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

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

$$x_{k+1} = A_d x_k + B_d u_k, $$

where $$A_d \in \mathbb{R}^{n x n}$$, and $$B_d \in \mathbb{R}^{n x m}$$,

the transient bound can be analyzed with the LMI below.

The Data
$$A_d \in \mathbb{R}^{n x n}$$, and $$B_d \in \mathbb{R}^{n x m}$$

The LMI:
The Euclidean norm of the state satisfies:

$$\lVert x_k \rVert _2^2 \leq \gamma^2 (\lVert x_0 \rVert _2^2 + \lVert u \rVert _{2k}^2), \forall k \in \mathbb{Z}_{\geq 0}$$

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


 * $$P - \gamma 1 \leq 0, $$
 * $$\begin{bmatrix} P & 1 \\ * & \gamma 1 \end{bmatrix} \geq 0, $$
 * $$\begin{bmatrix} A_d^T P A_d - P & A_d^T P B_d \\ * & B_d^T B_d- \gamma 1 \end{bmatrix} \leq 0. $$

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

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.

Related LMIs

 * Output Energy Bound for Discrete-Time Autonomous LTI Systems