Multivariable Processesprada/multivariablesUK.pdf · Control of multivariable processes using SISO controllers RGA Control loop pairing Decoupling Control Multivariable Control

Multivariable Processes

Prof. Cesar de Prada ISA-UVA

[email protected]

Prof. Cesar de Prada ISA-UVA 2

Outline

Interaction Control of multivariable processes using SISO

controllers RGA Control loop pairing Decoupling Control Multivariable Control


v1

Hot Oil

v2

v3

L1

v7

v5 v6

Hot Oil

F1 T1 T3

T2

F2

T4T5

F3 T6

T8

F4

L1

v8

T7

P1F5

F6T9

How to determine the maximum number of variables that can be controlled in a process?

Degrees of freedom


A basic requirement : Number of valves (actuators) ≥ number of controlled variables

v1

Hot Oil

v2

v3

L1

v7

v5 v6

Hot Oil

F1 T1 T3

T2

F2

T4T5

F3 T6

T8

F4

L1

v8

T7

P1F5

F6T9

This is a necessary but not sufficient condition in order to satisfy the control aims!

Degrees of freedom


These cars

• Are they controllable independently?

• Does it exist interaction?

An example!!!

Interaction


Linked by a spring

Interaction

These cars




Rigid link

Interaction

These cars




Reactor

Reactor

TT

T

Coolant Product

Input output interaction in both variables

Open loop interaction / Not necessarily equal to closed loop interaction

u Reactant

AT


MIMO (Multi Input Multi Output) Systems

Process u1

u2

y1

y2

y3

=

)s(U)s(U

)s(G)s(G)s(G)s(G)s(G)s(G

)s(Y)s(Y)s(Y

2

1

3231

2221

1211

3

2

1Interaction

Directions

Independent Manipulated variables

Dependent Controlled variables


A process is said to be controllable/operable if the controlled variables can be kept in its set points in

steady state, in spite of the disturbances acting on the plant

Mathematically, for a process to be controllable, the gain matrix of the process should be able to be inverted, that is, its determinant should be K ≠ 0.

DKK

MVMV

KKKK

CVCV

2d

1d

2

1

2221

1211

2

1

+

=

Model of a 2x2 process

Controlability/Operability


A process is said to be controllable/operable if the controlled variables can be kept in its set points in

steady state, in spite of the disturbances acting on the plant

If the determinant K ≠ 0, then we can find values of the MV’s that maintain the CV’s on spite of the value of the DV’s. (Assuming they remain within the appropriate range).

−

=

−

DKK

CVCV

KKKK

MVMV

2d

1d

2

11

2221

1211

2

1Model of a 2x2 process

Controlability/Operability


FA, xA

FS, xAS = 0 FM, xAM

SssAs

AAA

ssAs

AAAM

SA

AAAM

SAMSAM

FFFxFF

FFFxx

FFxFx

FFFFFF

∆

+

−+∆

+

−=∆⇒

+=

∆+∆=∆⇒+=

22 )()()1(

In this blending process

• Can FM and xAM be controlled independently?

• Is there interaction among the process variables ?

Controlability

xA = cte.


FA, xA

FS, xAS = 0 FM, xAM

0)()(

)1()(

)( 222 ≠+−

=+−

−+

−=

SA

A

SA

AA

SA

AA

FFF

FFxF

FFxFKDet

Yes, the process is controllable!

Would it be controllable if xAS were different from zero?

∆∆

+

−

+

−=

∆∆

S

A

ssAs

AA

ssAs

AA

AM

M

FF

FFxF

FFFx

xF

22 )()()1(

11


Interaction

G11

G21

G12

G22 u2

u1 y1

y2

=

=

)s(U)s(U

)s(G)s(G)s(G)s(G

)s(Y)s(Y

2

1

2221

1211

2

1

Open loop

How does it behave in closed loop?


Interaction

G11

G21

G12

G22 u2

u1

y1

y2

R1

R2

w1

w2

Control with SISO controllers R


Closed loop

G11

G21

G12

G22 u2

u1

y1

y2

R1

R2

w1

w2

)yw(RG)yw(RGuGuGy

)yw(RG)yw(RGuGuGy

2222211121

2221212

2221211111

2121111

−+−==+=

−+−==+=

2222

22211

222

1212

22111

2121

111

1111

wRG1

RG)yw(RG1

RGy

)yw(RG1

RGwRG1

RGy

++−

+=

−+

++

=


Interaction

2121212222111

2121

121212222111

1212122221111

2222

22211

222

1212

111

2121

111

1111

wRGRG)RG1)(RG1(

RGwRGRG)RG1)(RG1(

RGRG)RG1(RGy

)wRG1

RG)yw(RG1

RGw(RG1

RGwRG1

RGy

−+++

−++−+

=

+−−

+−

++

+=

G11

G21

G12

G22 u2

u1

y1

y2

R1

R2

w1

w2

2222

22211

222

1212

22111

2121

111

1111

wRG1

RG)yw(RG1

RGy

)yw(RG1

RGwRG1

RGy

++−

+=

−+

++

=


Interaction (Loop 1)

2121212222111

2121

121212222111

1212122221111 w

RGRG)RG1)(RG1(RGw

RGRG)RG1)(RG1(RGRG)RG1(RGy

−+++

−++−+

=

G11

G21

G12

G22 u2

u1

y1

y2

R1

R2

w1

w2

w1 y w2 affect y1

If G12 or G21 are = 0 the closed loop dynamics is the one of a SISO system u1 --- y1

If R2 is commuted to manual the dynamics of loop 1 changes

2222111

2121

111

1111 w

)RG1)(RG1(RGw

)RG1(RGy

+++

+= 1

111

1111 w

)RG1(RGy

+=


Interaction

TT

u1

q T

Condensate

Fv u2

FT

u2

u1 T q


Reactor

Reactor

TT

T

Coolant Product

u Reactant

AT

Input output interaction in both variables

Open loop interaction


Reactor

Reactor

TT

T

Refrigerante Producto

TC

u Reactante

AT

AC

Input - output interaction in both variables

Closed loop interaction


Interaction

How to measure the degree of interaction? Is is possible to control the process using SISO

controllers? If so, which is the best pairing of input – output

variables?

u1

u2 y1

y2


Steady state gain matrix

=

2

1

2221

1211

2

1

uu

kkkk

yy

The steady state gain matrix is not a good measure of interaction:

It depends on the units of the different variables

It does not reflect the main characteristic associated to interaction: the change in gain in a control loop when other loop switch from auto to man or vice versa.


Bristol Relative Gain Array (RGA) Bristol 1966 McAvoy 1983

2221

1211

2

1

21

yy

uu

λλλλ

G

u1

u2

y1

y2

λ11

G

u1

u2

y1

y2

λ11 measures the change in gain between u1 and y1 in the experiments described below

im;cteyj

i

jk;cteuj

i

j,i

m

k

uy

uy

≠∀=

≠∀=

∂∂

∂∂

=λ


Best choice 1j,i =λ

0j,i =λ

∞=λ j,i 2.08.0y8.02.0y

uu

2

1

21

RGA

The RGA can be used to choose adequately the pairing of manipulated and controlled variables in MIMO systems, selecting those pairs with minimum interaction in steady state (or at any other frequency).

im;cteyj

i

jk;cteuj

i

j,i

m

k

uy

uy

≠∀=

≠∀=

∂∂

∂∂

=λ

0j,i <λ Instability


RGA

G

u1

u2

y1

y2

λ11

2221212

2121111

ukuk0yukuky

∆+∆==∆∆+∆=∆

21122211

2211

22

21122211

1111

22

21122211

ctey1

1

122

21121111

kkkkkk

kkkkk

k

kkkkk

uy

uk

kkuky

2

−=

−=λ

−=

∆∆

∆−∆=∆

=

21122211

2211

21122211

2112

21122211

2112

21122211

2211

2

1

21

kkkkkk

kkkkkk

kkkkkk

kkkkkk

yy

u u

−−−

−−

−


Example: Distillation Column

V

L

x1

x2

−

−9.99,89,89,9

xx

V L

2

1

Strong interaction associated to the pairing ( L x1) (V x2)

Instability with ( L x2) (V x1)

RGA

−−

=35.038.082.099.0

)0(G

% %

% %


RGA

T1)G(G)G()G(RGA −×=Λ=

=×=Λ

−

=

−=

−

−

4.06.06.04.0

)G(G)G(

1.03.02.04.0

G

4321

G

T1

1

Sum of elements of a RGA row or column is 1

RGA does not depend on the units or scaling of u and y

When dealing with asymmetric processes, the inverse matrix can be substituted by the pseudoinverse

Matlab RGA = G.*pinv(G)’


