9:15 why does everything ‘disappear’? I cannot see why it’s all zero except for filling in the equilibrium point. Could you elaborate? The rest of the video was very clear
Hello! Yes, exactly as @kuwt says. Just take that line and multiply everything out. We start with x2*m*g*l*sin(x1)+m*l^2*x2*(-(g/l)*sin(x1). But if I multiply out everything in the second term I get exactly the negative of the first term!
Very difficult question, since it depends on your level too. For nonlinear control though, my go to is Khalil's 'Nonlinear Systems'. It is quite mathematical, but gives a really broad and good introduction to a lot of nonlinear control topics
Sir can you share the MATLAB coding of how to calculate the Lyapunov exponents , bifurcation diagram and existence of chaos through Lyapunov exponents?
Good question! My area of expertise is large-scale systems. Two large-scale systems that have benefited from Lyapunov functions are internet congestion control (how to assign bandwidth to individual computers) and electrical power systems. For the internet, the stability of several internet protocols can be justified by Lyapunov functions. Similarly in power systems, how the grid will respond to faults (e.g. a transmission line breaking) is often assessed with Lyapunov functions (here regions of attraction are particularly important). However, I should also say that in applications, often the theoretical justification lags behind the engineering practice. In these examples I've described, understanding things in terms of Lyapunov functions and stability was not the first step. A more accurate (but still inaccurate...) description of events would be that practitioners found approaches that they knew would work in certain cases, and then wanted to understand if the same approaches could be used in different networks, or as the networks grew and evolved. The questions they were asking were at heart questions about stability of their systems, and so some of the most powerful stability tools were applied. When learning control (and mathematics in general) I would also encourage you to think of the things that you are learning not only in terms of how they are used (which is of course important), but also in terms of learning problem solving skills. In learning about Lyapunov functions, you are really learning about how qualitatively assess the behaviour of systems, and how this can be described mathematically. This skill is at least as valuable as the specifics of this particular method! Good luck with the rest of your studies!
Good question! The short answer is no, but I also suspect that in most cases, if you can get the stability argument working everywhere except at a point where the Lyapunov function is infinite, it will be possible to modify things a little bit to still conclude stability (e.g. slightly redefine the Lyapunov function around that point). Note also that these arguments are based on these regions omega, so if you pick this region to exclude the points where the Lyapunov function is infinite, everything works (though you can't get global stability this way). It might also be possible to make things work by treating things as functions onto the extended real line (en.wikipedia.org/wiki/Extended_real_number_line) - I've not seen this done, but maybe it's possible (though I guess maybe it get's hard to define gradients etc)!
Thank you for explaining so clearly
Fantastic explanation. Thanks
9:15 why does everything ‘disappear’? I cannot see why it’s all zero except for filling in the equilibrium point. Could you elaborate? The rest of the video was very clear
try eliminate the l in the 2nd term. The two terms cancel.
Hello! Yes, exactly as @kuwt says. Just take that line and multiply everything out. We start with x2*m*g*l*sin(x1)+m*l^2*x2*(-(g/l)*sin(x1). But if I multiply out everything in the second term I get exactly the negative of the first term!
@@richard_pates @kuwt thanks!
Great explanation thanks
Glad it was helpful!
really helpful thank you
hello thank you for the explanation i think lyapunov function is hard to find
You are not wrong! In general it is very hard indeed! But if you understand the idea, you are well on your way!
Which is the best book to read about Stability analysis
Very difficult question, since it depends on your level too. For nonlinear control though, my go to is Khalil's 'Nonlinear Systems'. It is quite mathematical, but gives a really broad and good introduction to a lot of nonlinear control topics
Hii can you lyapunov stability analysis for quadrotor?
Sir can you share the MATLAB coding of how to calculate the Lyapunov exponents , bifurcation diagram and existence of chaos through Lyapunov exponents?
These are not tools I have worked with I'm afraid. But they are very interesting topics for future videos - I will look into it!
could you please give some idea about practical applications of lyapunov functions and respective stability theorem?
Good question! My area of expertise is large-scale systems. Two large-scale systems that have benefited from Lyapunov functions are internet congestion control (how to assign bandwidth to individual computers) and electrical power systems. For the internet, the stability of several internet protocols can be justified by Lyapunov functions. Similarly in power systems, how the grid will respond to faults (e.g. a transmission line breaking) is often assessed with Lyapunov functions (here regions of attraction are particularly important). However, I should also say that in applications, often the theoretical justification lags behind the engineering practice. In these examples I've described, understanding things in terms of Lyapunov functions and stability was not the first step. A more accurate (but still inaccurate...) description of events would be that practitioners found approaches that they knew would work in certain cases, and then wanted to understand if the same approaches could be used in different networks, or as the networks grew and evolved. The questions they were asking were at heart questions about stability of their systems, and so some of the most powerful stability tools were applied. When learning control (and mathematics in general) I would also encourage you to think of the things that you are learning not only in terms of how they are used (which is of course important), but also in terms of learning problem solving skills. In learning about Lyapunov functions, you are really learning about how qualitatively assess the behaviour of systems, and how this can be described mathematically. This skill is at least as valuable as the specifics of this particular method! Good luck with the rest of your studies!
can a lyapunov function have as a result infinity?
Good question! The short answer is no, but I also suspect that in most cases, if you can get the stability argument working everywhere except at a point where the Lyapunov function is infinite, it will be possible to modify things a little bit to still conclude stability (e.g. slightly redefine the Lyapunov function around that point). Note also that these arguments are based on these regions omega, so if you pick this region to exclude the points where the Lyapunov function is infinite, everything works (though you can't get global stability this way). It might also be possible to make things work by treating things as functions onto the extended real line (en.wikipedia.org/wiki/Extended_real_number_line) - I've not seen this done, but maybe it's possible (though I guess maybe it get's hard to define gradients etc)!