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Multivariable Processes
Prof. Cesar de Prada ISA-UVA
prada@autom.uva.es
Prof. Cesar de Prada ISA-UVA 2
Outline
Interaction Control of multivariable processes using SISO
controllers RGA Control loop pairing Decoupling Control Multivariable Control
Prof. Cesar de Prada ISA-UVA 3
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
Prof. Cesar de Prada ISA-UVA 4
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
Prof. Cesar de Prada ISA-UVA 5
These cars
• Are they controllable independently?
• Does it exist interaction?
An example!!!
Interaction
Prof. Cesar de Prada ISA-UVA 6
Linked by a spring
Interaction
These cars
• Are they controllable independently?
• Does it exist interaction?
Prof. Cesar de Prada ISA-UVA 7
Rigid link
Interaction
These cars
• Are they controllable independently?
• Does it exist interaction?
Prof. Cesar de Prada ISA-UVA 8
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
Prof. Cesar de Prada ISA-UVA 9
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
Prof. Cesar de Prada ISA-UVA 10
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
Prof. Cesar de Prada ISA-UVA 11
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
Prof. Cesar de Prada ISA-UVA 12
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.
Prof. Cesar de Prada ISA-UVA 13
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
Prof. Cesar de Prada ISA-UVA 14
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?
Prof. Cesar de Prada ISA-UVA 15
Interaction
G11
G21
G12
G22 u2
u1
y1
y2
R1
R2
w1
w2
Control with SISO controllers R
Prof. Cesar de Prada ISA-UVA 16
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
++−
+=
−+
++
=
Prof. Cesar de Prada ISA-UVA 17
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
++−
+=
−+
++
=
Prof. Cesar de Prada ISA-UVA 18
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
+=
Prof. Cesar de Prada ISA-UVA 19
Interaction
TT
u1
q T
Condensate
Fv u2
FT
u2
u1 T q
Prof. Cesar de Prada ISA-UVA 20
Reactor
Reactor
TT
T
Coolant Product
u Reactant
AT
Input output interaction in both variables
Open loop interaction
Prof. Cesar de Prada ISA-UVA 21
Reactor
Reactor
TT
T
Refrigerante Producto
TC
u Reactante
AT
AC
Input - output interaction in both variables
Closed loop interaction
Prof. Cesar de Prada ISA-UVA 22
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
Prof. Cesar de Prada ISA-UVA 23
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.
Prof. Cesar de Prada ISA-UVA 24
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
Prof. Cesar de Prada ISA-UVA 25
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
Prof. Cesar de Prada ISA-UVA 26
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
−−−
−−
−
Prof. Cesar de Prada ISA-UVA 27
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
% %
% %
Prof. Cesar de Prada ISA-UVA 28
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)’
Prof. Cesar de Prada ISA-UVA 29
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
Prof. Cesar de Prada ISA-UVA 30
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
Prof. Cesar de Prada ISA-UVA 31
−−=
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.
Prof. Cesar de Prada ISA-UVA 32
Distillation Column
1
2
Prof. Cesar de Prada ISA-UVA 33
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
Prof. Cesar de Prada ISA-UVA 34
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
=−−
Prof. Cesar de Prada ISA-UVA 35
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
=−−
Prof. Cesar de Prada ISA-UVA 36
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
Prof. Cesar de Prada ISA-UVA 37
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
Prof. Cesar de Prada ISA-UVA 38
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
Prof. Cesar de Prada ISA-UVA 39
Head composition control with bottom impurity control in manual
IMC tuning λ=50 min. Kp = 2.19, Ti = 70.85 min.
Prof. Cesar de Prada ISA-UVA 40
Bottom impurity control with head composition control in manual
IMC tuning λ=35 min. Kp = -5.47, Ti = 45.2 min.
Prof. Cesar de Prada ISA-UVA 41
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
Prof. Cesar de Prada ISA-UVA 42
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
Prof. Cesar de Prada ISA-UVA 43
Head composition control with bottom impurity control in automatic
Switch from man to auto in the bottom impurity control loop
Process dynamics changes completely
Prof. Cesar de Prada ISA-UVA 44
RGA
−
−=
1.91.81.81.9
RGA
Prof. Cesar de Prada ISA-UVA 45
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
Prof. Cesar de Prada ISA-UVA 46
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
Prof. Cesar de Prada ISA-UVA 47
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
Prof. Cesar de Prada ISA-UVA 48
∆∆
+−
−+−
=
∆∆
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 +=
Prof. Cesar de Prada ISA-UVA 49
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
Prof. Cesar de Prada ISA-UVA 50
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
Prof. Cesar de Prada ISA-UVA 51
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
Prof. Cesar de Prada ISA-UVA 52
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
<
∏=
Prof. Cesar de Prada ISA-UVA 53
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
Prof. Cesar de Prada ISA-UVA 54
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
Prof. Cesar de Prada ISA-UVA 55
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
Prof. Cesar de Prada ISA-UVA 56
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
Prof. Cesar de Prada ISA-UVA 58
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
Prof. Cesar de Prada ISA-UVA 59
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
Prof. Cesar de Prada ISA-UVA 60
=
−−−
=
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
Prof. Cesar de Prada ISA-UVA 61
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
==
−=
Prof. Cesar de Prada ISA-UVA 62
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
Prof. Cesar de Prada ISA-UVA 63
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
Prof. Cesar de Prada ISA-UVA 64
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
Prof. Cesar de Prada ISA-UVA 65
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
Prof. Cesar de Prada ISA-UVA 66
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.
Prof. Cesar de Prada ISA-UVA 67
Interaction, Single loop PI
Controllers tuned independently with the other loop in manual
94.5 95
Prof. Cesar de Prada ISA-UVA 68
With decoupling
94.5 95
Same tuning
Prof. Cesar de Prada ISA-UVA 69
With decoupling
2.6 3
Prof. Cesar de Prada ISA-UVA 70
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
Prof. Cesar de Prada ISA-UVA 71
With SS decoupling
94.5 95
Same tuning
Prof. Cesar de Prada ISA-UVA 72
With SS decoupling
2.6 3
Prof. Cesar de Prada ISA-UVA 73
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.
Prof. Cesar de Prada ISA-UVA 74
Reactor FT
FT
FC
FC
TT AT
MBPC
Temp Conc.
Coolant
Product
u1 u2
Multivariable Predictive Control MPC
DMC, GPC, EPSAC, HITO, PFC,....
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