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

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

This LMI can be used to 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,.