title

Stability and Eigenvalues [Control Bootcamp]

description

Here we discuss the stability of a linear system (in continuous-time or discrete-time) in terms of eigenvalues. Later, we will actively modify these eigenvalues, and hence the dynamics, with feedback control.
Chapters available at: http://databookuw.com/databook.pdf
These lectures follow Chapter 8 from:
"Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz
Amazon: https://www.amazon.com/Data-Driven-Science-Engineering-Learning-Dynamical/dp/1108422098
Book Website: http://databookuw.com
Brunton Website: eigensteve.com

detail

{'title': 'Stability and Eigenvalues [Control Bootcamp]', 'heatmap': [{'end': 668.533, 'start': 581.963, 'weight': 0.836}, {'end': 794.095, 'start': 770.107, 'weight': 0.707}, {'end': 843.984, 'start': 827.97, 'weight': 0.711}, {'end': 912.847, 'start': 898.502, 'weight': 0.706}], 'summary': 'Covers eigenvalues and eigenvectors for linear systems, explaining their use in simplifying matrix computations and system solving. it also discusses system stability and eigenvalues, emphasizing the relationship between eigenvalues of a diagonal matrix and system stability, and discrete time systems and stability, addressing the transition from continuous to discrete measurements and the computation of future states using matrices.', 'chapters': [{'end': 74.953, 'segs': [{'end': 74.953, 'src': 'embed', 'start': 30.082, 'weight': 0, 'content': [{'end': 35.545, 'text': "that essentially defines a new coordinate system where it's easier to represent the dynamics and solve them.", 'start': 30.082, 'duration': 5.463}, {'end': 45.35, 'text': 'So in MATLAB very, very simple to get a matrix of eigenvectors T and a diagonal matrix D that has only the eigenvalues.', 'start': 35.625, 'duration': 9.725}, {'end': 50.834, 'text': 'this should be lambda n, only the eigenvalues on the diagonal and zeros everywhere else.', 'start': 45.35, 'duration': 5.484}, {'end': 59.321, 'text': "And then in terms of this matrix of eigenvectors and eigenvalues, it's easy to solve the system x at some later time.", 'start': 51.615, 'duration': 7.706}, {'end': 64.965, 'text': 't in terms of the initial condition and these relatively easy to compute matrices.', 'start': 59.321, 'duration': 5.644}, {'end': 74.953, 'text': "t, again, matrix of eigenvectors, e to the diagonal d times time, which is easier to compute because it's diagonal, and t inverse.", 'start': 65.105, 'duration': 9.848}], 'summary': 'New coordinate system simplifies dynamics solving in matlab', 'duration': 44.871, 'max_score': 30.082, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf030082.jpg'}], 'start': 3.894, 'title': 'Eigenvalues and eigenvectors for linear systems', 'summary': 'Discusses the use of eigenvalues and eigenvectors to define a new coordinate system for representing dynamics and solving linear systems of equations, simplifying matrix computations and future system solving.', 'chapters': [{'end': 74.953, 'start': 3.894, 'title': 'Eigenvalues and eigenvectors for linear systems', 'summary': 'Discusses the use of eigenvalues and eigenvectors to define a new coordinate system for representing dynamics and solving linear systems of equations, making it easier to compute matrices and solve the system at a later time.', 'duration': 71.059, 'highlights': ['The eigenvalues and eigenvectors of matrix A define a new coordinate system, making it easier to represent the dynamics and solve linear systems of equations.', "In MATLAB, it's simple to obtain a matrix of eigenvectors T and a diagonal matrix D containing only the eigenvalues, lambda n, on the diagonal and zeros elsewhere.", 'Using the matrix of eigenvectors and eigenvalues, it becomes easier to solve the system at some later time, t, in terms of the initial condition and these relatively easy to compute matrices.']}], 'duration': 71.059, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf03894.jpg', 'highlights': ['The eigenvalues and eigenvectors of matrix A define a new coordinate system, simplifying linear system solving.', 'In MATLAB, obtaining a matrix of eigenvectors T and a diagonal matrix D with eigenvalues is simple.', 'Using the matrix of eigenvectors and eigenvalues makes it easier to solve the system at a later time.']