MAE509 (LMIs in Control): Lecture 13, part A - Robust and Quadratic Stability

all right hello class and welcome to lecture 13. so in this relatively short lecture we will consider a special case of uncertainty and you remember there was a quite great variety of them in particular we'll leave off the lft for the moment and come back to it in lecture 14. and kendra consider the special case

where our uncertainties in our system parameters or state space system parameters have the form of additive afine polytopic in interval uncertainty so we have a nominal matrix well this applies to the b matrices as well where we've got uh some nominal parameter then some ais a say 1 delta 1 plus plus a let's say k

delta k where these uh so these uncertain parameters are athy right in our representation which is actually a little bit unusual but it's an important special case and the parameters themselves lie in either the polytopic right so the convex hull of several points right so particularly we usually take this to be the polytope

so the set of delta such that they add up to one and they're all positive greater than or equal to zero so infinite number of possible values of the parameter in that case and of course the other case is the interval and the interval uncertainty that's the case where all of the parameters lie

in an interval so delta minus to a delta plus those have an eyes and x2 so actually in point of fact um we will consider both cases of course but actually we'll actually show that the interval one will reduce it to the polytopic one and then we'll solve things for the polytopic one actually

um so the polytopic one is actually the more important one in this case although the application applies to the interval as well so we'll do this in two parts so the first part uh we'll introduce a couple new notions um i hate to say of stability but they're called stability so i'll just