Control Bootcamp: Overview

hi everyone I'm Steve Brenton and this is the first video lecture on a series I'm calling a bootcamp on control where I'm going to rapidly go through the highlights of optimal and modern control theory so this is going to include how to write down a system description of a control system with inputs and outputs in terms of a system of linear differential equations and now how to design controllers to manipulate the behavior of that system how to design estimators like the common filter so the diffuse limited sensors you could reconstruct various aspects of that system this is not meant to be an exhaustive in-depth treatment of the subject but really kept at a high level and my goal is to first of all get you familiar with the major types of optimal and modern control theory I want to teach you how to use these in MATLAB to actually work with a real system and what I also want to give you a feeling for is what in control theory is easy and what's still quite challenging today so that you can get up to speed on the real pressing needs of control theory today okay and again this is not exhaustive so you know if this is really important to you and you want to you know you like control theory and you want to go more into depth there's deeper treatments both on the math side and only applied design side okay and so I want to give you just a little bit of perspective I think about the world in terms of dynamical systems so systems of ordinary differential equations in terms of the state of your system and this has been an extremely successful viewpoint for modeling real world phenomenon okay so we model the fluid flow over a wing or the population dynamics in a city or the spread of a disease or the stock market climate planets moving around the solar system all of these are modeled as dynamical systems and this has been a very very successful framework to take in data from the real world and build models that you can use for prediction but often we want to go beyond just describing the system of interest and we want to actually manipulate the system actively to change its behavior and so that could be just imposing some control logic just setting inputs into the that system in a certain pre-planned way to manipulate it or you could actually measure that system and make decisions based on how the system is responding to what you're doing okay and so that's kind of the overarching view and control theory is that you have some dynamical system and interest maybe it's a pendulum or a crane that you want to make more stable you write down the system of equations and then you design some control policy that changes the behavior of your system to be more desirable okay so that's what we're going to talk about and so I want to begin by just talking about the various types of control that there are so there's lots of control that goes around all around us every day that is not active it's called passive control so I'm going to draw just a diagram of the different types of control so one type that's very common you see all the time is passive control okay so for example if you see a large 18-wheeler transport truck going down the highway and it has those streamlined tail sections that's a form of passive control that's passively causing the air around the truck to behave in a favorable way to reduce drag and if you can get away with passive control of your system that's actually great because you just have to design an up front and then there's no energy expenditure and hopefully you get the desired effective for example minimizing drag on a truck but passive control is typically not enough and so oftentimes we need to do something like active control and so active control essentially just means that this is control when we're actually pumping energy into the system to actively manipulate its behavior okay and there's lots and lots of different types of active control so one that I'm going to tell you about is open-loop this is probably the most common form of active control where essentially you have your system of interest and I'm just going to actually draw this as a block here so you have some system and the system has some input so I'm going to call them variable U and it has some outputs that are variable Y okay and so what open loop control does is it essentially reverse designs your system and inverts the dynamics to figure out exactly what is the perfect input u to get a desired output Y okay and so if I take something like a inverted pendulum so we know that if I if I am very careful I can stabilize this inverted pendulum but it in physics you'll learn that if you just pump this pendulum up and down at a high enough frequency it will naturally stabilize the dynamics okay so if my base just oscillates at a high frequency sine wave then the dynamics of this pendulum so the base is you may be why is the angle of this pendulum and my desired control is to make this pendulum essentially stay at vertical okay and so if I pump in energy in a pre-planned way I just make my hand go up and down in the sinusoid I can put in a sinusoid Y that I want okay and essentially that is open with control it's very commonly used essentially you think about your system you pre plan a trajectory and you just enact that control law okay but the downside of open-loop control is that you're always putting in energy to this to this u so in the inverted pendulum example I constantly have to be pumping this thing up and down sinusoidally and the minute I stop the stability it becomes unstable and it falls okay and so the idea is that what we can do is called closed-loop feed feedback control so closed-loop feedback control and essentially what this means is that we take sensors bring my pendants drying out we take sensors sensor measurements of what the system is actually doing and then somehow we build a controller I'm just going to call this a controller and we feed that back into our our input signal that can manipulate the system so for example in that inverted pendulum example as a human if I had a tall enough pendulum so it's slow enough I could actually measure with my eyes if it's starting to wobble and I could do much more subtle control so if you have ever played around with as a kid with a broomstick cat trying to stabilize it you know that you can actually get pretty good at it so that with very low energy input or very small hand motions you can stabilize this thing so that it doesn't fall okay and so that's the basic idea is that by measuring the output you can often do much much better than just feeding in kind of a pre-planned control law okay so sensor based feedback measuring the output and then feeding that back as the input is basically going to be the entire subject of what we're going to talk about in this control boot camp so closed-loop feedback control is the name of the game and that's that's most of what we're going to talk about now that's not to say that if you can design a good open loop or a good passive control there you know there are some times you would do that but in the systems we're going to be interested in closed loop feedback based on sensors is going to give dramatically better performance okay and so I want to talk a little bit about why you would have feedback so I just want to make a quick list why feedback because this is a very very important important topic in control theory so I want to motivate again just maybe in more concrete terms why would I actually measure the system and feed it back instead of just ignoring any measurements and using open-loop so why feedback over open-loop control okay so this is a question I always ask my class and I let them think for a little bit why would you actually want to have the sensors feeding back into your system okay so one answer that I get most often is maybe my system has some inherent uncertainty okay so if my system is uncertain so uncertainty is one of the main and enemies of open-loop control right so if I have this pendulum and I perfectly pre-planned what I want to do let's say that the pendulum is one centimeter taller or it's a little bit heavier or there's wind blowing or something like that then any kind of uncertainty in that system is going to make it so that my pre-planned trajectory is going to be suboptimal but if I measure the outputs and I realize that it's not doing what I want it to do I can adjust my control law even if I don't have a perfect model of my system okay so uncertainty is a big one another really important one is instability so with open-loop control I can never fundamentally change the behavior of the system itself so in the pendulum example I could pump in an amount of energy with the sinusoidal based motion that would force the system to kind of correct itself up to vertical but I'm not actually changing the systems dynamics itself the system still is unstable and has an unstable eigenvalue but when I have feedback control I can directly manipulate the actual dynamics of this closed-loop system and I can change the the dynamic properties I can change the eigenvalues of this closed-loop system okay and I'm going to show you that as the last example in this overview so the third thing that I think is really really neat is that with feedback control you can also reject disturbances in your system so let's say that I have some external disturbance D that's coming into my system and this happens all of the time so so for example let's say in my pendulum example there's a gust of wind so that's a disturbance that would be very hard for me to predict or model or measure so there's this gust of wind that comes and if I had an open lead strategy essentially it might not be able to correct for that gust of wind where is that gust of wind will pass through the system dynamics will be measurable through some sensor and if my feedback control is good enough I can actually correct for that disturbance so I think of uncertainty as internal system uncertainty kind of disturbances to my model and I think if disturbances as external or exogenous forcing of the system that may be too difficult or too costly or too complicated to to model or predict or measure okay and feedback essentially handles all of those basic issues that can handle disturbances that can handle uncertainty and it can fundamentally change the stability of your system to make it more or less stable by actually changing the eigenvalues of this closed-loop system and unfortunately open-loop can't do any of those things which is a huge drawback and I guess the fourth one is energy or efficiency so I'll just say efficient control so again in the case of the pendulum in the open-loop case I constantly had to pump this thing up and down so I was always putting energy in but in the case of sensor-based or elegant feedback control you can picture yourself trying to stabilize this broomstick if you're doing a really good job if you have a really good controller this thing is barely moving at all and so you almost have to put no energy in to correct it so effective sensor based feedback control is also much more efficient which is really really important in lots of applications so if you're going to send a rocket somewhere you better have an efficient controller because you don't want to be wasting fuel okay so the last thing I want to show you is just this idea of why you can change the fundamental system dynamic