}, {'end': 545.466, 'segs': [{'end': 121.227, 'src': 'embed', 'start': 76.592, 'weight': 3, 'content': [{'end': 79.873, 'text': "And so what we're going to talk about now is essentially stability.", 'start': 76.592, 'duration': 3.281}, {'end': 89.094, 'text': "So we're going to talk about the stability of the system, which basically means what does the system do as time goes to infinity??", 'start': 79.893, 'duration': 9.201}, {'end': 89.954, 'text': 'Does it blow up??', 'start': 89.154, 'duration': 0.8}, {'end': 91.915, 'text': 'Does it all go to zero??', 'start': 90.475, 'duration': 1.44}, {'end': 93.055, 'text': 'Does something weird happen??', 'start': 91.935, 'duration': 1.12}, {'end': 99.456, 'text': "Now there's whole classes on ordinary differential equations and linear algebra and dynamical systems.", 'start': 94.235, 'duration': 5.221}, {'end': 107.338, 'text': "Then we'll talk about phase portraits and saddles and sinks and centers and you know, all kinds of deep and very interesting stuff.", 'start': 99.936, 'duration': 7.402}, {'end': 112.361, 'text': "But we're just going to get kind of the bare minimum that you need to understand stability here.", 'start': 107.358, 'duration': 5.003}, {'end': 117.865, 'text': 'Okay, and it all has to do with this e to the diagonal matrix d times time.', 'start': 112.381, 'duration': 5.484}, {'end': 121.227, 'text': 'So this is where things happen as time goes to infinity or in this term.', 'start': 117.905, 'duration': 3.322}], 'summary': 'Discussion on stability in systems as time approaches infinity, using e to the diagonal matrix d times time.', 'duration': 44.635, 'max_score': 76.592, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf076592.jpg'}, {'end': 477.621, 'src': 'embed', 'start': 447.869, 'weight': 0, 'content': [{'end': 452.638, 'text': "And anywhere where the real part is positive, it's unstable.", 'start': 447.869, 'duration': 4.769}, {'end': 461.488, 'text': "And this is really simple to remember, is that if any of your eigenvalues have real part that's positive, it's unstable.", 'start': 454.963, 'duration': 6.525}, {'end': 463.99, 'text': "If they all have real part that's negative, it's stable.", 'start': 461.628, 'duration': 2.362}, {'end': 472.337, 'text': "And so a big part of what we're going to do in control systems is we're going to start with an A matrix where maybe we actually have some unstable dynamics.", 'start': 464.931, 'duration': 7.406}, {'end': 477.621, 'text': 'Maybe we have a couple of eigenvalues out here that are a little bit unstable.', 'start': 472.377, 'duration': 5.244}], 'summary': 'Unstable if eigenvalues have positive real part, stable if negative.', 'duration': 29.752, 'max_score': 447.869, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0447869.jpg'}, {'end': 525.703, 'src': 'embed', 'start': 495.046, 'weight': 1, 'content': [{'end': 499.89, 'text': 'So we can characterize what we want A to look like almost entirely based on its eigenvalues,', 'start': 495.046, 'duration': 4.844}, {'end': 503.774, 'text': 'because we want to drive them into the stable left half plane in the complex plane.', 'start': 499.89, 'duration': 3.884}, {'end': 506.997, 'text': 'Okay? Okay, make sense.', 'start': 503.794, 'duration': 3.203}, {'end': 507.297, 'text': 'so far?', 'start': 506.997, 'duration': 0.3}, {'end': 509.86, 'text': 'We have a dynamical system.', 'start': 508.22, 'duration': 1.64}, {'end': 518.501, 'text': 'the solution depends on the eigenvalues and eigenvectors, but its dynamics in time only depends on the eigenvalues in this diagonal matrix D.', 'start': 509.86, 'duration': 8.641}, {'end': 525.703, 'text': "If those eigenvalues, and they can be complex, if they have a real part that's positive, any of them, the system's unstable and it blows up.", 'start': 518.501, 'duration': 7.202}], 'summary': "Dynamical system's stability depends on eigenvalues in diagonal matrix d.", 'duration': 30.657, 'max_score': 495.046, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0495046.jpg'}], 'start': 76.592, 'title': 'System stability and eigenvalues', 'summary': 'Discusses the stability of systems over time, emphasizing behavior as time approaches infinity and its relationship with the matrix d, and explains how eigenvalues of a diagonal matrix affect system stability, with positive eigenvalues leading to instability and negative eigenvalues to stability.', 'chapters': [{'end': 121.227, 'start': 76.592, 'title': 'Stability of systems', 'summary': 'Discusses the stability of a system over