It was a trick… He actually didn’t have to mirror his writing left-right. Instead, the whole presentation was drawn in advance. Then he went over it erasing the pen marks while talking backwards. Finally, the recording was played in reverse. Voila! 😜
@@fburton8 Seeing your reply made me realize this could be done by aiming the camera at a mirror then writing normal behind glass in the reflection. Thanks! : )
The way you present the lyapunov stability criterion as v(x_k) > v(x_(k+1)) makes alot of sense compared to the way I learnt it with the continuous time approach where you have to show that the dot_V(x) function is a negative definite function for inputs other then 0. This leads me to a question, if you develop a continuous time control law that is satisfies the stability critera dot_V(x) < 0 for a continuous time system dot_x = f(x). Will that same control law reliability stabilize a discretized version of the same system like the one you presented? (X_(k+1) = X_(k)). Controller discretization baffles me, I practical know how to do it with ZOH but I really don't understand how the theorems of continuous dynamics translate to the digital world as well as they seem to.
I've had a bit of a go answering parts this when responding to your other question - but I just want to reiterate - great questions and not easy to answer! I hope to be able to make a video about this at some point though since these are clearly very important considerations!
What a great video!! I had a question about a specific Lyapunov Candidate function that I was unsure of how to interpret: Suppose we have a Lyapunov Candidate such that: for all x with V(x) = 1+c where c is a positive scalar, we have that V(f(x)) = 1 + c/2. If V(x) 1 will never converge to 0. Is this simply reflecting a locally stable fixed point at 0, or is there some other way to interpret this? Thanks!
Another question I have is, is it possible to derive the stability conditions on the eigenvalues of linear systems from the lyapunov stability theorems. If so does it insight (again about discretization) on why can the eigenvalues of a discretized system be on the RHP so long as they are less then one while for continuous time systems any RHP poles and zeros are destabilizing
You've asked a pair of really good questions - but they are also really difficult to answer in the comments section! Preserving stability under controller discretization would make a great topic for another video, and hinges on some really nice connections between the Laplace and Z-transforms (or the spectra of the differential and difference operators). But it might take me some time to get onto that, so let me try to help here a little bit. The condition in this video - namely that there exists a positive definite P such that P-A^TPA is completely equivalent to the eigenvalues of A being contained in the unit circle. If we are a little bit sneaky we can use this to deduce the continuous time results. To do this, we need two facts: 1. Given any invertible matrix Q, a second matrix P is positive definite if and only if Q^TPQ is positive definite. 2. a matrix M has eigenvalues in the LHP if and only if (M+I)(M-I)^(-1) has eigenvalues inside the unit circle. The first fact essentially follows since any invertible Q just defines a coordinate transformation. To see the second observe that if z is an eigenvalue of (M+I)(M-I)^(-1), there is a vector x so that x^T(M+I)(M-I)^(-1)=zx^T, which after rearranging gives that x^T(M-zM)=-(z+1)x^T, from which a bit more rearranging shows that (z+1)/(z-1) is an eigenvalue of M. Some algebra then shows that a complex number (z+1)/(z-1) lies in the LHP if and only if z lies in the unit circle (this is essentially the bilinear transformation). So how can we put things together. Well suppose our discrete time stability condition holds: P-A^TPA is positive definite, and so A must have eigenvalues in the unit circle. Then by fact 2 there must be a matrix M with eigenvalues in the LHP such that A=(M+I)(M-I)^(-1). Plugging this in gives: P-(M-I)^(-T)(M+I)^TP(M+I)(M-I)^(-1) is positive definite Then use 1 with Q=M-I to get: (M-I)^TP(M-I)-(M+I)^TP(M+I) is positive definite Rearranging then gives: -2(PM+M^TP) is positive definite which is precisely the continuous time Lyapunov condition. So we have now seen how to use discrete time stability results (namely discrete time Lyapunov functions and eigenvalue conditions) to deduce the corresponding continuous time conditions involving eigenvalues and continuous time Lyapunov functions! Hopefully this gives you some idea how to connect these two worlds. The key step was in establishing fact 2, and this mapping between the LHP and unit circle. As I mentioned above there are deeper connections involving operators and transforms, but hopefully this helps a little bit!
@@richard_pates thank you so much for such a detailed answer, it will take a me a bit of time to fully process what you've answered here. Ill keep an eye on your channel for any discretization vids, I would appreciate it. I took a few nonlinear control theory classes in graduate school. Now I design satellite control systems in Industry but I find that the goto controllers on most real word systems are still simple discrete PID loops formulated for quaternion math. In the past this is likely due to computational limits of OBCs, but now that's mostly not an issue for ADCS systems. I think that alot of the resistance to using more complex controllers such as LQR and H-inf is because of the need to discretize them and prove there stability before they can be deployed on orbit. I know it's a big gap in my knowledge and confidence.
@@NZ-fo8tp Don't despair - this is tricky stuff, and a big problem is that most of the methods for doing this are really hacks. Typically you are presented with two choices. 1. Create a discrete time approximation of the process you want to control (by for example replacing the derivative of your state with it's zero hold approximation), and then design a controller in discrete time 2. Design a continuous time controller, and then discretize it with, for example, zero order hold or Tustin's method. The fundamental problem is that neither of these approaches are really doing what you want, which is to design a discrete time controller for a continuous time plant. No wonder you are confused...! As a final comment, I can recommend that you look at the book 'Optimal Sampled-Data Control Systems' by Chen and Francis - you can easily find a pdf online. The authors are excellent, and in this book they tackle the actual problem, namely how to design an optimal discrete time controller for a continuous time process, as well as discuss approaches 1 and 2 above. You'll have to be a bit selective about how into detail you want to get - but I think at least reading a few of the chapters may clarify things a lot for you
@@richard_pates Thank you for the book recommendation! It's cool it's from my alma mater, I actually already have it downloaded, it was part of a recommended reading list during my masters at UofT. Admittedly, I did not give it a thorough read, I'll have to give it another look.
Well this was a journey back into the memories of my advanced control course! Thanks!
Great to hear!
Great video. It's a hugely complicated topic but you've motivated it and described it quite well for how technical it is
Well done setting up that presentation space and writing all of that backward!
It was a trick… He actually didn’t have to mirror his writing left-right. Instead, the whole presentation was drawn in advance. Then he went over it erasing the pen marks while talking backwards. Finally, the recording was played in reverse. Voila! 😜
@@fburton8 Seeing your reply made me realize this could be done by aiming the camera at a mirror then writing normal behind glass in the reflection. Thanks! : )
@@AlexanderQ689 You're welcome! :) Flipping the image would certainly be simpler.
I can confirm all the flipping is done with software. But I know some people do use the mirror solution!
Great video! Thank you so much!!
I learned Brouwers Fixpoint Theorem last week in my Analysis 3 course xD
Finally I know what other students in the department of control were actually doing :)
The way you present the lyapunov stability criterion as v(x_k) > v(x_(k+1)) makes alot of sense compared to the way I learnt it with the continuous time approach where you have to show that the dot_V(x) function is a negative definite function for inputs other then 0.
This leads me to a question, if you develop a continuous time control law that is satisfies the stability critera dot_V(x) < 0 for a continuous time system dot_x = f(x). Will that same control law reliability stabilize a discretized version of the same system like the one you presented? (X_(k+1) = X_(k)).
Controller discretization baffles me, I practical know how to do it with ZOH but I really don't understand how the theorems of continuous dynamics translate to the digital world as well as they seem to.
I've had a bit of a go answering parts this when responding to your other question - but I just want to reiterate - great questions and not easy to answer! I hope to be able to make a video about this at some point though since these are clearly very important considerations!
Very nice explanation
Beautiful
What a great video!! I had a question about a specific Lyapunov Candidate function that I was unsure of how to interpret:
Suppose we have a Lyapunov Candidate such that: for all x with V(x) = 1+c where c is a positive scalar, we have that V(f(x)) = 1 + c/2. If V(x) 1 will never converge to 0.
Is this simply reflecting a locally stable fixed point at 0, or is there some other way to interpret this?
Thanks!
