Prof.tangirala Directionality Interaction

Arun K. Tangirala (IIT Madras) Lecture Series, University of Alberta June 10, 2011

INTERACTION ASSESSMENT IN LINEAR MULTIVARIABLE PROCESSES USING DIRECTED

SPECTRAL DECOMPOSITION

Arun K. Tangirala

Dept. of Chemical Engineering, IIT Madras Chennai, Tamilnadu, India


OUTLINE

2


OUTLINE

Motivation

Review

RGA, Dynamic RGA, Sensi0vity func0ons

Directed Analysis

Main results

Benchmark for interac0on assessment

Interac0on quan0ca0on

Simulation study

Concluding Remarks

2


INTERACTIONS


INTERACTIONS

u1

y1

G11-Gc1

ey1

eu1


INTERACTIONS

u1 u2

y2y1

G11 G22

G21

G12-Gc2-Gc1

ey1

eu1 eu2

ey2

u1

y1

G11-Gc1

ey1

eu1


INTERACTIONS

Interaction: Effect felt in one loop due to changes (disturbances / setpoint changes) in other loops (typical of all MIMO system)

Impact of interaction:

Reduces performance

Can lead to instabili0es

Not necessarily harmful!

Problems of interest:

1. Quantify interactions

2. Relate interactions to a performance metric

u1 u2

y2y1

G11 G22

G21

G12-Gc2-Gc1

ey1

eu1 eu2

ey2

u1

y1

G11-Gc1

ey1

eu1


WHATS THE USE?

Controller Design

For MIMO systems, two possible control congura0ons - (i) decentralized (mul0loop) controllers and (ii) mul0variable controllers

Control Loop Performance Assessment

A typical performance metric is variance. Can we determine the contribu0ons of interac0ons to variance?

4

Design controllers for multivariable systems such that interactions are at a minimum or even better, beneficial

For a given multivariable closed-loop system, assess the strength of interactions and relate it to performance


MEASURING INTERACTIONS

Relative Gain Array (Bristol, 1966)

Dynamic RGA (Witcher & McAvoy, 1977, Other researchers, later years)

Loop under study remains open

Outputs of all other loops held at their set-points

Both measures assume perfect control

5

ij =(yi/uj)all loops open

(yi/uj)all other loops closed except loop (yi - uj)

steady state

= KKT

(s) = G(s)G(s)T


INTERACTION MEASURES

Generalized Dynamic Relative Gain (Huang et al, 1993)

Actual controllers rather than perfect controllers are used

Loop decomposition method (Zhu and Jutan, 1996)

Takes into account all perturba0ons; Rela0ve interac0on, Absolute interac0on

Joint stationary representation approach (Seppala et al, 2002)

Use the mul0variate impulse response func0on (the H matrix in the VMA representa0on)

Semi-quan0ta0ve approach to interac0on

Performance RGA (Skogestad et al)

Deni0on based on exact factoriza0on of sensi0vity func0on; limited interpreta0on

Several other methods

6


WHAT DO WE DESIRE?

The interaction measure should quantitatively correspond to a performance metric (e.g., variance)

We should be able to compute from models (for controller design)

We should be able to estimate it from data (for performance assessment)

7


INTERACTION & DIRECTIONALITY

Interaction is a directed phenomenon (direction matters)

Effect felt (by an output) through indirect pathways arising due to connections with other loops

Dierence between a SISO loop and a MIMO control system

The direct & indirect transfer functions play a key role in quantifying interaction

Basic question:

8

What are the contributions of the indirect pathways to the variance of a closed-loop output?


DIRECTED VARIANCE DECOMPOSITION

Variance decomposition in frequency domain:

Set up

so that

Spectral Factorization (jointly stationary process driven by white-noise):

9

xx() = H()eH()

2yi =1

2

yiyi() d

xx() =

y1y1() y1u1() y1ym() y1um()u1y1() u1u1() u1ym() u1um()

......

.... . .

......

......

.... . .

