# stability/controllability of a finite time linear quadratic regulator

If I have a linear quadratic regulator (quadratic objective function with linear dynamics) and use an open loop control - are there stability/controllability requirements on the system? I know that there would be these requirements if I were to implement a closed loop control but I can't find any corresponding requirements for open loop.

• For open loop both the controller/input filter and the system have to be stable, in order to have a stable open loop system. But are you sure that you have an open loop system? Nov 23 '16 at 0:38

How do you have an LQR with open loop control?

I know that there would be these requirements if I were to implement a closed loop control

A linear quadratic regulator is basically a multi input version of a state feedback controller; your $Q$ and $R$ matrices determine the "cost" or priority each input should take.

With the $Q$ and $R$ matrices, you solve the matrix Riccati equation to get the optimizing matrix $P$. That matrix then allows you to solve for the optimal control gain matrix $K$.

Then $K$ is used to calculate the control input, $u = -K\hat{x}$, just like a state feedback controller.

The LQR controller is a multi-input state feedback controller. To say that you want an "open loop LQR controller" is not technically correct, because again you do have feedback: your current state estimate $\hat{x}$.

Any criteria you have for "closed loop control" applies to an LQR controller. The only difference is that what you're calling "closed loop" refers to either direct measurement of the system states or the use of observers to correct your estimate. Either way, you're either controlling based on an uncorrected state estimate or a corrected state estimate, but you are always controlling based on a state estimate.