This is so good. Slotine and Khalil are so general. This example ties it altogether. Had a Eureka moment here, I kept thinking your E had to be a ball. Now I realise its almost a rectangular region or slice on the x1 axis at x2=0. Now I see how it can come out of E if it still has energy.
The theorem says that M is the largest invariant set in E, but what you described is positive invariance. Invariance is for all t, not just for nonnegative t.
Hmm - this is something I don't really have any experience with I'm afraid. Some search terms that might help are Lyapunov-Krasovskii functionals, and IQC analysis of time-delay systems. This is a tricky area - good luck!
Thank you for the video! I was wondering if you could give a better explanation of what 'Largest set M' means or maybe give an example where the set M is not just one point. Thank you very much
Hello! I can try - though perhaps I can also recommend that you watch the video 'An Example Using LaSalle', since it might help clarify things. You really want M to contain a single point - and if it doesn't you will probably not be able to conclude stability of an equilbrium (rather you may have a limit cycle, or perhaps 'almost everywhere' stability instead - but the analysis starts get much more technical now). But I'll try to explain. The objective here is to try to extend Lyapunov arguments to functions that don't quite satisfy the condition \dot(V(x))
Good question - and very difficult to answer! This set omega acts as a kind of underlying region on which everything is defined. The reason you need it is mainly mathematical. There are some extra technical difficulties if you try and prove results on the entire state-space (you need some extra conditions on the growth of your Lyapunov function in these cases) - and sometimes you have systems with singularities (or want to pick Lyapunov functions with singularities), and then you want to pick omega so that it is sort or focusing on the part of the state-space around the equilibrium point where everything is defined. I think you can get pretty far by just ignoring the role of omega - there should be no problems as long as you are doing 'sensible things', and the main ideas are all in the sets E and M, and the idea of a Lyapunov function. If you want to write nice mathematical proofs however this is something you'll need to worry about!
Hello! Sorry for any confusion, but V(x) does need to be positive definite (ie V(x)>0 for all x except when the point x equals the equilibrium point). The only difference when applying LaSalle is how we treat grad(V).f(x). When using the normal Lyapunov stability argument, we need grad(V).f(x)
wow!
Am Emmanuel Akata from Nigeria. ur teaching is awesome and Worth recommendation.
Thanks!
Very clear and understandable explanation ! Thank you.
Amazing! Thank you for this!
This is so good. Slotine and Khalil are so general. This example ties it altogether. Had a Eureka moment here, I kept thinking your E had to be a ball. Now I realise its almost a rectangular region or slice on the x1 axis at x2=0. Now I see how it can come out of E if it still has energy.
Very happy to hear it!
Good explanation! It was very helpful
The theorem says that M is the largest invariant set in E, but what you described is positive invariance. Invariance is for all t, not just for nonnegative t.
Thanks! Great explanation!
it is really helpful . thank you
Happy to hear it - you're welcome!
Thank you for the vid. Can you give me some tips on constructing a Lyapunov Functional. Given a time delay system.
Hmm - this is something I don't really have any experience with I'm afraid. Some search terms that might help are Lyapunov-Krasovskii functionals, and IQC analysis of time-delay systems. This is a tricky area - good luck!
Thank you for the video! I was wondering if you could give a better explanation of what 'Largest set M' means or maybe give an example where the set M is not just one point. Thank you very much
Hello! I can try - though perhaps I can also recommend that you watch the video 'An Example Using LaSalle', since it might help clarify things. You really want M to contain a single point - and if it doesn't you will probably not be able to conclude stability of an equilbrium (rather you may have a limit cycle, or perhaps 'almost everywhere' stability instead - but the analysis starts get much more technical now).
But I'll try to explain. The objective here is to try to extend Lyapunov arguments to functions that don't quite satisfy the condition \dot(V(x))
Very helpful! One question: How the set omega was chosen?
Good question - and very difficult to answer! This set omega acts as a kind of underlying region on which everything is defined. The reason you need it is mainly mathematical. There are some extra technical difficulties if you try and prove results on the entire state-space (you need some extra conditions on the growth of your Lyapunov function in these cases) - and sometimes you have systems with singularities (or want to pick Lyapunov functions with singularities), and then you want to pick omega so that it is sort or focusing on the part of the state-space around the equilibrium point where everything is defined. I think you can get pretty far by just ignoring the role of omega - there should be no problems as long as you are doing 'sensible things', and the main ideas are all in the sets E and M, and the idea of a Lyapunov function. If you want to write nice mathematical proofs however this is something you'll need to worry about!
very interesting lecture. question please: why lyapuove function V(x) may not be positive definite in lasalle theorem?
Hello! Sorry for any confusion, but V(x) does need to be positive definite (ie V(x)>0 for all x except when the point x equals the equilibrium point). The only difference when applying LaSalle is how we treat grad(V).f(x). When using the normal Lyapunov stability argument, we need grad(V).f(x)
there is a confusion in talking from 4:30 to 4:40