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Decentralized control

Sigurd Skogestad

Department of Chemical Engineering

Norwegian University of Science and Tecnology (NTNU)

Trondheim, Norway

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Outline

• Multivariable plants

• RGA

• Decentralized control

• Pairing rules

• Examples

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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

B

2221

1211

B

D

Δx2outputonΔL1inputofEffect

0.10.1

0.10.2

1.51.6

0.020.01

1.01.1

0.020.031.51.6

0.950.94

1.01.1

0.950.97

0gg

0gg0G

ΔV

ΔL0G

Δx

ΔY

40s1

0.1

40s1

0.150s1

0.1

50s1

20.

G

:dynamicsincludealsoCan

s

B

D

x

y

(Time constant 50 min for yD)

(time constant 40 min for xB)

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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

G

y1

y2

g12

g21

g11

g22u2

u1

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 11 1 12 2

2 21 1 22 2

" "

( )

( )

Open loop

y s g s u s g s u s

y s g s u s g s u s

-

= +

= +

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Consider Effect of u1 on y1

1) “Open-loop” (C2 = 0): y1 = g11(s)·u1

2) Closed-loop” (close loop 2, C2≠0)

Change caused by “interactions”

( ) uæ ö× ÷ç ÷= -ç ÷ç ÷ç + ×è ø

12 21 21 11 1

22 2

g g Cy g s

1 g C

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Limiting Case C2→∞ (perfect control of y2)

( )( )

CL

×= = = =

- -

def1 1 RGAOL 11

1112 21 12 211 1

1122 11 22

y /μ g 1RelativeGain λ

g g g gy /μ g 1g g g

( ) uæ ö÷ç ÷= -ç ÷ç ÷çè ø

12 211 11 1

22

g gy g s

g

( )( )

( )( )

( )( )

( )( )

u

é ùê úê úé ù ê úê ú= = = ê úê ú ê úë û ê úê úë û

1 1 1 2OL OL

1 1 1 211 12 CL CL

21 22 2 1 2 2OL OL

2 1 2 2CL CL

y /u y /u

y /u y /uλ λRGA Λ

λ λ y y /u

y u y /u

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

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12RGA

2

0.10.2

0.10.11

1λ,

0.10.1

0.10.2G

0.5

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Example from before

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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 closed-loop 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 0.1G o

0.1 0.1

1 1 2

2 1 2

y = 0.2 u -0.1 u

y = 0.1u -0.1u

× ×

u

u

u

u

1 1

2 2

1 2

2 1

2 -1RGA =

-1 2

Only acceptablepairings :

y

y

Not recommended :

y

y

é ùê úê úë û

«

«

«

«

With integral action :

Negative RGA individual

loop unstable + overall system unstable

when loops saturate

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(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

example

1 1 91 90G= RGA

0.9 0.91 90 91

Large (BAD!)

21

12 12

0.9

1Relativechange - makesmatrixsingular!

1ˆThen g g 1 0.91

90

l=

æ ö÷ç= + =÷ç ÷çè ø

11.1%

90

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Singular Matrix: Cannot take inverse, that is, decoupler hopeless.

Control difficult

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Exercise. Blending process

• Mass balances (no dynamics)– Total: F1 + F2 = F– Sugar: F1 = x F

(a) Linearize balances and introduce: u1=dF1, u2=dF2, y1=F1, y2=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 u1=F1

water u2=F2

y1 = F (given flowrate)

y2 = x (given sugar fraction)

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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– 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

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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 closed-loop 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

TC

??1 2

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Off-diagonal pairings....

– Performance quite poor, but it works because of the “hidden” feedback loop g12 g21 k1 k2!!

– No failure tolerance

Controller design difficult. After some trial and error:

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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....

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

Closed-loop response (delay neglected):

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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

– OK performance,

– but no failure tolerance

if loop 2 fails

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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)

• 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

u1 = V

u2 = L

Feed (d)y2 = level (h)

y1 = pressure (p)

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Separator example....

38 0 20 40 60 80 100 120 140 160 180 200

-0.5

0

0.5

1

1.5

2

2.5

Time

y

0 20 40 60 80 100 120 140 160 180 200-0.5

0

0.5

1

1.5

Time

y

Separator example: Simulations (delay = 1)

Diagonal

Off-Diagonal

y1

y2

y1

y2

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Separator example: Simulations (delay = 5)

0 100 200 300 400 500 600 700 800 900 1000-4

-2

0

2

Time

y

0 100 200 300 400 500 600 700 800 900 1000-1

0

1

2

3

Time

y

Diagonal

Off-Diagonal

y1

y2

y1

y2

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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

• 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

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Example of Iterative RGA

Correct pairing

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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?

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

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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