MAE509 (LMIs in Control): Lecture 14, part B - LMIs for Robust Stability and Control using the LFT

all right class um so welcome back to uh really this is still part a because remember we're dividing this into the uh the parametric um unstructured case which we're doing now in the structured case which is really part b so this is the still part a right so just i'll just write that still

part a although i'm labeled i'll label it when i post these as part b so hopefully that won't cost too much confusion but it seemed like a good stopping point to give people a break so i decided to make the stopping point any case so remember we just introduced the s procedure which gives us a way of enforcing

conditions like x transpose f x greater than x 0 for all x such that x transpose g x greater than or equal to 0. so we are now going to apply this to the quadratic stability condition which is remember if and only if where we have to enforce this lmi right so this will be our f

uh on x and q and here we'll make this r this is just z like the the concatenated thing there so it's just z for all let's call it z such that z transpose something z now is less than or equal to zero so this will be our actually negative g matrix because

this is there's a negative inequality there so this is our f over here and this is our g right here so now we can apply the uh s procedure right to say that um here this is uh our f uh is pau is actually negative in this case from negativity um for all

uh x and q x and q such that