Steve Brunton: "Dynamical Systems (Part 1/2)"

all right thank you some and to see all of you here and to talk about one of my absolute favorite topics and dynamical systems so I love dynamical systems it's why I am where I am today is to study things that move and change in time and so I prepared this this tutorial and tomorrow's tutorial on dynamical systems to have a pretty healthy mix kind of intro material that lots of you will know with some advanced material that maybe some of you don't know also a lot of old historical material mixed in with the new and up-and-coming techniques that people are using and so I really really want you to ask me questions and interrupt and tell me what you're interested in because basically I've organized a bunch of little mini tutorial lectures and I can mix and match them if you find one area more interesting or less or if your background is deeper or shallower and so on and so forth so please do interact with me and I'm going to ask you questions and I'm going to stack I'm going to expect you to like raise your hand and tell me how much dynamical systems you know and things like that because that's going to help me tailor the material a little bit better okay so dynamical systems I'm gonna mix a little bit on on the screen and then on the board if do some by hands my definition of a dynamical system is very very simple it's any system that evolves or changes in time according to some rules so pretty much anything you can think of is a dynamical system okay you can definitely think of things that are not a dynamical system your chair is not a dynamical system until you sit in it then it is for that moment that it moves a dynamical system where you have earthquakes for that so it can be forced and become a dynamical system it has a spring constant okay so anything the changes in time according to some rule is a dynamical system and I'm gonna give you a lot of cool examples that motivate me in a daily in my daily work so again a quick show of hands how many of you work with dynamical systems on a kind of daily or weekly basis it's part of your bread and butter beautiful how many of you have you know learned about dynamical systems in school but maybe haven't used them for a long time this is kind of you know undergrad material okay excellent how many of you really couldn't write down the solution to an OD e if I woke you up at 2:00 in the morning and forced you to okay good good so I'm gonna I'm gonna have a little bit of something for everybody all right it might happen so okay good so I'm just going to give you some examples of dynamical systems I actually realized that at the beginning all of my pictures were pictures and in reality I should be showing you movies because dynamical systems are essentially movies of systems evolving in time so I love rollercoasters I think we should all go to Magic Mountain sometimes while we're here fluid flows are high dimensional nonlinear dynamical systems that describe for example in this case the evolution of vorticity or how much the fluid swirls at each point in space and time I couldn't find a good movie but imagine your brain is an incredibly complex multi scale dynamical system it's constantly changing States so if you're paying attention to what I'm saying you're in one state of mind if you're kind of relaxing and zoning out that's a different state of mind and the evolution of those thought processes and how you learn and interact with the world is an extremely interesting dynamical system something important to mention about the brain and this is a theme that I'm going to bring into this entire discussion on dynamical systems is that I think of dynamical systems most often in terms of engineering objectives like optimization and control so you can't just study a brain in isolation people tried to do this you take a dead fish from the supermarket you can study its brain but you don't learn that much okay what you want to study is how this brain uses its robot body to interact with the world that's much more interesting so dynamical systems generally are interfacing with some external environment that might be causing it to have stochastic forcing slow varying parameters and maybe the the dynamical system wants to achieve some some task there's also other interesting examples of high dimensional dynamical systems like disease modeling and spread climate science social networks these are all high dimensional nonlinear multi scale dynamical systems that evolve in space and time because that's going to be my theme high dimensional nonlinear multi scale dynamical systems that evolve in space and time okay this is one of my favorite images from Philip Dutoit who is a PhD student at Caltech and he basically showed these invariant manifolds in the ocean this is again it should be a movie but each frame takes a supercomputer maybe a week at the time he computed this to compute so the ocean is a dynamical system the atmosphere is a dynamical system and there are extremely important and kind of societally relevant things that matter from dynamical systems like we want to be able to predict the future evolution of the dynamical system we want to be able to put some uncertainty bounds on our future prediction if I take actual measurements of my system and I have a model I might want to be able to reconcile those two that's like the calman filter and again eventually I might want to change the optimization of a system I want to I want to change what I'm doing to the system to optimize or control its behavior okay this is again one of my favorite videos I do a lot with inverted pendulum because they're really fun and they're really easy to analyze so this is a SpaceX rocket and I claim that this is basically an inverted pendulum right you're putting force at the bottom of a long stick and in this case you know they're going to to try to land the inverted pendulum and this is a pretty interesting and challenging control problem so the model has to be pretty accurate and my model prediction has to be fast enough that I can use this in closed loop for real-time feedback control in this case to slowly and gently land a rocket okay a human Walker is a very interesting control system so I am approximately some double or triple pendulum with spring mass torque dampers in my body and I'm neutrally stable so if you shot novocaine into my legs I'd fall over okay but that neutral stability allows me to move very very efficiently so that's also an important aspect there are instabilities in the system right if you pushed me and I had I wasn't using the muscles in my foot and leg I just fall over but that instability is what allows me to walk very very efficiently because all I have to do is kick myself just a tiny bit in the unstable direction and I get a lot of transport essentially for free okay so that's another concept that we're going to use a lot is instability and dynamical systems I like to think of the kind of astrophysics applications and motivations of dynamical systems so a lot of what I'm going to tell you today and tomorrow is going to be about parallels between how we learn basic physics F equals MA Newton's law general relativity physics which are dynamical systems and parallels with how the state of the art and machine learning and artificial intelligence are evolving today okay so there are these deep parallels between the the giants of classical astrophysics Kepler Newton Tomei and others and kind of these different perspectives on machine learning that we're using today so I'm going to bring up a little bit of that in and again this picture looks static it doesn't look like a dynamical system it's not a movie but for example and also if we measure these objects they don't move that fast right like as far as we're concerned they're essentially stationary we're rotating of course but if you measure their redshift then you can start seeing that everything is actually moving incredibly fast and accelerating according to these basic basic laws of motion okay so this is data that has been the underpinning of all of our knowledge of dynamical systems for for thousands of years was was essentially started here okay so I love dynamical systems they're super interesting all kinds of fun things you can do with