View source for LMIs in Control/Stability Analysis/Discrete Time/Polytopic Uncertainty/Open-Loop Robust Stability

== ''' The System''' ==
Consider the set of matrices<br/>
<math>\bold A =  \{ \bold A _d ( \alpha ) \in\mathbb R^{n \times n} | \bold A _d ( \alpha ) = \Sigma^{n}_{i=1} \alpha \bold A _{d,i}, \bold A _{d,i}\in\mathbb R^{n \times n}, \alpha _i \in\mathbb R _{\geq 0} \},   \Sigma^{n}_{i=1} \alpha _{i} =1 \}</math>,<br/>


== '''The Data''' ==
The discrete-time LTI system <math>\bold x _{k+1} = \bold A _d ( \alpha ) \bold x _{k+1}</math> is asymptotically stable for all <math>  \bold A _d ( \alpha ) \in\mathbb  \bold A </math> if there exists <math>\bold P \in\mathbb S^n </math> ,<math>i = 1, ..., n </math> , and <math>\bold G \in\mathbb R^{ n \times n} </math> , where <math>\bold P _i > 0 , i = 1 , ... , n </math> , such that <br/>

<math>\begin{bmatrix} \bold P _i & \bold P A ^T  _{d,i} \bold G ^T \\
* & \bold G + \bold G ^T - \bold P _i  \end{bmatrix} < 0. </math> <br/> , <math> i = 1 , ... , n </math> 



== '''Implementation''' ==
This is used to get open-loop stability.


== '''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}}