time, emphasizing the behavior as time approaches infinity and its relationship with the matrix d. it also mentions the topics of phase portraits, saddles, sinks, and centers in the context of stability.', 'duration': 44.635, 'highlights': ['The chapter emphasizes the stability of the system over time, particularly its behavior as time approaches infinity and its relationship with the matrix d.', 'It briefly mentions the topics of phase portraits, saddles, sinks, and centers, indicating that these will be covered in more depth in later discussions.', 'The discussion will cover the minimum understanding required for stability, indicating a focused approach to the topic.']}, {'end': 545.466, 'start': 122.745, 'title': 'Eigenvalues and stability in dynamics', 'summary': 'Explains how the eigenvalues of a diagonal matrix affect the stability of a system, where if any of the eigenvalues have a positive real part, the system becomes unstable and blows up, while if all eigenvalues have negative real parts, the system is stable and decays to zero as time goes to infinity.', 'duration': 422.721, 'highlights': ['The eigenvalues of a diagonal matrix determine the stability of a system; if any eigenvalue has a positive real part, the system becomes unstable and blows up, while if all eigenvalues have negative real parts, the system is stable and decays to zero as time goes to infinity. eigenvalues, stability, system dynamics, positive real part, negative real part', 'The exponential growth or decay of the system is determined by the real part of the eigenvalues, where a positive real part leads to exponential growth and instability, while a negative real part leads to exponential decay and stability. exponential growth, exponential decay, real part of eigenvalues, stability, instability', "The system's stability can be visualized in the complex plane, where eigenvalues with negative real parts correspond to stability in the left half plane, while eigenvalues with positive real parts correspond to instability. complex plane, stability visualization, eigenvalues, positive real part, negative real part", "The system's stability can be influenced by manipulating the eigenvalues through control inputs, aiming to drive the eigenvalues into the stable left half plane of the complex plane. influence of control inputs, eigenvalue manipulation, stability, left half plane"]}], 'duration': 468.874, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf076592.jpg', 'highlights': ['The eigenvalues of a diagonal matrix determine the stability of a system; if any eigenvalue has a positive real part, the system becomes unstable and blows up, while if all eigenvalues have negative real parts, the system is stable and decays to zero as time goes to infinity.', "The system's stability can be influenced by manipulating the eigenvalues through control inputs, aiming to drive the eigenvalues into the stable left half plane of the complex plane.", "The system's stability can be visualized in the complex plane, where eigenvalues with negative real parts correspond to stability in the left half plane, while eigenvalues with positive real parts correspond to instability.", 'The chapter emphasizes the stability of the system over time, particularly its behavior as time approaches infinity and its relationship with the matrix d.', 'The discussion will cover the minimum understanding required for stability, indicating a focused approach to the topic.']}, {'end': 1165.944, 'segs': [{'end': 668.533, 'src': 'heatmap', 'start': 572.799, 'weight': 0, 'content': [{'end': 581.963, 'text': "But in a physical system, if I have an experiment or if I am recording data from a system, I don't have a continuous measurement x.", 'start': 572.799, 'duration': 9.164}, {'end': 586.805, 'text': 'What I do is I measure now and now and now and now at these fixed delta t increments.', 'start': 581.963, 'duration': 4.842}, {'end': 593.428, 'text': 'So what I actually have in practice, generally speaking, is a dynamical system.', 'start': 588.446, 'duration': 4.982}, {'end': 598.75, 'text': 'x at time k plus 1 equals some new matrix.', 'start': 593.428, 'duration': 5.322}, {'end': 603.735, 'text': "I'm going to call it a tilde times x at the previous time k.", 'start': 598.75, 'duration': 4.985}, {'end': 605.358, 'text': "And let's be super explicit here.", 'start': 603.735, 'duration': 1.623}, {'end': 612.409, 'text': 