LMIs in Control/pages/Discrete-Time Detectability

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 $$(A_d,B_d,C_d,D_d)$$ $$ \begin{align} &A_d \in \bf{R^{n*n}}, &B_d \in \bf{R^{n*m}}, &C_d \in \bf{R^{p*n}}, &D_d \in \bf{R^{p*m}} \;\\ \end{align}$$

The Data
The matrices: System $$ (A_d,B_d,C_d,D_d), P, W $$.

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

$$ \begin{align} P \in {S^{n}}; W \in {R^{m*n}}\;\\ &P>0\\ \begin{bmatrix}P & A_d^TP+C_d^TW\\
 * & P\end{bmatrix}&>0,\\

L = P^{-1}W

\end{align}$$

Conclusion:
The system is detectabe iff there exits a $$P$$, such that $$P>0$$. The matrix $$A_d+LC_d$$ is Schur with $$L = P^{-1}W$$

Implementation
A link to CodeOcean or other online implementation of the LMIMATLAB Code

Related LMIs
- Continuous time Detectability