# Is there any nonlinear LQR controller?

There is a nonlinear LQE observer. It's more referred as Extended Kalman Filter(EKF). It's quite easy, just linearize matrix $A$ and $C$ in the estimated vector $\hat{x}$ before you find the kalman gain matrix $K$ from the Riccati equation.

But is ther a nonlinear LQR controller?

Can I just linearize matrix $A$ and $B$ in the estimated vector $\hat{x}$ before I find the control law gain matrix $L$ from the Riccati equation? But still have the same $Q$ and $R$ weighting matrices?

• There are a few different variations I guess. I would suggest starting with looking at iLQG or tvLQR. – Steve Heim Aug 1 '17 at 8:16
• Cannot find anything about tvLQR. But iLQG looks interesting. It's a nonlinear LQR ? – Daniel Mårtensson Aug 1 '17 at 9:43
• tvLQR is "time-varying LQR", see this application and this reference (10.2.2). iLQG is "iterated LQG", where you are basically following along a nominal trajectory, and applying LQG along that trajectory by linearizing as you go. Of course if your controller causes you to deviate, your linearization is no longer accurate, so you iterate along the new nominal trajectory, re-linearizing. – Steve Heim Aug 1 '17 at 9:51
• What would happen if I linearize $f(x,u)$ in the estimated state vector to update the control lag $L$ ? – Daniel Mårtensson Aug 1 '17 at 10:07
• I think model predictive control would be a general extension of LQR. However finding the global minimal solution in general might be hard, because the nonlinearities can make the problem non-convex. – Kwin van der Veen Aug 4 '17 at 3:17