Model Input

Bipartite structural graph

The functionality of fault-diangosis revolves around structural representation of mathamatical system models. According to this methodology, the system model is abstracted into a qualitative bipartite graph representation where:

• Every equation is assigned a vertex in the “equations” set
• Every variable is assigned a vertex in the “variables” set
• A variable which participates in an equation is connected to it with an edge

Traditionally, no other information is encoded into the structural graph. However, more modern approaches require a bit more information to produce valid and causal results. For that reason:

• Variables vertices are embelished with information on whether they are known, measured, inputs, outputs or residual generators.
• Edges are embelished with information on their calculation cost and whether they represent a differentiation, an integration or a non-solvable variable-equation relation
• All derivative variables must be declared through explicit differentiation equations

Model input format

Currenlty fault-diagnosis only accepts structural model input, instead of mathematical equations. Examples of structural model specificaiton can be found in the GraphPool folder.

Each line of the model description file represents one equation. The indentifier fault at the start of the line signifies that the equation is subject to a fault. All subsequent words, space-separated are unique variable names, except for the following identifiers:

• msr sets the following variable to a known measurement
• inp sets the following variable to a known input
• ni restricts this equation to be evaluated for this variable
• int signifies the use of that variable in an explicit differentiation as integration, i.e. this is the derivative of the state
• dot signifies the use of that variable in an explicit differentiation is differentiaton, i.e. this is the original state

Below is a short model description. g007 is the name of the model. The separation of equations to the con, der and msr cells is only visual and serves better readability purposes.

classdef g007 < model
    %% Linear T.I. airplane model found in
    % Izadi-Zamanabadi, R. (2002).
    % Structural analysis approach to fault diagnosis with application to fixed-wing aircraft motion.
    % Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301), 5, 3949–3954. doi:10.1109/ACC.2002.1024546

    % x1dot = a11 x1 + a13 x3 + a14 x4 + a16 x6
    % x2dot = a21 x1 + a22 x2 + a23 x3 + a27 x7
    % x3dot = a31 x1 + a33 x3 + a36 x6
    % x4dot = x2
    % x5dot = x3 + a55 x5
    % x6dot = a66 x6 + b61 u1
    % x7dot = a77 x7 + b72 u2
    % y1 = x1
    % y2 = x4
    % y3 = x5
    methods
        function this = g007()
            this.name = 'g007';
            this.description = 'Linear T.I. airplane model found in "Structural analysis approach to fault diagnosis with application to fixed-wing aircraft motion"';

            con = [...
                {'dot_x1 x1 x3 x4 x6'};...
                {'dot_x2 x2 x1 x3 x7'};...
                {'dot_x3 x1 x3 x6'};...
                {'dot_x4 x2'};...
                {'dot_x5 x3 x5'};...
                {'fault dot_x6 x6 inp u1'};...
                {'fault dot_x7 x7 inp u2'};...
                ];

            der = [...
                {'dot x1 int dot_x1'};...
                {'dot x2 int dot_x2'};...
                {'dot x3 int dot_x3'};...
                {'dot x4 int dot_x4'};...
                {'dot x5 int dot_x5'};...
                {'dot x6 int dot_x6'};...
                {'dot x7 int dot_x7'};...
                ];

            msr = [...
                {'fault msr y1 x1'};...
                {'fault msr y2 x4'};...
                {'fault msr y3 x5'};...
                ];

            this.constraints = [...
                {con},{'c'};...
                {der},{'d'};...
                {msr},{'s'};...
                ];

            this.coordinates = [];

        end

    end

end