'x of k equals this continuous x evaluated at k delta t.', 'start': 605.378, 'duration': 7.031}, {'end': 615.687, 'text': 'Okay?. So I take some system.', 'start': 612.409, 'duration': 3.278}, {'end': 618.629, 'text': 'it might have an underlying continuous dynamics,', 'start': 615.687, 'duration': 2.942}, {'end': 629.258, 'text': "but I'm going to measure it at these individual delta t instances of time and there's some discrete update that tells me if the state was x at time k,", 'start': 618.629, 'duration': 10.629}, {'end': 634.742, 'text': "here's what the state's going to be at the next delta t at time k plus 1..", 'start': 629.258, 'duration': 5.484}, {'end': 647.791, 'text': "And it's also nice just to write down what this a tilde is in terms of a, so this is also super simple, a tilde is just e to the a delta t.", 'start': 634.742, 'duration': 13.049}, {'end': 655.114, 'text': 'Okay, and we spend a lot of time just now deriving that that if I had some dynamics and I wanted to figure out what it is at the next, you know,', 'start': 647.791, 'duration': 7.323}, {'end': 655.994, 'text': 'plus delta t.', 'start': 655.114, 'duration': 0.88}, {'end': 660.316, 'text': 'in the future all I have to do is map it by e to the a delta t.', 'start': 655.994, 'duration': 4.322}, {'end': 664.198, 'text': 'Okay, so these two matrices are related intimately by the matrix exponential.', 'start': 660.316, 'duration': 3.882}, {'end': 668.533, 'text': "But we're not going to use that so much here.", 'start': 666.392, 'duration': 2.141}], 'summary': 'Recording data from a dynamical system involves discrete measurements at fixed time intervals, updating the state using a matrix equation.', 'duration': 25.951, 'max_score': 572.799, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0572799.jpg'}, {'end': 682.339, 'src': 'embed', 'start': 655.994, 'weight': 1, 'content': [{'end': 660.316, 'text': 'in the future all I have to do is map it by e to the a delta t.', 'start': 655.994, 'duration': 4.322}, {'end': 664.198, 'text': 'Okay, so these two matrices are related intimately by the matrix exponential.', 'start': 660.316, 'duration': 3.882}, {'end': 668.533, 'text': "But we're not going to use that so much here.", 'start': 666.392, 'duration': 2.141}, {'end': 676.676, 'text': "What's really interesting is we have a notion of stability in continuous time, so having to do with the real part of the eigenvalue.", 'start': 669.453, 'duration': 7.223}, {'end': 682.339, 'text': "In discrete time, there's also a notion of stability, but it's a little bit different.", 'start': 677.457, 'duration': 4.882}], 'summary': 'Matrix exponential relates matrices; stability in continuous & discrete time.', 'duration': 26.345, 'max_score': 655.994, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0655994.jpg'}, {'end': 727.136, 'src': 'embed', 'start': 700.476, 'weight': 2, 'content': [{'end': 707.663, 'text': 'Given that initial condition, I can compute the whole trajectory or the whole set of measurements for all future times just by multiplying by a tilde.', 'start': 700.476, 'duration': 7.187}, {'end': 715.751, 'text': 'So if I have x naught, then x1 is just a tilde times x naught.', 'start': 708.865, 'duration': 6.886}, {'end': 716.773, 'text': "That's exactly what this says.", 'start': 715.792, 'duration': 0.981}, {'end': 720.596, 'text': 'So if I want x1, it equals a tilde x naught.', 'start': 716.973, 'duration': 3.623}, {'end': 727.136, 'text': "If I want x2, it's equal to a tilde x1, which is a tilde squared x naught.", 'start': 721.573, 'duration': 5.563}], 'summary': 'Using the initial condition x naught, the whole trajectory can be computed for future times by multiplying by a tilde.', 'duration': 26.66, 'max_score': 700.476, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0700476.jpg'}, {'end': 800.677, 'src': 'heatmap', 'start': 770.107, 'weight': 0.707, 'content': [{'end': 772.528, 'text': 'A tilde is just.', 'start': 770.107, 'duration': 2.421}, {'end': 778.85, 'text': "I forget if it's T inverse or T like T inverse, D, T.", 'start': 772.528, 'duration': 6.322}, {'end': 782.811, 'text': 'And A tilde squared is T inverse D squared T.', 'start': 778.85, 'duration': 3.961}, {'end': 786.973, 'text': 'And A tilde to the N is just T inverse D to the N T.', 'start': 782.811, 'duration': 4.162}, {'end': 794.095, 'text': 'And so the only thing that really gets multiplied to all of these powers is this diagonal matrix D again.', 'start': 788.553, 'duration': 5.542}, {'end': 800.677, 'text': 'So in the continuous time case, we saw that the only thing that got exponentiated was this matrix D.', 'start': 794.615, 'duration': 6.062}], 'summary': 'A tilde and its powers are defined in terms of matrix t and diagonal matrix d.', 'duration': 30.57, 'max_score': 770.107, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0770107.jpg'}, {'end': 843.984, 'src': 'heatmap', 'start': 806.849, 'weight': 3, 'content': [{'end': 819.043, 'text': "And so if we have some eigenvalue, let's say lambda, well here it's just lambda, here it would be lambda squared, lambda cubed, lambda to the n.", 'start': 806.849, 'duration': 12.194}, {'end': 826.612, 'text': 'So, if I follow what happens to an eigenvalue as I map it through the system, squared, cubed, fourth, all the way to the nth,', 'start': 819.043, 'duration': 7.569}, {'end': 833.975, 'text': "what we find is that it's actually the radius of this eigenvalue that either grows or decays.", 'start': 827.97, 'duration': 6.005}, {'end': 839.56, 'text': "So in the complex plane, what we have, I'm just going to draw a little complex plane here.", 'start': 834.716, 'duration': 4.844}, {'end': 843.984, 'text': 'Complex plane, remember real and imaginary.', 'start': 841.202, 'duration': 2.782}], 'summary': 'Eigenvalues, such as lambda, undergo radius growth or decay in the complex plane as mapped through the system.', 'duration': 27.126, 'max_score': 806.849, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0806849.jpg'}, {'end': 923.053, 'src': 'heatmap', 'start': 898.502, 'weight': 0.706, 'content': [{'end': 905.763, 'text': 'So what happens is the radius either gets bigger or smaller, but the angle just kind of rotates around and around and around.', 'start': 898.502, 'duration': 7.261}, {'end': 912.847, 'text': 'So if this radius is bigger than 1, this lambda to the n blows up as n goes to infinity.', 'start': 906.943, 'duration': 5.904}, {'end': 923.053, 'text': "Just like in the continuous time case, if this lambda has a radius that's bigger than 1, this will blow up, as n goes to infinity,", 'start': 912.947, 'duration': 10.106}], 'summary': 'The radius varies, but lambda to the power of n blows up if the radius is bigger than 1.', 'duration': 24.551, 'max_score': 898.502, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0898502.jpg'}, {'end': 991.795, 'src': 'embed', 'start': 941.904, 'weight': 4, 'content': [{'end': 949.625, 'text': "So now what I'm going to do is I'm going to draw some unit circle where everything inside has a radius less than 1.", 'start': 941.904, 'duration': 7.721}, {'end': 951.046, 'text': 'So this is my unit circle.', 'start': 949.625, 'duration': 1.421}, {'end': 955.667, 'text': 'Inside, everything is stable in discrete time.', 'start': 951.966, 'duration': 3.701}, {'end': 965.759, 'text': 'So if my eigenvalues of A tilde live inside this unit circle where radius is less than 1, this system is absolutely stable.', 'start': 957.051, 'duration': 8.708}, {'end': 974.507, 'text': "And if my eigenvalues, if even one of my eigenvalues has a real part that's bigger than 1, everything out here is unstable.", 'start': 966.9, 'duration': 7.607}, {'end': 983.126, 'text': 'Okay?. Because if I have even one of those eigenvalues with a real part bigger than 1,', 'start': 976.863, 'duration': 6.263}, {'end': 987.868, 'text': 'then this term blows up to infinity and this chain will blow up to infinity.', 'start': 983.126, 'duration': 4.742}, {'end': 991.795, 'text': "And so I'm glossing over an important step.", 'start': 989.654, 'duration': 2.141}], 'summary': 'Unit circle: stable if eigenvalues inside, unstable if real part of eigenvalues > 1', 'duration': 49.891, 'max_score': 941.