Simple PID Controller with Simulink

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Jeff Dillon 2022-9-27

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评论： Aquatris 2023-7-30

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.

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Jeff Dillon 2023-7-29

编辑：Jeff Dillon 2023-7-29

@Aquatris Great suggestion. Also I'll probably need to convert to Newtons so 14 * 4.45. And this block would go just before the transfer function?

Aquatris 2023-7-30

That kinda depends on the model of your system. For linear system with sufficiently fast response, the difference might be negligible. But there might be cases where the difference is significant, as when you put the block before the transfer function, the states of your system moves faster than the observed output (e.g., you apply step input to the system, the system output becomes 10 in 0.1 sec, the output takes 5 sec to reach that equilibrium due to rate limiter, you send 0 input 2 sec after the step input, at this point the rate limiter output value is around 4, and transfer function output is 10. within that 0.1 sec your transfer function takes to reach equlibrium, the observed output will still rise until your transfer function output is less than the observed output. Is this desired behaviour? If so then you should put it after transfer function, however if it is not, then you should put it before transfer function)

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Answer 1

Sam Chak 2022-9-28

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在 MATLAB Online 中打开

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:

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Sam Chak 2023-6-19

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Hi @Jeff Dillon

I'm no expert in delayed systems. Your interpretation of time-delay may be different from mine.

To approximately describe the transient period of 5 seconds (from 0 to almost 100%) in a mathematical model, you really need to find out whether the jet turbine (actuator) behaves linearly or nonlinearly. If it behaves like a first-order linear system, then the model in time-domain looks like the following.

Note that I didn't use the transfer function because I don't want you to get confused.

tspan = linspace(0, 10, 10001);

x0 = 0;

[t, x] = ode45(@odefcn, tspan, x0);

% Solution Plot

plot(t, x, 'linewidth', 1.5),

grid on, xlabel('t'), ylabel('x')

% check the output of jet turbine at 5 seconds.

idx = find(t == 5); % find the location of t = 5

jet_output = x(idx) % insert this location to the function almost like f(t)

jet_output = 0.9975

% 1st-order linear system

function xdot = odefcn(t, x)

u = 1; % input reference

Ts = 5; % transient time

tau = Ts/6; % time constant parameter in the model

xdot = (- x + u)/tau;

end

Sam Chak 2023-6-21

编辑：Sam Chak 2023-6-21

在 MATLAB Online 中打开

Hi @Paul

Other than the input-output characteristic of the jet turbine, I have also asked if @Jeff Dillon can identify if the actuator has a nominal model like

[linear] or

[nonlinear]. I believe that the jet turbine itself has an internal control mechanism that regulates the throttle so that it can produce the output linearly with time t as shown.

At the moment, I don't know about model for the throttle dynamics, but I guess that the ODE is nonlinear. The following is not a model for the jet turbine, but an example of a nonlinear model (mathematically made, does not exist in real world) for a system that produces the 100% output at approximately 5 seconds for the input

. The input-output characteristic is nonlinear in my example.

tspan = linspace(0, 10, 10001);

x0 = [0 0.001];

[t, x] = ode45(@odefcn, tspan, x0);

% Plot

plot(t, x(:,1), 'linewidth', 1.5),

grid on, xlabel('t, (sec)'), ylabel('Throttle')

% Dynamics of actuator

function xdot = odefcn(t, x)

xdot = zeros(2, 1);

u = 1.0; % throttle percentage

p1 = 2.8488; % tuning parameter

p2 = 9.2/(p1*u);

xdot(1) = x(2);

xdot(2) = p2*(1 - 2*x(1)/u)*x(2); % nonlinear ODE

end

Jeff Dillon 2023-6-21

编辑：Jeff Dillon 2023-6-21

I'm looking for an update to the Simulink plant model, not MATLAB code. I'm using the Simulink model in real-time with the actual hardware. There is no regulator inside the turbine. It accepts direct PWM input just like the motor on a quadcopter. A quadcopter motor also has a slight response delay, this is no different. A quadcopter motor has to spin the propeller and the delay is caused by the inertia of the propeller. It takes about about a 1/4 of a second for the quad rotor system to spin up to full throttle. The jet turbine is no different, you are making this unnecessarily difficult. When you provide a full throttle PWM input to the jet engine, it instructs the fuel pump to send more fuel. It then takes 5 seconds for the internal RPM of the jet turbine to reach full throttle. There is no regulator, it's just physics. For the turbine, the internal compressor spins at 10,000 RPM at zero throttle, and 140,000 RPM at full throttle when provided sufficient fuel from the fuel pump. If possible, please provide a corresponding update to the Simulink plant model, which is currently using a transfer function.

Sam Chak 2023-6-22

Hi @Jeff Dillon

What do you mean "is not the approach I wish to take"?

The ODE approach mentioned in my comment to @Paul is simply an example of a model for a hypothetical nonlinear actuator. It is not intended to be a model for the Jet Turbine. I apologize if you interpreted it as unnecessarily difficult for unknown reasons. That was not my intention.

What approach are you suggesting?

I don't want to make the ODE approach appear complicated for you. That's why suggest consulting with @Paul.

How do I use the differential equations that you provided in Simulink?

The ODEs were never intended for the Jet Turbine itself. They were used to demonstrate to @Paul my attempt at modeling the dynamics and generating the input-output characteristics. Therefore, they are not applicable to your Simulink modeling.

