Structural analysis

Supervisory control slides

Control Engineering Department

2006

Structural analysis in FTC development

Components

Component'sabnormalfunction

Component'snormal

function

FMECA(Hazard analysis)

Structural analysis

Faults to behandled

MonitorableParts

Non-monitorable

Parts

Remedialaction selection

&

DetailedFDI design

Knowledge representation Knowledge formulation and manipulation

Abstraction Implementation & analysis Decision & design

Structural analysis - background

• Analysis of the structural properties of the models, i.e., properties which are independent of the actual values of the parameters. Only links between the variables and relations, which are derived from the operating model, are represented in this analysis.

• The links are independent of the operating model and are thus independent of the form under which this operating model is expressed (qualitative or quantitative data, analytical or non-analytical relations).

Structural analysis approach

7

6

5

4 3

2

F D I (d e ta iled ) d esig n

A n a ly tica l redu n da n cyre la tion s (A R R )-q u an tita tive

F u n ctio n a lre la tion s/equ a tion s

Stru c tu ra l (qu a lita tive )equ a tio n s

In c id en cem atrix

D i-g ra phC an on ica ldecom po sitio nM a tch in g

A n a ly tica l redu n da n cyre la tion s (A R R )-q u a lita tive

D eta iledin form a tio n

F D I d esig n

1

Structural graph representation

• Purpose:– To find the monitorable (over-determined) part of the system.

• Matching:– is a set of pairs (fj,xi) such that any constraint fj FX belongs to one

pair. FX set of constraints which are applied at least on one unknown variable.

• Starting point: Hypothesis: An unknown variable xi is computed, using the constraint fj under the assumption that all other variables are known.

Structural model

• System considered as a set of components

S =Each composing a set of relations fi between a set of variables

zj, j=1,....,n.

Structured model - 1

Structured model - II

Incidence matrix I

A* is not necessarily the same as AT. It highlights the property of calculability.

where

Example 1Nonlinear diff. Eq.:

Can be rewritten to the following diff.algebraic equations:

Can then be represented by following relations:

Example II

LTI system:

Can be rewritten to diff. Alg. Eqs.

Is represented bythe following incidencematrix

An Example: P-control system

b ) F u n ctio n a l rep resen ta tio n

_ S et p o in t

C o n tro llerS u b system /co m p o n en tD y n am ics

S en so r

uc re fkts

D y n am icsP rop ortion a l

co n tro llerm

-

a ) B lo ck rep resen ta tio n

S en so r

Used information: variables with non-zero coefficients that are involved in equations/relations.

Incidence matrix of the P-control system

m

f3

c

f2

c

f1

c

uc

f3

c

f2

c

f1

c

ref m uc

1 1 1

1 1

1 1

a) Di-graph b) Incidence matrix

ref

F

K

Z

X

Matching and calculability property

u c u c

a ) P o ss ib le m a tch in g b ) Im p o ss ib le m a tch in g

f2 f2

u c

f2 1 M

c ) M a trix rep re sen ta tio n o f sy s tem ’s s tru c tu re

x 2

b ) F u n c tio n a l re la tio n

f

f 1 M

c ) M a trix rep re sen ta tio n o f sy s tem ’s s tru c tu re

x 2

x 1

f

x1

x 1x 2

a ) P o ss ib le m a tch in g

-kk

Canonical decomposition

K n o w n * *\ * *

0 0

0

*

*

*

*\ *

*

U n k n o w n ( )X

Con

straints

F x

f1

fkfk+ 1

fk+ p

fk+ p + 1

fn

x 1 x l

x l+ 1 x l+ p

x l+ p + 1 x m

B-

B+

B0

f3

c

f2

c

f1

c

ref m u c

1 1 1

1 1

1 1

F

X X

*

*B+

Canonical decompositionDecomposition of the P-control system

Matching: P-control system

f3

cf2

cf1

c

u cf3

c

f2

c

f1

c

ref m u c

1 1 1

1 M

1 1

C a u sa l m a tch in g In c id en ce m a tr ix

*

*\ *

* *\ *

*

B +

B 0

Sensor Fault representation

Two different views are considered

Actuator fault representation

Actuator malfunctionsCompletely

Actuator functionsbut incorrectly(with bias)

Fault detection and isolation

Any minimal over-determined subsystem results in an ARRexpression:

Residuals are obtained

Fault isolation: look athow different faults affect different residuals

Software prototype

A dedicated prototype software toolbox is been developedIn the Cont. Eng. Dept. (AAU). The prototype can be downloaded from

http://www.control.auc.dk/ftc/html/software_tools.html

Example (linear):

LTI model of an wing dynamic of an aircraft: