LMIs in Control/Stability Analysis/Discrete Time/Transient Impulse Response Bound

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

The Data
$$\bold A_d \in\mathbb R^{n \times n}$$, $$\bold B_d \in\mathbb R^{n \times 1}$$, and $$\bold C_d \in\mathbb R^{p \times n}$$, and it is assumed that $$\bold A_d$$ is invertible. Let $$\bold z_k = \bold C_d A ^{k-1} _d B_d $$. be the unit impulse response of the system. The Euclidian Norm of the impulse response satisfies the following LMI.

The LMI
$$\left \vert \left \vert z_k \right \Vert \right \vert _2 \leq \gamma, \forall k \in \mathbb{Z}_{\geq 0}$$

if there exist $$\bold P \in\mathbb S^n $$ and $$\gamma \in\mathbb R_{>0}$$  where  $$\bold P > 0 $$, such that

$$\begin{bmatrix} \bold P & \bold C_d^T \\
 * & \gamma \bold 1 \end{bmatrix} \geq 0. $$

$$\begin{bmatrix} \bold P & \bold P A^{-1} _d B_d \\
 * & \gamma \end{bmatrix} \geq 0. $$

and $$ A^T _d P A_d - P \leq 0. $$

Implementation
This can be implemented in any LMI parser such as YALMIP which can implement a solver like Mosek to return a solution.

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