Control of Multiple-Input, Multiple-Output Processes

Multiloop controllers• Modeling the interactions• Relative Gain Array (RGA)• Singular Value Analysis (SVA)• Decoupling strategiesC

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Control of multivariable processes

In practical control problems there typically are a number of process variables which must be controlled and a number which can be manipulated

Example: product quality and through put must usually be controlled.

• Several simple physical examples are shown in Fig. 18.1.

Note "process interactions" between controlled and manipulated variables.

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SEE FIGURE 18.1in text.

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• Controlled Variables:

• Manipulated Variables:

BDBD hhPxx and ,,,,

BD Q and ,Q,R,B,D

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In this chapter we will be concerned with characterizing process interactions and selecting an appropriate multiloop control configuration.

If process interactions are significant, even the best multiloop control system may not provide satisfactory control.

In these situations there are incentives for considering multivariable control strategies

Definitions:

•Multiloop control: Each manipulated variable depends on only a single controlled variable, i.e., a set of conventional feedback controllers.

•Multivariable Control: Each manipulated variable can depend on two or more of the controlled variables.

Examples: decoupling control, model predictive control

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Multiloop Control Strategy•Typical industrial approach•Consists of using n standard FB controllers (e.g. PID), one for

each controlled variable.

•Control system design1. Select controlled and manipulated variables.2. Select pairing of controlled and manipulated variables.3. Specify types of FB controllers.

Example: 2 x 2 system

Two possible controller pairings:U1 with Y1, U2 with Y2 …orU1 with Y2, U2 with Y1

Note: For n x n system, n! possible pairing configurations.

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Transfer Function Model (2 x 2 system)

Two controlled variables and two manipulated variables(4 transfer functions required)

Thus, the input-output relations for the process can be written as:

)()(

)(),(

)(

)(

)()(

)(),(

)(

)(

222

221

1

2

122

111

1

1

sGsU

sYsG

sU

sY

sGsU

sYsG

sU

sY

PP

PP

)()()()()(

)()()()()(

2221212

2121111

sUsGsUsGsY

sUsGsUsGsY

PP

PP

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Or in vector-matrix notation as,

where Y(s) and U(s) are vectors,

And Gp(s) is the transfer function matrix for the process

)()()( sUsGsY P

)(

)()(,

)(

)()(

2

1

2

1

sU

sUsU

sY

sYsY

)s(G)s(G

)s(G)s(G)s(G

22P21P

12P11PP

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Control-loop interactions

• Process interactions may induce undesirable

interactions between two or more control loops.

Example: 2 x 2 system

Control loop interactions are due to the presence

of a third feedback loop.• Problems arising from control loop interactions

i) Closed -loop system may become destabilized.

ii) Controller tuning becomes more difficult

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Block Diagram Analysis

For the multiloop control configuration the transfer function between a controlled and a manipulated variable depends on whether the other feedback control loops are open or closed.

Example: 2 x 2 system, 1-1/2 -2 pairingFrom block diagram algebra we can show

Note that the last expression contains GC2 .

),()(

)(11

1

1 sGsU

sYP

222

2211211

1

1

1)(

)(

PC

CPPP GG

GGGG

sU

sY

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(second loop open)

(second loop closed)

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Figure 18.6 Stability region for Example 18.2 with 1-1/2-2 controller pairing

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Figure 18.7 Stability region for Example 18.2 with 1-2/2-1 controller pairing

Relative gain array

• Provides two useful types of information:

1) Measure of process interactions

2) Recommendation about best pairing of

controlled and manipulated variables.

• Requires knowledge of s.s. gains but not

process dynamics.

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Example of RGA Analysis: 2 x 2 system• Steady-state process model,

The RGA is defined as:

where the relative gain, ij, relates the ith controlled variable and the jth manipulated variable

2221212

2121111

UKUKY

UKUKY

2221

1211RGA

gain loop-closed

gain loop-openij

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Scaling Properties:

i) ij is dimensionless

ii)

For 2 x 2 system,

Recommended Controller Pairing

Corresponds to the ij which has the largest positive value.

0.1j

iji

ij

211112

2211

211211 1,

KKKK

1

1

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In general:1. Pairings which correspond to negative pairings should not be selected.2. Otherwise, choose the pairing which has ij closest

to one.

Examples:Examples: Process Gain Relative Gain Matrix, : Array, :

22

11

K0

0K

0K

K0

21

12

22

1211

K0

KK

2221

11

KK

0K

10

01

01

10

10

01

10

01

K

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211112

2211

211211 1,

KKKK

1

1

Recall, for 2X2 systems...

2221212

2121111

UKUKY

UKUKY

EXAMPLE:EXAMPLE:

29.229.1

29.129.2

25.1

5.12

KK

KKK

2221

1211

Recommended pairing is Y1 and U1, Y2 and U2.

EXAMPLE:EXAMPLE:

64.036.0

36.064.0

25.1

5.12K

Recommended pairing is Y1 with U1, Y2 with U2.

