Control Bootcamp: Linear Quadratic Gaussian (LQG)

okay welcome back so now I'm going to tell you about one of the things that I think is neatest about linear optimal control design which is the fact that if I have a linear system that is controllable and observable I can develop an optimal full state feedback control using lqr okay I can also develop an optimal common filter or linear quadratic estimator using a and C and what's amazing is that when I combine them they remain optimal so I choose this lqr to place the eigenvalues of the closed-loop system here I choose the common filter games to place the eigenvalues of my estimator dynamics and it's not obvious but it's very cool that when I combine these I retain those same dynamics okay so this combined system where I estimate the full state then I do lqr given Gaussian white noise disturbances and measurement noise this is called the linear quadratic Gaussian control and in practice we call it lq g l q g so it's again linear control that minimizes a quadratic cost function and here it's given gaussian disturbances for my full state estimation so lq g is a pretty neat control it's essentially just combining a common filter and an lq r it does have some limitations so there are some fundamental issues that when you combine these things sometimes your LQG system can suffer from robustness issues sometimes your system can be like arbitrarily fragile meaning if this has a little bit of system uncertainty or some nonlinear dynamics you didn't model this could in general become unstable even if you think it should be stable that's something we'll talk about later and that actually motivates what's called robust control which is optimal and a difference but for now let's just combine these and make sure that the dynamics of the combined system has the same eigenvalues we think it should have so what I want to do is remember first of all we have epsilon equals x minus x hat and we had some epsilon dot dynamics that I'll remind you of in a minute but what I want to check is that X dot so DDT of X is equal to all of this stuff okay it's equal to ax plus B u but remember U is minus K so u equals minus kr X hat we're using the estimate of X because we don't have access to the full states times our our lq our gain matrix so this is really a minus b KR x hat okay plus some disturbances okay and so now what we get to do is think to ourselves well what is X hat well X hat I what I can do is I can essentially say X hat is equal to X minus X minus X hat sounds reasonable I think and so I'm going to plug this in here this is just an identity I'm going to plug that in here and I'm going to get a X minus BK are X minus minus is plus BK are X minus X hat plus WD no big deal okay and if you recall the dynamics of our epsilon if I looked at epsilon dot I'm pretty sure I got something like a minus my calman filter gains times C epsilon and I didn't actually derive this for the case with disturbance and noise but if you do what you get is a plus W D so this w D essentially feeds through and then I also get a minus KS times my sensor noise so you can just verify I do this on some pencil and paper and verify that you get these terms but it's pretty simple and so now what I want to do is I want to combine these two so I want to build a new state space equation D DT of my true state X this is the actual state of the actual system I want you to be stabilized and the error of my estimation I also want this to be stabilized so X is my honest-to-goodness full state epsilon is my estimation error so I want both of these to be stable and if I write this down what I have is okay so I'm going to write this down as a matrix times hope that I have enough room here times X and epsilon okay so X dot equals a minus BK x kr is a column and filter I also have a plus B K epsilon so there's a B K R here and then there is I'm not going to have another matrix times my disturbances and noise so to some extent in my calming filter system the disturbances and noise are kind of the exogenous inputs to my system so there I'm writing them kind of like an input and in this equation I had a V an identity and a 0 here okay because disturbance feeds through to X dot but not sensor noise okay and now my epsilon dot equation I have this doesn't depend at all on X which is amazing so 0 then I have a minus calm and filter see ok so this is getting pretty cool and then I have an identity on my disturbance and a minus KF on my noise and so this is really really remarkable when I take and I actually combined my full se Testament from my common filter with an L Q are optimal full state feedback control I can get I can derive the dynamics for X and epsilon and what I get is the eigenvalues of this coupled system because this is a diagonal I get the eigenvalues of this matrix and the eigenvalues of this matrix so what's really cool is even when I combine these systems the eigenvalues of my full state X are still stabilized by my lqr controller so this is my kind of lqr eigenvalues their optimal from lqr and my estimator dynamics retain the exact same calman filter eigenvalues my calmer filter or lq e eigenvalues and so in control theory this is called the separation principle separation principle which essentially means I can design my lqr and my lqe controllers and estimators in isolation separately and when I combine them the combined system retains the desirable properties of each of them this is really really cool it just works out because of the linear algebra so we can actually take a real system with limited measurements limited actuation and we can build both of these and put them together and so that's what we're going to do next and for some cases we're going to find that this does have the robustness issues so we're going to generalize this optimal control framework to try to find controllers that are optimal with respect to other cost functions like maybe I want to penalize fragility and which will define later okay thank you

Channel: Steve Brunton

Views: 45,720

Keywords: Control, Control theory, Linear algebra, Eigenvalues, Closed-loop, Feedback, Observability, Gramian, Applied Math, Matlab, Estimation, Kalman filter, LQG, Optimal control, LQR

Id: H4_hFazBGxU

Length: 8min 34sec (514 seconds)

Published: Mon Feb 06 2017