PART - 4
KNU/EECS/ELEC 835001
Multivariable Control
for MIMO processes
Multivariable Control
Dr. Kalyana C. Veluvolu
Outline - Module 5.4
Decoupler Design for MIMO processes
– Ideal Decoupler
» Simplified Decoupler
» Generalized Decoupler
– Limitation of Decoupler
– Simpler Decoupling
» Partial Decoupling
» Steady-state Decoupling
– Effects of MV Constraints
– Ill Conditioned Process
» Degeneracy
» Singular Value Decomposition
» Decoupling Based on SVD
Multivariable Control
Dr. Kalyana C. Veluvolu
2
Multi-loop vs. Multivariable Control
•
multi-loop - use of several single-loop controllers (e.g., PID) on
pairs of manipulated/controlled variables
•
multivariable - make control adjustments decisions jointly
considering all outputs simultaneously
•
Multi-loop control configurations are typically used as a base
control configuration and reside in the Distributed Control
System (DCS).
» e.g., flow control, temperature control, pressure control
Multivariable control configurations typically require additional
computational capability, and sit over a base multi-loop control
configuration, sending setpoints to the multi-loop controllers.
Multivariable Control
Dr. Kalyana C. Veluvolu
3
Multi-loop vs. Multivariable Control
Under the multi-loop control strategy, each controller gci operates according to:
u i = g ci ( y di − y i ) = g ci ε i
u1 = f 1 (ε 1 , ε 2 , ε n )
u 2 = f 2 (ε 1 , ε 2 ,ε n )
u 3 = f 3 (ε 1 , ε 2 ,ε n )
Multivariable Control
y1,sp
y2,sp
G11(s)
y1
G21(s)
Multivariable
Controller
G12(s)
u2
Dr. Kalyana C. Veluvolu
G22(s)
y2
+
+
=
u n = f n (ε 1 , ε 2 , ε n )
u1
++
Multivariable controller must decide on ui, not using only εi, but using the
entire set, ε1, ε2,, ...,εn,;. Thus, the controller actions are obtained from
4
Principles of Decoupling
Main loop y1 — u1 , y2 — u2,…, yn — un, couplings
desirable for control
Cross-couplings, yi — uj (i ≠ j)
undesirable; loop interactions
Eliminates the effect of the undesired cross-couplings
improve control performance.
Objective is to compensate for interactions by cross-couplings
not to “eliminate” the cross-couplings; impossibility, require
altering the physical nature of the system.
Multivariable Control
Dr. Kalyana C. Veluvolu
5
Simplified Decoupling
Two compensator blocks gI1 and gI2.
Controller outputs v1 and v2, actual control on the process u1 and u2.
Without the compensator,
u1 = v1 and u2 = v2, and
the process model
y1 = g11u1 + g12 u 2
y1 = g 21u1 + g 22 u 2
Compensator, Loop 2
“informed” of changes in
v1 by g12, u2 is adjusted.
The same for Loop 1
Multivariable Control
Dr. Kalyana C. Veluvolu
6
Design Simplified Decoupler
y1 = g 11u1 + g 12 u 2
y 2 = g 21u1 + g 22 u 2
⇒
g I1 = −
⇒
u1 = v1 + g I 1v 2
u 2 = v 2 + g I 2 v1
⇒
y1 = ( g 11 + g 12 g I 2 )v1 + ( g 12 + g 11 g I 1 )v 2
y 2 = ( g 21 + g 22 g I 2 )v1 + ( g 22 + g 21 g I 1 )v 2
g 12
g 11
gI2 = −
y1 = ( g11 −
g 12 g 21
)v1
g 22
y 2 = ( g 22 −
g12 g 21
)v 2
g 11
Multivariable Control
g 21
g 22
Dr. Kalyana C. Veluvolu
7
Difficulties for Simplified Decoupler
larger than 2 x 2, decoupling become tedious.
3x3 , six compensator.
NxN: (N2-N) compensators.
Multivariable Control
Dr. Kalyana C. Veluvolu
8
Generalized Decoupling
MIMO process
u = GI v
y =Gu
⇒ y = GG I v
To eliminate interactions, y to v : a diagonal matrix; GR(s).
GG I = G R (s )
⇒
y = G R ( s) v
Choose GI such that
G I = G −1GR ( s)
Selected to provide desired decoupled behavior with the simplest form
A commonly employed choice
GR ( s) = Diag[G( s)]
Multivariable Control
Dr. Kalyana C. Veluvolu
9
Relation Between the Two Schemes
Simplified decoupling
y = GG I v
2 x 2 and 3 x 3 system, the compensator transfer function matrix:
 1
GI = 
g I 2
 1
G I =  g I 21
 g I 31
g I1 
1 
For the desired GI , task is to find
g Iij
g I 12
1
g I 32
g I 13 
g I 23 
1 
to make GGI diagonal
General decoupling
Final diagonal form GGI specified as GR, then GI can be derived.
Multivariable Control
Dr. Kalyana C. Veluvolu
10
Example Distillation Column
− 18.9e −3s 

