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=== Fault Modelling ===
The figure to the right shows a plant controlled by a controller in a standard control loop.
The nominal linear model of the plant is
<math>\begin{cases}\dot{\mathbf{x}}_f & = \mathbf{A}_f\mathbf{x}_f + \mathbf{B}_f\mathbf{u},\\▼
\mathbf{y}_f & = \mathbf{C}_f\mathbf{x}_f,\end{cases}</math>▼
<math>\begin{cases}\dot{\mathbf{x}} & = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}\\
where the index <math>f</math> indicates that the system is faulty. This approach models multiplicative faults by modified system matrices. Not all matrices need to change after every fault. In particular, actuator faults are represented by the input matrix <math>\mathbf{B}_f</math>, sensor faults are represented by the output map <math>\mathbf{C}_f</math>, and internal plant faults are represented by the system matrix <math>\mathbf{A}_f</math>. Alternative scenarios model faults as an additive external signal <math>\mathbf{f}</math>,▼
\mathbf{y} & = \mathbf{C}\mathbf{x}\end{cases}</math>
The plant subject to a fault (indicated by a red arrow in the figure) is modelled in general by
<math>\begin{cases}\dot{\mathbf{x}}_f & = \mathbf{A}\mathbf{x}_f + \mathbf{B}\mathbf{u} + \mathbf{E}\mathbf{f},\\▼
\mathbf{y}_f & = \mathbf{C}_f\mathbf{x}_f + \mathbf{F}\mathbf{f}.\end{cases}</math>.▼
▲where the
Alternative scenarios model faults as an additive external signal <math>\mathbf{f}</math>:
▲<math>\begin{cases}\dot{\mathbf{x}}_f & = \mathbf{A}\mathbf{x}_f + \mathbf{B}\mathbf{u} + \mathbf{E}\mathbf{f}
The upper part of the figure shows a supervisory loop consisting of fault detection and isolation (FDI) and a reconfiguration which changes the loop by
# choosing new input and output signals from
# changing the controller internals (including dynamic structure and parameters),
# adjusting the reference input <math>\mathbf{w}</math>.
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