PART - 2
KNU/EECS/ELEC 835001
Interaction Analysis
Multivariable Control
Dr. Kalyana C. Veluvolu
KNU/EECS/ELEC 835001
Outline - Module 5.2
Interaction Analysis
–
–
–
–
–
–
Interaction Measure of Control Loops
Relative Gain Array (RGA)
Loop Paring Based on RGA
RGA Calculation
Niederlinski index
Extension
» Loop Paring for Nonlinear Process
» Loop Paring for Process with Integrator
» Loop Paring for Non-square Matrix
– Timescale decoupling
– Other pairing methods
Multivariable Control
Dr. Kalyana C. Veluvolu
2
KNU/EECS/ELEC 835001
Interaction Measure of Control Loops
• One manipulated variable affects more than one
controlled variable
• transfer function matrix - non-diagonal structure
2x2 system
y1 ( s ) = g11 ( s ) m1 ( s ) + g12 ( s ) m2 ( s )
y2 ( s ) = g 21 ( s ) m1 ( s ) + g 22 ( s ) m2 ( s )
• transmission interaction:
exists when change in one
controller setpoint affects output
through the actions of other
controller(s)
Multivariable Control
Dr. Kalyana C. Veluvolu
3
KNU/EECS/ELEC 835001
Interaction Measure: Experiment 1
Unit step change in m1 with all loops open
Both y1 and y2 change,
At steady state, the change in y1
as a result of m1 be ∆y1m;
∆y1m = K11
As no other input variable change, and all the control loops
are opened, there is no feedback control involved.
Multivariable Control
Dr. Kalyana C. Veluvolu
4
KNU/EECS/ELEC 835001
Interaction Measure: Experiment 2
Unit step change in m1 with Loop 2 closed
Due to the change in m1:
1. y1 changes because of g11, and y2 change because g21,
2. Loop 2 manipulating m2 until y2 is restored to its initial state.
3. The changes in m2 return to affect y1 via the g12 .
Changes in y1 from two sources
Direct influence of m1 on y1 ; ∆y1m
The retaliatory action from Loop 2
counter m1 on y2 say, ∆y1r.
Multivariable Control
Dr. Kalyana C. Veluvolu
5
KNU/EECS/ELEC 835001
Relative Gain
Closed-Loop: The net change in y1, ∆y1* given
∆y1 * = ∆y1m + ∆y1r
At steady state, It is given by
 K K 
*
∆y1* = K11 1 − 12 21  = K11
 K11 K 22 
An interaction measure
∆y
λ11 = 1m
∆y1 *
Multivariable Control
λ11 =
∆y1m
∆y1m + ∆y1r
Dr. Kalyana C. Veluvolu
6
KNU/EECS/ELEC 835001
Steady-State Process Gain Matrix
•
•
use steady-state gain matrix to evaluate process behaviour - interaction
useful gain info for determining control loop pairings
Distillation Example - Gain Matrix
 0.0747e−3s

G(s) =  12s +−133
.
e .s
01173
 117
. s +1
− 0.0667e−2s 

15s + 1 
− 01253
.
e−2s 
10.2s + 1 
Relative magnitudes, signs
0.0747 − 0.0667
K = G(0) = 

.
− 01253
.
 01173

Multivariable Control
Dr. Kalyana C. Veluvolu
7
KNU/EECS/ELEC 835001
Relative Gain Array
Relative gain - ratio of gain with other loops open to other loops closed
mathematical definition:
λij =
∂yi
∂u j
∂yi
∂u j
uk constant , k ≠ j
=
gain with other loops open
gain with other loopsclosed
yk constant , k ≠i
matrix of relative gains - describes steady-state interaction behaviour
λ12 
λ
Λ =  11

λ
λ
 21
22 
Multivariable Control
 λ11
λ
Λ =  21
 ...

 λ n1
Dr. Kalyana C. Veluvolu
λ12
λ 22
...
λn2
... λ1n 
... λ 2 n 
... ... 

... λ nn 
8
KNU/EECS/ELEC 835001
RGA Calculation by Principle
2 x 2 system steady-state model:
 ∂y1 


= K11
 ∂m1  all loops open
y1 = K11m1 + K12 m2
y 2 = K 21 m1 + K 22 m2
m2 must take to keep y2 = 0 in changes in m1;
m2 = −
K 21
m1
K 22
y1 = K11m1 −
K12 K 21
m1
K 22
 ∂y1 

K K 


= K11 1 − 12 21 
K11 K 22 
 ∂m1  loop 2 closed

Multivariable Control
 ∂y1 


 ∂m1  all loops open
1
=
 ∂y1 
 K12 K 21 


1 −

m
∂
 1 loop 2 closed
 K11 K 22 
Dr. Kalyana C. Veluvolu
9
KNU/EECS/ELEC 835001
RGA Calculation by Principle
λ12 , λ21 , and λ22
Let
ζ =
K12 K21
K11K 22
⇒
λ 11 =
1
1−ζ
 ∂y1 
 K K  K
= K11* = K11  1 − 12 21  = 11


 ∂m1 loop 2 closed
 K11 K 22  λ11
λ22 = λ11 =
1
1− ζ
λ12 = λ 21 =
−ζ
1−ζ
RGA for the 2 x 2 system :
 λ 1− λ
Λ=
λ 
1 − λ
Multivariable Control
Dr. Kalyana C. Veluvolu
10
KNU/EECS/ELEC 835001
RGA Calculation by Matrix Method
K : steady-state gain matrix of G(s)
lim G ( s) = K
s →0
R: transpose of the inverse of K:
R = (K -1 )T
RGA can be obtained from
λij = K ij ⊗ rij
⊗ an element-by-element multiplication
Multivariable Control
Dr. Kalyana C. Veluvolu
11
KNU/EECS/ELEC 835001
RGA Calculation Example
RGA: wood and binary distillation column
 12.8e − s

G ( s ) = 16.7 s−+7 s1
 6.6e
10.9 s + 1
12.8 − 18.9
K = G(0) = 

 6.6 − 19.4
term-by-term multiplication of Kij
and rij gives the RGA as:
Multivariable Control
− 18.9e −3 s 

21.0 s + 1 
− 19.4e −3 s 
14.4 s + 1 
0.053 
 0.157
R = ( K −1 )T = 

 − 0.153 − 0.104
 2.0 − 1.0
Λ=

− 1.0 2.0 
Dr. Kalyana C. Veluvolu
12
KNU/EECS/ELEC 835001
Properties of the RGA
•
•
•
•
 λ11
λ
Λ =  21
 ...

λn1
λ12 ... λ1n 
λ 22 ... λ 2 n 
...
λn 2
scale-independent - from definition
row or column elements sum to 1
May be sensitive to errors in the gains - be careful using
empirical transfer functions
Closed-loop steady-state gain and open-loop gain
K ij * =
1
λij
... 

... λ nn 
...
K ij
Negative relative gain for I-j:
Change in yi in response to mj opposite in sign for other loops are
open and closed.
The input/output pairings are potentially unstable and be avoided.
Multivariable Control
Dr. Kalyana C. Veluvolu
13
KNU/EECS/ELEC 835001
Interpretation of the RGA
K ij * =
1
λij
K ij
Look at both sign and magnitude
λij < 0
λ ij = 0
-- gain reversal occurs when other loops are closed - undesirable
–instability due to incorrect controller action
-- open loop gain is zero - control action depends on other
control loops being closed – not select
0 < λij < 1 -- gain amplification - loop gain increased if other loops are closed
λij = 1
-- no interaction: open-loop gain unaffected by other loops
–desirable - one-way interaction still possible
λij > 1
-- attenuation of open-loop gain when other loops are closed
λij → ∞ -- loss of control when other loops are closed - gain goes to zero
–undesirable
Multivariable Control
Dr. Kalyana C. Veluvolu
14
KNU/EECS/ELEC 835001
Interpretation of the RGA - Distillation Example
•
relative gain array
 0.0747e − 3s

G ( s) =  12 s +−13.3s
 0.1173e
 11.7 s + 1
− 0.0667e − 2 s 

15s + 1 
− 0.1253e − 2 s 
10.2 s + 1 
K ij * =
1
λij
K ij
 6.09 − 5.09
Λ=

− 5.09 6.09 
pairing - use u1-y1 and u2-y2 loops
–avoid negative relative gains
Multivariable Control
Dr. Kalyana C. Veluvolu
15
KNU/EECS/ELEC 835001
The Niederlinski Criterion
test for instability based on steady-state information
•
arrange transfer function matrix so that pairings are on diagonal
•
stable - process elements must be open-loop stable
•
rational transfer functions
» transfer functions consist of ratios of polynomials in “s”
» i.e., no deadtime, which introduces exponentials
•
proper transfer functions - order of denominator greater than order
of numerator
•
individual control loops are closed-loop stable
Multivariable Control
Dr. Kalyana C. Veluvolu
16
KNU/EECS/ELEC 835001
Niederlinski Criterion
• closed-loop multi-loop system is unstable if
G(0)
NI ≡
n
< 0 i.e.,
∏ g (0)
ii
i =1
det(G(0))
<0
product
of




 diagonal elements 
• sufficient condition to identify instability
– if condition is not satisfied, we have failed to detect
instability (as opposed to confirming stability) --> this
does NOT imply stability
• use as a screening tool - identify problem cases
Multivariable Control
Dr. Kalyana C. Veluvolu
17
KNU/EECS/ELEC 835001
Example
Evaporator process for producing alumina:
0.046 
 0.117


G p ( s) =  5s + 1 35s + 1 
0.128 − 0.050


18s + 1 30 s + 1 
0.117 0.046 
K=

0.128 − 0.05
relative gain is 0.5 for each pairing
- no preference indicated
re-arrangement of gain matrix isn’t necessary given 1-1, 2-2 pairing
(paired transfer functions are on main diagonal)
Niederlinski test:
det( G (0))
n
- instability is NOT detected
∏g
=
− 0.0117
=2>0
− 0.0059
ii
i =1
•Note - no time delays, assuming integral control, proper transfer
functions, open-loop stable
Multivariable Control
Dr. Kalyana C. Veluvolu
18
KNU/EECS/ELEC 835001
Example
Loop pairing for a 3 x 3 system
1
5 3 1
K = G (0) =  1 1 3 1 


1 1 3
 1
 10 − 4.5 − 4.5
Λ = − 4.5
1
4.5 
− 4.5 4.5
1 
By RGA, the 1-1/2-2/3-3 pairing is recommended, but
3
| K | = -0.148
∏K
ii
i =1
 5  1  1   5 
=     =  
 3  3  3   27 
It is an unstable configuration
Multivariable Control
Dr. Kalyana C. Veluvolu
19
KNU/EECS/ELEC 835001
Example
another possible pairing of 1-1/2-3/3-2, put the relative element into diagonal
1
5 3 1
K = G (0) =  1
1 1 3


 1 1 3 1 
Now
K =
⇒
NI =
Notice the
change of K
4
27
4 / 27 4
=
5/3
45
system is stable
Multivariable Control
Dr. Kalyana C. Veluvolu
20
KNU/EECS/ELEC 835001
Loop Paring for Nonlinear Systems
With available process model, two approaches used to
obtain RGA’s for nonlinear systems
1. By first principles, using steady-state version of nonlinear model, it
is possible to obtain analytical expressions.
2. By linearizing the nonlinear model around a specific steady state
and using the approximate K matrix to obtain the RGA.
The second approach is usually less tedious than the first
one, and is more popular.
Multivariable Control
Dr. Kalyana C. Veluvolu
21
KNU/EECS/ELEC 835001
Example by first principles
Total Mass and Component A Mass Balance:
F= m1+m2
 ∂F 


m
∂
 1
x=
both loops open
m1
m1 + m2
=1
Upon closing the second loop, setting x = x*
m2 =
m1
− m1
x*
 ∂F 


 ∂m1 
m
F = m1 + 1 − m1
x*
1
=
second loop closed
x*
Multivariable Control
Final result
 x * 1 − x *
1 − x *
x * 

 ∂F 

 both loops
∂m1 

λ=
 ∂F 

 sec ond loop
 ∂m1 
Dr. Kalyana C. Veluvolu
open
1
= x*
1/ x *
closed
22
KNU/EECS/ELEC 835001
Example Stirred Mixing
Tank
linearized around steady-state operating
point, hs, Ts.
1


K
)
 Ac ( s +
A
h
2
c
s

G ( s) = 
(TH − Ts )

 Ac hs ( s + K )

