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

Output Energy Bound for Discrete-Time 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 operating in discrete time.

## The System

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

${\displaystyle {\mathbf {x}}_{k+1}={\mathbf {A}}_{d}{\mathbf {x}}_{k}}$,

${\displaystyle {\mathbf {y}}_{k}={\mathbf {C}}_{d}{\mathbf {x}}_{k}}$,

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

## Determining an Upper Bound

The output of this system will satisfy

${\displaystyle \left\vert \left\vert {\mathbf {y}}\right\vert \right\vert _{2k}\leq \gamma \left\vert \left\vert {\mathbf {x}}_{0}\right\vert \right\vert _{2},\forall k\in \mathbb {Z} _{\geq 0}}$

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

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

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

${\displaystyle {\begin{bmatrix}{\mathbf {A_{d}^{T}PA_{d}}}-{\mathbf {P}}&{\mathbf {C_{d}^{T}}}\\*&-\gamma {\mathbf {I}}\end{bmatrix}}\leq 0}$.

## Conclusion

Given an autonomous system operating in discrete-time conditions, the parameter ${\displaystyle \gamma }$ can be used to determine the largest feasible bound on the output of that system.