Example

−

−

=×=Λ

=

=

−

04.004.1014.005.091.0

yy

)G(G)G(

u u u 0.08-0.140.260.02-0.35-0.46

pinv(G) uuu

5.040132

yy

2

1T1

321

3

2

1

2

1

y1 must be paired with u1

y2 must be paired with u2


Example RGA

−−=

40.51.5427.141.10.317.163.45.308.16

G

−−−

−

03.295.008.045.097.041.048.199.050.1

yyy

u u u

3

2

1

321

The only admissible SISO pairing is:

y1 ---- u1 y2 ---- u2 y3 ---- u3

With a higher interaction in the third loop

RGA

RGA


−−=

40.51.5427.141.10.317.163.45.308.16

G

−−−

−

03.295.008.045.097.041.048.199.050.1

yyy

u u u

3

2

1

321

RGA

im;cteyj

i

jk;cteuj

i

j,i

m

k

uy

uy

≠∀=

≠∀=

∂∂

∂∂

=λ

Notice that, if λ > 1, when changing from auto to man, the resulting gain will be larger than before and, likely, the loop will tend to oscillate. By the contrary, if λ < 1, will provide a slower response.


Distillation Column

1

2


Open loop experiment

50 55

1s60e648.0

1seKG

s7.21

11

sd11

11

11

+=

+τ=

−−

1s7.84e815.0

1seKG

s4.34

21

sd21

21

21

+=

+τ=

−−

1

2


Open loop experiment

47.9 53

1s3.54e894.0

1seKG

s6.21

12

sd12

12

12

+−

=+τ

=−−

1s9.41e236.0

1seKG

s61.6

22

sd22

22

22

+−

=+τ

=−−


Open loop Model

FOPD Step response models

Both loops open

1s60e648.0

1seKG

s7.21

11

sd11

11

11

+=

+τ=

−−

1s7.84e815.0

1seKG

s4.34

21

sd21

21

21

+=

+τ=

−−

1s3.54e894.0

1seKG

s6.21

12

sd12

12

12

+−

=+τ

=−−

1s9.41e236.0

1seKG

s61.6

22

sd22

22

22

+−

=+τ

=−−


Watch the experiment!

1s8.72e99.0

1seKG

s7.22

11

sd11

11

11

+=

+τ=

−−

Change of 0.5%

1s65.66e38.0

1seKG

s9.30

21

sd21

21

21

+=

+τ=

−−

In non-linear systems, smaller input changes can provide better results

50 51


Plan well the experiment

1s67.66e82.0

1seKG

s36.22

12

sd12

12

12

+−

=+τ

=−−

1s02.57e35.0

1seKG

s5.4

22

sd22

22

22

+−

=+τ

=−−

47.9 49


RGA

−−

=236.0815.0894.0648.0

K

−

−=

265.0265.1265.1265.0

RGA

−

−=

1.91.81.81.9

RGA

−−

=35.038.082.099.0

K


Head composition control with bottom impurity control in manual

IMC tuning λ=50 min. Kp = 2.19, Ti = 70.85 min.


Bottom impurity control with head composition control in manual

IMC tuning λ=35 min. Kp = -5.47, Ti = 45.2 min.


Control loop interaction

CONTROLProcess 1

INTERACT 12PV1

response toCO2

INTERACT 21PV2

response toCO1

CONTROLProcess 2

PROCESS 11PV1

response toCO1

PROCESS 22PV2

response toCO2

+- ++

+- ++

1y

2y

1setpointy

2setpointy

1u

2u


Open loop response G between 1-1 with the bottom impurity control in automatic

Dynamics has changed completely! Gain G11 changed from 0.99 to 0.11 and the type of response is different

λ11 = 0.99/0.11 = 9

50 51


Head composition control with bottom impurity control in automatic

Switch from man to auto in the bottom impurity control loop

Process dynamics changes completely


RGA

−

−=

1.91.81.81.9

RGA


Open loop response G22 with the head composition control in automatic

Dynamics has changed completely! Gain G22 changed from -0.35 to -0.043and the type of response is different

λ22 =

-0.35/(-0.043)

= 8.1

47.9 49


Bottom impurity control with head composition control in automatic

Switch from man to auto in the head composition control loop

Dynamics has changed completely

Example: Mixing two streams


AT FT

F , x

F1 , x1

F2 , x2

u1

u2 21 FFF +=

Global balance:

Composition balance:

