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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

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

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

• $PA+AP\leq 0$

Proof

• The proof follows same procedure as the proof for transient output Bound for Autonomous LTI systems, but in this case taking $X_{0}=B$as the initial condition that yields the result $X^{T}(T)PX(T)\leq B^{T}PB$.

• Using the non-strict Schur complement, the matrix inequality in ${\begin{bmatrix}P&C^{T}\\*&-\gamma 1\end{bmatrix}}\geq 0.$ is equivalent to $B^{T}PB\leq \gamma$. Substituting this and ${\begin{bmatrix}P&C^{T}\\*&-\gamma 1\end{bmatrix}}\geq 0$ into $X^{T}(T)PX(T)\leq B^{T}PB$ gives the desired result.

The System

For the single-input multi-output discrete-time LTI system with state-space realization,

${\begin{aligned}\ x_{k+1}&=A_{d}x_{k}+B_{d}u_{k}\\y_{k}&=C_{d}x_{k}\\\end{aligned}}$

where $A\in \mathbb {R} ^{n\times n}$, $B_{d}\in \mathbb {R} ^{n\times 1}$ and $C_{d}\in \mathbb {R} ^{p\times n}$ and it is assumed that $A_{d}$ is invertible. It is also considered that $Z_{k}C_{d}A_{d}^{k-1}B_{d}$ be the unit impulse response of the system.

The Data

$A\in \mathbb {R} ^{n\times n}$, $B_{d}\in \mathbb {R} ^{n\times 1}$ and $C_{d}\in \mathbb {R} ^{p\times n}$

The LMI

If the Euclidean norm of the impulse response satisfies. $\lVert z_{k}\rVert _{2}\leq \gamma ,\forall k\in \mathbb {Z} _{\geq 0}$ and if there exist $P\in \mathbb {S} ^{p}$ and $\gamma \in \mathbb {R} _{>0}$,where P > 0, such that

• ${\begin{bmatrix}P&PA_{d}^{-1}B_{d}\\*&\gamma \end{bmatrix}}\geq 0,$

• ${\begin{bmatrix}P&C_{d}^{T}\\*&\gamma 1\end{bmatrix}}\geq 0,$

• $A_{d}^{T}PA_{d}-P\leq 0.$

Proof

• The proof follows same procedure as for transient output bound for Discrete time autonomous LTI sysyems,but taking $X_{0}=A_{d}^{-1}B_{d}$as the initial condition, so that the unit impulse response matching the free response $Z_{k}=C_{d}A_{d}^{k}X_{0}$.

• This yields the result$X_{k}^{T}PX_{k}\leq B_{d}^{T}A_{d}^{-T}PA_{d}^{-1}B_{d}$.

• Using the non-strict Schur complement, the matrix inequality ${\begin{bmatrix}P&PA_{d}^{-}1B_{d}\\*&\gamma \end{bmatrix}}\geq 0,$ is equivalent to the inequality $B_{d}^{T}A_{d}^{-}TPA_{d}^{-}1B_{d}\leq \gamma$.Substituting this and ${\begin{bmatrix}P&C_{d}^{T}\\*&\gamma 1\end{bmatrix}}\geq 0,$ into $X_{k}^{T}PX_{k}\leq B_{d}^{T}A_{d}^{-T}PA_{d}^{-1}B_{d}$.gives the desired result.

Conclusion

The above LMIs can be used to analyze the Transient Impulse Response Bound and analyze the Discrete-Time 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,.