MAE509 (LMIs in Control): Lecture 13, part B - LMIs for Quadratic Stability and Stabilization

all right class welcome back to uh part b of this relatively short lecture on uh parametric uncertainty specifically polytopic and interval uncertainty in part a of the lecture we introduced two concepts uh robust uh stability and quadratic stability uh robust stability being defined in terms of the representation of the system in that the eigenvalues of the

of the system are negative for all possible values of the uncertainty and alternatively quadratic stability which required the existence of a common lyapunov function for all possible systems within the uncertain set and that uh the epinope function condition was expressed specifically in terms of an lmi so an lmi however which has to hold

over a set over an uncertainty set um in this way the notions of quadratic stability and stability over a polytope or stability over a an interval are somewhat different than the lft representations where we shuffle off the uncertainty into a block and try and deal with it separately rather in the in this case we are simply

looking for a lyapunov function which holds for every possible value of the system right so we have a set of systems and we want to search for a lyapunov function which works for all of the values in that system set now i said this was expressed as an lmi of course but it's

really expressed as an infinite dimensional lmi in that uh it has to hold for all right for all values of in this case delta a although we'll have different representations of that for all values of delta a in a set and the set is is infinite right there's an infinite number of points so if it's a polytope for example

right there's uh there's the vertices but then there's every point of delta a inside the polytope so it's it has to hold an infinite number of points and so the logical question is how does one enforce such a condition so the first result in this part b is a really the only theoretical result or the only sort of