them and I'm gonna start walking you through an overview of what I'm gonna talk about for the next two tutorials today and tomorrow and then we're gonna jump into a couple of examples just kind of baby dynamical systems where we can look at a few interesting things that you might do with them and then I'm gonna start building up the modern data-driven dynamical systems perspective that's probably most relevant for all of us how do you actually discover dynamical systems from data how do you find the right coordinates where they become simple things like that okay any questions before I jump in everyone's ready okay good how many of you love dynamical systems okay good I hope I hope all of you love dynamical systems by the end of this this workshop okay so when I say a system that changes or evolves in time of course I need to care to describe the state of the system so I'm going to start with a differential equation in terms of some variable X I'm going to use underbar to denote that this is a vector of states and I have DDT the time derivative of that state X is equal to some function of X of time of some control input U and probably of some parameters B and so I want to just go a little bit slowly through all of this so that all of us are on the same page so X is the state of my system and generally when I say state I mean the minimal description the minimal set of variables I need to describe my system okay so if I have a cannonball I need its position and its velocity and three components so that's six variables I need to characterize okay if I have a pendulum then I need to characterize the system by its angle theta but also by its velocity its angular velocity theta dot I need to know if it's going this way or this way that's the minimal state of the system and I would append those as terms and X yeah so for now I'm talking about a system that continuously evolves in time according to this differential equation but there is another formulation in terms of discrete time where I maybe measure my system at a fixed interval or my system is seasonal like if I want to measure the bunny population in Seattle every March that's a discrete time dynamical system so essentially you could write that as XK plus 1 exit at time k plus 1 equals some function of the previous state yeah so this is discrete time this is continuous time and you can kind of often go back and forth between these ok good and when I say that the state is the minimal state used to describe the system often what I mean here is that my future state or the time derivative of my state only depends on the current measurements I have at time T now so that's that's very important is that this X is measured instantaneously now and I'm assuming that all of the information I have now is what I need to predict the system in the future so that's true for the pendulum right if I know theta and theta dot right now I can tell you what's going to happen in the future but that might not be a true of the stock market okay so if I just give you the stock prices right now I need to know if they were going up or going down I need to know what Trump was tweeting I need to know all kinds of stuff about the past to inform that so that's not a minimal state of the system okay so I'm assuming that there is no history dependence here ok good now in general this this function f these are what I call the dynamics and in some sense I think of this as a vector field because it's a vector of values at every point in space and time and it allows me to to move along like like a vector field and generally the dynamics encode all of the physics and constraints of the system so if I have conservation of energy if I have symmetry if I know that my my pendulum is is forced to be on a circle all of that comes into these dynamics f okay this is where you encode your knowledge of physics and constraints these dynamics can change in time so maybe I should draw a little picture of a pendulum okay so that's my pendulum and it's moving according to gravity but if there's an earthquake these dynamics are gonna change in time okay everything if I if the gravity of Earth changes slightly if I do this on a spaceship or on an airplane then these dynamics are going to change slowly in time you is often my actuation or control inputs so this is usually what I think of is I have control over you I get to do something to the system to change its behavior maybe I can move the base of this pendulum maybe I can apply a magnetic field maybe I can do something like that and then finally beta contains all of the parameters that are relevant so the mass the length the strength of gravity things like that okay and this is the basic basic formulation of a dynamical system that we can use to describe all of the systems I showed you they're much more complicated but that's that's the basic underpinning yeah question you absolutely could fold all of these coefficients beta into the coefficients of F but what I'm really interested often is how the qualitative dynamics of F change as I vary these parameters so I want to look at the explicit dependence of F on these parameters yeah so for example I want to know yeah does my stability change at some point if I change the length of the pendulum arm things like that so it's really important for analysis design and control it's just essentially for guarantees I want to understand that my system is not gonna break if I change these parameters you could ask that question but that might not be the most relevant question to ask yep you had a question yeah so this is a great great point and I was gonna wait to talk about this but maybe I'll talk about it now about deterministic versus probabilistic systems so I'll give you just a couple of maybe in a couple of minutes I'll tell you about that the probabilistic analogue of this yeah so there are some assumptions X is the state and it could live in a vector space it could live in our n X could live on some manifold it could be you know the fluid flow on the surface of Earth like I would need to consider the the topology or that sorry the the geometry of the manifold I'm working on if I was for example in simulating hurricanes yeah so usually I'm going to to punt and just say that this can be embedded in a sufficiently high vector space I'm just gonna work on vectors because it's easy and that's that's often what we do but sometimes you actually need to take into account the geometry of the manifold that the state lives on for example I'm implicitly I map licit implicitly assuming that theta or X here lives on a circle rather than in a continuous line okay great questions other questions so you right now I'm gonna say is a function of T that I get to play with but sometimes I'm going to build a little widget that measures X and feeds it back and that will be feedback control great in the back yeah you can all if you as a function of X then this all becomes just a function of X yeah that's right so again I'm separating out the things I have control over versus the things I don't have control over versus the things that I might want to be robust to changes in and time they're all really really good good points okay I'm gonna get to the probabilistic description in a minute okay and I'm gonna talk right now about some of the challenges of these systems so so the modern challenges that we have that I'm gonna repeat these over and over for all of this lecture and and the one tomorrow so you're gonna hear this a bunch for me so I don't think of these as problems I think of them as challenges the first most fundamental challenge is that usually we don't know F for the systems I actually care about okay so one we don't know F now you might be saying to yourself we know lots of s we know F equals MA we know quantum mechanics we know general relativity we know all of these basic equations navier's stokes equations that describe how physical systems evolved that's true we don't have an equation for the brain we don't have an equation for how disease spreads we can't agree on equations for climate okay and even the equations we do have access to the Schrodinger's equation and the navier's stokes equations that describe these complex systems we want and interact with those equations are not at a resolution