View source for 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<br/>
<math>\bold x_{k+1} = \bold A_d \bold x_k + \bold B_d u_k</math>,<br/>
<math>\bold y_{k} = \bold C_d \bold x_k</math>

== '''The Data''' ==
<math>\bold A_d \in\mathbb R^{n \times n}</math>, <math>\bold B_d \in\mathbb R^{n \times 1}</math>, and <math>\bold C_d \in\mathbb R^{p \times n}</math>, and it is assumed that <math>\bold A_d</math> is invertible. Let <math>\bold z_k = \bold C_d A ^{k-1} _d B_d </math>. be the unit impulse response of the system. The Euclidian Norm of the impulse response satisfies the following LMI. 

== '''The LMI''' ==
<math>\left \vert \left \vert z_k \right \Vert \right \vert _2 \leq \gamma , \forall k \in 
\mathbb{Z}_{\geq 0}</math><br/>

if there exist <math>\bold P \in\mathbb S^n </math> and  <math>\gamma \in\mathbb R_{>0}</math>  where  <math>\bold P > 0 </math>, such that <br/>

<math>\begin{bmatrix} \bold P & \bold C_d^T \\
* & \gamma \bold 1 \end{bmatrix} \geq 0. </math> <br/>

<math>\begin{bmatrix} \bold P & \bold P A^{-1} _d B_d \\
* & \gamma  \end{bmatrix} \geq 0. </math> <br/>

and <math> A^T _d P A_d - P \leq 0. </math> 


== '''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]]

== '''References''' ==

{{cite book|title=LMI Properties and Applications in Systems, Stability, and Control Theory|last1=Caverly|first1=Ryan|date=2021|last2=Forbes|first2=James}}

{{BookCat}}