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

Output Energy Bound for Autonomous LTI Systems

Autonomous systems are initialized under a given set of initial conditions, and then run without any additional inputs. It is useful to know ahead of time the bounds such a system will operate within. This analysis can be used to determine the upper bound on the output of a given autonomous LTI system.

## The System

Consider the continuous-time, LTI autonomous system with state space representation

${\displaystyle {\mathbf {\dot {x}}}={\mathbf {Ax}}}$,

${\displaystyle {\mathbf {y}}={\mathbf {Cx}}}$,

where ${\displaystyle {\mathbf {A}}\in \mathbb {R} ^{n\times n},{\mathbf {C}}\in \mathbb {R} ^{p\times n}}$ and ${\displaystyle {\mathbf {x}}(0)={\mathbf {x}}_{0}.}$

## Determining an Upper Bound

The output of this system will satisfy

${\displaystyle {\sqrt {\int _{0}^{T}{\mathbf {y^{T}y}}dt}}=\left\vert \left\vert {\mathbf {y}}\right\vert \right\vert _{2T}\leq \gamma \left\vert \left\vert {\mathbf {x}}_{0}\right\vert \right\vert _{2},\forall T\in \mathbb {R} _{\geq 0}}$

if there exists some matrix ${\displaystyle {\mathbf {P}}\in \S ^{p}}$and scalar ${\displaystyle \gamma \in \mathbb {R} _{>0}}$ such that

${\displaystyle {\mathbf {P}}>0}$,

${\displaystyle {\mathbf {P}}-\gamma {\mathbf {I}}\leq 0}$,

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

## Conclusion

Given an autonomous system with an initial operating condition, the parameter ${\displaystyle \gamma }$ can be used to determine the largest feasible bound on the output of that system.