21.0s + 1 
− 19.4e −3 s 
14.4s + 1 
 12.8e − s

G ( s) = 16.7 s−+7 s1
 6.6e
10.9s + 1
simplified decoupler
(16.7 s + 1)e −2 s
g I 1 = 1.48
21.0s + 1
gI2
(14.4s + 1)e −4 s
= 0.34
10.9s + 1
actual implementation
(16.7 s + 1)e −2 s
u1 = v1 + 1.48
v2
21.0 s + 1
(14.4 s + 1)e −4 s
u 2 = v 2 + 0.34
v1
10.9 s + 1
Multivariable Control
Dr. Kalyana C. Veluvolu
11
Example Distillation Column
Generalized decoupling:
 12.8e − s

G R ( s ) = 16.7 s + 1

0

∆=

0

−3 s 
− 19.4e 
14.4 s + 1 
 − 19.4e −3 s
1
G −1 ( s ) =  14.4 s +−17 s
∆  − 6.67e
 10.9 s + 1
18.9e −3 s 

21.0 s + 1
−s
12.8e 
16.7 s + 1 
g
G I =  I 11
 g I 21
g I 12 
g I 22 
− 248.32(21.0s + 1)(10.9s + 1)e −4 s + 124.74(16.7 s + 1)(14.4s + 1)e −10 s
(21.0s + 1)(10.9s + 1)(16.7 s + 1)(14.4s + 1)
g I 11 =
− 248.32( 21.0 s + 1)(10.9 s + 1)
124.74(16.7 s + 1)(14.4s + 1)e −6 s − 248.32(21.0s + 1)(10.9s + 1)
− 366.66(16.7 s + 1)(10.9s + 1)e − 2 s
124.74(16.7 s + 1)(14.4s + 1)e −6 s − 248.32(21.0s + 1)(10.9 s + 1)
84.48(21.0s + 1)(14.4s + 1)
=
124.74(16.7 s + 1)(14.4 s + 1)e −6 s − 248.32(21.0s + 1)(10.9s + 1)
g I 12 =
g I 21
The actual implementation:
Multivariable Control
g I 22 = g I 11
u1 = g I 11v1 + g I 12 v 2
u 2 = g I 21 v1 + g I 22 v 2
Dr. Kalyana C. Veluvolu
12
Comparison of the Two Methods
Simplified decoupling: “equivalent” open-loop decoupled system

 12.8e − s
g12 g 21 
18.9 × 6.6(14.4 s + 1)e −7 s 
v1 = 
v1
y1 =  g11 −
−
g
s
s
s
(
16
.
7
1
)
19
.
4
(
21
.
0
1
)(
10
.
9
1
)
+
+
+
22





 − 19.4e −3 s 18.9 × 6.6(16.7 s + 1)e −9 s 
g12 g 21 
v 2 = 
v 2
y 2 =  g 22 −
−
g
(
14
.
4
s
1
)
12
.
8
(
21
.
0
s
1
)(
10
.
9
s
1
)
+
+
+


11


much more complicated than GR specified in the Generalized decoupling
Difficult to tune controller
Generalized decoupling:
tuning and performance better than for Simplified decoupling
complicated decoupler
Multivariable Control
Dr. Kalyana C. Veluvolu
13
Limitations in Application
Perfect decouple if model perfect - impossible in practice.
The simplified decoupling similar to feedforward controllers
realization problems, time delay elements
Perfect dynamic decouplers based on model inverses.
can only be implemented if inverses causal and stable.
2 x 2 compensators, gI1 and gI2 must be causal (no e+αs terms) and stable
time delays in g11 smaller than time delays in g12
time delays in g22 smaller than time delays in g21
g11 and g22 no RHP zeros
g12 and g21 must no RHP poles
Multivariable Control
Dr. Kalyana C. Veluvolu
14
Implementation
Adding delays to the inputs u1, u2, ..., un, by define: G m = GD
e − d11s