Can you show the updated Simulink diagram that is using the differential equations in the plant model?

I apologize for the confusion. I didn't save the Simulink file 9 months ago. However, if you are interested in working with nonlinear ODEs (without using transfer functions), I recommend checking out this video:

https://www.youtube.com/watch?v=A_AZd09Rml8

How did you generate the images you just shared?

They are plotted as a time-dependent piecewise algebraic function, without involving derivatives or ODEs. It is a simple math formula that describes the saturation characteristics.

Are they outputs of the Simulink model?

No, they are directly plotted in MATLAB. However, I assume these are the step response plots that @Paul requested from you. I hope they help Paul guide you in the next steps of the non-ODE modeling process in Simulink.

Jeff Dillon 2023-6-22

This problem doesn't seem much different than in a typical motor control system where the motor doesn't respond instantly to control input but instead follows a ramp

Using ChatGPT:

Me: I have a Simulink model where a jet turbine is mounted vertically and used to balance a VTOL system. The model is working using a PID controller but assumes that the jet turbine responds to control input instantly. In reality, the turbine takes 5 seconds to go from zero to full throttle, linearly. How would I modify the plant model which currently uses a transfer function to accommodate for this response delay?

ChatGPT: To modify your plant model in Simulink to accommodate for the response delay of the jet turbine, you can introduce a time delay block in the control loop. This time delay block will simulate the 5-second delay before the turbine responds to the control input. Here's how you can do it:

Open your Simulink model.
Locate the transfer function block that represents the jet turbine dynamics. This block should be connected to the PID controller.
Remove the existing transfer function block representing the turbine dynamics.
Add a time delay block in its place. You can find this block in the Simulink Library Browser under the "Discontinuities" category.
Double-click on the time delay block to open its parameters dialog box.
Set the time delay value to 5 seconds. This value represents the delay between the control input and the response of the turbine.
Connect the output of the time delay block to the rest of the model, as it was connected before with the transfer function block.
Save your Simulink model and run the simulation to observe the effect of the time delay on the system's response.

By adding the time delay block in the control loop, you are accounting for the actual response time of the jet turbine. This delay will introduce a more realistic behavior into your model, considering the 5-second time it takes for the turbine to go from zero to full throttle.

Me: Well, it's not really a 5 second delay before the turbine responds. It responds quickly, but takes 5 seconds to achieve full throttle. The thrust begins to increase immediately upon control input, but takes time to ramp up.

ChatGPT: I apologize for the misunderstanding. If the turbine responds quickly but takes 5 seconds to achieve full throttle, you can modify your plant model accordingly. Instead of using a simple time delay block, you can use a ramp-up function to simulate the gradual increase in thrust. Here's how you can do it:

Remove the time delay block that was added previously (if you have already added it).
Add a ramp-up function to your Simulink model. You can use a combination of blocks to achieve this effect. Here's an example of how you can construct the ramp-up function:a. Use a Unit Delay block (found in the "Discrete" category in the Simulink Library Browser) to hold the previous control input value.b. Subtract the previous control input from the current control input to obtain the change in input.c. Use an Integrator block (found in the "Continuous" category) to integrate the change in input over time.d. Multiply the output of the integrator by a gain value to control the rate of ramp-up.e. Add the output of the ramp-up block to the initial control input.
Adjust the gain value in the ramp-up block to control the rate at which the thrust ramps up. You may need to tune this value to match the desired time it takes for the turbine to achieve full throttle.
Connect the output of the ramp-up block to the rest of your Simulink model.

By using this ramp-up function, you are simulating the gradual increase in thrust over time, which aligns with the behavior of the turbine taking 5 seconds to achieve full throttle

Jeff Dillon 2023-7-17

编辑：Jeff Dillon 2023-7-17

@Sam Chak Your reply (including others, now deleted) are not relevant. I've explained in good detail regarding the input/response behavior of a jet engine. You don't need to know how it works for the same reason you don't need to know how a battery works at the electro-chemical level in order to model it's behavior. I'm not looking for a ramp controller. I simply stated that the throttle response of the jet turbine is slow. It's exactly like turning on a heater (with a step input) and the heater takes 7 seconds to warm up to full temperature, linearly. Or when you step on the gas in an automobile, it takes 7 seconds to achieve full speed. I need a PID controller that ACCOMODATES this response behavior, not induces it. And you ignored my last comment regarding the use of EDF's and apparently did not view the video I shared, the problem now allows instantaneous response using electric fans, similar to a quadcopter controller. The jet engine can be completely ignored, it's only used in steady state to offset the majority of the vehicle mass with the remainder of the control done by the electric fans. Google for "EDF electric ducted fan"

Jeff Dillon 2023-7-19

@Sam Chak I get that you may be a little embarrassed on your misunderstanding of the problem statement and is why you removed your posts, hopefully it's clear now! Is there anyone else at Mathworks that you can ask? You had presented a non-linear MATLAB method previously that would likely apply, however you were not able to show it in the context of an updated Simulink model. It's ok. FYI, I'm following the course here http://apmonitor.com/pdc/index.php/Main/ArduinoTemperatureControl on Control Systems, hopefully I'll be able to answer my own questions in due time! Thanks for your help so far.

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