Stability via MLF: Continuity conditions: unclear notation

[bukvoj]

I am sorry, perhaps I missed something, but...
at the very bottom of the page
https://hurak.github.io/hys/stability_via_multiple_lyapunov_function.html
in the continuity conditions of the Lyapunov function on the boundary of regions $\Omega_i$, $\Omega_j$:

I am not quite sure where the terms $q_i$, $r_i$, $q_j$, $r_j$ came from.

Do they come from the Lyapunov function which has the form $V(x) = x^T P x + q^T x + r$? If so, perhaps clarifying that would be helpful because earlier on the very same page, the Lyapunov function was defined simply as $V(x) = x^T P x$.
Or do they come from somewhere else? If so, how do I find them?

Thank you very much...

[hurak]

Indeed, when I discuss the continuity conditions on the boundary between the two regions, I use an equation for more general (still quadratic) Lyapunov function. I did mention the need to consider these more general quadratic functions in the previous paragraph (on affine subspaces), but it is true that I did not keep consistency of notations: V = [x; 1]^T [P P12; P21 P22] [x; 1] vs V = [x; 1]^T [P q; q^T r] [x; 1]. The motivation for introducing the latter notation was that I started to need yet another lower index to distinguish between the two Lyapunov-like functions. Indeed keeping the notation consistent (and yet readable) is a challenge (not only) for me.

Is this explanation sufficient?

I will leave the issue open until I improve the text. Obviously I shall not only fix the notation, but I should also add some more explanation. Indeed, the text gradually evolves from sketchy notes.

[bukvoj]

Awesome, thank you. I just wanted to make sure I understand it correctly. :)