I have to define the choice of parameters I have chosen to create an LQR controller for a drone, and I have written the following:
- High penalties in the Q matrix mean that the state will try to stabilise the target with the least changes of states. Since elevation is a highly coupled state, the coefficients associated with the elevation states are the largest. This ensures that the desired elevation is reached without affecting the other states as much. However this response comes to the expense of a longer response time. Conversely we want to have a relatively fast response for travel, hence the coefficients associated with travel are the smallest since we are trying to achieve a faster response.
- High penalties in the R matrix result in the actuators trying to stabilise the state with the least energy as possible, whilst low values for the Qu penalties result in the controller stabilising the system with a lot of actuator energy. In our case the actuator energy refers to the thrust of the fan.
I struggle to understand the impact of the Q matrix on the overall system. Is someone able to clarify? Many thanks
In the general setting, what matters is the relative value between $Q$ and $R$, not their absolute value. For simplicity of exposition, let $R=\rho I$ and, in this case, the LQR cost becomes
$$\int_0^\infty [x(s)^TQx(s)+\rho u(s)^Tu(s)]ds=\rho\int_0^\infty \left[x(s)^T\dfrac{Q}{\rho}x(s)+u(s)^Tu(s)\right]ds.$$
So, now, if $Q/\rho\gg1$$, then the state is more penalized that the input and the minimization of the cost will tend to make the state to zero as quickly as possible so as to set that part to zero by letting the input taking large values.
On the other hand, if $Q/\rho\ll1$, then the control input will be more penalized in its amplitude and as a result, the state will converge more slowly to zero.