2211 x Fx Fx F +=

2ss

221

2111

ss2

21

212

2221

221122112

21

2211121

2121

2211

F)F F(

)xx( FF)F F(

)xx(Fx

F)F F(

)x Fx F(x)F F(F)F F(

)x Fx F(x)F F(x

FFF F Fx Fx Fx

∆+−

−∆+−

=∆

∆+

+−++∆

++−+

=∆

∆+∆=∆++

=

Linearization in a certain steady state point

Example: Mixing two streams


∆∆

+−

−+−

=

∆∆

2

1

ss2

21

211

ss2

21

212

FF

11)F F(

)xx( F)F F(

)xx(F

Fx

Steady state gain matrix

Eliminating F2 between both equations:

2

211 xx

xx . F F−−

= FF

xxxx

xxxx

1 1

21

2

2

21F,F 1

=−−

=

−−

=λ

2211222121112211121112 xFxFxFxFxFxFFxxFxFFxxFxFFxFx +=++−=+−=⇒−=−

21 FFF += 2211 x Fx Fx F +=


Mixing two streams

RGA :

Which is the best pairing between manipulated and controlled variables? What influences the answer?

1F 2F

FF1

FF1

FF11−

FF11−

F

x

1F 2F


Mixing two streams

Case F1 = 10, F = 15 1F 2F

FF1

FF1

FF11−

FF11−

F

x

1F 2F

Case F1 = 3, F = 15

1F 2F

2.0F

x

1F 2F

8.0

2.08.0

1F 2F

67.0F

x

1F 2F

67.0

33.0

33.0


RGA(jω)

))j(G(RGA ω

RGA was originally formulated for the steady state case (zero frequency), but the same concept can be applied to the operation of the process at any other frequency to measure the interaction during transients.

Niederlinski stability theorem


Let’s assume that inputs and outputs of a multivarible system have been ordered so that y1 is controlled with u1, y2 is controlled with u2, etc. and each pair is controlled with a regulator having integral action, then, the closed loop is unstable if:

0)0(G

))0(Gdet(n

1iii

<

∏=


Singular values

selected areGGy GG of seigenvaluelargest m)min(l,k the m),G(l fi

)GG()GG()G(

**

*i

*ii

=

×

λ+=λ+=σ

)G()G()G( i σ≤λ≤σ

Eigenvalues of G:

)G()G( number condition )G(max )G(min ii σ

σ=σ=σσ=σ

)G(max)G( ii

λ=ρSpectral radious:

How to compute the eigenvalues of a non-square matrix? Singular values

0IG =λ−

)G(1)G( 1−σ=σ


SVD

[ ]

σ

σ

=Σ=

∗

∗

∗

∗

m

2

1

v...vv

000000000...0000

u...uuVUGn

1

l21

1v v v1u uu *VV UUals)(orthonorm matricesunitary )mm(V),ll(U

2iijji2iijji1-*1- =δ==δ===

××∗∗

Columns of U: ui output singular vectors: unitary eigenvectors of GG*

Columns of V: vi input singular vectors: unitary eigenvectors of G*G


SVD

iii uGvUVVUGVVUG

σ=Σ=Σ=⇒Σ=

∗∗

One input in the direction vi gives an output in the direction ui

σi gives the gain of G in the direction vi

2i

2i2ii2ii2ii

2i2i

vGv

Gv)G(uu

1u,1v sA

==σ=σ=σ

==

G vi ui

Directions computed using SVD are orthogonals


Interaction using SVD

Opposite to the RGA, the SVD depend on the scaling of the matrix, so, in oreder to obtin sensibles results, the G matrix should be scaled to units used in the controller

First column of V (row of V*) provides the combination of controller moves with the largest effect on the controlled variables, these ones changing in the direction given by the first column of U. The second column of V provides the second largest effect, etc.These gains are given by the singular values σi

∗Σ= VUG

Pairing variables with SVD

Prof. Cesar de Prada ISA-UVA 57 2output and 2input toingcorrespond ones thelueslargest va thebeing

column) (secondgain largest second with therepeated is procedure Then the1.output - 1input is pairing drecommende theso 1,output

withassociated 0.872,- is luelargest va theoutput, in the and 1input

with associated is 0,794- luelargest va theinput, in the 0.490-0.872-

direction

output theand 0.608-0.794-

direction input ebetween th appearsgain largest

794.0608.0608.0794.0

272.000343.7

872.0490.0490.0872.0

VUG

2345

G

−−−

−

−−=Σ=

=

∗∗

G u y

Selecting variables with Cn


G u y

)G()G( Cnumber condition n σ

σ==

Multivariable systems with large condition number presents pairs with very large or very small gains that will make difficult the design of a good control system. So, may be that only a subset of the inputs and outputs should be selected for control. This subset can be selected using the Cn

