# LMIs in Control/Stability Analysis/Discrete Time/Transient Output Bound for Discrete-Time Autonomous LTI Systems

## The System

Consider the discrete-time LTI system with state-space realization
${\displaystyle {\mathbf {x}}_{k+1}={\mathbf {A}}_{d}{\mathbf {x}}_{k}}$,
${\displaystyle {\mathbf {y}}_{k}={\mathbf {C}}_{d}{\mathbf {x}}_{k}}$

## The Data

${\displaystyle {\mathbf {A}}_{d}\in \mathbb {R} ^{n\times n}}$ and ${\displaystyle {\mathbf {C}}_{d}\in \mathbb {R} ^{p\times n}}$

## The LMI

The Euclidean norm of the output satisfies
${\displaystyle \left\vert \left\vert y_{k}\right\Vert \right\vert _{2}\leq \gamma \left\vert \left\vert x_{0}\right\Vert \right\vert _{2},\forall k\in \mathbb {Z} _{\geq 0}}$
if there exist \bold P \in\mathbb S^n and \gamma \in\mathbb R_{>0}, where \bold P > 0, such that
${\displaystyle {\mathbf {P}}-\gamma {\mathbf {1}}\leq 0}$, ${\displaystyle {\begin{bmatrix}{\mathbf {P}}&{\mathbf {C}}_{d}^{T}\\*&\gamma {\mathbf {1}}\end{bmatrix}}\geq 0.}$ ${\displaystyle {\mathbf {A}}_{d}^{T}{\mathbf {P}}{\mathbf {A}}_{d}-{\mathbf {P}}\leq 0}$

## Implementation

This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.

## Conclusion

By using this LMI the transient state bound can be analyzed for a given autonomous LTI system.