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 y D from 0.95 to 0.97, and x B from 0.02 to 0.03” “Increasing boilup V from 1.5 to 1.6 changes y D from 0.95 to 0.94, and x B from 0.02 to 0.01” Steady-State Gain Matrix (Time constant 50 min for y D ) (time constant 40 min for x B )

4 Simplest multivariable controller: Decentralized control

5 Multivariable decoupling control K diag DyDy G DuDu K K = D u K diag D y yu

6 Most common: input decoupling K = D K diag 1.Dynamic decoupling, D = G(s) -1 2.Steady-state decoupling, D = G(0) -1 3.Approximate decoupling, D = G(jw) -1 4.Desirable: Keep diagonal elements in GD unchanged –GD = G diag 5.«Simplified decoupling»: Diagonal elements in D are 1. –GD = ([G -1 ] diag ) -1 –similar to feedforward control 6.«Inverted» decoupling (Shinskey). Combines 4&5 7.One-way decoupling K diag D G

7 One-way Decoupling c2c2 g 11 g 12 g 21 g 22 y1y1 y2y2 r 1 -y 1 r 2 -y 2 u2u2 u1u1 c1c1 Process: y1 = g11 u1 + g12 u2 (1) y2 = g21 u1 + g22 u2 (2) Consider u2 as disturbance for contol of y1. Think «feedforward»: Adjust u1 to make y1=0. (1) gives u1 = - (g12/g11) u2 -g 12 /g 11 GD=one-way coupled process

8 Simplified Decoupling: Using standard implementation (Seborg) c2c2 g 11 g 12 g 21 g 22 y1y1 y2y2 r 1 -y 1 r 2 -y 2 u2u2 u1u1 c1c1 Decoupler has 1’s on diagonal: D = [1 -g12/g11; -g21/g22 1] ….but note that diagonal elements of decoupled process are different from G. Problem for tuning! Process: y1 = g11 u1 + g12 u2 Decoupled process: y1 = (g11-g21*g12/g22) u1’ + 0*u2’ Similar for y2. -g 12 /g 11 -g 21 /g 22 GD=decoupled process= ([G -1 ] diag ) -1 u’ 1 u’ 2

9 Simplified Decoupling: Using Shinskey’s improved «inverted» implementation c2c2 g 11 g 12 g 21 g 22 y1y1 y2y2 r 1 -y 1 r 2 -y 2 u2u2 u1u1 c1c1 -g 12 /g 11 -g 21 /g 22 decoupled process = G diag u’ 1 Note: Follows directly from “feedforward” thinking Advantage: Decoupled process has same diagonal elements as G. Easy tuning! Proof: y 1 = g 11 u 1 + g 12 u 2, where u 1 = u 1 ’ – (g 12 /g 11 )u 2. Get for decoupled process: y 1 = g 11 u’ 1 + 0* u 2 ’ y 2 = 0 * u’ 1 + g 22 u 2 ’ u’ 2

10 Decentralized control: 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

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

12 Decentralized control: Design (tuning) of each controller k i (s) 1.Fully coordinated design –can give optimal –BUT: requires full model, so may as well use multivariable controller –not used much in practice (process control) 2.Independent design –Base design on “paired element” –Can get failure tolerance –Not possible for interactive plants. RGA useful tool 3.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

13 Analysis of Multivariable processes INTERACTIONS: Caused by nonzero g 12 and/or g 21 Process Model 2x2 y1y1 y2y2 g 12 g 21 g 11 g 22 u2u2 u1u1

14 Consider effect of u 1 on y 1 1)“ Open-loop ” (C 2 = 0): y 1 = g 11 (s) · u 1 2)“ Closed-loop ” (close loop 2, C 2 ≠0): Change caused by “ interactions ” C2C2 y1y1 y2y2 g 12 g 21 g 11 g 22 u2u2 u1u1

15 Limiting Case C 2 →∞ (perfect control of y 2 ) How much has “gain” from u 1 to y 1 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

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

17 (1)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!) Use of RGA: 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 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)

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19 (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!)

