Finding the right PD values for a quadcopter simulator

Question

I am working on this short assignment where I have to find the right PD values for a simulated quadcopter. I have to reach 0.9 meters under 1 second with an overshoot of less than 5%. This is my code:

function [ u ] = pd_controller(~, s, s_des, params)
%PD_CONTROLLER  PD controller for the height
%
%   s: 2x1 vector containing the current state [z; v_z]
%   s_des: 2x1 vector containing desired state [z; v_z]
%   params: robot parameters

% u = 0;

error = s_des - s;
z_des = 0.9;

Kp = 0.5;
Kv = 1.0;

u = params.mass*(z_des + Kp*error(1) + Kv*error(2) + params.gravity);

end

So far this is my output:

I've tried many different value combinations and so far this is the closest that I could get.

Answer 1

This problem can be conveniently solved by resorting to a state-feedback controller as the one depicted below.

In this case, we have that $P=\frac{1}{m \cdot s^2}$, $x=\left[ z, \dot{z} \right]^T$, $K=\left[k_p, k_d\right]$, $k_f=\frac{k_p}{m}$. Then, we need to apply the feed-forward term $v=mg$ to counteract the gravity.

We end up with the transfer function of the closed-loop system:

$$ \frac{y(s)}{r(s)}=\frac{\frac{k_p}{m}}{s^2+\frac{k_d}{m}s+\frac{k_p}{m}}. $$

This is a well-known and well-studied second-order system given in the form

$$ \frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2}, $$

where the parameters $\omega_n$ and $\xi$ are expressed as functions of the gains $k_p$, $k_d$ and the mass $m$ as $\omega_n=\sqrt\frac{k_p}{m}$ and $\xi=\frac{k_d}{2\sqrt{m \cdot k_p}}$.

To meet the given requirements, have a look at the information you can peruse for example at https://www.javatpoint.com/control-system-time-response-of-second-order-system, where the overshoot and the settling-time are provided by the following expressions: $$ \begin{cases} e^\frac{\pi\xi}{\sqrt{1-\xi^2}}=0.05 \\ \frac{4}{\xi\omega_n}=1 \end{cases}. $$

The gains $k_p$ and $k_d$ can be thus readily determined from the formulas above through substitution.

For a unitary mass $m=1$ kg, it comes out that $k_d=8$ Ns/m and $k_p=k_f=33.596$ N/m.

Here's the step response:

The step response is generally given with respect to a unitary input step; however, the system is linear and thus this behavior is preserved also for an input step of amplitude $0.9$ m.

To conclude, your controller's output shall be:

u = params.mass*params.gravity - kp*z - kd*v_z + kf*z_des;

Answer 2

You are controlling a second order system. Set both kp and kv to 4.

If the system does not reach your desired altitude then there are additional problems with the code.