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

## The System

For a discrete-time LTI system with a state-space representation of:

${\displaystyle x_{k+1}=A_{d}x_{k}+B_{d}u_{k},}$

where ${\displaystyle A_{d}\in \mathbb {R} ^{nxn}}$, and ${\displaystyle B_{d}\in \mathbb {R} ^{nxm}}$,

the transient bound can be analyzed with the LMI below.

## The Data

${\displaystyle A_{d}\in \mathbb {R} ^{nxn}}$, and ${\displaystyle B_{d}\in \mathbb {R} ^{nxm}}$

## The LMI:

The Euclidean norm of the state satisfies:

${\displaystyle \lVert x_{k}\rVert _{2}^{2}\leq \gamma ^{2}(\lVert x_{0}\rVert _{2}^{2}+\lVert u\rVert _{2k}^{2}),\forall k\in \mathbb {Z} _{\geq 0}}$

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

• ${\displaystyle P-\gamma 1\leq 0,}$
• ${\displaystyle {\begin{bmatrix}P&1\\*&\gamma 1\end{bmatrix}}\geq 0,}$
• ${\displaystyle {\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}\\*&B_{d}^{T}B_{d}-\gamma 1\end{bmatrix}}\leq 0.}$

if x0 = 0 and u is a unit-energy input (${\displaystyle \lVert u\rVert _{2k}\leq 1,\forall k\in \mathbb {Z} _{\geq 0}}$), then the above LMIs ensure that ${\displaystyle \lVert x_{k}\rVert _{2}\leq \gamma ,\forall k\in \mathbb {Z} _{\geq 0}}$

## Conclusion

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

## Implementation

The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.