Another question I have is, is it possible to derive the stability conditions on the eigenvalues of linear systems from the lyapunov stability theorems. If so does it insight (again about discretization) on why can the eigenvalues of a discretized system be on the RHP so long as they are less then one while for continuous time systems any RHP poles and zeros are destabilizing
You've asked a pair of really good questions - but they are also really difficult to answer in the comments section! Preserving stability under controller discretization would make a great topic for another video, and hinges on some really nice connections between the Laplace and Z-transforms (or the spectra of the differential and difference operators). But it might take me some time to get onto that, so let me try to help here a little bit. The condition in this video - namely that there exists a positive definite P such that P-A^TPA is completely equivalent to the eigenvalues of A being contained in the unit circle. If we are a little bit sneaky we can use this to deduce the continuous time results. To do this, we need two facts:
1. Given any invertible matrix Q, a second matrix P is positive definite if and only if Q^TPQ is positive definite.
2. a matrix M has eigenvalues in the LHP if and only if (M+I)(M-I)^(-1) has eigenvalues inside the unit circle.
The first fact essentially follows since any invertible Q just defines a coordinate transformation. To see the second observe that if z is an eigenvalue of (M+I)(M-I)^(-1), there is a vector x so that x^T(M+I)(M-I)^(-1)=zx^T, which after rearranging gives that x^T(M-zM)=-(z+1)x^T, from which a bit more rearranging shows that (z+1)/(z-1) is an eigenvalue of M. Some algebra then shows that a complex number (z+1)/(z-1) lies in the LHP if and only if z lies in the unit circle (this is essentially the bilinear transformation).
So how can we put things together. Well suppose our discrete time stability condition holds:
P-A^TPA is positive definite,
and so A must have eigenvalues in the unit circle. Then by fact 2 there must be a matrix M with eigenvalues in the LHP such that A=(M+I)(M-I)^(-1). Plugging this in gives:
P-(M-I)^(-T)(M+I)^TP(M+I)(M-I)^(-1) is positive definite
Then use 1 with Q=M-I to get:
(M-I)^TP(M-I)-(M+I)^TP(M+I) is positive definite
Rearranging then gives:
-2(PM+M^TP) is positive definite
which is precisely the continuous time Lyapunov condition. So we have now seen how to use discrete time stability results (namely discrete time Lyapunov functions and eigenvalue conditions) to deduce the corresponding continuous time conditions involving eigenvalues and continuous time Lyapunov functions! Hopefully this gives you some idea how to connect these two worlds. The key step was in establishing fact 2, and this mapping between the LHP and unit circle. As I mentioned above there are deeper connections involving operators and transforms, but hopefully this helps a little bit!
@@richard_pates thank you so much for such a detailed answer, it will take a me a bit of time to fully process what you've answered here.
Ill keep an eye on your channel for any discretization vids, I would appreciate it. I took a few nonlinear control theory classes in graduate school. Now I design satellite control systems in Industry but I find that the goto controllers on most real word systems are still simple discrete PID loops formulated for quaternion math. In the past this is likely due to computational limits of OBCs, but now that's mostly not an issue for ADCS systems. I think that alot of the resistance to using more complex controllers such as LQR and H-inf is because of the need to discretize them and prove there stability before they can be deployed on orbit. I know it's a big gap in my knowledge and confidence.
@@NZ-fo8tp Don't despair - this is tricky stuff, and a big problem is that most of the methods for doing this are really hacks. Typically you are presented with two choices.
1. Create a discrete time approximation of the process you want to control (by for example replacing the derivative of your state with it's zero hold approximation), and then design a controller in discrete time
2. Design a continuous time controller, and then discretize it with, for example, zero order hold or Tustin's method.
The fundamental problem is that neither of these approaches are really doing what you want, which is to design a discrete time controller for a continuous time plant. No wonder you are confused...! As a final comment, I can recommend that you look at the book 'Optimal Sampled-Data Control Systems' by Chen and Francis - you can easily find a pdf online. The authors are excellent, and in this book they tackle the actual problem, namely how to design an optimal discrete time controller for a continuous time process, as well as discuss approaches 1 and 2 above. You'll have to be a bit selective about how into detail you want to get - but I think at least reading a few of the chapters may clarify things a lot for you
@@richard_pates Thank you for the book recommendation! It's cool it's from my alma mater, I actually already have it downloaded, it was part of a recommended reading list during my masters at UofT. Admittedly, I did not give it a thorough read, I'll have to give it another look.
Subtítulos please 🙏🙏