Decentralized control
Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Tecnology (NTNU)
Trondheim, Norway
1
Outline
•
•
•
•
•
2
Multivariable plants
RGA
Decentralized control
Pairing rules
Examples
MIMO (multivariable case)
Distillation column
“Increasing L from 1.0 to 1.1 changes yD
from 0.95 to 0.97, and xB from 0.02 to 0.03”
“Increasing V from 1.5 to 1.6 changes yD
from 0.95 to 0.94, and xB from 0.02 to 0.01”
Steady-State Gain Matrix
 ΔYD 
 ΔL 

  G0 

 ΔV 
 Δx B 
 0.97  0.95 0.94  0.95 
g12 0   1.1  1.0
1.6  1.5  0.2  0.1



g 22 0   0.03  0.02 0.01  0.02  0.1  0.1 
 1.1  1.0
1.6  1.5 
Effect of input 1ΔLon output 2 Δx B 
g
G0    11
g 21
Can also include dynamics :
G s 
3

 0. 2
  1  50s
 0.1

 1  40s

0.1 

1  50s 
0.1 


1  40s 
y D
x B
(Time constant 50 min for yD)
(time constant 40 min for xB)
Analysis of Multivariable processes
What is different with MIMO processes to SISO:
The concept of “directions” (components in u
and y have different magnitude”
Interaction between loops when single-loop
control is used
INTERACTIONS
Process Model
u1
y1
g11
" Open - loop "
g12
g21
G
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u2
g22
y1 (s ) = g11 (s )u1 ( s ) + g12 (s )u2 (s )
y2
y2 (s ) = g 21 (s )u1 ( s ) + g 22 (s )u2 (s )
Consider Effect of u1 on y1
1) “Open-loop” (C2 = 0): y1 = g11(s)·u1
2) Closed-loop” (close loop 2, C2≠0)
æ
ö
g12g21 ×C2 ÷
ç
÷u1
y1 = çg11 (s)÷
çè
1+ g22 ×C2 ÷
ø
Change caused by
“interactions”
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Limiting Case C2→∞ (perfect control of y2)
æ
g g ö
÷
y1 = ççg11 (s)- 12 21 ÷
u
÷
÷1
çè
g22 ø
How much has “gain” from u1 to y1 changed by
closing loop 2 with perfect control?
Relative Gain =
def
(y1/μ1 )OL
×g11
1
=
=
= λRGA
11
(y1/μ1 )CL g - g12 g21 1- g12 g21
11
g22
The relative Gain Array (RGA) is the matrix
formed by considering all the relative gains
éλ
RGA = Λ = ê 11
êλ 21
ë
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é(y1/u1 ) (y1/u2 ) ù
OL
OL ú
ê
ê
λ12 ù ê(y1/u1 )CL (y1/u2 )CL ú
ú
ú= ê
ú
ú
λ 22 û ê(y2 u 1 )OL (y2 /u2 )OL ú
ê
ú
ê(y2 u1 )CL (y2 /u2 )CL ú
ë
û
g11 g22
Example from before
 0.2
G
0.1
 0.1
 0.1 
, λ 11 
1
2
0.1 0.1
1
0.2 0.1



0.5
 2
RGA  
 1
 1
2
Property of RGA:
Columns and rows always sum to 1
RGA independent of scaling (units) for u and y.
Note: RGA as a function of frequency is the most
important for control!
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Use of RGA:
(1) Interactions
From derivation: Interactions are small if
relative gains are close to 1
Choose pairings corresponding to RGA
elements close to 1
Traditional: Consider Steady-state
Better: Consider frequency corresponding to closedloop time constant
But: Avoid pairing on negative steady-state relative
gain – otherwise you get instability if one of the loops
become inactive (e.g. because of saturation)
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Example:
0.2  0.1 
G o  

