LMIs in Control/pages/FDI Filter Design

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

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

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

The Data

The state space matrices $(A,B_1,B_2,C)$ are required to be known.

The Optimization LMI

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

$\begin{array}{cc}& {\text{min}}_{\mathrm{\Phi },\mathrm{\Theta },\gamma }\gamma ,\\ & \text{subj. to: }\mathrm{\Phi }>0,\\ & \left[\begin{array}{cccc}{A}^{\mathrm{\top }}\mathrm{\Phi }+\mathrm{\Phi }A+C^{\mathrm{\top }}{\mathrm{\Theta }}^{\mathrm{\top }}+\mathrm{\Theta }C& \mathrm{\Phi }{B}_1& \mathrm{\Theta }& I\\ *& -\gamma I& 0& 0\\ *& *& -\gamma I& 0\\ *& *& *& -\gamma I\end{array}\right]<0\\ \end{array}$

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

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:

https://github.com/smhassaan/LMI-Examples/blob/master/FDI_Filter_example.m

Related LMIs

H-infinity Optimal Filter

H-infinity Optimal Observer

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.