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

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## Introduction

Wile in the case of autonomous systems, they only need initial input constraints and cannot be changed using external inputs; the non-autonomous systems on the other hand can have changes happen to its behavior using a controller.

## The System

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

## The Data

${\displaystyle {\mathbf {A}}_{d}\in \mathbb {R} ^{n\times n}}$, ${\displaystyle {\mathbf {B}}_{d}\in \mathbb {R} ^{n\times m}}$, ${\displaystyle {\mathbf {C}}_{d}\in \mathbb {R} ^{p\times n}}$, ${\displaystyle {\mathbf {D}}_{d}\in \mathbb {R} ^{p\times m}}$

## Determining the bound

The output of this system must satisfy
${\displaystyle \sum _{i=0}^{k}{\mathbf {y}}_{i}^{T}{\mathbf {y}}_{i}=\left\vert \left\vert {\mathbf {y}}\right\vert \right\vert _{2k}^{2}\leq \gamma ^{2}(\left\vert \left\vert {\mathbf {x}}_{0}\right\vert \right\vert _{2}^{2}+\left\vert \left\vert {\mathbf {u}}\right\vert \right\vert _{2k}^{2},\forall k\in \mathbb {Z} _{\geq 0}}$
if there exists some matrix ${\displaystyle {\mathbf {P}}\in \S ^{p}}$and scalar ${\displaystyle \gamma \in \mathbb {R} _{>0}}$, where ${\displaystyle {\mathbf {P}}>0}$, such that
${\displaystyle {\mathbf {P}}-\gamma {\mathbf {I}}\leq 0}$,

${\displaystyle {\begin{bmatrix}{\mathbf {PA}}+{\mathbf {A^{T}P}}&{\mathbf {PB}}&{\mathbf {C^{T}}}\\*&-\gamma {\mathbf {I}}&{\mathbf {D}}^{T}\\*&*&-\gamma {\mathbf {I}}\end{bmatrix}}\leq 0}$.

## Implementation

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

## Conclusion

Given a non-autonomous system with initial operating conditions and a controller, the parameter ${\displaystyle \gamma }$ can be used to determine the feasible bound on the output of that system.