0.1  0.1
é2 - 1 ù
ú
RGA = ê
êë-1 2ú
û
Only acceptablepairings :
u1 « y1
u2 « y 2
Not recommended :
u1 « y 2
u2 « y 1
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y1 = 0.2 ×u1 -0.1×u 2
y 2 = 0.1u1 -0.1u 2
With integral action :
Negative RGA  individual
loop unstable + overall system unstable
when loops saturate
(2) Sensitivity measure
But RGA is not only an interaction measure:
Large RGA-elements signifies a process that is
very sensitive to small changes (errors) and
therefore fundamentally difficult to control
Large (BAD!)
example
1 
1
G= 

0.9 0.91

91  90 
RGA  

 90 91
1
 1.1%
90
Relativechange = -
1
l 21
makes matrix singular!
æ 1ö
Then gˆ 12 = g12 çç1 + ÷
= 0.91
÷
çè 90 ÷
ø
0.9
10
Singular Matrix: Cannot take inverse, that is,
decoupler hopeless.
Control difficult
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Exercise. Blending process
sugar u1=F1
water u2=F2
•
Mass balances (no dynamics)
–
–
(a)
(b)
(c)
(d)
(e)
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y1 = F (given flowrate)
y2 = x (given sugar fraction)
Total:
Sugar:
F1 + F2 = F
F1 = x F
Linearize balances and introduce: u1=dF1, u2=dF2, y1=F1, y2=x,
Obtain gain matrix G (y = G u)
Nominal values are x=0.2 [kg/kg] and F=2 [kg/s]. Find G
Compute RGA and suggest pairings
Does the pairing choice agree with “common sense”?
Decentralized control
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Two main steps
• Choice of pairings (control configuration selection)
• Design (tuning) of each controller
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Design (tuning) of each controller ki(s)
•
Fully coordinated design
– can give optimal
– BUT: requires full model
– not used in practice
•
Independent design
– Base design on “paired element”
– Can get failure tolerance
– Not possible for interactive plants (which fail to satisfy our three pairing rules – see
later)
•
Sequential design
–
–
–
–
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Each design a SISO design
Can use “partial control theory”
Depends on inner loop being closed
Works on interactive plants where we may have time scale separation
Effective use of decentralized control requires
some “natural” decomposition
• Decomposition in space
– Interactions are small
– G close to diagonal
– Independent design can be used
• Decomposition in time
– Different response times for the outputs
– Sequential design can be used
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Independent design: Pairing rules
Pairing rule 1. RGA at crossover frequencies. Prefer pairings such that
the rearranged system, with the selected pairings along the diagonal,
has an RGA matrix close to identity at frequencies around the closedloop bandwidth.
Pairing rule 2. For a stable plant avoid pairings ij that correspond to
negative steady-state RGA elements, ij(0)· 0.
Pairing rule 3. Prefer a pairing ij where gij puts minimal restrictions on
the achievable bandwidth. Specifically, its effective delay ij should be
small.
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Example 1: Diagonal plant
• Simulations (and for tuning): Add delay 0.5 in each input
• Simulations setpoint changes: r1=1 at t=0 and r2=1 at t=20
• Performance: Want |y1-r1| and |y2-r2| less than 1
• G (and RGA): Clear that diagonal pairings are preferred
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Diagonal pairings
Get two independent subsystems:
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Diagonal pairings....
Simulation with delay included:
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Off-diagonal pairings (!!?)
Pair on two zero elements !! Loops do not work independently!
But there is some effect when both loops are closed:
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Off- diagonal pairings for diagonal plant
• Example: Want to control temperature in two completely different
rooms (which may even be located in different countries). BUT:
– Room 1 is controlled using heat input in room 2 (?!)
– Room 2 is controlled using heat input in room 1 (?!)
TC
2
1
TC
22
??
Off-diagonal pairings....
Controller design difficult. After some trial and error:
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–
–
Performance quite poor, but it works because of the “hidden” feedback loop g12 g21 k1 k2!!
No failure tolerance
Example 2: One-way interactive (triangular) plant
• Simulations (and for tuning): Add delay 0.5 in each input
• RGA: Seems that diagonal pairings are preferred
• BUT: RGA is not able to detect the strong one-way interactions (g12=5)