that's useful for us as practitioners so solving the navier-stokes equation as a PDE makes great movies but it's very very hard to use that for real time optimization and control so I want a reduced description that's useful for the things I care about so I would argue even in the cases where we do have the governing physics we might not have a description in terms of an effective state that's useful for us in our daily lives and so so I'm thinking of F as my model of reality and this model doesn't have to be perfect there's always a balance between accuracy and can comprehend is and how complex or slow or cumbersome or detailed the model is and so there are different levels of resolution even if I have the Navy or Stokes equation very accurate very cumbersome I might want to find a reduced order model that I can use for my quadrotor control loop okay a question good okay so a lot of what we're going to talk about in this entire three-month workshop is based on the fact that we don't know F and we're going to discover it from data or effective reduced order models okay - is that F is often nonlinear okay so nonlinear F and depending on which which day of the week it is I might alternate which of these is a bigger problem for me okay so non-linearity is what keeps us all in business fundamentally you have job security we all have job security for our entire lives presumably because F is nonlinear in most cases that are interesting okay so when I say nonlinear versus linear differential equation how many of you have a gut feeling of what I mean how many of you would like me to just very quickly remind you why linear and nonlinear is important okay so a linear differential equation is something so if it was linear meaning X dot is equal to some matrix a times X or or if X was a scalar just a single value then X dot equals lambda X then I know everything about the solution of these differential equations if I know a or lambda so X at any time T in the future I hope can people see when I'm writing ya is simply e to the lambda T times my initial condition X measured at time 0 so if I have an initial measurement of my system I can tell you what that state will be at all future times just by multiplying it by e to the lambda T for a scalar system or by the matrix e to the matrix eighty times my vector of initial conditions and there's some linear algebra for how you actually compute this thing but this is completely computable and I can tell you the solution at any time in the future I can develop control laws with one line of Python code everything in life becomes easier and the reason is is because solutions superpose if I have two solutions of my differential equation and it's linear I can add them up and that's also a solution to my differential equation all of that goes out the window if my system has even a tiny amount of non-linearity an epsilon non-linearity destroys all of the theory we have for for generic theory of differential equations it's unsolvable in general okay kind of a mess to be honest those are the two big ones and then I'll talk about a few more there's not doesn't end there and please do again interrupt high dimensionality is a huge problem so high dimensionality that's just to say that X is in RN but n is a really big number like a million or a billion or a trillion so if I want to model the climate or model the weather or model the spread of disease I need millions or billions of states that I'm measuring to know how the system is going to evolve from one time step to the next it's very expensive very cumbersome and you can't use it that easily for real-time computations so you've seen this consistently in these molecular dynamics simulations and is ridiculously large but we know from for example the theory of gases that Avogadro's number is not the actual number of degrees of freedom I care about right I care about things like temperature and pressure you know there are a few often leading order quantities that you can extract from that high dimensional data and it's actually the dynamics and those coordinates that matter and so these are all really related even if I knew the equations for gas dynamics of an individual gas particle I might need to learn pressure and temperature so that I could learn the effective thermodynamic equations on those variables okay so high dimensionality is a huge challenge that goes hand-in-hand with multi scale so the gas dynamics is actually a great example of something that's super high dimensional but not that multi scale I can kind of homogenized over you know over the small scales and I get a few equations but there are lots of systems that are inherently multi scale we believe the brain is multi scale we believe that turbulence is multi scale meaning that if you look at a picture of the system and you zoom in and you zoom in there are structures that matter at every size scale and also at every time scale so think of a human your cells certainly matter but you don't think about your cells on a daily basis your organs matter right you want to keep those healthy your cells and your organs rely on your DNA which rely on your basic fundamental chemistry so it's multi scale in space it's also multi scale in time right now again you're paying attention to the few words I just said you're trying to connect those to something I said five minutes ago but you also have you know days and weeks and months you have a lifetime of time skills that are presumably relevant for you okay so multi scale is that's what I'm talking about you can't just extract one set of equations at one scale and this is really probably the major complexity that modern methods in optimization data science and machine learning are opening up most rapidly I would say we're really starting to be able and with with just large-scale computations okay we have chaos chaos is super interesting I'm going to give you a whole like ten minute thing on chaos how many of you again have a gut feeling of what chaos is how many don't all right good other things there's noise so I really should probably have said this is plus you know some kind of stochastic forcing some exogenous forcing that I can't measure and I don't know about and I might not even know what it's statistics are so noise and forcing and then the last one is the fact that what I told you here does anyone see any big problems with this maybe I'll let you decide what's what's the big glaring problem with what I wrote down all of this is hard but I'm assuming I'm making a big assumption here yeah so steady state and transient so transients is also very important I want to be able to describe oftentimes steady-state behavior and transient behavior but I'm thinking something maybe even more fundamental yeah that what so the model does exist yeah okay the derivative exists again we could go to discrete time these are good yeah in the back ok so it's a marina nailed it so I'm assuming that I have X the full state of the system that I actually am going to describe everything in terms of which is great for theorists and computational ist's but I don't often know what the right thing to measure is of an experiment so assuming that I am actually measuring the variables that matter to describe my system at the dynamics is a huge fundamental assumption that we usually make and don't even think about it so I'm gonna lump that all into latent variables and I mean so many things by this and I'm overloading this term maybe there are hidden variables that I'm not measuring that matter and I have to estimate them and I don't even know they exist maybe I measure far too many variables like in the gas dynamics and I want to distill that down to two or three or four variables that actually matter maybe I don't know what I should be measuring and writing my state in terms of this is in some sense coordinates and most of the history of mathematical physics is driven by seven one and two okay Fourier transform trying to find coordinates to make a system simpler okay everything that Kepler and Tico Bray and told me did was essentially trying to find coordinates so that they could get dynamics that actually made sense culminating in Newton's second law okay and maybe I'll write that equation this is important enough to write down that