Simple PID Controller with Simulink

I want to develop a simple PID controller for hovering, please see the attachment. The mass M (10 lbs) will start at h0 = 0 and the goal is to hover at h (6’ for example). The input to the plant is the throttle (from 0 to 100%, 0 to 20 lbs thrust) and the output is the altitude. So the desired altitude is the reference value and the equation of motion is simply f=ma. Could someone help me with the Simulink diagram? This not a homework problem, I'm helping to develop a VTOL propulsion system with undergrad students at USC and this would help us to jumpstart the project.

Answer 1

[Sam Chak]

Hi @Jeff Dillon

Not a VTOL person, but from the free-body diagram, Newton's 2nd Law can be applied with the basic assumptions of relatively low altitude compared to the Earth radius, and air resistance is neglected. The basic equation of motion is given by

where the weight of VTOL system, . Rearranging that

.

Since the accelation of the VTOL is the second-order derivative of the altitude with respect to time, , then

.

Taking the Laplace transform with zero initial conditions, you get

which can be rearranged to obtain the transfer function from plant inputs to plant output

Here, is the command input and is the disturbance.

If can be accurately estimated, then you can design

to counter the effect of the disturbance, and is the PID thing that you need to design

m    = 10*0.45359237;       % mass of VTOL
g    = -9.80665;            % gravitational acceleration
W    = m*g                  % weight of VTOL

W = -44.4822

Gp   = tf(1, [m 0 0])

Gp =
 
      1
  ---------
  4.536 s^2
 
Continuous-time transfer function.

Lmax = 20*4.4482216153;     % max lift force available
Umax = Lmax + W             % design limit for U(s) or output of Gu(s)

Umax = 44.4822

% Design of U(s)
Gc   = pidtune(Gp, 'PIDF', 0.95)

Gc =
 
             1            s    
  Kp + Ki * --- + Kd * --------
             s          Tf*s+1 

  with Kp = 0.76, Ki = 0.0338, Kd = 4.2, Tf = 0.193
 
Continuous-time PIDF controller in parallel form.

Gcl  = minreal(feedback(Gc*Gp, 1))

Gcl =
 
           4.961 s^2 + 0.8757 s + 0.03861
  ------------------------------------------------
  s^4 + 5.181 s^3 + 4.961 s^2 + 0.8757 s + 0.03861
 
Continuous-time transfer function.

Gu   = minreal(feedback(Gc, Gp))

Gu =
 
         22.5 s^4 + 3.972 s^3 + 0.1752 s^2
  ------------------------------------------------
  s^4 + 5.181 s^3 + 4.961 s^2 + 0.8757 s + 0.03861
 
Continuous-time transfer function.

t    = linspace(0, 50, 5001);
hd   = 6*0.3048;            % desired altitude
u    = hd*heaviside(t);
% Check if output of Gcl tracks hd
lsim(Gcl, u, t), grid on

% Check if output of Gu exceeds 44.4822
lsim(Gu, u, t/10), grid on

The Simulink model should look like this:

[Jeff Dillon]

This works perfect! Now if I want to test this on real hardware which outputs the altitude and accepts the lift force (via ROS2), where would I place these two blocks?

[Sam Chak]

Good to hear that, @Jeff Dillon. If you find the solution in MATLAB/Simulink is helpful, please consider accepting ✔ and voting 👍 the Answer. Thanks!

Regarding the ROS 2, I think you need to ask the hardware experts or MathWorks staff by opening a new question.

[Jeff Dillon]

Sorry for the confusion. I already have the ROS2 blocks and the hardware working, you can ignore that part. I was just wondering where in your model I would interface to the hardware. Where would I add the real-world input/outputs.

[Paul]

@Sam Chak

heavisde(0) = 1/2 (unless changed via sympref), which probably isn't what's desired in this example. I don't think it changes any of the conclusions, but thought I'd point it out because it does affect the responses at t = 0 and shortly after.

[Sam Chak]

Thanks @Paul for the highlight.

OP can replace these 3 lines

u    = hd*heaviside(t);
lsim(Gcl, u, t), grid on
lsim(Gu, u, t/10), grid on

with

opt  = stepDataOptions('StepAmplitude', hd);
step(Gcl, t, opt), grid on
step(Gu, t/10, opt), grid on

or replace only

u    = hd*heaviside(t);

with

Gr     = tf(hd, 1);
[u, t] = step(Gr, t);

@Jeff Dillon, regarding the interface, perhap these 3 articles are useful for you.

https://www.mathworks.com/discovery/hardware-in-the-loop-hil.html

https://www.mathworks.com/help/simscape/ug/what-is-hardware-in-the-loop-simulation.html

https://www.mathworks.com/help/slrealtime/gs/real-time-simulation-and-testing.html

[Jeff Dillon]

The insight here as been amazing, thanks! Yes, I've done real-time HIL in the past including this fun project https://aeropendulum.arizona.edu/ . In my case, can I just replace the Transfer Function with the actual Plant block?

[Sam Chak]

Hi @Jeff Dillon

Thanks for showing the AeroPendulum Profject and the applications of MATLAB. 👍 +1

Yes, you can replace it with actual Plant block. And you may need to tune the PID to get the satisfactory performance.

[Jeff Dillon]

I have a completed demo with both simulation and the plant, how do I embed quality images inline?

[Sam Chak]

Click the "photo" icon (in the INSERT tab), if you want to embed the image into this forum thread.

There are display settings for you to adjust the size of the displayed image.

[Jeff Dillon]

Here is my first draft of the implementation, it will be interesting to compare actual vs simulated. I'm using ROS2 and UDP to compare latency. If ROS2 proves fast enough, I'll use it throughout.