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0941904.jpg'}, {'end': 1114.685, 'src': 'embed', 'start': 1084.731, 'weight': 6, 'content': [{'end': 1089.994, 'text': 'in continuous time or discrete time, completely depends on the eigenvalues of your matrix.', 'start': 1084.731, 'duration': 5.263}, {'end': 1096.216, 'text': "In continuous time, if the eigenvalues of your matrix have a negative real part, it's always stable.", 'start': 1090.994, 'duration': 5.222}, {'end': 1100.458, 'text': "If a single one of these eigenvalues has a positive real part, it's unstable.", 'start': 1096.737, 'duration': 3.721}, {'end': 1107.722, 'text': 'And in discrete time, the analog is that all of my eigenvalues of A tilde have to be inside the unit circle.', 'start': 1101.479, 'duration': 6.243}, {'end': 1114.685, 'text': 'They have to have radius or magnitude less than 1, and if a single one of them is outside that unit circle, the system is unstable.', 'start': 1107.742, 'duration': 6.943}], 'summary': 'Eigenvalues determine stability in continuous and discrete time systems.', 'duration': 29.954, 'max_score': 1084.731, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf01084731.jpg'}, {'end': 1165.944, 'src': 'embed', 'start': 1143.726, 'weight': 8, 'content': [{'end': 1148.19, 'text': 'But I just want you to know that these are kind of you know, given the continuous time system,', 'start': 1143.726, 'duration': 4.464}, {'end': 1152.513, 'text': "there's an equivalent discrete time system and I can look at the stability in either of these.", 'start': 1148.19, 'duration': 4.323}, {'end': 1159.659, 'text': "Okay, so in the next time, in the next part, we're going to look at how do you get linear systems from nonlinear equations,", 'start': 1153.394, 'duration': 6.265}, {'end': 1165.944, 'text': "and we're going to start thinking about what happens when you add plus BU and see when it's controllable.", 'start': 1159.659, 'duration': 6.285}], 'summary': 'Analyzing stability of continuous and discrete time systems, and transitioning to linear systems from nonlinear equations.', 'duration': 22.218, 'max_score': 1143.726, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf01143726.jpg'}], 'start': 546.126, 'title': 'Discrete time systems and stability', 'summary': 'Discusses discrete time systems in control theory, emphasizing the transition from continuous measurements to discrete measurements, the relationship between matrices a and a tilde, stability in discrete time, computation of future states using a tilde to the power of n, behavior of eigenvalues in the complex plane, and the importance of eigenvalues in determining the stability of continuous and discrete time systems.', 'chapters': [{'end': 805.319, 'start': 546.126, 'title': 'Discrete time systems in control theory', 'summary': 'Discusses discrete time systems in control theory, emphasizing the transition from continuous measurements to discrete measurements, the relationship between matrices a and a tilde, stability in discrete time, and the computation of future states using a tilde to the power of n.', 'duration': 259.193, 'highlights': ['The transition from continuous measurements to discrete measurements is explained, where the system is measured at fixed delta t increments, resulting in a dynamical system. ', 'The relationship between matrices a and a tilde is discussed, with a tilde being defined as e to the power of a delta t, highlighting the mapping of dynamics into the future using matrix exponential. ', 'The notion of stability in discrete time is introduced, showing the computation of future states using a tilde to the power of n, emphasizing its simplicity and usefulness. ']}, {'end': 991.795, 'start': 806.849, 'title': 'Eigenvalues and stability in complex plane', 'summary': 'Discusses the behavior of eigenvalues in the complex plane, showing that as the radius of an eigenvalue grows beyond 1, it leads to instability, whereas a radius smaller than 1 leads to stability in discrete time systems.', 'duration': 184.946, 'highlights': ['Eigenvalue can be represented in the complex plane as a radius and an angle, where the radius determines the growth or decay of the eigenvalue.', 'If the radius of an eigenvalue is greater than 1, the eigenvalue blows up as n goes to infinity, leading to instability in the system.', 'In discrete time systems, if all eigenvalues of A tilde live inside the unit circle with a radius less than 1, the system is stable.']