usually I don't actually I don't just have this differential equation what I often have is some kind of an output equation of a measurement equation so usually what I'll have is some y equals G of X and again it could depend on you it could depend on T you could depend on lots of stuff I'm just going to say G of X where Y is my measurement of the system usually that's all I get to see is my measurement of the system Y and I might not I might or I might not know what G is okay for a long time did we know that the temperature G was the average kinetic energy of the molecules right we might not know what G is we might just know that there's a measurement we get all right and there's uncertainty in everything in beta in T my measurements of T have uncertainty my measurements of X my measurements and everything has uncertainty that's a challenge - okay good that's a lot these are a lot of challenges but that's good we are gonna be in business for a long time we've got lots and lots of work to do in modeling these complex dynamical systems that are multi scale where we don't know the right coordinates where there's non-linearity where we need to reduce descriptions for optimization and control okay so run with me so far good any questions about this yeah no absolutely not so most of our continuum descriptions of systems like if we if I have a conservation law like conservation of mass momentum and energy of this gas I write down as an infinite dimensional conference conservation law and then I discretize it to simulate i discretize into a finite dimensional end so very yeah and in fact a lot of what I'm going to tell you is taking a state X lifting it to infinite dimensional's infinite dimensional state often allows me to linearize my nonlinear dynamics so if you know about the kernel trick in machine learning this is just like the kernel trick but for dynamical systems okay good other questions yeah sorry I'm gonna call that coordinates yeah so if I'm 80% right I'm happy so maybe okay I could give you a few example uses of these models a few of the techniques I'm gonna talk about and then I'm gonna give you your probabilistic model unless who would like to see the probabilistic model first who would like to see the uses of the model and the techniques first sorry because they're outvoted okay so the uses of the model so I've already described most of these I'm just gonna kind of summarize them again the primary uses of the model are prediction and what I mean by prediction is of the future I don't have predict the past we'd often test our methods by acting like we didn't know the past and then predicting it but really predicting the future and this could be a deterministic prediction for example in the pendulum I can do deterministic predictions but in statistical systems or systems with large uncertainty or chaos I might want to do a statistical forecast I might want to tell you the probability of being in a state or the distribution of my state as it evolved in time so this could be an ensemble or a statistical prediction often I'll want to use these models for design or optimization so for example I think we've heard a lot of cool examples of drug design the design of osmotic membranes for desalination in my field of fluid mechanics we think of design optimization for things like Formula one cars and super yachts and aircraft and transportation vehicles so we want to reduce drag increase ground force and things like that okay so actually a lot in my field of these reduced order models are used for design and optimization of Formula One and super yachts hundreds and hundreds of millions of dollars of simulations probably Saturn 5 Rockets you know out the wazoo to design and optimize Formula one cars and super yachts which i think is kind of fun but also yeah it's a luxury so design and optimization things that also matter are where you put your winglets on an airplane so that it has a few more percentage fuel efficiency that translates to billions of dollars and and tons of of carbon ok control I'm going to talk a lot more about control and when I say control how many of you know what I mean usually what I mean as I measure the system I make some decision based on that measurement of the system and I change you to manipulate the system to some behavior in real time it's basically real time optimization and in fact the lines between optimization and control are becoming blurred with supercomputers and GPUs and an onboard computation and probably the thing that gets left out the most in engineering and that I think is really important is understanding we don't just model to control we also want to understand the world we want to gain insight we want to understand why these are the right coordinates and why this is the physics and communicate and understand because that will in fact help us do better design prediction and control if we actually understand the system and so there's a few aspects of understanding and I more or less speaking to the machine learning people in a lot of this what I mean by that is I want my model to be interpretable and generalizable those are two of the most important things to me so I want my model to be interpretable and I want it to be generalizable and I might be willing to sacrifice some percentage points of accuracy for interpretability and generalizability because I'm going to gain later when I put this in my engineering loop so it's so vague and I can't tell you that my definition of interpretability is barely interpretive bowl and I guess what I mean by interpretability is you can analyze make reason of communicate distill down you know I think of F equals MA as a fairly interpretive 'mad 'l kepler's ellipsis we're also interprete bowl Newton's second law is the ultimate generalizable model you can learn about it from a falling Apple you can use it to design a rocket to put people on the moon okay so F equals MA is kind of my model at least of generalizable interprete Buhl's a little bit more subtle right usually what I mean by interpretable is sparse or low rank I want there to be a few things I have to look at and communicate because I'm a human and for me more than 3d is Heidi okay and then some of the techniques I'm going to talk it out and these are really techniques for what I'm going to call modern dynamical systems so I'm going to talk a little bit more about classical dynamical systems but modern dynamical system is really trying to solve those bullets based on data I want to point out all of the uses of my model except for maybe understanding our optimization problems obtaining the model from data is an optimization problem I want to find the best parameters beta or the best model f that describes my my data maybe with a penalty on complexity of f that's an optimization problem this is all optimization with data and when I oversimplify the world all I think machine learning is optimization with data it's a set of improved techniques for optimization and regression based on data all of these are Automation and regression based on data so they go perfectly with emerging tools in machine learning okay so the techniques I'm going to talk about are go-to for the last and this is not new at all since the 60s has been regression techniques so the calman filter is essentially a linear regression that reconciles measurements of data in perfect measurements with an imperfect model so regression and I'm gonna break this up into linear and nonlinear and then there's other variants you can add penalties by Gil arised regression sparsity regularized regression there is an increasing movement to use neural networks and deep learning for these tasks so I'll certainly talk a bit about that and there's pros and cons of each of these neural networks as you know have greater expressive power but often less interpretability and generalizability regression models are very very communicative all interpretable and often generalizable if you regularize them but they don't have the same expressive power okay so generalizability really is about regularizing your problem and you can regularize both of these of these these techniques and then a third area that i don't do as much with but i think is super interesting and it started off this whole data driven discovery in