Example: Mixing streams


AT FT

F , x

F1 , x1

F2 , x2 u2

For F1 = 3, F2 = 2 x1 = 0.7 x2 = 0.2 The steady state is : F 5, x = 0.5

−=

+−

−+−

=11

25/5.125/1

11)F F(

)xx( F)F F(

)xx(F)0(G

ss2

21

211

ss2

21

212

−−−

=

−0.70680.7075-0.7075-0.7068-

0707.0001.4143

01.09999.09999.001.0

)1106.004.0

(svd

Cn = 1.4143/0.0707 = 20.0

u1

=

−−−

=

0.70680.7075-0.7075-0.7068-

V

01.09999.09999.001.0

U

=

−−−

=

0.70680.7075-0.7075-0.7068-

V

01.09999.09999.001.0

U

Example: mixing streams


=

−−−

=

0.70680.7075-0.7075-0.7068-

V

01.09999.09999.001.0

U

AT FT

F , x

F1 , x1

F2 , x2 u2

Largest singular value 1.4143

Second largest singular value 0.0707

x controlled with F2

F controlled with F1

u1

Example: mixing streams


AT FT

F , x

F1 , x1

F2 , x2 u2

=

−=

6.04.04.06.0

RGA

1106.004.0

)0(G

u1

RGA analysis recommends the same pairing: (but not always)

x controlled with F2

F controlled with F1

25.01*04.0

det(G(0))index kiNiederlins

1106.004.0

)0(G

==

−=


MULTILOOP vs Centralized

L

F

T

A

Multiloop: several independent PID controllers

One single multivariable

controller

L

F T

A

Controlador

Centralizado

Designing control systems in multivariable processes


Decoupling

G

u1 y1

y2

R1 w1

R2 w2

u2

D

Find a matrix D such that GD behaves as a diagonal (or quasi diagonal) matrix, so that the interaction is cancelled


Steady state decoupling

G

u1 y1

y2 G(0)-1

R1 w1

R2 w2

u2

If D is chosen as αG(0)-1 ,then G(s) G(0)-1 is diagonal in steady state, so that there is no interaction at equilibrium. This decoupler is very easy to compute and implement because G(0)-1 = inverse of the steady state gain matrix


Control structure with Decoupling

CONTROLProcess 1

CONTROLProcess 2

DECOUPLER 12PV1

decoupled fromCO2

DECOUPLER 21PV2

decoupled fromCO1

+- ++ ++

+- ++ ++

1feedbacku

2feedbacku

1totalu

2totalu

1y

2y

1setpointy

2setpointy

)(12 sD

)(21 sD

)(11 sG

)(21 sG

)(12 sG

)(22 sG

2decoupleu

1decoupleu

PROCESS 11PV1

response toCO1

INTERACT 21PV2

response toCO1

PROCESS 22PV2

response toCO2

INTERACT 12PV1

response toCO2


2x2 Multivariable Decoupling

It requires 4 dynamical models: – Process 11 (how CO1 influences PV1) – Interact 12 (how CO2 influences PV1) – Interact 21 (how CO1 influences PV2) – Process 22 (how CO2 influences PV2)

These models must be obtained from validated

experimental data. Loop decoupling is not use very often because it

requires a certain effort in terms of modelling, tuning and maintenance, and MPC can provide better performance spending similar resources.


Interaction, Single loop PI

Controllers tuned independently with the other loop in manual

94.5 95


With decoupling

94.5 95

Same tuning


With decoupling

2.6 3


SS Decoupling

−−

=

−−

=−

−

37.2888.1049.2303.10

35.038.082.099.0

)0(G1

1

G

u1 y1

y2

R1 w1

R2 w2

u2

−−

37.2888.1049.2303.10


With SS decoupling

94.5 95

Same tuning


With SS decoupling

2.6 3


Multivariable Control

G

u1 y1

y2 w1

R w2 u2

The controller receives signals from all controlled variables (and perhaps measurable disturbances) and computes simultaneously control actions for all actuators taking into account the interactions.


Reactor FT

FT

FC

FC

TT AT

MBPC

Temp Conc.

Coolant

Product

u1 u2

Multivariable Predictive Control MPC

DMC, GPC, EPSAC, HITO, PFC,....

Documents

Multivariable Processesprada/multivariablesUK.pdf · Control of multivariable processes using SISO controllers RGA Control loop pairing Decoupling Control Multivariable Control