good explanation of svd

7/28/2019 Good Explanation of sVd

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CHAPTER 6:

Decentralized Control(Multi-loop control)

Introduction

The first step of designing the decentralized is to determine the control

structure and control configuration.

Decent r al i zed cont r ol i s al ways conduct ed on squar esyst em, i . e. number of MV equal number of CV.

The process consists ofN MV's and M CV's, therefore, there are three

cases:

IfN = M (Square system), then control loop configuration, i.e.input output pairing, should be determined.

There are several possible configurations of control loops. Thenumber of different loop configuration increase rapidly withN:

ForN= 3 we have 3! = 6 different loop configurationsForN= 4 we have 4! = 24 different loop configurations

ForN= 5 we have 5! = 120 different loop configurations

IfN>M, therefore, we need to extract the bestMMV's to be usedwith the MCV's. This is called control structure design. Having

determined the best structure, we need to go back to step 1 and

determine the loop configuration. The remaining r = N Minputscan be used in split-range or left for emergencies.

IfN< M, then there are r = M N control variables can not becontrolled. In this case, the r controlled variables that have thelowest priority should be taken out of the control objective list or

controlled through override scheme. For the remainingNCV's, the

loop configuration should be determined, i.e. step 1. Non-squareRGA is useful to role out some outputs.


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1. Control loop configuration

Once all MV's and CV's are determined, we need to decide how they are

going to be interconnected through control loops. This means what output

measurement will actuate a given MV, or what MV will be used toregulate a given output measurement.

There are large numbers of loop pairing, choosing the "best"

configuration is a critical task. Various criteria can be used to select the

bestpairing:

Use plant experience and physical reasoning, qualitative method Use a quantitative method

The two common quantitative methods are:

RGA method, which determine the control configuration that yieldcontrol loops with minimum interaction.

SVD method1.1 Loop pairing using the RGA

Now we will consider how the RGA may be used as a guide for selection

of input/output pairs that lead to minimum interaction among controlloops. The interpretation of the values of the RGA can be classified

according to the following categories:

1. ij = 1, indicates that open loop gain between yi and uj isidentical to the closed-loop gain. Loop i will not be subject toretaliatory actions from other control loops when they are closed.

Thus, uj can controlyi without interference from other control loop.Pairing recommendation: Pairingyi and u will therefore be ideal.

2. ij = 0, indicates that open-loop gain betweenyi and uj is zero.This means uj has no direct influence on yi.Pairing recommendation: Do not pairyi with uj.

3. 0 < ij < 1, indicating the open-loop gain between yi and uj issmaller than the closed-loop gain. Since the closed-loop gain is the

sum of the open-loop gain and the retaliatory effect from the other

loops, the loops are definitely interacting.

Pairing recommendation: if possible avoid pairing yi with ujwheneverij = 0.5.


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4. ij > 1, indicating that the open-loop gain between yi and uj islarger than the closed-loop gain. The loops interact, and the

retaliatory effect from the other loops acts in opposition to the main

effect ofu jandyi.Pairing recommendation: where possible, do not pairyi with uj if

ijtakes a very high value, e.g. >25.

5. ij < 0, indicating that open-loop and closed-loop gains betweenyi and uj have opposite signs. The loops interact, and the retaliatory

effect from the other loops is not only in opposition to the main

effect, but also the more dominant of the two effects.

Pairing recommendation: avoid pairing yi with uj.

The foregoing discussion leads to the following rule:

RGA RULEA: pair input and output variables that have positive RGA

elements and closets to one.

NIEDERLINSKY INDEX

Even though pairing Rule A is usually sufficient in most cases; it does not consider

the stability of the resulting control structure. Therefore, it is necessary to check the

stability of the resulted control structure. This can be according to the Niederlinskytheorem.

Consider the multivariable system whose input and output variables have been

paired as follows: , resulting in a transfer function

model of the form:

nn

nnmymymy ,,, 2211 K

Guy =

In this model, each element ofG, gii, is rational and open loop stable. Furthermore,

assume there are no individual feedback controllers with integral action and eachcontroller is stable when the othern-1 loops are open. When all loops are closed, the

system will be unstable for all possible values of controller parameters (Structurally

monatomic unstable), if the Niederlinsky index, Ndefined in the following equation

is negative.

=

=

==

n

i

ii

SS

n

i

ii K

K

g

GN

11


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RULEB: any pairing is unacceptable if it leads to a control system for

which the Niederlinsky index is negative.