20 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

21 Cookbook for pairing and RGA 1.Indentify the n y’s (CVs) and n u’s (MVs) for which you are uncertain about the pairings 2.Find the transfer matrix, y = G(s) u (preferably with dynamics) 3.Compute RGA as a function of frequency, RGA = G(jw) * G(jw) -T 4.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?) 5.Pairings: Consider the steady-state RGA (s=0). AVOID pairing on negative RGA-elements!! 6.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 7.Pairings: Prefer pairing on variables with good controllability (=small effective delay). Pair “close”! 8.If you cannot find satisfying pairings, then consider multivariable control, for example, decoupling or MPC.

22 Example Only diagonal pairings give positive steady-state RGAs!

23 Exercise. Blending process Mass balances (no dynamics) –Total: F 1 + F 2 = F –Sugar: F 1 = x F (a)Linearize balances and introduce: u 1 =dF 1, u 2 =dF 2, y 1 =F 1, y 2 =x, (b)Obtain gain matrix G (y = G u) (c)Nominal values are x=0.2 [kg/kg] and F=2 [kg/s]. Find G (d)Compute RGA and suggest pairings (e)Does the pairing choice agree with “common sense”? sugar u 1 =F 1 water u 2 =F 2 y 1 = F (given flowrate) y 2 = x (given sugar fraction)

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25 Example 3.10: Separator (pressure vessel) Pairings? Would expect y 1 /u 1 and y 2 /u 2 But process is strongly coupled at intermediate frequencies. Why? Frequency-dependent RGA suggests opposite pairing at intermediate frequencies u 1 = z V u 2 = z L Feed (d) y 2 = level (h) y 1 = pressure (p)

26 Separator example....

27 Separator example: Simulations (delay = 1) Diagonal Off-Diagonal y1y1 y2y2 y1y1 y2y2 BEST

28 Separator example: Simulations (delay = 5) Diagonal Off-Diagonal y1y1 y2y2 y1y1 y2y2 BEST

29 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! –Linear valve: q L = z L C V ( ¢ p / ½ ) 1/2 –New u2 = qL –How and Why? –Let -> denote “has an effect on” –Original: u 1 =z V -> V -> p -> qL=f(z L,p) -> y 2 =h –With flow controller for liquid flow: qL is independent of p, so u 1 has no effect on y 2. –What about flow controller for vapor flow V? Does not help because we still have u 2 =L -> h -> Volume vapor -> y 1 =p

30 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 g ij puts minimal restrictions on the achievable bandwidth. Specifically, its effective delay  ij should be small.

31 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

32 Example 1: Diagonal plant (“two rooms”) Simulations (and for tuning): Add delay 0.5 in each input Simulations setpoint changes: r 1 =1 at t=0 and r 2 =1 at t=20 Performance: Want |y 1 -r 1 | and |y 2 -r 2 | less than 1 G (and RGA): Clear that diagonal pairings are preferred

33 Diagonal plant: Diagonal pairings Get two independent subsystems:

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

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

36 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 ?? 1 2

37 Off-diagonal pairings.... –Performance quite poor, but it works because of the “hidden” feedback loop g 12 g 21 k 1 k 2 !! –No failure tolerance. WOLULD NOT USE IN PRACTICE! Controller design difficult. After some trial and error:

38 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: 1.It may work, but still be far from optimal 2.Use only decentralized control if it does not require a lot of effort to tune 3.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): 1.As easy 2.Much better performance

39 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 (g 12 =5)

40 Diagonal pairings One-way interactive:

41 Diagonal pairings.... With  1 =  2 the “interaction” term (from r 1 to y 2 ) is about 2.5 Need loop 1 to be “slow” to reduce interactions: Need  1 ¸ 5  2 Closed-loop response (delay neglected):

42 Diagonal pairings.....

43 Off-diagonal pairings Pair on one zero element (g 12 =g 11 * =0) BUT pair on g 21 =g * 22 =5: may use sequential design: Start by tuning k 2 *

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45 Offdiagonal: –OK performance, –but no failure tolerance if loop 2 fails Comparison of diagonal and off-diagonal pairings

46 Example 3: Two-way interactive plant -Already considered case g 12 =0 (RGA=I) -g 12 =0.2: plant is singular (RGA= 1 ) -will consider diagonal parings for: (a) g 12 = 0.17, (b) g 12 = -0.2, (c) g 12 = -1 Controller: with  1 =5 and  2 =1

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

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

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50 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!

51 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

52 Example of Iterative RGA Correct pairing