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
${𝐱}_{k+1}={𝐀}_d{𝐱}_k+{𝐁}_d{u}_k$,
${𝐲}_k={𝐂}_d{𝐱}_k$

The Data

${𝐀}_d\in {ℝ}^{n\times n}$, ${𝐁}_d\in {ℝ}^{n\times 1}$, and ${𝐂}_d\in {ℝ}^{p\times n}$, and it is assumed that ${𝐀}_d$ is invertible. Let ${𝐳}_k={𝐂}_d{A}_d^{k-1}{B}_d$. be the unit impulse response of the system. The Euclidian Norm of the impulse response satisfies the following LMI.

The LMI

${\left||z_k\Vert \right|}_2\le \gamma ,\mathrm{\forall }k\in {\mathbb{Z}}_{\ge 0}$

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

$\left[\begin{array}{cc}𝐏& {𝐂}_d^T\\ *& \gamma \mathrm{𝟏}\end{array}\right]\ge 0.$

$\left[\begin{array}{cc}𝐏& 𝐏A_d^{-1}{B}_d\\ *& \gamma \end{array}\right]\ge 0.$

and $A_d^{T}P{A}_d-P\le 0.$

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