1. Decentralized control
Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Tecnology (NTNU)
Trondheim, Norway

2. Outline Multivariable plants RGA Decentralized control Pairing rules
Examples

3. Multivariable control problem
Distillation column
“Increasing reflux L from 1.0 to 1.1 changes yD from 0.95 to 0.97, and xB from 0.02 to 0.03”
“Increasing boilup 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
(Time constant 50 min for yD)
(time constant 40 min for xB)

4. Simplest multivariable controller: Decentralized control

5. Multivariable decoupling controlK y u Dy
Kdiag Du G
K = Du Kdiag Dy

6. Most common: input decoupling
K = D Kdiag
Dynamic decoupling, D = G(s)-1
Steady-state decoupling, D = G(0)-1
Approximate decoupling, D = G(jw)-1
Keep diagonal elements in GD unchanged – similar to feedforward control
One way decoupling
Kdiag D G

7. One-way decoupling ”Feed-forward” implementation u’1 u1 y1 r1-y1 c1
-g12/g11
g g12
g g22
r2-y2 y2 c2
u’2 u2
Consider u2 as measured disturbance on y1
y1 = g11 u1 + g12 u2
Want y1=0: 0 = g11 u1 + g12 u2
-> ”feedforward”: u1 = -g12/g11 u2
Decoupled process:
y1 = g11 u’1
y2 = g21 u’1 + (g22 – g21g12/g11) u’2
Note: y2/u2 changes

8. (Two-way) decoupling: ”Feed-forward” implementation
GD = decoupled process u1 y1
r1-y1 c1
-g12/g11
g g12
g g22
-g21/g22
r2-y2 y2 c2 u2
1. ”Feedforward”: Decoupler has 1’s on diagonal: D = [1 -g21/g22; -g21/g22 1]
General decoupler with 1’s on diagonal: D = G-1 (G-1diag)-1
… but note that diagonal elements of decoupled process GD change !
2. To keep (GD)diag = Gdiag we need more complicated decoupler: D= G-1 Gdiag

9. Effective use of decentralized control requires some “natural” decomposition
Decomposition in space = Non-interactive
Interactions are small (RGA useful tool)
G close to diagonal
Independent design can be used
Decomposition in time
Different response times for the outputs are allowed
Usually: At least factor 5 different
Sequential design can be used

10. Two main steps Choice of pairings: RGA useful tool
Design (tuning) of each controller

11. Decentralized control: Design (tuning) of each controller ki(s)
Fully coordinated design
can give optimal
BUT: requires full model, so may as well use multivariable controller
not used in practice
Independent design
Base design on “paired element”
Can get failure tolerance
Not possible for interactive plants. RGA useful tool
Sequential design
Each design a SISO design
Can use “partial control theory”
Depends on inner loop always being closed
Works on interactive plants if time scale separation for outputs is OK

12. Analysis of Multivariable processes
Process Model 2x2
INTERACTIONS: Caused by nonzero g12 and/or g21 y1 u1
g11
g12
g21 y2 u2
g22

13. Consider effect of u1 on y1
g12
g21
g11
g22 u2 u1
“Open-loop” (C2 = 0): y1 = g11(s)·u1
“Closed-loop” (close loop 2, C2≠0):
Change caused by “interactions”

14. Limiting Case C2→∞ (perfect control of y2)
How much has “gain” from u1 to y1 changed by closing loop 2 with perfect control?
The relative Gain Array (RGA) is the matrix formed by considering all the relative gains
Example from before

15. Property of RGA:
Columns and rows always sum to 1
RGA independent of scaling (units) for u and y.

16. Use of RGA: Interactions
RGA-element ()> 1: Smaller gain by closing other loops (“fighting loops” gives slower control)
RGA-element () <1: Larger gain by closing other loops (can be dangerous)
RGA-element () negative: Gain reversal by closing other loops (Oops!)
Rule 1. Avoid pairing on negative steady-state relative gain – otherwise you get instability if one of the loops become inactive (e.g. because of saturation)
Rule 2. Choose pairings corresponding to RGA-elements close to 1
Traditional: Consider Steady-state
Better (improved Rule 2): Consider frequency corresponding to closed-loop time constant

18. (2) Sensitivity measure
But RGA is not only an interaction measure:
Large RGA-elements signals a process that is fundamentally difficult to control.
Close to singular and sensitive to small errors
Large (BAD!)

19. Implication: If the RGA-elements are large, especially at “control”
Singular Matrix: Cannot take inverse, that is, decoupler hopeless.
Control difficult
Implication: If the RGA-elements are large, especially at “control”
frequencies, then the process is fundamentally difficult to control.
Need to change the process! For example, change MVs or CVs

20. Cookbook for pairing and RGA
Indentify the n y’s (CVs) and n u’s (MVs) for which you are uncertain about the pairings
Find the transfer matrix, y = G(s) u (preferably with dynamics)
Compute RGA as a function of frequency, RGA = G(jw) * G(jw)-T
General: If there are large RGA-elements (>10-15) in the frequency range important for control, then the process is fundamentally difficult to control. Consider modifying the process (change MVs or CVs?)
Pairings: Consider the steady-state RGA (s=0). AVOID pairing on negative RGA-elements!!
Pairings: Prefer pairing on RGA-elements close to 1 at control frequencies.
Check: Want small element magnitudes of RGA-Ip, where Ip is matrix with 1’s at selected pairings
Pairings: Prefer pairing on variables with good controllability (=small effective delay). Pair “close”!
If you cannot find satisfying pairings, then consider multivariable control, for example, decoupling or MPC.

