User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Transient State Bound for Non-Autonomous LTI Systems

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

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

where $$A \in \mathbb{R}^{n x n}$$, $$B \in \mathbb{R}^{n x m}                               $$ and x(0) = x0,

the transient bound can be evaluated with the following LMI.

The Data
$$A \in \mathbb{R}^{n x n}$$, $$B \in \mathbb{R}^{n x m}                               $$ and x(0) = x0.

The LMI:
The Euclidean norm of the state satisfies:

$$\lVert x(T) \rVert _2^2 \leq \gamma^2 (\lVert x_0 \rVert _2^2 + \lVert u \rVert _{2T}^2), \forall T \in \mathbb{R}_{\geq 0}$$

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


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

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

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.

Related LMIs

 * Output Energy Bound for Autonomous LTI Systems