......

ymy1() ymu1() ymym() ymum()umy1() umu1() umym() umum()

x =y1 ym u1 um

T


WHAT DOES H CONTAIN? DTF

The DTF (Saito and Harashima, Kaminski and Blinowska) is the normalized hij() and is a non-parametric quantity by definition

Its estimation, however, is carried out using a vector auto-regressive (VAR) modelling of the time-series

10

Directed Transfer Function (DTF)hij() is the net transfer function from the (white-noise) innovations in xj to the ith variable xi (in that direction)

H

ey1ey2eu1eu2

y1y2u1u2


ESTIMATION OF DTF

Construct VAR / VMA model

VAR models are easier to construct since LS es0ma0on methods can be used. Then,

Each element hij(): Total effect of the jth source on the ith variable

The transfer function hij() consists of direct and indirect components

11

x[k] =p

r=1

Ar x[k r] + e[k] OR x[k] =q

r=1

Hre[k r] + e[k]

H() = A1() =

h11() . . . h1m()h21() . . . h2m()

.... . .

...hm1() . . . hmm()

; A() = Ip

r=1

Arerj =

a11() . . . a1m()a21() . . . a2m()

.... . .

...am1() . . . amm()

hij() = hD,ij() + hI,ij() ; hD,ij() =aij()det(Mji())

det(A())i = j


DIRECT(ED) ENERGY TRANSFERS

The total energy transfer is the sum of direct, indirect and an interference terms

Interferences occur due to phase dierences between direct and indirect transfers

They can be either construc0ve or destruc0ve depending on the phase dierence

The term |hii()|2 quantifies the fraction of energy received by xi[k] from its own driving force (and due to unaccounted sources)

For analysis purposes, fix (or force)

12

|hij()|2 = |hD,ij()|2 + |hI,ij()|2 + 2|hD,ij()||hI,ij()|cos(D() I())

e = Inn


PATHWAYS FOR TRANSFER OF ENERGY

13

Total transfer function

Ej() Xi()

Direct transfer function

Ej() Xi()

Indirect transfer function

Indirect energy transfer

Direct energy transfer

Sejej () Sxixi()

Interference effect

Interference effect


A 2X2 SYSTEM

14

u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2


A 2X2 SYSTEM

Decompose spectrum of y1:

14

u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2


A 2X2 SYSTEM


14

y1y1() = |hy1y1()|2 + |hy1u1()|2 + |hy1y2()|2 + |hy1u2()|2 Interaction eects

u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2


A 2X2 SYSTEM


14


u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2

Due to past of y1 and unaccounted variables


A 2X2 SYSTEM


14

|hy1u1()|2 = |hD,y1u1()|2 + |hI,y1u1()|2 + hIF,y1u1() Interaction eects


u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2



A 2X2 SYSTEM


14



Feedback and interaction dependent

u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2



A 2X2 SYSTEM


14



Feedback and interaction dependent

u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2


Idea is to arrive at an interaction and feedback invariant term


DIRECT AND INDIRECT TRANSFER FUNCTIONS

15

hy1y1 =1 +G22Gc2

; hy1u1 =G11(1 +G22Gc2)

+G12G21Gc2

hy1y2 =

G12Gc2 ; hy1u2 =

G12

where = (1 +G11Gc1)(1 +G22Gc2)G12G21Gc2Gc1

u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2

hD,y1u1 =G11(1 +G22Gc2)

= G11hy1y1

hI,y1u1 =G12G21Gc2

=G12G21Gc21 +G22Gc2

hy1y1


SENSITIVITY FUNCTION & THE H MATRIX

16

whereGij() is the transfer function connecting the out-put yi and the input uj . The interaction experienced inith loop, ( yi paired with uj) from all other loops is

i() =yiyi()

|hyiyi()|2 |hD,yiuj ()|

2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)

Contol engineers find it convenient to relate interactionto sensitivity function as it plays a role in controllerdesign and performance assessment. Also it will allow tocompare qualitatively with existing measures.

3.1 Relationship with Sensitivity Function

To observe the connection between multivariate sensi-tivity function matrix S0 and H matrix, we begin with

y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)

The VARmodel for the process in terms of transfer func-tions is given by

I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)

Assuming all output disturbances to be white, by virtueof definition, the sensitivity function S0yy() can berelated to the multivariate frequency response matrixH(). By comparing (28) and (30), the multiloop sen-sitivity function given by

S0yy = (Imm +GpGc)1 (31)

Here we consider a multivariable process Gp and a de-centralized (diagonal) controller Gc.

Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)

Knowing the transfer function matrices Gp and Gc,the process can be represented in a theoretical VARform as given in (29) and A() and H() can be calcu-lated. The term hy1y1() is S0(1, 1) = S011(). Similarrelationships holds for other elements as well, so thathyiyj () = S0ij() . Thus, for the ith loop of a decentral-ized control system,

yiyi()|S0ii()|2

=

Interactionfeedback invariant

1 + |Gij |2 +

Interactionfeedback dependent

i() (34)

The equations (25) / (26) / (34) thus decomposes thenormalized output spectrum into two components, (i)an interaction and feedback invariant (first two terms onRHS i.e. (1 + |Gij()|2) which depends on the chosencontrol loop pairing and (ii) an interaction and feedbackdependent (last term on RHS i.e. i()) term which isdepended on pairing as well as the controller settings.

The LHS of (34) can be nicely interpreted as the spec-trum of the output of the ith loop filtered by the inverseof the diagonal of the multiloop sensitivity function,

Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()

Thus, i() represents the contribution to the varianceof the filtered output.

Equation (34) attracts some nice interpretations. It iswidely known that filtering with the inverse of the sen-sitivity function provides the open loop system [1]. Inthis work, the output spectrum filtering is implementedon the inverse of the diagonal matrix of the sensitivityfunction. This amounts to opening up of that particularloop while keeping all other loops closed.

Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)

S0yiyj = ijth element of the sensitivity matrix of the

equivalent process with ith loop open andS = Imm +GpGc

Proof:

S0yiyjS0yiyi

=cofactor(Syjyi)cofactor(Syiyi)

7


i() =yiyi()


2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)




y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)


I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)




Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)


yiyi()|S0ii()|2

=


1 + |Gij |2 +


i() (34)



Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()



Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)



Proof:

S0yiyjS0yiyi


7


i() =yiyi()


2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)




y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)


I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)




Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)


yiyi()|S0ii()|2

=


1 + |Gij |2 +


i() (34)



Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()



Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)



Proof:

S0yiyjS0yiyi


7



Multi-loop sensitivity for a 2 x 2 system:

16

Syy0 = (I+GpGc)

1 =1

1 +Gc2G22 G12Gc2G21Gc1 1 +Gc1G11

=

S0y1y1 S

0y1y2

S0y2y1 S0y2y2



i() =yiyi()


2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)




y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)


I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)




Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)


yiyi()|S0ii()|2

=


1 + |Gij |2 +


i() (34)



Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()



Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)



Proof:

S0yiyjS0yiyi


7


i() =yiyi()


2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)




y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)


I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)




Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)


yiyi()|S0ii()|2

=


1 + |Gij |2 +


i() (34)



Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()



Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)



Proof:

S0yiyjS0yiyi


7


i() =yiyi()


2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)




y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)


I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)




Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)


yiyi()|S0ii()|2

=


1 + |Gij |2 +


i() (34)



Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()



Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)



Proof:

S0yiyjS0yiyi


7



Multi-loop sensitivity for a 2 x 2 system:

16

Syy0 = (I+GpGc)

1 =1

1 +Gc2G22 G12Gc2G21Gc1 1 +Gc1G11

=

S0y1y1 S

0y1y2

S0y2y1 S0y2y2



i() =yiyi()


2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)




y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)


I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)




Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)


yiyi()|S0ii()|2

=


1 + |Gij |2 +


i() (34)



Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()



Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)



Proof:

S0yiyjS0yiyi


7


i() =yiyi()


2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)




y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)


I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)




Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)


yiyi()|S0ii()|2

=


1 + |Gij |2 +


i() (34)



Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()



Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)



Proof:

S0yiyjS0yiyi


7


i() =yiyi()


2

|hyiyi()|2 1

=yiyi()

|hyiyi()|2 (1 + |Gij()|2) (27)




y

u

=

S0yy S0yuS0uy S0uu

eyeu

= H e (28)


I GpGc I

y

u

=

eyeu

(29)

=y

u

=

I GpGc I

1 eyeu

= H e (30)




Gp =

G11 G12 G1mG21 G22 G2m...

