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

Outline Multivariable plants RGA Decentralized control Pairing rules Examples

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)

Simplest multivariable controller: Decentralized control

Multivariable decoupling control K y u Dy Kdiag Du G K = Du Kdiag Dy

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

One-way decoupling ”Feed-forward” implementation u’1 u1 y1 r1-y1 c1 -g12/g11 g11 g12 g21 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

(Two-way) decoupling: ”Feed-forward” implementation GD = decoupled process u1 y1 r1-y1 c1 -g12/g11 g11 g12 g21 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

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

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

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

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

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”

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

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

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

(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!)

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

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.

Example Only diagonal pairings give positive steady-state RGAs!

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”?

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

Separator example....

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

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

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

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.

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

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

Diagonal plant: Diagonal pairings Get two independent subsystems:

Diagonal pairings.... Simulation with delay included:

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

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

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!

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

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)

Diagonal pairings One-way interactive:

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

Diagonal pairings.....

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*

Comparison of diagonal and off-diagonal pairings 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

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

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

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!

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

Example of Iterative RGA Correct pairing

The remaining slides are for the PhD course only

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

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

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

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

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.

Example PRGA: Distillation see book