LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-Time Transient

Transient Impulse Response Bound

The System
For a single-input multi-output continuous-time LTI system with state-space realization



\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y&=Cx\\ \end{align}$$

where $$A \in \mathbb{R}^{n \times n}$$, $$B \in \mathbb{R}^{n \times 1}$$ and $$C \in \mathbb{R}^{p \times 1}$$.

The Data
$$A \in \mathbb{R}^{n \times n}$$, $$B \in \mathbb{R}^{n \times 1}$$ and $$C \in \mathbb{R}^{p \times 1}$$.

Also it is assumed that Z(t)=C$e^{\text{At}}$ B be the unit impulsive response of the system.

The LMI:
If the Euclidean norm of the impulse response satisfies. $$\lVert z(T) \rVert \leq \gamma, \forall T \in \mathbb{R}_{\geq 0}$$ and if there exist $$P \in \mathbb{S}^p$$ and $$\gamma \in \mathbb{R}_{> 0}$$,where P > 0, such that


 * $$\begin{bmatrix} P & PB \\ * & \gamma \end{bmatrix} \geq 0, $$


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


 * $$PA+AP \leq 0 $$

Conclusion
This LMI can be used to analyze the Transient Impulse Response Bound for the given system

Implementation
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,.