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

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

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

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

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