadtime and unusual dynamic behavior

Using the least square formulation made it possible to solve complex multivariable control calculations quickly

DMC BLOCK DIAGRAM

• At each time step control is computed by solving an openloop optimization problem for the prediction horizon

• First value of the computed control sequence is applied• At the next time step, new system is re-computed

DYNAMIC MATRIX CONTROL - EXAMPLE Consider the following furnace example

(Cutlet & Ramaker) MV (manipulated variable)

Fuel flow FIC DV(dynamic variable)

Inlet temperature TI CV(control variable)

Outlet temperature TIC

TICFIC

FpTi

Fuel

ProcessHeater

Process Flow

6

CONT… The furnace DMC

model is defined by its dynamic coefficients a and b

Response to step change in fuel, a

Response to step change in inlet temperature, b

658.0

653.0

640.0

622.0

590.0

540.0

465.0

340.0

240.0

0.0

,

986.0

949.0

904.0

836.0

736.0

600.0

414.0

214.0

086.0

014.0

ba

7

CONT… DMC Dynamic

coefficents Response to step

change in fuel, a Response to step

change in inlet temperature, b

Fuel Coeff icients ai

0

0.5

1

1.5

0 5 10

Fuel Coeff icients ai

Inlet Temperature Coefficients bi

0

0.5

1

0 5 10

Inlet TemperatureCoefficients bi

8

DYNAMIC MATRIX CONTROL – MATRIX COMPUTATION The DMC prediction may be calculated from those

coefficients and the independent variable changes

3

2

1

3

2

1

2121

123123

1212

11

3

2

1

00

0000

DV

DV

DV

MV

MV

MV

bbbaaa

bbbaaa

bbaa

ba

CV

CV

CV

CV

iiiiiii

9

DYNAMIC MATRIX CONTROL – FEED FORWARD PREDICTION Feedforward prediction is enabled by moving the

DV to the left hand side

3

2

1

21

123

12

1

13

2

1

3

2

1

0

00

MV

MV

MV

aaa

aaa

aa

a

DV

b

b

b

b

CV

CV

CV

CV

iiiii

10

CONT… Controller definition

Prediction horizon = 30 time steps

Control horizon = 10 time steps

InitializationSet CV prediction

vector to current outlet temperature

Calculate error vector

0

0

30

1

CV

CV

CV

CV

30

1

30

1

30

1

CV

CV

SP

SP

e

e

11

LEAST SQUARE APPROACH Least squares formulation including move

suppression

0

0

000

0

00

0

000

0

0

0

000

10

2

1

30

2

1

10,10

9,9

2,2

1,1

21282930

1

123

12

1

e

e

e

MV

MV

MV

aaaa

a

aaa

aa

a

Move suppression

12

DYNAMIC MATRIX CONTROLLER CYCLE

DMC Controller stepsCalculate moves using least square solutionUse predicted fuel moves to calculate changes to outlet

temperature and update predictionsShift prediction forward one unit in timeCompare current predicted with actual and adjust all 30

predictions (accounts for unmeasured disturbances)Calculate feedforward effect using inlet temperatureSolve for another 10 moves and add to previously

calculated moves

13

FURNACE RESPONSE Furnace Example Temperature

Disturbance DT=15 at t=0

Three Fuel Moves Calculated

Furnace Temperature Response

-15

-10

-5

0

5

10

15

0 5 10 15

Time

Te

mp

era

ture

575

600

625

Fuel_0Temp_0Fuel_1Temp_1Fuel_2Prediction

14

DYNAMIC MATRIX CONTROL-RULES Constrain max MV movements during each time

interval Constrain min/max MV values at all times Constrain min/max CV values at all times Drive to economic optimum Allow for feedforward disturbances

