Pardon me to upload the picture without writing as it is long. My question is while defining switching sequence why they are considering system realization, by realization are they talking about minimal realization? Also, the definition is same as this? a switching signal is $\sigma: (0,\infty)\to P=\{1,2,\dots, N\}$ is a piecewise constant without accumulation of jumps. I shall be highly grateful if someone make me understand their definition. 
2026-02-22 19:30:00.1771788600
Understanding Definition of Switching Sequence
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In this context the word "realization" is referring to the pair $(A_i,B_i)$. So by using a different $r(t)=i$ the realization of the system dynamics changes.
It can also be noted that in this context having a "minimal realization" would imply that the pair $(A_i,B_i)$ would be controllable for any $i$. In that case one would never have to switch, since any $i$ would allow you to stabilize the system.