21. Example
Only diagonal pairings give positive steady-state RGAs!

22. Exercise. Blending process
sugar u1=F1
y1 = F (given flowrate)
y2 = x (given sugar fraction)
water u2=F2
Mass balances (no dynamics)
Total: F1 + F2 = F
Sugar: 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”?

24. Example 3.10: Separator (pressure vessel)
u1 = zV
y1 = pressure (p)
Feed (d)
y2 = level (h)
u2 = zL
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

25. Separator example....

26. Separator example: Simulations (delay = 1)
Diagonal y2
BEST y1
Off-Diagonal y2

27. Separator example: Simulations (delay = 5)
Diagonal y1
Off-Diagonal
BEST y2

28. Separator example
Close on diagonal if small delay (delay=1) (“high frequencies”)
Close on off-diagonal if larger delay (delay=5) (“intermediate frequencies”)
Could consider use of multivariable controller
BUT NOTE IN THIS CASE: May eliminate one of the interactions by implementing a flow controller!
How and Why?
Let -> denote “has an effect on”
Original: u1=zV -> V -> p -> L=f(zL,p) -> y2=h
With flow controller for liquid flow: L is independent of p, so u1 has no effect on y2.
What about flow controller for vapor flow V?
Does not help because we still have u2=L -> h -> Volume vapor -> y1=p

29. Summary pairing rules (mainly for independent design)
Pairing rule 1. For a stable plant avoid pairings ij that correspond to negative steady-state RGA elements, ij(0)· 0.
Pairing rule 2. 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 closed-loop bandwidth.
Pairing rule 3. Prefer a pairing ij where gij puts minimal restrictions on the achievable bandwidth. Specifically, its effective delay ij should be small.

30. Decentralized control: More details
Most plants can be controlled using decentralized control, but this does not mean that it is a good strategy
If you need a process good model G to tune the decentralized controller, then you may as well use multivariable control, eg. MPC
In the following: Some examples that show that decentralized control can work for any plant

31. Example 1: Diagonal plant (“two rooms”)
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

32. Diagonal plant: Diagonal pairings
Get two independent subsystems:

33. Diagonal pairings....
Simulation with delay included:

34. Off-diagonal pairings (!!?)
Pair on two zero elements !! Loops do not work independently!
But there is some effect when both loops are closed:

35. 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

36. Off-diagonal pairings....
Controller design difficult. After some trial and error:
Performance quite poor, but it works because of the “hidden” feedback loop g12 g21 k1 k2!!
No failure tolerance. WOLULD NOT USE IN PRACTICE!

37. Conclusion “two-room” example
Completely wrong pairing was made workable
Demonstrates that decentralized control can be made to work (with a good enegineer) even for very interactive plants
Lessons decentralized control:
It may work, but still be far from optimal
Use only decentralized control if it does not require a lot of effort to tune
If the plant is interactive and a lot of effort is needed to make decentralized control work (to obtain model), then you may as well use multivariable control (e.g. MPC):
As easy
Much better performance

38. 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)

39. Diagonal pairings
One-way interactive:

40. 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

41. Diagonal pairings.....

42. 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*

44. Comparison of diagonal and off-diagonal pairings
OK performance,
but no failure tolerance
if loop 2 fails

45. 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

46. Typical: Pairing on large RGA-elements gives interactions and sluggish (slow) response
when both loops are closed

47. Typical: Pairing on RGA-elements between 0 and 1 gives interactions and fast response
when both loops are closed (fast -> be careful about instability)

49. Conclusions decentralized examples
Performance is often OK with decentralized control (even with wrong pairings!)
However, controller design becomes difficult for interactive plants (where pairing on RGA close to identity is not possible)
and independent design may not be possible
and failure tolerance may not be guaranteed
Consider multivariable control if interactions are large!

50. Sometimes useful: Iterative RGA
For large processes, lots of pairing alternatives
RGA evaluated iteratively is helpful for quick screening
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

51. Example of Iterative RGA
Correct pairing

52. The remaining slides are for the PhD course only

53. Stability of Decentralized control systems
Question: If we stabilize the individual loops, will the overall closed-loop system be stable?
E is relative uncertainty
is complementary sensitivity for diagonal plant

54. Stability of Decentralized control systems
Question: If we stabilize individual loops, will overall closed-loop system be stable?
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

55. Mu-Interaction measure
Closed-loop stability if
At low frequencies, for integral control )

56. 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

57. 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.

58. Example PRGA: Distillation
see book