15

CONT… Restrict computed MV move sizes (move

suppression) Relative weighting of MV moves Relative weighting of CV errors (equal concern

errors) Minimize control effort

16

CONT… For linear differential equations the process output

can be given by the convolution theorem

k

j

kjjk

k dMVaCVCV0

101

001

01 dMVaCVCV 11

10

202 dMVaMVaCVCV

221

12

03

03 dMVaMVaMVaCVCV

17

CONT… Breaking up the summation terms into past and

future contributions

1Nk

kj

jjlk

lklk MVaCVCV

kkk MVaCVCV

1

11

112

22 kkkk MVaMVaCVCV2

31

23

33 kkkkk MVaMVaMVaCVCV

18

CONT… Let N=number future moves, M=time horizon to

reach steady state, then in matrix form

1

1

11

11

12

1

2

1

2

1 0

00

Nk

k

k

MMM

NMMM

NN

Mk

k

k

Mk

k

k

MV

MV

MV

aaa

aaa

aaa

aa

a

CV

CV

CV

CV

CV

CV

19

CONT… Setting the predicted CV value to its setpoint and

subtracting the past contributions, the “simple” DMC equation results

ΔMVAe

MkMkS

kkS

kkS

CVCV

CVCV

CVCV

22

11

20

CONT… Predicted response is determined from current

outlet temperature, predicted changes and past history of the MVs and DVs

Desired response is determined by subtracting the predicted response from the setpoint

Solve for future MVs -- Overdetermined systemLeast square criteria (L2 norm)Very large changes in MVs not physically

realizableSolved by introduction of move suppression

21

SCALING OF RESIDUALS To scale the residuals, a weighted least squares problem is

posed

• For example, the relative weights with two CVs

2

2

1

2

1eΔMVAW Min

2

2

2

1

1

1

2

1

0

0

w

w

w

w

w

w

W

22

DMC-RESTRICTION To restrict the size of

calculated moves a relative weight for each of the MVs is imposed

2

2

2

1

1

1

0

0

r

r

r

r

r

r

R

2

2

1

2

1

0

eΔMV

R

AWMin

23

CONT… Subject to linear constraints

The change in each MV is within a “step” bound

HI

HI

HI

Nk

k

k

LO

LO

LO

MV

MV

MV

MV

MV

MV

MV

MV

MV

1

1

I

24

CONT… Subject to linear constraints

Size of each MV step for each time interval

1

1

1

1

111

0

11

001

Nk

k

k

Nk

k

k

MV

MV

MV

MV

MV

MV

MV

MV

MV

MV

LI

25

CONT… Subject to linear constraints

MV calculated for each time interval is between high and low limits

MV

MV

MV

MV

MV

MV

MV

MV

MV

MV

MV

MV

MV

MV

MV

HI

HI

HI

Nk

k

k

LO

LO

LO

1

1

LI

26

CONT… Subject to linear constraints

CV calculated for each time interval is between high and low limits

HI

HI

HI

Nk

k

k

Mk

k

k

LO

LO

LO

CV

CV

CV

MV

MV

MV

CV

CV

CV

CV

CV

CV

1

12

1

A

27

LINEAR PROGRAM SOLUTION

The following LP subproblem is solved where the economic weights are know a priori

HILO

HILO

CVMVgMVgMVgCV

CVMVgMVgMVgCV

2*

323*

222*

1212

1*

313*

212*

1111

*33

*22

*11 MVMVMV Minimize

Subject to:

28

MODIFIED DMC The original dynamic matrix is modified

Aij is the dynamic matrix of the ith CV with respect to the jth MV,

3*

3

2*

2

1*

1,

100

010

001

MVMV

MVMV

MVMV

t

t

t

2

1

232221

131211

e

e

e

AAA

AAA

A

29

DMC TUNING RULES Approximate the process dynamics of all

controller output to measured process variables

30

CONT… Select the sampling time as close as possible to

Compute the prediction horizon, P and model horizon, N

31

CONT... Compute the control horizon , M

Select the controlled variable weights to scale process variable units to be the same

Compute the move suppression coefficients

32

CONT…

• Implement DMC using the traditional step response of the actual process and the initial values of parameters

33

RESPONSE OF DMC Performance of DMC using tuning rules T=100, P=N=198, λ=3.53

34

REFERENCES

• Tuning guidelines for DMC- Danielle Dougherty and Douglas J. Cooper-Ind. Eng. Chem. Res. 2003, 42, 1739-1752

• http://web.stanford.edu/class/archive/ee/ee392m/ee392m.1056/Lecture14_MPC.pdf

• www.che.utexas.edu/course/che360/lecture_notes

• tx.technion.ac.il/~dlewin/054414/LECTURE_12.pdf

35

THANK YOU