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Diagonal pairings
One-way interactive:
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Diagonal pairings....
Closed-loop response (delay neglected):
With 1 = 2 the “interaction” term (from r1 to y2) is about 2.5
Need loop 1 to be “slow” to reduce interactions: Need 1 ¸ 5 2
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Diagonal pairings.....
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Off-diagonal pairings
Pair on one zero element (g12=g11*=0)
*
BUT pair on g21=g 22=5: may use sequential design: Start by tuning k2*
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Comparison of diagonal and off-diagonal
pairings
–
–
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OK performance,
but no failure tolerance
if loop 2 fails
Example 3: Two-way interactive plant
- Already considered case g12=0 (RGA=I)
- g12=0.2: plant is singular (RGA=1)
- will consider diagonal parings for: (a) g12 = 0.17, (b) g12 = -0.2, (c) g12 = -1
Controller:
with 1=5 and 2=1
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Conclusions decentralized examples
• Performance is OK with decentralized control (even with wrong
pairings!)
• However, controller design becomes difficult for interactive plants
– and independent design may not be possible
– and failure tolerance may not be guaranteed
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Example 3.10: Separator (pressure vessel)
u1 = V
y1 = pressure (p)
Feed (d)
u2 = L
• Pairings? Would expect y1/u1 and y2/u2
• But process is strongly coupled at
intermediate frequencies. Why?
• Frequency-dependent RGA suggests
opposite pairing at intermediate
frequencies
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y2 = level (h)
Separator example....
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Separator example: Simulations (delay = 1)
1.5
y1
y
1
0.5
Diagonal
y2
0
-0.5
0
20
40
60
80
100
Time
120
140
160
180
200
2.5
2
y
1.5
y1
1
Off-Diagonal
0.5
y2
0
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-0.5
0
20
40
60
80
100
Time
120
140
160
180
200
Separator example: Simulations (delay = 5)
2
y1
0
y
y2
Diagonal
-2
-4
0
100
200
300
400
500
Time
600
700
800
900
1000
3
2
y
y1
Off-Diagonal
1
0
y2
-1
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0
100
200
300
400
500
Time
600
700
800
900
1000
Separator example
• BUT NOTE: May easily eliminate interactions to y2 (level) by simply
closing a flow controller on u2 (liquid flow)
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Iterative RGA
• For large processes, lots of pairing alternatives
• RGA evaluated iteratively is helpful for quick screening
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• Converges to “Permuted Identity” matrix (correct pairings) for
generalized diagonally dominant processes.
• Can converge to incorrect pairings, when no alternatives are dominant.
• Usually converges in 5-6 iterations
Example of Iterative RGA
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Correct pairing
Stability of Decentralized control systems
• Question: If we stabilize individual loops, will the overall closed-loop
system be stable?
• E is relative uncertainty
•
is complementary sensitivity for diagonal plant
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Stability of Decentralized control systems
• Question: If we stabilize individual loops, will overall closed-loop
system be stable?
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Let G and
have same unstable poles, then closed-loop system stable if
Let G and
have same unstable zeros, then closed-loop system stable if
Mu-Interaction measure
Closed-loop stability if
At low frequencies, for integral control
)
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Decentralized Integral Controllability
• Question: If we detune individual loops arbitrarily or take them out of
service, will the overall closed-loop system be stable with integral
controller?
• Addresses “Ease of tuning”
• When DIC - Can start with low gains in individual loops and increase
gains for performance improvements
• Not DIC if
• DIC if
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Performance RGA
• Motivation: RGA measures two-way interactions only
• Example 2 (Triangular plant)
• Performance Relative Gain Array
• Also measures one-way interactions, but need to be calculated for
every pairing alternative.
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Example PRGA: Distillation
• see book
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