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

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

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${\textstyle e^{\text{At}}}$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,.