the modern era is the use of genetic programming by HOD Lipson and others don't ask me to spell genetic programming so essentially learning dynamical systems and conservation laws with genetic programming okay now there are so I said that all of these things are optimization problems machine learning is a growing body of optimization techniques and regression techniques based on data with fancy and improving regularization but that's not the entire story dynamical systems have extra knowledge that is very very important when you're building these these models using machine learning so for example I know about conservation laws I know that mass is conserved and energy is conserved in certain systems I know about constraints and symmetries so for example if you trained a neural network to predict the motion of a pendulum and you gave it the X in the Y position I bet you'll get a model that is pretty accurate and gives you pretty good sinusoidal motion but I think you're gonna be hard-pressed to guarantee its to be stable and stays on the constraint of the pendulum arm staying on a circle fixed length that's the kind of thing we know from our basic physics lots and lots of constraints that are hard to get out of a basic machine learning architecture without baking it in ahead of time okay conservation constraints symmetries stability is a huge one and so on and so forth ok ok good I should stop and ask for questions for a minute yeah I mean these are essentially some of the things I'm gonna talk about there's lots more but I'm gonna break it up into these basic categories and there there is more regression techniques neural network techniques and let's just say other machine learning techniques or modelling F for learning coordinate systems where the system becomes simple for handling high dimensionality for for things like that ya know I mean that that's the modern industrial landscape has been defined by expert humans writing down equations of motion from expert knowledge first principles and then designing a control law and then implementing that in hardware so it works we have aircraft we have automobiles like that is how the modern world is built but there are roadblocks that we hit when we try to apply those methods from a complex system like an aircraft or a car or a construction equipment to an extremely complex system like a human brain or a disease system or the climate so absolutely that works if you can do it by first principles by hand write down the control a lot like that is what we've been doing for the last forever yeah yep yeah so I think of that as a another form of a regulatory feedback where the control has some state that is accumulating information some dynamical system and then those two systems regulate each other yeah so the closed-loop system might have different dynamics than without control you can change the stability with control of these systems so if I have let's say a pendulum you can a picture and holding a broomstick and if I don't move my hand it's unstable but if I look at the broomstick and I move my hand when I see it wobbling I can if I'm lucky stabilize it for a little while okay yeah yeah so there is so the theory of differential equations I'm gonna again oversimplify we understand everything about linear systems so if I have a linear system that a matrix I take its eigenvalues and its eigenvectors and it tells me what directions in state space are important what x directions are important and what their dynamics are that's a very useful analogy in in more complex dynamical system so so there's six points a fixed point means yeah I'm gonna give you some basic dynamical systems overview and maybe I'll do that sooner sooner than later so I'm just going to talk about a few basic things this is gonna be super fast just a refresher of words you should know and if you don't please ask me okay and there's no real order to this to be honest with you so I told you that if I have X dot equals ax this is a matrix this is my state then X of T equals e to the a TX not so this is a linear system and there's some there's one fixed point at the origin so so let's say I have X dot equals f of X a fixed point is defined as some point X bar so that F at X bar equals zero and what that means is that DDT of X at X bar is zero I have no motion my system is fixed so if I take my pendulum in the down position that is a fixed point of the system technically my pendulum in the up position is also a fixed point is just unstable if you perfectly balanced it in a vacuum it might stay there for a little while but if you breathed on it it would go unstable and start swinging okay so fixed points are very very very important in dynamical systems often we want to model the behavior at the fixed point to see if the system will go unstable we linearize these non-linear dynamics around this fixed point and see if my eigenvalues are stable or unstable okay so what I would do is I would look at X I would say it's you know I'd look in some little Delta X neighborhood let's say I have a fixed point X bar what I would do is I'd look at some little epsilon neighbor of that neighborhood of that and I would plug this into my differential equation and I would try to derive a differential equation for Delta X dot and it turns out a legal the Jacobian of F evaluated at X bar dotted with Delta X plus higher order terms you just get this from the Taylor series and this is like an a matrix so if I have a fixed point of a very high dimensional super complicated nonlinear system if i zoom in to that fixed point i can linearize the dynamics approximate for small Delta X neighborhood the dynamics look linear you get that from the Taylor series I skipped one step where I plugged this in and I Taylor series expanded and this is the leading order term in the Taylor series and then what I do is I look at that linear dynamical system near that fixed point to see if it has any unstable eigenvalues and eigenvectors and if it's unstable then that means that weird stuff can happen and I can get carried away from this neighborhood and my system will go into other behaviors that might not be beneficial if they're all stable then that means that this fixed point at least is going to locally these trajectories are going to go towards this fixed point so a concrete example I think about in fluid flows is the boundary layer over your wing so this is your wing surface this is the boundary layer which basically says fast flow is up here and the flow has to accelerate up to that speed from the surface to the free stream so this is called the boundary layer and a laminar boundary layer is what you want this is a low drag laminar boundary layer but if you have any turbulence or roughness or anything at all in a modern aircraft what's going to happen is your boundary layer is going to become really turbulent and messy and unsteady and you're gonna have high drag and it's going to bump the people around on the plane and it's going to cause fatigue all kinds of bad stuff so what you might want to do is analyze the stability around this is a fixed point it's an unstable fixed point I might want to understand why it's unstable how its unstable and then control to stabilize it so that I avoid this on my wing okay yeah so a lot of what we think about how many of you know about phase portraits how many how many don't okay so phase portraits are kind of how I think about going from linear to nonlinear systems so I'm just going to draw a really really simple one so let's say I have a differential equation X double dot equals X minus X cubed so this is force equals mass times acceleration this is my acceleration and this is a force from a potential this is a gradient of some potential fee that looks like this so the way I want you to think about this there's literally a wire and there's a bead on that wire in a gravitational potential and if I let that be go you can picture it moving around this surface yeah it's like a roller coaster so I can write down X dot equals some velocity and V dot equals X minus X cubed and then what I do with this system is I think what are all of the fixed points so step one is I list the fixed points and the fixed points so for X dot and V dot to be zero I need V to be zero and I need X minus X cube to be zero x times one minus x squared to equal zero so I have three fixed points X can be 0 X can be 1 X can be minus 1 so I have 1 0 minus 1 0 and 0 0 this is X and this is V then what I would do is I would linearize these equations I'd compute this there's a partial derivative of F with respect to X at these fixed points and I would get an a matrix and I would look at its eigen values and eigen vectors to see if it's stable and so then what you can do in X and V coordinates I have a fixed point here at 0 0 I have a fixed point here at 1 0 and I have a fixed point here at negative 1 0 and this is extremely consistent with your intuition that if I started the bead right here with zero velocity would stay there that's what zero zero means if I started the bead right here with zero velocity it would stay there and if I started it here but then when I compute the linearized equations about these three fixed points what I find is that this fixed point is stable so if I add a little bit of energy to the system it basically stays just oscillating same with this one but if I look at this fixed point it's unstable it'll have a positive real part of one of its eigenvalues and what that means is that if I push the system a little bit of off equilibrium it's going to move really far really fast away from that equilibrium and so those directions if I computer the eigenvectors would look like this and then there's a conceptual leap where I can complete these circles because I know what's happening at all of these fixed points and I have some intuition for the system so this is a phase portrait you've probably seen these before okay if I add friction these turn into little sinks and everything spirals into them and so that's really I mean I'm gonna again oversimplify a little bit but that's kind of the state of the arts linear systems are easy I know everything about this system so if I have a nonlinear system what I do is I try to find all of the fixed points where I can linearize and then I go and do the thing I understand in these regions and I try to connect the dots that's really what we do in dynamical systems okay okay good any other questions so linearization stability and I'm glossing over I'm gonna see me most of you know that stability is determined by the eigenvalues and eigenvectors of a so if you need you know to brush up on that that'll be helpful how many of you have heard about bifurcations and normal forms okay how many of you like feel pretty comfortable at bifurcations many fewer I'm not gonna belabor this but it is really important and I think it's something that is going to be increasingly important in the context of machine learning is thinking about remember there are parameters that might might change in the system so I don't always just care about about a system with fixed parameters often what I want No is how will the system change if I change the parameters of this system does it does it go from stable to unstable that's important for me to know if I'm designing an airplane if my airplane is stable at one velocity in the air and it becomes unstable if I increase that velocity by ten miles per hour that's a pretty bad design okay and so bifurcations I'm just going to give a really simple example here so I'm going to take X dot equals beta X minus X cubed okay so I get to vary beta from negative infinity to positive infinity and I want to know how the system changes as I vary beta okay how many of you've seen this example so again I can look at the fixed points of the system I say the fixed points are when x times beta minus x squared equals zero so I always have a fixed point at x equals zero and then for positive beta so beta greater than zero I have a fixed point at x equals plus or minus root beta okay and again I can compute the stability of each of these fixed points by basically computing the derivative of this with respect to X and plugging in these numbers okay so if I looked at you know x equals zero then my derivative of my vector field with respect to X is just well my derivative is always beta minus three x squared and if I evaluated at x equals zero I plug x equals zero in here then this equals beta which might be positive might be negative if x equals plus or minus root beta then this will equal beta minus three beta which is minus 2 beta and I said this was only for beta positive and so a negative to beta is definitely a negative number which means this is always a stable fixed point and so what you can do is you can plot these trajectories on the beta line and let's say this is beta equals zero and the y axis I'm gonna say is the fixed points of X so if beta is negative there's one fixed point there's only one value that satisfies this when beta is negative it's x equals zero and for beta negative the fixed point at x equals zero is stable because it's eigen value is stable I get basically X dot equals theta X locally near x equals zero and so I have a stable fixed point as I increase beta through zero I pick up two additional fixed points plus and minus beta so I have two new fixed points that branch out x equals plus beta root beta and x equals minus root beta and I claim because I plugged in that that value and I get a negative number that eigen value is stable and so these branches are stable fixed points and the fixed point at x equals zero becomes unstable when beta goes positive because now my eigen value is positive and this grows blows up exponentially and so now this becomes an unstable that's just what a dashed line means in dynamical systems and so this is like the simplest example you can cook up where you have a very simple differential equation but now there's some parameter you get to play around with and as you go from negative values of that parameter to positive values really weird qualitatively different things happen so a bifurcation is one of these points where the dynamics qualitatively change you get new fixed points your stability changes things like that and these are what you're looking for in lots and lots of engineering systems like a seizure might be described by a bifurcation in the dynamics your aircraft might go through a bifurcation of the dynamics where the instability like the flutter on the wing starts to amplify and the wing rips off this happened a lot in the 30s actually happened to the a380 before they redesigned it bifurcations happen everywhere so it's really important to understand what parameters actually change with with your system and what the sensitivity of your dynamics are to those parameters okay that was really fast that's all I'm gonna tell you about the background of dynamical systems okay any any questions so the words you need to know linear systems eigenvalues and eigenvectors stability of fixed points which are determined by those eigenvalues phase portraits for nonlinear systems and bifurcation bifurcations and normal forms okay good so now I think I'm going to tell you a little bit more about kind of what I think of as modern modern dynamical systems and this is going to set up what we're going to talk about in the next the next part of this okay yep yeah yes so there's a lot of theory I'm completely glossing over that's very very interesting and so there's something called the Hartman Grubman theorem which basically says depending on certain properties of my dynamical system if i zoom in close enough to the to a fixed point of a nonlinear system I can describe it by its linearization and so it looks linear if i zoom in enough yeah so there's a whole theory called normal form transformations which allow you to change your coordinates to make that window of validity bigger and bigger and bigger and that's actually what they did what they've been doing in space mission design for a long time there are a bunch of fixed points in the solar system and what they do are these normal form transformations to make them as linear as possible for as large of a radius as possible yeah any other questions okay so some things I'm gonna talk about our kind of this progression of complexity from increasing non-linearity from linear systems where we know everything and notice I put the pendulum here this is linear for very small amplitudes it becomes nonlinear as you make the amplitude bigger kind