1.1.2 Examples

Example 1: Consider the mixer process, which can be modeled at steady

state as follows:

21 FFF +=

F

FFx 21

+=

MixerF

1, x

1

F2, x

2

F, x

Fi gur e 1: Mi xer exampl e

The outputs are F, x

The inputs are F1, F2Create the steady state transfer function (i.e., linearize):

11

=

F

F

12

=

F

F

F

x

F

Fx

F

F

F

x )()( =

==

1122

2

1

F

x

F

xF

F

F

F

x =

=

=

22

1

2

Therefore:


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=

2

1

1

11

F

F

F

x

F

xx

F

The operating condition is F= 200 mole/h, x = 0.6

=

32

10001000K

=

6040

4060

..

..

The RGA recommend pairing Fwith F1 andx with F2.

Mixer

F1, x

1

F2, x

2

F, x

FTFC

CTCC

Fi gur e 2: Mi xer under f eedback cont r ol

Because all relative gains are close to 0.5, the control loop interaction will

be serious.

Example 2: The relative gain for a 4X4 refinery distillation column is

given as follows:

=

91919000322150

19127031431350

1541286042900110

164008015009310

....

....

....

....

The recommended pairing is,y1-u1,y2-u4,y3-u2,y4-u3


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Example 3: Consider the following system:

=

3

111

1311

113

5

K

=

15454

54154

545410

..

..

..

The recommended pairing is 1-1/2-2/3-3. According to the Niederlisnky

rule:

1450.)det( == KK

27

5

3

1

3

1

3

53

1

===

))()((

i

iiK

Therefore;

03

1


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Which corresponds to:

=

13

11

3

111

113

5

K

Computing the Niederlinski index gives:

N= 4/45 stable looping

1.1.3 Shortcomings

The above RGA pairing method ignores process dynamics. It has been

shown that if the transfer function has very large time delay or time

constant relative to the others, steady state RGA analysis provide anincorrect recommendation.

Example: Consider the following transfer function:

++

++

=

2

1

2

1

110

2

1

51

1

51

110

2

u

u

s

e

s

e

s

e

s

e

y

y

ss

ss

.

.

=

251512

..K

=

640360

360640

..

..

The recommended pairing is 1-1/2-2.

However, the off-diagonal elements indicates that y1

responds ten times

faster to u2 than u1 because their relative time constant.


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Computer simulation indicates that the opposite pairing is better

performance.

Show SIMLNK simulation


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1.2 Singular value decomposition

1.2.1 Definition:

The singular value decomposition of a matrix K results in three

component matrices as follows:

TVUK =

whereK: is an nxn matrix

U: is an nxn orhtonormal matrix, with its column is called left

singular vectorV: is an mxm orhtonormal matrix, with its column is called right

singular vector

: is nxm diagonal matrix of scalars called the singular values thatare organized in decending order.

1.2.2 Physical interpretation:

K is the steady state gain matrix, contains the sensitivity of each

measured variable (sensor) to change in the manipulated variable.

It is very important that elements of K be scaled.

u

yK

=

A good physical scaling should give:

MVof%range

span%sensor=K

U = U1:U2: Un provides the most appropriate coordinate for

viewing the process sensor. The first column indicates the easiest

sensor direction in which the system can be changed by the MV.

V = V1:V2: Vm provides the most appropriate coordinate for

viewing the MV. The first column of VT

indicates the combination

of control action that has the most effect on the system.

= diag(1, 2, m) provide ideal decoupled gain of the open-loopprocess. The ratio of the maximum singular value to the minimum

singular value (max/min) is the condition number.


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The condition number is a measure of the difficulty of the decoupled

multivariable control problem.

Large condition number indicates that it is difficult or impossible to

accomplish all the control objectives.

1.2.3 Example

Consider the mixer of two different temperature streams:

FT TTF

1, T

h

F2

, Tc

Hot water

Cold water Fi gur e 3: Mi xi ng t hermal st r eams

The linearized model is given as follows:

=

2

1

F

F

F

T

m

mK

For an operating condition of F1 = 10 gpm, F2 = 20 gpm, Th = 100o

F, Tc =65

oF, the steady state gain matrix have the following numerical values:

=

000.1000.1

3889.07778.0K

Which decomposes to:

= 276.0961.0

961.0276.0U

=

809.0587.0

587.0809.0V

=

803.00

0453.1


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7.10.803

1.4531 ==

Assume the feed changed by the amount shown in the circle. The effect on the outputs is shown by the ellipse, which indicates

the following:

The major effect, which corresponds to the first column U1,increases both outputs but with more emphasis to Fm.

The minor effect, which corresponds to the first column U2,decreases Tm and increases Fm.

The second effect is minor compared to the first one because 2

good explanation of svd

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