.... . .

...

Gm1 Gm2 Gmm

(32)

Gc=

Gc1 0 00 Gc2 0...

.... . .

...

0 0 Gcm

(33)


yiyi()|S0ii()|2

=


1 + |Gij |2 +


i() (34)



Yi,f () =1

S0ii()Yii()

= yifyif () =1

|S0ii()|2yiyi()



Proposition 1

S0yiyjS0yiyi

= S0yiyj (35)



Proof:

S0yiyjS0yiyi


7

S0y1y1 S

0y1y2

S0y2y1 S0y2y2

=

hy1y1 hy1y2hy2y1 hy2y2

The multi-loop output sensitivity is identical to the output sub-block of H


INTERACTION FACTORIZATIONSpectrum can now be expressed as

The quan0ty can be nega%ve, posi%ve or zero depending on the interference term

The interference term can cancel out the remaining terms only at select (nite) frequencies but not over a range of frequencies

17

01() = |S011|21()

The quantity 01() vanishes at all if and only if G12 = 0

01() = 0, G12 = 0

01()

Main Result: Factorization

y1y1 = |S011|2(1 + |G11|2)+ |S011|2|1 +G22Gc2|2|G12|2

|G21Gc2|2 + 2|G11|

G21G12S2 |Gc2|cos+ 1 + |Gc2|2

Interaction eects 01()

1() = |1 +G22Gc2|2|G12|2

|G21Gc2|2 + 2|G11| G21G12S2

|Gc2|cos+ 1 + |Gc2|2


GRAPHICAL REPRESENTATION

18

Graphical representation

u1 u2

y2y1

G11 G22

G21

G12

-Gc2-Gc1

ey1

eu1 eu2

ey2

Signal flow graph of 2 2 system

ey1

!

eu1

S011G11

-G21Gc2 !

eu2

-Gc2

ey2

y1

S2Loop 2

Direct transfer

Indirect transfer

G12

Transmission of eects of noise source through

direct and indirect paths

G. Sebastian and A. K. Tangirala (IIT-M) Interaction Analysis July 28, 2010 15 / 26


INTERACTION AND FEEDBACK INVARIANCE

19


INTERACTION AND FEEDBACK INVARIANCEFor any ith loop of a decentralized control loop

19

1

|S0ii()|2yiyi() =

Interactionfeedback invariant 1 + |Gii()|2 +


i()



For a SISO loop, the interac0on term vanishes and the result is an iden0ty

19

1

|S0ii()|2yiyi() =



i()




LHS is the spectrum of the output filtered by its sensitivity function

19

1

|S0ii()|2yiyi() =



i()





It can be calculated both from the knowledge of transfer func0ons as well as from data

19

1

|S0ii()|2yiyi() =



i()





It can be calculated both from the knowledge of transfer func0ons as well as from data

19

1

|S0ii()|2yiyi() =



i()

yifyif () = 1 + |Gii|2 = interaction eects are absent;yifyif () 1 + |Gii|2 = interaction eects are detrimental / beneficial

The invariant term is useful in detecting and quantifying the extent of interactions for design and assessment.


FILTERING BY THE INVERSE OF SENSITIVITY

20



Filtering the output by inverse of diagonal of sensitivity function Opening up the corresponding loop while keeping other loops closed

20




20

u1 u2

y2y1

G11 G22

G21

G12-Gc2-Gc1

ey1

eu1 eu2

ey2




20

u1 u2

y2y1

G11 G22

G21

G12-Gc2-Gc1

ey1

eu1 eu2

ey2

y1[k]

S0y1y1(q1)

Y1()

S0y1y1()




20

u1 u2

y2y1

G11 G22

G21

G12-Gc2-Gc1

ey1

eu1 eu2

ey2

u1 u2

y2y1

G11 G22

G21

G12-Gc2

ey1

eu1 eu2

ey2

y1[k]

S0y1y1(q1)

Y1()

S0y1y1()