of weakly nonlinear systems where there's a few fixed points but it's relatively periodic or quasi periodic to full-blown chaotic systems like the double pendulum the Lorenz system and things like that which I'll talk about a little later in practice what we often do in dynamical systems is if we have the equations of motion F what we'll do is we will integrate trajectories through that dynamical system so using modern numerical integrators like runge-kutta fourth order runge-kutta or Euler what I can do is I can take these dynamics and I can cook up some scheme to step X forward one Delta T at a time one small time step at a time and so in this case here what I've done is I've taken a grid of initial conditions in white and I've integrated them along some fluid velocity field this is flow past a wing this is like a mosquito wing or a moth wing and you can see a lot about the dynamics based on where those particles go so for example you can color code regions where a lot of stretching occurs or there's a lot of sensitivity and you need to measure or control in those those locations okay so a couple things I want to point out and I'm gonna end soon in modern dynamical systems the first thing I want you to know is that if you want to know more about any of the stuff I've been talking about and any of this stuff I will talk about we just wrote a book data-driven science and engineering and this is not just a shameless plug for the book I have put up a complete copy of the PDF on my website so you can get it for free you don't have to buy it if you like it you can but you can get the entire PDF here and it has everything I'm going to talk about and it goes into a lot more depth about how you linearize around fixed points why eigenvalues are important all of this stuff so if you feel like I rushed over that and you didn't catch it chapter 7 is gonna walk you through all of it okay and databook u-dub calm we actually have videos for every section of every chapter of the book so all of these are on on youtube if you want to like walk through really really carefully all of this information so slightly shameless plug but a lot of free material out there if you think I went too fast okay so the things I think about that I'm going to talk about for the rest of my time today and then for the entire tutorial tomorrow are really these basic themes and so I'm gonna take from those challenges one through seven on the board I'm gonna pick my favorite ones and that's what we're going to talk about okay the first one is that often equations are unknown or partially known or the form of the equations is not useful for me and so we're going to talk a lot about model discovery using these techniques of linear and nonlinear regression genetic programming neural networks and so on and so forth the other aspect that I'm going to focus a lot on is the aspect of non-linearity so again non-linearity immediately hampers our ability to cook up generic simple solutions and predict for arbitrary times in the future and so what I'm going to talk about are ways that people are using data-driven methods to discover affected coordinate systems that simplify your system now simplify means very different things to different people okay for me as a control theorist if I can linearize my dynamics that's like the ultimate simplification because I know everything about linear systems sometimes making my systems sparse would be a simplification because there's less terms I have to analyze and think about and again the history of mathematical physics is defined by moments where people had epiphanies where they came up with the coordinate transformation that explained their system and made their system simple okay so if you think about the Fourier transform the Fourier transform is my favorite example of this bori didn't do this for no reason he cooked up the Fourier transform to diagonalize the heat equation it is the coordinate transformation that makes the heat equation simple and in his case simple means linear diagonal decoupled and sparse okay and it's with us today and so increasingly we are able to discover these coordinate transformations and represent them in these more powerful representative architectures like neural networks and then the last one is the challenge of high dimensionality so high dimensionality often obscures our understanding of the dynamics think about the gas dynamics example it took a long time to go from individual atoms to a thermodynamic description and but we know that patterns exist in most systems we care about and that facilitates reduction okay so I have a little bit more time I guess I have ten minutes are there any questions before I go into more like more technical stuff yeah yes so so so this should be simplify and then my definition of what simplify means is it changes every day of the week sometimes simple means mean field sometimes simple means higher-dimensional but linear sometimes simple means probabilistic it really you know whatever I can get an effective model that allows me to solve some of my objectives of you know design optimization and control like that's that's what I want sorry I never got to probabilistic description is of dynamics and I might not have a chance to you at this moment so I'm gonna do a choose-your-own-adventure there are a few things I can talk about I could talk about chaos and give you some cool movies or I could talk about control and give you some cool movies how many of you would like control good how many of you would like chaos all right control has it and I'll start off next lecture with with with chaos okay good so this is an experiment built in my lab by Karthik Amon this is a double pendulum on a cart that you can actuate and you can essentially swing this thing up and stabilize it again there's a fixed point with both of them up but it's unstable but by measuring the system there's little encoders that measure the angles here and here I can measure that and with a model of the system I can feed that back and change the dynamics and stabilize it pretty cool so that's one of the things you would want to do with a model so how many of you have some expertise on control and feel pretty comfortable with it how many would like to learn more about it beautiful okay so there's two there's a lot of dichotomies in control that you can talk about and I'm gonna just give you the the most kind of naive division of these there is active versus passive so passive means you're not putting any energy into the system it's a stop sign a stop sign is a passive control active would be a stoplight okay so you're actively putting energy into the system and changing the control input so a stop sign is passive stop light is active and then in terms of active control there's open loop and closed loop open loop just means that you have a pre-planned control strategy that you hope works so my stoplight just turns red for a couple you know 30 seconds then it turns green for 30 seconds then it turns red then it turns green that's open loop that's not how it works in Los Angeles there's measurements in the road to see what the traffic flow is to make it faster but a normal traffic system with a simple dumb stoplight is just open-loop the controller does some pre-planned thing to give the the desired output behavior that you want and you'll get a better job if your city if you have a model of your system okay so another example I think of is like a toaster oven if you want toast at a certain level of doneness of toasted miss your toaster oven basically just sets a timer and cooks it for that long and usually it comes out about right okay there's a model for how fast things toast but then if you put another piece of toast in right after what happens gets a little burnt because the toaster is hot and your models wrong okay unless you have a super fancy toaster like my grad students just bought in which case it measures the temperature and changes the the time for that system so there's a lot of challenges and problems with open-loop it relies on a good model but it fundamentally can't stabilize unstable dynamics so in my pendulum on a cart example there's no way of stabilizing that system without actually taking measurements of the system there's no perfect pre planned