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

The System
Consider the discrete-time LTI system with state-space realization $$\bold x_{k+1} = \bold A_d \bold x_k$$, $$\bold y_{k} = \bold C_d \bold x_k$$

The Data
$$\bold A_d \in\mathbb R^{n \times n}$$ and $$\bold C_d \in\mathbb R^{p \times n}$$

The LMI
The Euclidean norm of the output satisfies $$\left \vert \left \vert y_k \right \Vert \right \vert _2 \leq \gamma \left \vert \left \vert x_0 \right \Vert \right \vert _2, \forall k \in \mathbb{Z}_{\geq 0}$$ if there exist \bold P \in\mathbb S^n and \gamma \in\mathbb R_{>0}, where \bold P > 0, such that $$\bold P - \gamma\bold 1 \leq 0$$, $$\begin{bmatrix} \bold P & \bold C_d^T \\ $$\bold A_d^T \bold {P} \bold A_d - \bold P \leq 0$$
 * & \gamma \bold 1 \end{bmatrix} \geq 0. $$

Implementation
This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.

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

Related Links

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