To understand this, recall

Filtering the output of a SISO loop by the inverse of sensi0vity opens up the loop (discounts for / cuts o the feedback)

Filtering the output vector of a MIMO system by the inverse of sensi0vity matrix is equivalent to opening up all loops

20

u1 u2

y2y1

G11 G22

G21

G12-Gc2-Gc1

ey1

eu1 eu2

ey2

u1 u2

y2y1

G11 G22

G21

G12-Gc2

ey1

eu1 eu2

ey2

y1[k]

S0y1y1(q1)

Y1()

S0y1y1()


ILLUSTRATION: 2 X 2 SYSTEM

21

Gp =

z1

10.4z1z1

10.2z1z1

10.1z1z1

10.3z1

Gc =

0.40.2z1

1z1 00 0.40.2z

11z1

Gc =

0.40.2z1

1z1 00 0.20.1z

11z1

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ILLUSTRATION: 2 X 2 SYSTEM

Interaction effects are negative (reduced variance) in frequency ranges at the cost of positive (larger variance) effects in other frequency ranges

Lesser interaction effect in loop 1 is achieved at the cost of larger settling times in loop 2

Change in loop 2 controller produces interaction effects only in loop 1 (in the invariance domain)

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Gp =

z1

10.4z1z1

10.2z1z1

10.1z1z1

10.3z1

Gc =

0.40.2z1

1z1 00 0.40.2z

11z1

Gc =

0.40.2z1

1z1 00 0.20.1z

11z1

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!

%

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!$

01/2-/*34

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NORMALIZED INTERACTIONS

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Kc = 0.4 and KI = 0.2 in both loops

0 0.5 1 1.5 2 2.5 32

0

2

4

6

mag

nitu

de

loop 1

0 0.5 1 1.5 2 2.5 32

0

2

4

6

mag

nitu

de

loop 2

frequency

K=K0/|hy1y

1|2

zero lineK0 = absolute interactionK/(1+|G112

Kc = 0.2 and KI = 0.1 in loop 2

0 0.5 1 1.5 2 2.5 32

0

2

4

6

mag

nitu

de

loop 1

0 0.5 1 1.5 2 2.5 32

0

2

4

6

mag

nitu

de

loop 2

frequency

K=K0/|hy1y

1|2

zero lineK0= absolute interaction

K/(1+|G11|2)

The normalized interaction term should be as close to zero as possible


CONCLUDING REMARKS

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CONCLUDING REMARKS

Quantification of interaction with respect to variance

Factoriza.on of interac.on term has been derived

Existence of an invariant term in the filtered domain has been established

Contributions of interaction to output variance involves indirect energy transfer as well as interference

Interference term can cause posi0ve or nega0ve interac0on depending on the phase dierence

Sensitivity function and the conditional sensitivity function are the keys

Road ahead: A single index for interaction, quantify margins, proof that negative valued interaction is beneficial,

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REFERENCES

Kaminski M. and Blinowska, K. (1991). A new method of the description of the information flow in the brain structures. Biological Cybernetics, 65, 203-210.

Gigi, S. and Tangirala, A.K. (2010). Quantitative analysis of directional strengths in jointly stationary linear multivariate processes. Biological Cybernetics, 103(2), 119-133.

Priestley, M. (1981). Spectral analysis and time series. Academic Press, London

Gevers, M. and Anderson, B. (1981). Representations of jointly stationary stochastic feedback processes. Int. J. Control 33(5), 777-809.

Lutkepohl, H. (2005). New introduction to multiple time series analysis. Springer, New York.

Gigi, S. and Tangirala, A.K. (2010). Frequency-domain quantification of interactions in MIMO systems using directed variance decomposition. In: 5th international symposium on Design, Operation and Control of Chemical Processes, PSE-Asia, Singapore.

Seppala, C.T., Harris, T.J. and Bacon, D.W. (2002). Time series methods for dynamic analysis of multiple controlled variables. Journal of Process Control, 12:257276.

Zhu, Z-X., and Jutan, A. (1996). Loop decomposition and dynamic interaction analysis of decentralized control systems. Chemical Engineering Science., 51(12), 3325-3335.

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Prof.tangirala Directionality Interaction

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