Diagnosis and FTC by Prof. Blanke [pdf] - NTNU

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Models of dynamical systems
• Notation for nonlinear system with faults
The general, nonlinear case reads:
dx() t
= gx ( ( t), u( t), d( t), f( t)),
x(0) = x0
dt
y() t = h( x(), t u(), t d(),()) t f t
where x is state vector, u is input vector,
d is disturbance vector and f(t)
a fault vector.
Models of dynamical systems
Mogens Blanke – Spring 2006
We obtain a linar model in a point of operation
xudf , , , by Taylor expansion around x = x- x
dx dx dx ∂g∂ ∂ ∂
= + = +
g
+
g
+ g
gxudf ( , , , ) x u d+ f,
dt dt dt ∂x∂u∂d∂f x(0) = x0...
∂h ∂h ∂h ∂
= + = ( , , , ) + + + h
y y y h x u d f x u d + f
∂x ∂u ∂d ∂f
or
x = Ax + Bu + E +
xd Fxf y = Cx + Du + E +
yd Fyf In the sequel, we ommit the x notation and just write x
Mogens Blanke – Spring 2006
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