trajectory that can stabilize an inverted pendulum so that if you blow on it it doesn't fall because it can't know that you're gonna blow on it okay and it tends to over react to noise and disturbances and it's not robust so the solution what we all do is we measure the output of the system line and we feed that back into the controller so now the controller is actually measuring the response seeing if the system actually did what you told it to do and then it changes its control input to modify the system based on this kind of discrepancy between what it thinks the model should be doing and the measurement of the actual system okay so this is a very very standard feedback control 101 is that you get a lot of benefit just by measuring your system kind of obvious but again you're now reconciling a discrepancy between your model of the system and the output of what the system actually does okay so with feedback control you can stabilize on stable dynamics you can compensate for uncertainties you can handle sensor noise and disturbances so if you're in a windy room you can stabilize the pendulum on a cart and you can push on the pendulum a little and the system will react okay these are all the things that control is good for okay now the standard industry control there's two standard algorithms that almost every industry in the world uses the first one is PID proportional integral derivative that's been around forever and the second one is model predictive control so most like construction equipment and you know chemical plants and advanced process in manufacturing uses model particular control advanced robotics uses model predictive control because it allows you to use a model that you have for your system even when the model is nonlinear and there's constraints and there's you know things you have to like if I can only control so much or my robot arm can't hit a wall I can build those constraints into this model predictive control optimization and so the basic idea here is you might guess it's based on a model so what model predictive control does is you have some goal that you want to achieve you want to drive your system from here to here and so what you do is you set up this really heavy optimization where I try a bunch of different control laws over a finite time horizon I see which control law in my model gets me closest to where I want to be and I take one step in that direction that's this first step here then I reinitialize everything I rerun a bunch of optimizations with these new control laws I take the one that gets me closest to where I want to be and I take another step in that direction and another step and another step and I can build in constraints I can do this with nonlinear systems and it's extremely robust to relatively poor models all kinds of stuff and this is the the backbone of like modern industrial control design is model predictive control yeah what do I mean I basically try a bunch of U of T's in open-loop often like I just I cook up a bunch of U of T U of T functions of things I could try for control and I picked the one that does the best and I take I apply that control offer a little bit of time it would pick which it would pick it yeah it would basically decide what bangbang sequence gets me closest to the objective and then it would start doing that bangbang sequence for a little while and then it would rerun that optimization in kind of a moving window yeah so that it's really robust because even if your dynamics aren't that great if you're moving in the right direction model predictive control will keep moving in the right direction you had a question so there's a lot of parameters you get to choose which is your prediction horizon for your model how long you apply that control law before you reoptimize and the ultimate horizon for your set point and those are all depending on how much computation you have onboard how fast the time scales are because this is all limited by how fast you can run this optimization okay so this is one of those cases where you want you don't want the best model possible if it's really expensive you want the best model that gets you close to the right direction as fast as possible because you're running this real time optimization this massive optimization and so what people actually do now is build essentially deep model predictive controllers where they use deep learning for the model very very fast model evaluations and then you wrap this model predictive control around a deep learning architecture I do want to point out that in some cases you actually need a pretty good model so this is what happens when you do this on a bad model actually it's not even that bad of a model it's a pretty good model and this is what happens when you do it on a good model so you have to really get a good models in some cases in others it's not so important and so what's happening nowadays there's two major fields where machine learning is entering control one is deep model predictive control so using deep learning to learn the model that you then use for model predictive control and the other one is in reinforcement learning so how many of you have worked with reinforcement learning so this is the engine for alphago and for the computers that can be people at chess and things like that where essentially if you have a complex let's see if this starts so I'm just gonna play this in the background this is learning how to play an Atari game you've probably seen this before chess is a pretty tough game to learn a control strategy to win because there are so many combinatorially many board configurations you as a human you have rules of thumb to tell if you're winning or not like if my son takes my queen he's probably winning it's like the simplest thing is just like count the points on the board that's one number to assess if a board is good or not what deep reinforcement learning does is it essentially uses deep neural networks to learn a value landscape of what is good or bad on your board or in your video game so it learns the same kinds of intuitive proxies for what is a good board strata to a strategic position versus a bad strategic position which is would normally be combinatorially expensive to enumerate okay in this case I really like this example because eventually this deep reinforcement learning learns the advanced human trick have you seen this before yeah so it learns the advanced human trick of trying to weaken one side and then it solves it all instantaneously for it so this is kind of amazing the the computer actually learned is this kind of an epiphany it learned the the strategy that you know an advanced human we have learned over a long period of time you can also use this for robots playing catch this one's kind of cool I'm going to we soon with a few trials this is a model where they've actually incorporated some rudimentary physics into the learning procedure so if you do reinforcement learning with no model and you just try to learn the value landscape it's extremely inefficient so download Google like reinforcement learning pendulum stabilization and run it on your computer it'll run for eight hours before it learns how to stabilize the pendulum unless you give it some basic physics and then it can learn after maybe 3040 trials I don't forget exactly how many it is but that's about how many a kid actually takes to learn this task so it's pretty impressive towards human level performance if you bake in some some physics knowledge and then this is when I always show Boeing executives which is this great compilation video of all of the failures of reinforcement and you know big basic robotic control so we still have a long way to go these bodies don't understand themselves they don't understand their own physics okay so I'm gonna end there what we're going to talk about on the next tutorial is mostly how we learn coordinates and how we learn models from data using increasingly sophisticated regression and optimization techniques with data okay thank you [Applause] you

Channel: Institute for Pure & Applied Mathematics (IPAM)

Views: 21,736

Keywords: steve brunton, machine learning, dynamical systems, ucla, ipam, math, mathematics, physics

Id: 2VBN_dJZLWc

Length: 77min 38sec (4658 seconds)

Published: Tue Oct 08 2019