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EXAMPLE: Thermal Mixing SystemEXAMPLE: Thermal Mixing System

The RGA can be expressed in terms of the manipulated variables:

Note that each relative gain is between 0 and 1. Recommendedcontroller pairing depends on nominal values of W,T, Th, and Tc.

See Exercise 18.16

ch

hc

c

hc

h

hc

h

hc

c

WW

WWW

WWW

WWW

WWW

T

W

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EXAMPLE: Ill-conditioned Gain Matrix

y = Ku

2 x 2 process y1 = 5 u1 + 8 u2

y2 = 10 u1 + 15.8 u2

specify operating point y, solve for u

-1

Adj

det

K= =

Ku K y y

11 22 11 2211

11 22 12 21

RGA : det

K K K K

K K K K K

effect of det K → 0 ?

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RGA for Higher-Order Systems:

For and n x n system,

Each ij can be calculated from the relation

Where Kij is the (i,j) -element in the steady-state gain matrix,

And Hij is the (i,j) -element of the .

Note that,

1 2

1 11 12 1

2 21 22 2

1 1

n

n

n

n n n nn

U U UY

Y

Y

ijijij HK

UKY

T1KH

HK

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EXAMPLE: HydrocrackerEXAMPLE: Hydrocracker

The RGA for a hydrocracker has been reported as,

Recommended controller pairing?

4321

4

3

2

1

919.1900.0030.2215.0

910.1270.0314.3135.0

154.1286.0429.0011.0

164.0080.0150.0931.0UUUU

Y

Y

Y

Y

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Singular Value AnalysisK = W VT

is diagonal matrix of singular values(1, 2, …, r)

The singular values are the positive square roots of the eigenvalues of

KTK (r = rank of KTK)W,V are input and output singular vectors Columns of W and V are orthonormal. Also

WWT = IVVT = I

Calculate , W, V using MATLAB (svd = singular value decomposition)Condition number (CN) is the ratio of the largest to the smallest singular value and indicates if K is ill-conditioned.

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CN is a measure of sensitivity of the matrix properties to changes in a specific element.

Consider

(RGA) = 1.0

If K12 changes from 0 to 0.1, then K becomes a singular matrix, which corresponds to a process that is hard to control.

RGA and SVA used together can indicate whether a process is easy (or hard) to control.

K is poorly conditioned when CN is a large number (e.g., > 10). Hence small changes in the model for this process can make it very difficult to control.

1 0

10 1

K

10.1 0( ) = CN = 101

0 0.1K

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Selection of Inputs and Outputs

• Arrange the singular values in order of largest to smallest and look for any σi/σi-1 > 10; then one or more inputs (or outputs) can be deleted.

• Delete one row and one column of K at a time and evaluate the properties of the reduced gain matrix.

• Example:

0.48 0.90 0.006

0.52 0.95 0.008

0.90 0.95 0.020

K

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CN = 166.5 (σ1/σ3)

The RGA is as follows:

Preliminary pairing: y1-u2, y2-u3,y3-u1.

CN suggests only two output variables can be controlled. Eliminate one input and one output (3x3→2x2).

0.5714 0.3766 0.7292

0.6035 0.4093 0.6843

0.5561 0.8311 0.0066

W

1.618 0 0

0 1.143 0

0 0 0.0097

0.0541 0.9984 0.0151

0.9985 0.0540 0.0068

0.0060 0.0154 0.9999

V

2.4376 3.0241 0.4135

1.2211 0.7617 0.5407

2.2165 1.2623 0.0458

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Alternative Strategies for Dealing with Undesirable Control Loop Interactions

1. "Detune" one or more FB controllers.2. Select different manipulated or controlled variables. e.g., nonlinear functions of original variables3. Use a decoupling control scheme.4. Use some other type of multivariable control scheme.

Decoupling Control Systems

Basic Idea: Use additional controllers to compensate for processinteractions and thus reduce control loop interactions

Ideally, decoupling control allows setpoint changes to affect onlythe desired controlled variables.

Typically, decoupling controllers are designed using a simple process model (e.g. steady state model or transfer function model)

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Design Equations:

We want cross-controller, GC12, to cancel out the effect of U2 on Y1.Thus, we would like,

Since U2 0 (in general), then

Similarly, we want G21 to cancel the effect of M1 on C2. Thus, werequire that...

cf. with design equations for FF control based on block diagramanalysis

12 11 2 12 2 0P PT G U G U

1212

11

P

P

GT

G

21 22 1 21 1

2121

22

0P P

P

P

T G U G U

GT

G

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Process Interaction

Corrective Action (via “cross-controller” or “decoupler”).Ideal Decouplers:

1212

11

2121

22

( )( )

( )

( )( )

( )

P

P

P

P

G sT s

G s

G sT s

G s

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Alternatives to Complete Decoupling

• Static Decoupling (use SS gains)• Partial Decoupling (either GC12 or GC21 is set equal to zero)

Variations on a Theme:

•Partial Decoupling:Use only one “cross-controller.”

•Static Decoupling:Design to eliminate SS interactionsIdeal decouplers are merely gains:

•Nonlinear DecouplingAppropriate for nonlinear processes.

1212

11

2121

22

P

P

P

P

KT

K

KT

K

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