Ac hs


K
Ac ( s +
)
2 Ac hs 

(TC − Ts )

K
)
Ac hs ( s +
Ac hs 
 2 hs
2 hs 
1

G(0) =  (TH − Ts ) (TC − Ts ) 
K

hs
hs 

Multivariable Control
1
y1= liquid level;
y2 = tank temperature
m1 = hot stream flowrate;
m2 = cold stream flowrate
 TC − Ts
 T −T
H
Λ= C
(
T
T
−
−
H
s)

 TC − TH
− (TH − Ts ) 
TC − TH 

TC − Ts 
TC − TH 
Dr. Kalyana C. Veluvolu
23
KNU/EECS/ELEC 835001
Example Stirred Mixing Tank
TH = 650C
Ts > 400C (Ts = 550C)
0.8 0.2
Λ=

0.2 0.8
 TC − Ts
 T −T
H
Λ= C
 − (TH − Ts )
 TC − TH
TC = 150C
Ts <400C (Ts = 250C)
0.2 0.8
Λ=

0.8 0.2
Ts = TH
Ts = 400C
1 0
Λ=

0 1 
Multivariable Control
− (TH − Ts ) 
TC − TH 

TC − Ts 
TC − TH 
0.5 0.5
Λ=

0.5 0.5
Dr. Kalyana C. Veluvolu
24
KNU/EECS/ELEC 835001
Systems with Integrator
• RGA involves steady-state information, for integrator
processes, alternative way given by example

1.318e −2.5 s

20s + 1
G ( s) = 
0.038(182 s + 1)

 ( 27 s + 1)(10 s + 1)(6.5s + 1)
− e −4 s 

3s 
0.36 
s 
−I 

1
.
318
K = lim G (s) = lim 
3 
s − >0
I − >∞ 

0.038 0.36I 
0.97 0.03
Λ=

0.03 0.97
Making
I=
1
s


1
λ = lim 
I −>∞
 1 + 0.038 × 0.333I
1.138 × 0.36 I






1-1/2-2 pairing is recommended.
Multivariable Control
Dr. Kalyana C. Veluvolu
25
KNU/EECS/ELEC 835001
Loop Pairing for Underdefined Systems
Fewer inputs than outputs, not all the outputs can be controlled. Decide which
outputs are important; paired with the m inputs; the remaining not controlled.
 0.66e −2.6 s

 y1   6.7 s −+6.15 s
 y  =  1.11e
 2   3.25s + 1
 y3   − 33.68e −9.2 s

 8.15s + 1
 0.66e −2.6 s
 y1   6.7 s + 1
 =
  
− 9 .2 s
 y3   − 33.68e
 8.15s + 1

y2 =

− 0.0049e − s

9.06s + 1
 m
−1.2 s
− 0.012e
 1 
 m2 
7.09s + 1
0.87(11.61s + 1)e − s 

(3.89s + 1)(18.8s + 1) 


  m1 
 m 
0.87(11.61s + 1)e − s   2 
(3.89 s + 1)(18.8s + 1) 
− 0.0049e − s
9.06 s + 1
Steady-state gain matrix of subsystem
− 0.0049
 0.66
K =
0.87 
− 33.68
~
RGA
 1.4 − 0.4

− 0.4 1.4 
λ=
−0.012e−1.2 s
1.11e−6.5s
m1 +
m2
3.25s + 1
7.09s + 1
Multivariable Control
Dr. Kalyana C. Veluvolu
26
KNU/EECS/ELEC 835001
Loop Pairing for Overdefined Systems
More input control fewer outputs in more than one way. Possible pairing
Determine all possible square subsystems and RGA, pick best RGA.
 0.5e −0.2 s
 y1   3s + 1
 y  =  0.004e −0.5 s
 2 
 1.5s + 1
 0.5e −0.2 s
 y1   3s + 1
 y  =  0.004e −0.5 s
 2 
 1.5s + 1
0.07e −0.3s
2. 5 s + 1
− 0.003e −0.2 s
s +1
0.04e −0.03s   m 
 1
2. 8 s + 1   m 
− 0.001e −0.4 s   2 
m 
1.6 s + 1   3 
 0.5e −0.2 s
 y1   3s + 1
 y  =  0.004e −0.5 s
 2 
 1.5s + 1
0.04e −0.03s 
2.8s + 1   m1 
− 0.001e −0.4 s  m3 
1.6 s + 1 
0.758 0.242
Λ2 = 

