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

## The System

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

${\displaystyle {\dot {x}}=Ax+Bu}$

where ${\displaystyle A\in \mathbb {R} ^{nxn}}$, ${\displaystyle B\in \mathbb {R} ^{nxm}}$ and x(0) = x0,

the transient bound can be evaluated with the following LMI.

## The Data

${\displaystyle A\in \mathbb {R} ^{nxn}}$, ${\displaystyle B\in \mathbb {R} ^{nxm}}$ and x(0) = x0.

## The LMI:

The Euclidean norm of the state satisfies:

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

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

• ${\displaystyle P-\gamma 1\leq 0,}$
• ${\displaystyle {\begin{bmatrix}P&1\\*&\gamma 1\end{bmatrix}}\geq 0,}$
• ${\displaystyle {\begin{bmatrix}PA+A^{T}P&PB\\*&-\gamma 1\end{bmatrix}}\leq 0.}$

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

## 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.