}, {'end': 1165.944, 'start': 991.896, 'title': 'System stability and eigenvalues', 'summary': 'Discusses the importance of eigenvalues in determining the stability of continuous and discrete time systems, with a focus on the condition for stability and the relation between discrete and continuous time systems.', 'duration': 174.048, 'highlights': ['The stability of a system in continuous time depends on the eigenvalues of the matrix, where negative real parts indicate stability and positive real parts indicate instability. In continuous time, if the eigenvalues of the matrix have a negative real part, the system is stable. If a single eigenvalue has a positive real part, the system is unstable.', 'In discrete time, all eigenvalues of the A tilde matrix must be inside the unit circle, with a magnitude less than 1 for stability. In discrete time, all eigenvalues of A tilde must have a magnitude less than 1. If a single eigenvalue is outside the unit circle, the system is unstable.', 'The equivalence of stability analysis in continuous and discrete time systems, with the discrete time system being useful for understanding certain concepts such as controllability. Given a continuous time system, there exists an equivalent discrete time system, and stability can be analyzed in either system. Discrete time systems are particularly useful for understanding concepts like controllability.']}], 'duration': 619.818, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/h7nJ6ZL4Lf0/pics/h7nJ6ZL4Lf0546126.jpg', 'highlights': ['The transition from continuous to discrete measurements is explained, resulting in a dynamical system.', 'The relationship between matrices a and a tilde is discussed, highlighting the mapping of dynamics into the future using matrix exponential.', 'The computation of future states using a tilde to the power of n is introduced, emphasizing its simplicity and usefulness.', 'Eigenvalue can be represented in the complex plane as a radius and an angle, determining the growth or decay.', 'If the radius of an eigenvalue is greater than 1, the eigenvalue blows up as n goes to infinity, leading to instability.', 'In discrete time systems, if all eigenvalues of A tilde live inside the unit circle with a radius less than 1, the system is stable.', 'The stability of a system in continuous time depends on the eigenvalues of the matrix, indicating stability or instability.', 'In discrete time, all eigenvalues of the A tilde matrix must be inside the unit circle, with a magnitude less than 1 for stability.', 'The equivalence of stability analysis in continuous and discrete time systems, with the discrete time system being useful for understanding certain concepts like controllability.']}], 'highlights': ['The eigenvalues and eigenvectors of matrix A define a new coordinate system, simplifying linear system solving.', 'Using the matrix of eigenvectors and eigenvalues makes it easier to solve the system at a later time.', 'In MATLAB, obtaining a matrix of eigenvectors T and a diagonal matrix D with eigenvalues is simple.', 'The eigenvalues of a diagonal matrix determine the stability of a system; if any eigenvalue has a positive real part, the system becomes unstable and blows up, while if all eigenvalues have negative real parts, the system is stable and decays to zero as time goes to infinity.', "The system's stability can be influenced by manipulating the eigenvalues through control inputs, aiming to drive the eigenvalues into the stable left half plane of the complex plane.", "The system's stability can be visualized in the complex plane, where eigenvalues with negative real parts correspond to stability in the left half plane, while eigenvalues with positive real parts correspond to instability.", 'The transition from continuous to discrete measurements is explained, resulting in a dynamical system.', 'The relationship between matrices a and a tilde is discussed, highlighting the mapping of dynamics into the future using matrix exponential.', 'The computation of future states using a tilde to the power of n is introduced, emphasizing its simplicity and usefulness.', 'Eigenvalue can be represented in the complex plane as a radius and an angle, determining the growth or decay.', 'If the radius of an eigenvalue is greater than 1, the eigenvalue blows up as n goes to infinity, leading to instability.', 'In discrete time systems, if all eigenvalues of A tilde live inside the unit circle with a radius less than 1, the system is stable.', 'The stability of a system in continuous time depends on the eigenvalues of the matrix, indicating stability or instability.', 'In discrete time, all eigenvalues of the A tilde matrix must be inside the unit circle, with a magnitude less than 1 for stability.', 'The equivalence of stability analysis in continuous and discrete time systems, with the discrete time system being useful for understanding certain concepts like controllability.']}