0.242 0.758
 0.843 0.157
Λ1 = 

0.157 0.843
 0.07e −0.3 s
 y1   2.5s + 1
 y  =  − 0.003e −0.2 s
 2 

s +1
Multivariable Control
0.07e −0.3 s 

2.5s + 1   m1 
− 0 .2 s 

− 0.003e
  m2 

s +1
0.04e −0.03 s 
2.8s + 1  m2 
− 0.001e −0.4 s   m3 
1.6s + 1 
− 1.4 2.4 
Λ3 = 

 2.4 − 1.4
Dr. Kalyana C. Veluvolu
27
KNU/EECS/ELEC 835001
Loop Pairing without Process Model
1.
Experimentally determine the steady-state gain matrix K, by
implementing step changes in the process input variables, one at a time, the
steady-state gain between the ith output variable and the jth input variable:
kij =
∆yij
∆m j
Once this gain matrix is known, we can easily generate the RGA.
2.
RGA element λij can be determined upon performing two experiments;
1. determines the open-loop steady-state gain by measuring the response
of yi to input mj, when all the other loops are opened;
2. in the second experiment, all the other loops are closed ---- using PI to
ensure that there will be no steady-state offsets — and the response of yi
to input mj is redetermined.
Multivariable Control
Dr. Kalyana C. Veluvolu
28
KNU/EECS/ELEC 835001
Timescale Decoupling
Timescale decoupling occurs
when the fast loop responds so fast that the effect of the slow
loop appears as a constant disturbance;
the slow loop does not respond at all to the high-frequency
disturbances coming from the fast loop.
This means that we can safely pair loops with large
differences in closed-loop response times even when the
RGA is not favorable.
Multivariable Control
Dr. Kalyana C. Veluvolu
29
KNU/EECS/ELEC 835001
Example of Timescale Decoupling
2 
 2
10 s + 1
s +1 
G ( s) = 
1
−4 


 s + 1 10 s + 1
0.8 0.2
Λ=

0.2 0.8
By RGA, the y1 — m1 / y2 — m2 pairing
The closed-loop response for a unit setpoint change in y1 using this pairing
( K C 1 = 4,τ I 1 = 0.5, K C 2 = −4,τ I 2 = 0.3)
Reverse Pairing: the open-loop time
constants on diagonal are only 1 minute
( K C 1 = 10,τ I 1 = 0.3, K C 2 = 20,τ I 2 = 0.3)
Multivariable Control
Dr. Kalyana C. Veluvolu
30
KNU/EECS/ELEC 835001
RIA BASED LOOP PARING
Relative interaction: the increment of the process gain after all
other loops are closed and gain in the same loop when all other
loops are open
ϕij =
(∂yi / ∂u j ) all other loops close except for loop yi −u j − (∂yi / ∂u j ) all loops open
(∂yi / ∂u j ) all loops open
Compared with the definition of the RGA
ϕij =
Multivariable Control
1
λij
−1
Dr. Kalyana C. Veluvolu
31
KNU/EECS/ELEC 835001
RI Interaction Measure
ϕij = 0 ⇒ no interaction;
ϕij =
1
λij
−1
Kij * = (ϕ ij + 1) Kij
ϕij > 0 ⇒ interaction in same direction as interaction-free process
ϕij > 1.0 ⇒ interaction dominates over interaction-free process gain
ϕ ij < 0 ⇒ in the reverse direction as interaction-free process gain
ϕij < −1.0 ⇒ reverse interaction dominates over process gain.
(1) All the RIA elements are closest to 0;
Pairing Rules
(2) NI is positive;
(3) All the RIA elements are greater than -1;
(4) RIA elements close to -1 are avoided.
Multivariable Control
Dr. Kalyana C. Veluvolu
32
KNU/EECS/ELEC 835001
Other Loop Pairing Methods
Dynamic RGA (DRGA) based loop pairing
Performance RGA (PRGA) based loop pairing
Generalized relative dynamic gain (GRDG) based loop
pairing
Structural singular value based loop pairing
Relative Energy Array Based loop Pairing
etc
Multivariable Control
Dr. Kalyana C. Veluvolu
33