# LMIs in Control/pages/FDI FilterDesign

FDI Filter Design For Systems With Sensor Faults: an LMI

Systems with faulty sensors are a very common type of systems. In many cases, redundancy is added in the form of additional sensors, but in certain cases it could be a costly solution. For general linear system models, the LMI in this section can be utilized to design state estimators which can detect and isolate faulty sensor readings in order to mitigate their effects.

## The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}w(t)+B_{2}u(t)\\y(t)&=Cx(t)+v(t)\end{aligned}}}

where ${\displaystyle x(t)\in \mathbb {R} ^{n}}$ is the state, ${\displaystyle u(t)\in \mathbb {R} ^{m}}$ is the control input, ${\displaystyle w(t)\in \mathbb {R} ^{r}}$ is the process noise, ${\displaystyle y(t)\in \mathbb {R} ^{p}}$ is the output and ${\displaystyle v(t)\in \mathbb {R} ^{q}}$ is the measurement noise.

## The Data

The state space matrices ${\displaystyle (A,B_{1},B_{2},C)}$ are required to be known.

## The Optimization LMI

The following LMI is used to design the Fault Detection and Isolation (FDI) filter:

{\displaystyle {\begin{aligned}&{\text{min}}_{\Phi ,\Theta ,\gamma }\;\gamma ,\\&{\text{subj. to: }}\Phi >0,\\&\quad \quad {\begin{bmatrix}A^{\top }\Phi +\Phi A+C^{\top }\Theta ^{\top }+\Theta C&\Phi B_{1}&\Theta &I\\*&-\gamma I&0&0\\*&*&-\gamma I&0\\*&*&*&-\gamma I\end{bmatrix}}<0\\\end{aligned}}}

Then the filter is ${\displaystyle K=\Phi ^{-1}\Theta }$.

## Conclusion:

The LMI designed in this section is used to design filters that can work on systems that are prone to sensors getting damaged or faulty.

## Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem: