Model Predictive Control

welcome back today I want to tell you about a powerful optimization strategy for feedback control called model predictive control also known as MPC so MPC fits right here in this feedback control diagram and essentially what MPC does it's a very flexible procedure where if you have a system model you run a set of you run forecasts of this model forward in time for different actuation strategies you and you optimize over the control and put you over a short time period and essentially determine your immediate next control action based on that optimization once you've applied that immediate next control action then you reinitialize your optimization over you move your window over you re optimized to find your next control inputs you enact that and this thing essentially keeps marching that that window forward and forward in time so it's an optimization strategy based on a model of the system I'm going to kind of explain a little bit how it works here so the the basic idea let's say I have some time and I have let's say that this is my desired output and this is my control signal u and let's say that there's some kind of a set point that I'm trying to reach so there's some kind of goal that I'm trying to reach I want my system to go up up here and let's say I start here okay so essentially what I'll do with model predictive control if I am starting at some point let's call this let's say that I have the whole state X and this is at time K so X at K and let's say y equals x just to make this easier then what I'll do is I will start my system I'll initialize it at time at X at time K XK and then I'll run an optimization procedure over the inputs to try to find what is the best u over some short window that gets any closer to my desired objective okay so there will be some some window of hi and let's call this my horizon that's what we call this in MPC so I've got some time horizon and what I'm going to do is I'm going to optimize you over that time horizon okay and maybe what that looks like is some kind of strategy like this maybe I'm just making something up maybe that's the you that comes out of the optimization procedure that I run here then what I do is I essentially just lock in the first value of that optimizing control law and I see where my system goes so now my system maybe goes there then what I do is I shift my horizon over one delta T and I realize so now maybe maybe this system didn't go exactly where I thought so instead of doing this now I think that my optimal control should be a little bit more aggressive something like this I'm just making this up again okay and so then what you do is you essentially pick that next optimizing control lock that in and see where your system goes and so over time you can essentially watch your system step forward and at every new time step you reassess what the optimal optimizing control input you should be over some kind of moving horizon and you only enact that next instant control law that next step of control and see where your system goes and you can essentially get very very very good control even for strongly nonlinear systems that that achieve your desired objective okay so if I write this control down you would actually say that my control K my control signal K at time a little X K would be whatever my optimal can control U is at time step K plus-1 based on time step K ok so I'm going to decode this for you because I found I found this a little confusing I found this a little confusing the first time so u K plus 1 of X K basically means this is my optimal short time control strategies short time control starting at XK okay so I initialize this optimization at XK I figure out this long trajectory of use that would optimize my objective function starting at that initial condition given this model and then I only implement the first time step this is the first time step so I run a simulation and I optimize my control over many time steps but I only implement the first time step that's my control law at time K then my X moves to XK plus 1 I realize an entire new control trajectory starting at XK plus 1 and again I implement the first time step of that new optimized control strategy it steps me forward and so on and so forth so every time step I'm rerunning this entire optimization forward in time over some short time window to see what is the best possible you given my new state and what the dynamics actually did ok so that's the basic idea of model predictive control now why is this so powerful I think I think this is so powerful for a number of reasons one is that you can impose constraints so I can have constraints on the state of my system so basically I can do a control so that I never hit some boundary so I stay within some bounds but I can also constrain my input which is very very important so in a lot of control strategies that we've seen up until now the actuation is this kind of continuous variable and often times your your controller would make unrealistic demands on what you could do so for example if I'm you know designing a cruise controller if I design my controller badly then it might command use which are unphysical it might say go a million miles an hour for a fraction of a second and then slow down what model predictive control can do is it can essentially put bounds what your control signal can actually do so when it's running this optimization loop for every time step when this optimisation is running it can essentially constrain the optimization to not hit these these max or min values okay so that's really cool it can run these these hard constraints or soft constraints on the state or on the input it works for nonlinear systems and there's a few ways you can think about this so let's imagine that I had a linear system but there were some disturbances or noise or something like that so I can I can determine the optimal control signal for a linear system using something like linear quadratic regulator and and I could use that for my MPC short-time optimization and then if there's some disturbance or some changing parameters of my system those will compensate over time if I have a nonlinear system what I could do is I could take my nonlinear model and I could linearize it about that particular state and then I could do kind of linear optimization so maybe what I'm trying to get at here is that optimization is the heart of MPC and PC is built around an optimization every single time step I'm Ryo maizing this control strategy U and then just picking the first time step and reinitializing and starting over so this is a little bit expensive because I'm optimum running this full optimization at every time step okay and this is enabled by increasingly fast computers and fast Hardware okay so this is essentially relies on fast Hardware to be able to run this optimization loop instead of usually if I do something like a linear quadratic regulator or a common filter I run my optimization offline on a you know a big computer and then I take those results and I hard-code them on my my fast online control system NPC is different it relies on the fact that you have fast computational hardware to run this optimization online at every time step okay now we know that there are known optimal solutions for you for linear systems so oftentimes model predictive control will be applied to linearized equations of motion even if I have a nonlinear system maybe I'll compute the linearized dynamics and every point X and I'll feed that into a linear optimization because that's fast but increasingly now that computers are getting faster and faster and faster it's becoming increasingly possible to actually directly optimize the nonlinear equations of motion if you have them okay so kind of the big picture here is that now instead of doing our control optimization one time offline like we used to do with linear quadratic regulator x' we can do this optimization continuously at every time step of our system and we can modify our control behavior if our system starts to deviate or if the dynamics start to change now this has driven a tremendous amount of interest in system identification because this relies on having a system model so because of the power of model predictive control we're very very interested in getting good system models that we can actually use so non linear and linear models of our system that we can control linear is the easiest and fastest to optimize so a lot of model predictive controls are wrapped around linear models and of course there's this feedback so if my system doesn't do what my model says then I just reinitialize and make a new prediction so model predictive control is more flexible than traditional linear control methods because if the system starts to move or deviate or doesn't do what you what you think it should you reinitialize your optimization at every new time step okay so that's really important is that this is more flexible sometimes what people do is linear parameter varying so essentially what I do is I have a family of linear models so instead of just X dot equals X plus bu what I'll do is I'll have a sum parameter mu and B of some parameter mu and as my parameter mu changes in time I can still run a linear optimization but I'm running this linear optimization on different a and B matrices as my trajectory of all's in time okay and then now of course since we're getting faster and faster computers and our latency is is getting lower and lower it's actually becoming feasible in some situations to run this online optimization on the full nonlinear dynamical system okay so that's the overview of model predictive control in a nutshell what we're going to talk about later is how we can use modern system identification techniques so kind of based on data-driven optimization and machine learning to build these models from data and then use those with model predictive control to stabilize and track reference trajectories for strongly nonlinear systems okay thank you

Channel: Steve Brunton

Views: 125,251

Rating: 4.9747963 out of 5

Keywords: Control theory, Machine learning, Data science, Dynamical systems, Machine learning control, Matlab, Optimization, Control, Model reduction, System identification, Reduced order models, Regression, Feedback control, Applied mathematics, Linear algebra, Balanced model reduction, balanced truncation, singular value decomposition, dimensionality reduction, Dynamic mode decomposition, Koopman theory, Model predictive control

Id: YwodGM2eoy4

Length: 12min 13sec (733 seconds)

Published: Sun Jun 10 2018