dynamics and change the stability with feedback control okay so the basic property that we're going to or the basic mathematical architecture we're going to be working with in this class is going to be a stay space system of ordinary differential equations so we're going to have a state variable X X as a vector that describes all of the quantities of interest in my system so for example in my pendulum it could be the angle and angular velocity it could be two states if I have you know an airplane going through the sky it could be the three the position vector XY and Z and also its its rotation angles and their derivatives okay so it could be like a six degree of freedom or twelve state twelve component vector X and so what we're going to look at is the system X dot equals ax so we're going to start with linear systems of equations that describe how those states interact with each other okay and so I'm going to assume that we're all pretty comfortable with this linear systems of OD e so for example we know that the solution of this is X of T equals e to the matrix say T times X at time x zero okay so we know how the system behaves we know that if a has any eigenvalues with a positive real part then the system will be unstable and if all of the eigen values have negative real part then these have stable dynamics that they go to zero as time goes to infinity but what we're going to do in control theory is we're going to add plus B U so we're going to add this ability to actuate or manipulate our system okay so we're going to say that U is our actuator it's the thing we can its our control knob okay so it could be in the case of the pendulum it could be the position of the base or it could be the voltage onto a motor that controls something but this is the knob that we get to turn to try to stabilize our system and B tells you how this control knob directly affects the time rate of change of my state okay and down the road we're going to look at another extension where we're actually going to measure only certain aspects of the state so we're going to measure so linear combination of the state X and this might actually be a limited set of measurements we might not measure all of this the state of its high-dimensional and we might only have access to those few sensor measurements in Y but for now let's just talk about the top equation so if I assume that I can measure everything in the system and in this case of the pendulum as a human I have a pretty good estimate of its vertical position and how fast it's moving so let's say I can measure all of X then we can develop a control law let's say u equals minus some matrix K times X okay so I'm just going to say let's posit a basic control law that my control input U is going to be some matrix times X just some constant constant times the components of X when I plug this in so this is this is really sensor based feedback where y equals x okay in this case we're assuming that y equals x we can measure all of our state and we're going to feed that back into a control law which is minus K u equals minus K times X and we're going to try to modify the dynamics so if you plug u equals minus KX into our dynamics we basically get and let's make another color here we basically get X dot equals ax and then minus B K X okay so B is maybe a tall vector the same or set of vectors the same height as X K it's kind of the transpose size of that and so this is a matrix of size n by n if X is an n-dimensional state and so this equals a minus BK times X so notice that by by measuring the state in this case we're measuring the full state X and feeding that back to the control u through this law u equals minus a X we're able to actually change the dynamic matrix so now we have a new dynamical system X dot equals a minus BK times X and so it's actually the eigenvalues and of this matrix that tell you if the system is stable so I can have a really originally unstable system like this inverted pendulum and by measuring the state and feeding it back to my control knobs I get to move I can stabilize the dynamics I can actually make the system asymptotically stable okay and so figuring out when you can do this so this doesn't work for all systems and for all measurements and for all actuators so figuring out when the system is controllable and how to design this case so that it is well controlled are going to be the subjects of the next couple of lectures okay but really really important feedback solve all of these fundamental problems if I have an uncertainty in my system I can compensate for it by measuring what's actually happening and feeding that back if I have an instability in my system I can actually change the dynamics with this feedback and you can't really do that with open-loop I can also account for external disturbances like a gust of wind that might have been really hard to measure and could totally throw off your pre-planned trajectory but if you measure what's happening you can account for and correct for that and finally feedback control is efficient if you're doing effective feedback control to stabilize a system then the more effective you are the less energy you have to put in okay so this should be a really exciting set of lectures I'm really hoping to get you up to speed quickly and with MATLAB examples so that you can control these systems you can design controllers to actually manipulate your system to do what you want it to do okay thank you

Channel: Steve Brunton

Views: 249,166

Keywords: Control, Control theory, Feedback, Closed-loop, Open-loop, Uncertainty, Modeling, Linear systems, Linear algebra, ODE, Dynamical systems

Id: Pi7l8mMjYVE

Length: 19min 31sec (1171 seconds)

Published: Mon Jan 23 2017