Discrete-Time Detectability

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Discrete-Time LTI systems can be made detectable using observer gain L, which can be found using LMI optimization, such that the close loop system is detectable.

The System

Discrete-Time LTI System with state space realization ${\displaystyle (A_{d},B_{d},C_{d},D_{d})}$
${\displaystyle {\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}}$

The Data

The matrices: System ${\displaystyle (A_{d},B_{d},C_{d},D_{d}),P,W}$.

The Optimization Problem

The following feasibility problem should be optimized:

Maximize P while obeying the LMI constraints.
Then L is found.

The LMI:

Discrete-Time Detectability

The LMI formulation

${\displaystyle {\begin{aligned}P\in {S^{n}};W\in {R^{m*n}}\;\\&P>0\\{\begin{bmatrix}P&A_{d}^{T}P+C_{d}^{T}W\\*&P\end{bmatrix}}&>0,\\L=P^{-1}W\end{aligned}}}$

Conclusion:

The system is detectabe iff there exits a ${\displaystyle P}$, such that ${\displaystyle P>0}$. The matrix ${\displaystyle A_{d}+LC_{d}}$ is Schur with ${\displaystyle L=P^{-1}W}$

Implementation

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIs

- Continuous time Detectability

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.

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