D( s) = 


 0
e − d 22 s
0 




e − d nn s 
Simplified decoupling: requiring the smallest delay in each row on the diagonal,
designed by using Gm.
Generalized decoupling: use modified process Gm so that GI=(GD)-1GR are
causal which requiring that GR-1(GD) have the smallest delay in each row on the
diagonal.
Multivariable Control
Dr. Kalyana C. Veluvolu
15
Example: Distillation Column
add a time delay of 3 minutes to the input u1:
 12.8e −4 s

G ( s ) =  16.7 s −+101s
 6.67e
 10.9 s + 1
− 18.9e −3s 

21.0 s + 1 
− 19.4e −3 s 
14.4 s + 1 
Smallest delay in each row is not on diagonal,
simplified decoupling compensator becomes:
(16.7 s + 1)e s
g I 1 = 1.48
21.0 s + 1
Design D(s) to add a time delay of 1 minute to the input u2, i.e.:
1 0 
D( s) = 
−s 
0 e 
 12.8e

Gm = GD =  16.7 s −+101s
 6.67e
 10.9 s + 1
−4 s
Multivariable Control
− 18.9e 

21.0s + 1 
− 19.4e − 4 s 
14.4 s + 1 
−4 s
Dr. Kalyana C. Veluvolu
(16.7 s + 1)
21.0s + 1
(14.4s + 1)e −6 s
= 0.34
10.9s + 1
g I 1 = 1.48
gI2
16
Example: Distillation Column
 12.8e −4 s

G ( s ) =  16.7 s −+101s
 6.67e
 10.9s + 1
− 18.9e −3s 

21.0s + 1 
− 19.4e −3s 
14.4s + 1 
g I 1 = 1.48
(16.7 s + 1)e s
21.0 s + 1
As time prediction term much small than time constant, drop prediction
g I 1 = 1.48
(16.7 s + 1)
21.0 s + 1
Effective time constant of g12 and g11 are similar
16.7 + 4 ⇔ 21 + 3
Steady-state decoupling
g I 1 = 1.48
Multivariable Control
Dr. Kalyana C. Veluvolu
17
Partial Decoupling
Consider partial decoupling if
some of the loop interactions are weak
some of the loops need not have high performance
Partial decoupling focused on a subset of control loops
interactions are important, and/or
high performance control is required.
Consider partial decoupling for 3x3 or higher systems
main advantage: reduction of dimensionality.
Multivariable Control
Dr. Kalyana C. Veluvolu
18
Partial Decoupling Example
Grinding circuit analysis
Least sensitive variables y2
Most interaction: Loops 1 and 3,
Decouplers: loops 1 and 3,
Loop 2 without decoupling.
 119

 y1   217 s + 1
 y  =  0.00037
 2   500 s + 1
 y3   930
 500 s + 1
the transfer function matrix for the subsystem
153
337 s + 1
0.000767
33s + 1
− 667e −320s
166 s + 1
− 21 
10 s + 1  u1 
− 0.00005   
 u2
10 s + 1   
− 1033  u 3 
47 s + 1 
− 21 
 119
 y1   217 s + 1 10 s + 1  u1 
 y  =  930
− 1033  u 3 
 3 

 500 s + 1 47 s + 1
using the simplified decoupling approach
21
0.176(217 s + 1)
;
g I 1 = 10s + 1 =
119
10s + 1
217 s + 1
Multivariable Control
930
0.0(47 s + 1)
g I 3 = 500s + 1 =
1033
500s + 1
47 s + 1
Dr. Kalyana C. Veluvolu
19
Steady-State Decoupling
Steady-state decoupling: uses steady-state gain of transfer function
2 x 2 system
Simplified steady-state decoupling
Generalized steady-state decoupling
g I1 = −
K12
K
, g I 2 = − 21
K11
K 22
GI = K −1 K R
Very easy to design and implement, first technique to try;
ideal decoupler only if dynamic interactions persistent
big performance improvements with very little work or cost
most often applied in practice.
Multivariable Control
Dr. Kalyana C. Veluvolu
20
Example Distillation Column
12.8 − 18.9
K=

 6.6 − 19.4
⇐
 12.8e − s

G ( s ) = 16.7 s−+7 s1
 6.6e
10.9s + 1
− 18.9e −3s 

21.0 s + 1 
−3s
− 19.4e 
14.4s + 1 
Simplified steady-state decoupling
gI1 = −
−18.9
= 1.48,
12.8
gI 2 = −
6.6
= 0.34
−19.4
⇒
u1 = v1 + 1.48v 2
u 2 = v 2 + 0.34v1
Generalized steady-state decoupling
0 
12.8
KR = 
− 19.4
 0
Multivariable Control
⇒
 2.01 2.97
GI = 

0.68 2.01
⇒
u1 = 2.01v1 + 2.97 v 2
u 2 = 0.68v1 + 2.01v 2
Dr. Kalyana C. Veluvolu
21
Effect of Inputs Constraints
Always existing constraints on the process input variables
valves cannot go beyond full open or full shut
heaters cannot go beyond full power or zero power, etc.
Decoupling ok, if controller outputs not reached constraints
Even one input reaches a constraint, (reset windup)
control system no longer function decoupling
extremely poor (or even unstable) responses
Multivariable Control
Dr. Kalyana C. Veluvolu
22
Input Constraints
Example
Closed-loop response
of Y1 and Y2.
Simplified steady-state decoupler for the WB
Distillation Column with Kc1 =0.30, 1/τI1 =
0.307, Kc2 = - 0.05, 1/τI2 = 0.107.
g I1 = −
− 18.9
= 1.48,
12.8
gI2 = −
Unconstrained
manipulated
variables, u1, u2,
6. 6
= 0.34
− 19.4
If u, 0≤ u1 ≤ 0.15, the closed-loop response
is very poor once the reflux valve is full open
and the system becomes unstable.
Manipulated
variables u1,
u2 when 0≤ u1
≤0.15.
Response of Y1
and Y2 with
constrained u1, 0≤
u1≤0.l5.
Multivariable Control
Dr. Kalyana C. Veluvolu
23
Sensitivity to Model Error
K: system steady-state gain matrix:
y = Ku
Generalized decoupler
u = GI v
G I = K −1 K R
⇒
y = KK −1 K R v = K R v
∆K - error in the estimate of the steady-state gain matrix, then
−1
=
+
∆
y
(K
K)K
K Rv
⇒
y = (K +∆K)u
−1
= K R v +∆KK
⇒ ∆y = ∆KK −1 K R v ⇒ ∆y =
Multivariable Control
K Rv
⇒
∆K Adj(K)K R v
|K|
Dr. Kalyana C. Veluvolu
24
RGA and Model Error
∆y =
∆K Adj(K)K R v
|K|
If |K|very small, its reciprocal will be very large
Small modeling errors will cause very large errors in y
Small changes in controller output v result in large errors in y
Decoupling difficult: input/output variables are paired on very large
RGA values; system will also be very sensitive to modeling errors.
λij =
K ij C ij
Cij cofactor of Kij
|K|
Multivariable Control
Dr. Kalyana C. Veluvolu
25
Example Heavy Oil Fractionator
Transfer function model
 4.05e −27 s

G ( s ) =  50 s +−18 s
 4.06e
 13s + 1
1.20e −27 s 

45s + 1 
1.19 
19 s + 1 
⇒
4.05 1.20
K =

4.06 1.19
determinant very close to zero: decoupling very difficult.
| K |= −0.0525
RGA
 − 91.8 92.8 
Λ=

 92.8 − 91.8
Decoupling extremely difficult
small value of gain matrix determinant
large values of RGA elements
Multivariable Control
Dr. Kalyana C. Veluvolu
26
SVD and Condition Number
The matrix is said to be singular, if its determinant is zero,
Near singularity matrix: singular values
σ i = (λi ( A* A))1/ 2
i = 1, 2,..., n
Condition number: The ratio of the largest and smallest singular value
k=
σ max
σ min
Example: Heavy Oil Fractionator continued
4.05 1.20
K =

4.06 1.19
singular values σ1=5.978, σ2 = 0.00878
and a condition number:
κ = 680.778
clearly indicating serious ill-conditioning.
Multivariable Control
Dr. Kalyana C. Veluvolu
27