# LMIs in Control/Stability Analysis/Discrete Time/Transient Impulse Response Bound

## The System

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

## The Data

${\displaystyle {\mathbf {A}}_{d}\in \mathbb {R} ^{n\times n}}$, ${\displaystyle {\mathbf {B}}_{d}\in \mathbb {R} ^{n\times 1}}$, and ${\displaystyle {\mathbf {C}}_{d}\in \mathbb {R} ^{p\times n}}$, and it is assumed that ${\displaystyle {\mathbf {A}}_{d}}$ is invertible. Let ${\displaystyle {\mathbf {z}}_{k}={\mathbf {C}}_{d}A_{d}^{k-1}B_{d}}$. be the unit impulse response of the system. The Euclidian Norm of the impulse response satisfies the following LMI.

## The LMI

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

if there exist ${\displaystyle {\mathbf {P}}\in \mathbb {S} ^{n}}$ and ${\displaystyle \gamma \in \mathbb {R} _{>0}}$ where ${\displaystyle {\mathbf {P}}>0}$, such that

${\displaystyle {\begin{bmatrix}{\mathbf {P}}&{\mathbf {C}}_{d}^{T}\\*&\gamma {\mathbf {1}}\end{bmatrix}}\geq 0.}$

${\displaystyle {\begin{bmatrix}{\mathbf {P}}&{\mathbf {P}}A_{d}^{-1}B_{d}\\*&\gamma \end{bmatrix}}\geq 0.}$

and ${\displaystyle A_{d}^{T}PA_{d}-P\leq 0.}$

## Implementation

This can be implemented in any LMI parser such as YALMIP which can implement a solver like Mosek to return a solution.

## Conclusion

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

## References

Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.