sir 14:11 you have taken some terms common and made some constraints, for all problems can we do the same, I mean the coefficients may vary right, then how to consider that condition. Like how to take common at that situation. could you please clarify.
hello sir could you tell me that name of another method for generating Lyapunov's Krasovskii functional? can we use this method for generation of LK functional for delay system?
You taken the value of a21 as 2 directly at delta v equation in the third term where as variable in the first term of delta v equation. Similarly for v dot equation also.
Sorry, there's been a mistake. On expansion of expression, it should be (a12 - a22)x2^2. So we should be setting a12 = 2. However rest of the video is correct as it is scripted properly. Thanks for pointing out the mistake :)
Can the variable gradient method and Krasovski's theorem be used to generate lyapunov functions for MIMO systems (i.e., systems which are not autonomous)? And if so how would one go about doing this, as there are now input terms in the dynamics?
Hi Nitin, Sorry, I do not have the knowledge to answer your question. I haven't dealt with MIMO systems before. But I believe that the book 'Nonlinear Control System' by Alberto Isidori have many chapters dedicated for MIMO systems and their dynamics. Suggesting you to take a look at them :)
why there is a constant and variable part for aij in ∇v? they are constants, right? Also, can you suggest a good book for practising more problems for the topics in this playlist?
Please note that I said aij's are coefficients, not constants. We are not sure if the coefficient is a constant until we determine it. Hence we have to check if there's a variable part. Here are some books I refer, 1.Applied Nonlinear Control by Slotine and Li - amzn.to/2Ed8Rw6 2. Nonlinear Control Systems by Alberto Isidori - amzn.to/3l5VeQv 3. Nonlinear Systems by Hassan K Khalil - amzn.to/3aG0zsA 4. Modern Control Engineering by Mohandas K P - amzn.to/317zqfd
@@Topperly thank you 😊. By variable part do you mean aij's may be dependent on state variables? In 16:03 LHS is obtained as 2 by considering d/dx2(a21 X1) as zero in the term d/dx2(a21 X1 + 3x1^2+2x2) .if a21 is dependent on state variable X2, d/dx2 (a21x1 ) won't be equal to zero right? please correct me if I'm wrong.
The coefficients aij's do not depend on state variables. For example, a coefficient can be a12= 3 + m(t), where m may be some quantity that is time dependent. So you can see that here our coefficient has a constant part '3' and a variable part 'm(t)', okay?
Sorry, there's been a mistake. On expansion of expression, it should be (a12 - a22)x2^2. So we should be setting a12 = 2. However rest of the video is correct as it is scripted properly. Thanks for pointing out the mistake :)
Hey topperly, I've a numerical problem about krasovskii method Can you solve this? Determine the stability of the given system w.r.t. origin as an equilibrium point: x1= -3x1 + x2 x2= x1-x2-x2^3 There is a dot upon x1 & x2. Thank you
I've already solved a similar question in the video on Krasovskii's method. Here's the link to that video - ua-cam.com/video/mT4EwOqelIM/v-deo.html Also, I'll try to add this question in another video :)
In case someone wants to revisit Gradient and Curl, you can try these links from Khan Academy: 1) ua-cam.com/video/tIpKfDc295M/v-deo.html 2) ua-cam.com/video/eEwZeY51mT0/v-deo.html
Thank you 🙏🏼🙏🏼
Great job👏
Thank you! Your appreciation means a lot :)
very beautifully explained.. please upload the next video eagerly waiting.Thank you sir
Your appreciation means a lot! Thank you :)
sir 14:11 you have taken some terms common and made some constraints, for all problems can we do the same, I mean the coefficients may vary right, then how to consider that condition. Like how to take common at that situation. could you please clarify.
There's no common situation. You have to play around with constraints until you reach a solution :)
hello sir could you tell me that name of another method for generating Lyapunov's Krasovskii functional?
can we use this method for generation of LK functional for delay system?
I'm sorry! I didn't exactly understand your question.
As to second question, I'm not sure. I'll need to check before I answer.
You taken the value of a21 as 2 directly at delta v equation in the third term where as variable in the first term of delta v equation.
Similarly for v dot equation also.
Sorry, there's been a mistake. On expansion of expression, it should be (a12 - a22)x2^2. So we should be setting a12 = 2. However rest of the video is correct as it is scripted properly. Thanks for pointing out the mistake :)
Can the variable gradient method and Krasovski's theorem be used to generate lyapunov functions for MIMO systems (i.e., systems which are not autonomous)? And if so how would one go about doing this, as there are now input terms in the dynamics?
Hi Nitin,
Sorry, I do not have the knowledge to answer your question. I haven't dealt with MIMO systems before. But I believe that the book 'Nonlinear Control System' by Alberto Isidori have many chapters dedicated for MIMO systems and their dynamics. Suggesting you to take a look at them :)
why there is a constant and variable part for aij in ∇v? they are constants, right?
Also, can you suggest a good book for practising more problems for the topics in this playlist?
Please note that I said aij's are coefficients, not constants. We are not sure if the coefficient is a constant until we determine it. Hence we have to check if there's a variable part.
Here are some books I refer,
1.Applied Nonlinear Control by Slotine and Li - amzn.to/2Ed8Rw6
2. Nonlinear Control Systems by Alberto Isidori - amzn.to/3l5VeQv
3. Nonlinear Systems by Hassan K Khalil - amzn.to/3aG0zsA
4. Modern Control Engineering by Mohandas K P - amzn.to/317zqfd
@@Topperly thank you 😊.
By variable part do you mean aij's may be dependent on state variables?
In 16:03 LHS is obtained as 2 by considering d/dx2(a21 X1) as zero in the term d/dx2(a21 X1 + 3x1^2+2x2) .if a21 is dependent on state variable X2, d/dx2 (a21x1 ) won't be equal to zero right?
please correct me if I'm wrong.
The coefficients aij's do not depend on state variables. For example, a coefficient can be a12= 3 + m(t), where m may be some quantity that is time dependent. So you can see that here our coefficient has a constant part '3' and a variable part 'm(t)', okay?
@@Topperly okey 👍 Thanks
Can u upload some practice problems on lyapunov Functions...because my exams are there and I need to solve more problems as to answer correctly.
in 14:30 you had put a21=2 but you didn't use it in eqn V'=a21 X1^4-X2^2 ??? why ??
you mean a12=2 not a21=2
Thank you again Madam
Sorry, there's been a mistake. On expansion of expression, it should be (a12 - a22)x2^2. So we should be setting a12 = 2. However rest of the video is correct as it is scripted properly. Thanks for pointing out the mistake :)
@@Topperlythank you it's very helpful
Hey topperly, I've a numerical problem about krasovskii method
Can you solve this?
Determine the stability of the given system w.r.t. origin as an equilibrium point:
x1= -3x1 + x2
x2= x1-x2-x2^3
There is a dot upon x1 & x2.
Thank you
I've already solved a similar question in the video on Krasovskii's method. Here's the link to that video - ua-cam.com/video/mT4EwOqelIM/v-deo.html
Also, I'll try to add this question in another video :)
I've made a video which answers your question. Here's the link - ua-cam.com/video/nyW1F9YoCFE/v-deo.html
In case someone wants to revisit Gradient and Curl, you can try these links from Khan Academy:
1) ua-cam.com/video/tIpKfDc295M/v-deo.html
2) ua-cam.com/video/eEwZeY51mT0/v-deo.html