step response in 3 scenarios: setting the initial reference voltage, shifting to another output voltage level

hello friends,

A buck converter with a PID controller has a transfer function. .By applying three different scenarios, the figure below is obtained, the blue line is the solution to the problem after three scenarios. In the first stage, the transfer function is applied by applying a Vref=12v from t=0 to 2e-6. In the second scenario, from the t= 2e-6 to 4e-6, the reference voltage has decreased to 6 volts(Vref=6v), and in the third scenario, a disturbance voltage of 1 volt has been applied the system. I wrote the step response for a reference voltage of 12V, but I don't know how to apply the next two scenarios to form the graph.

% Define the numerator and denominator coefficients of the transfer function

umerator_coeffs = [0.3571 605.8 42.27]; % Adjust coefficients as per your scaling factor

denominator_coeffs = [0.0000001 0.3573 606.8 42.27]; % Adjust coefficients as per your transfer function

% Create a transfer function object

sys = tf(numerator_coeffs, denominator_coeffs);

step(12*sys); % Simulate step response

xlabel('Time');

ylabel('Step Response');

title('Step Response');

Answer 1

[Sam Chak]

Hi @mostafa jabbari

Please note that only the first two stages can be simulated since it's unclear through which channel the disturbance is introduced into the system. If the disturbance follows the block diagram provided below, then the information provided for Stage 3 is insufficient.

• Additionally, the transfer functions for Gp (buck converter) and Gc (PID controller) have not been provided, which makes it challenging to proceed with accurate simulations.
• Furthermore, it is unclear whether the disturbance in Stage 3 is in the form of an impulse signal or a continuous-time signal. Clarification on this point would be helpful in refining the simulation.
• Lastly, you mentioned that the reference voltage Vref = 6 is only sustained from time t = 2e-6 to 4e-6. Can I assume that it drops to zero starting from t = 4e-6 onwards?

Edit: The block diagram and code have been updated according to the additional information provided by the OP. As the closed-loop system is a Linear Time-Invariant (LTI) system, we can apply the superposition principle. This principle states that the net output response resulting from the reference and disturbance signals is the sum of the individual responses caused by each signal separately.

%% Redefine the value of Heaviside function H(u) at u = 0.
newVal = 1;
oldVal = sympref('HeavisideAtOrigin', newVal);

%% time vector for simulation
tEnd= 6e-6;
t   = linspace(0, tEnd, 6001);

%% Buck converter (plant), Gp
Gp  = tf(216000, [0.0006, 1, 6000])

Gp =
 
         216000
  ---------------------
  0.0006 s^2 + s + 6000
 
Continuous-time transfer function.

%% PID controller (compensator), Gc
Gc  = pid(16.8278, 1.1742, 0.00992)     % improper

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

  with Kp = 16.8, Ki = 1.17, Kd = 0.00992
 
Continuous-time PID controller in parallel form.

%% closed-loop system, Gcl
Gcl = feedback(Gc*Gp, 1)

Gcl =
 
        2143 s^2 + 3.635e06 s + 2.536e05
  ---------------------------------------------
  0.0006 s^3 + 2144 s^2 + 3.641e06 s + 2.536e05
 
Continuous-time transfer function.

%% reference voltage input
r   = 12*heaviside(t) - 6*heaviside(t - 2e-6);

%% signal that disturbs the response at the output channel
do  = 1*heaviside(t - 4e-6);

%% simulated time response
yr  = lsim(Gcl, r, t);  % output response due to reference
yd  = lsim(Gcl, do, t); % output response due to disturbance at the output channel
yt  = yr - yd;          % true output response
ym  = yt + do';         % measured output response
plot(t, [ym'; r; do]), grid on, ylim([0, 14])
xlabel('t'), ylabel('y(t)')
legend('Measured Output', 'Reference Input', 'Disturbance')

[mostafa jabbari]

Tanks for your anwser @Sam Chak

Yes, you are right, I should provide more information. In this section, you can see the system overview ,and how to enter the disturbance signal. You can also see the values of the PID controller parameters in this section . We also have the buck converter values as follows . Exactly my problem is the execution of three scenarios in a spisific time(t=6e-6). Will presenting these items help to solve the problem, dear friend? Now, if we can't implement the transfer function in three scenarios, how is the graph of the step function obtained with the input of the disturbance signal? if its helpful for you that paper mention this section for these three senarios( "The performance of the controllers is evaluated under three distinct conditions: setting the initial reference voltage, transitioning to a different output voltage level, and responding to a sudden disturbance in the output voltage. Initially, the output voltage level is set at 12V (Vref = 12V). Once the initial voltage is established, the reference voltage level is then reduced from 12V to 6V at t = 2×10^−6. At t = 4×10^−6, a sudden positive 1V change occurs in the converter output voltage, representing a significant disturbance that requires rapid correction. This sudden deviation at the output, with the reference voltage at 6V, constitutes more than a 16% variation from the desired output. This disturbance is simulated as a step increase of +1V at the converter output.")

[Paul]

Hi Sam,

Don't forget with the default sympref that

heaviside(0)

ans = 0.5000

Probably doesn't make much of a difference in this problem, but still.

[Sam Chak]

Hi @Paul,

I wanted to express my gratitude for your advice. It reminded me of something important that I tend to overlook when using Heaviside functions. In the past, when I designed path-following signals for some robots by "stitching" together piecewise functions, I encountered peculiar spikes at the transition points.

Hi @mostafa jabbari

Based on the new info provided, I have updated the code to demonstrate how to obtain the measured output response using the Superposition principle. Please let me know if this solution aligns with what you are seeking and if it corresponds to the result presented in the paper.

[mostafa jabbari]

@Sam Chak@Paul

Tanks a lot, your code is corroct.

[Sam Chak]

Hi @mostafa jabbari

I'm glad to hear that it worked correctly. If you found the example on Superposition Principle helpful, please consider clicking 'Accept' ✔️ on the answer and voting 👍 for it.

By the way, you have another unclosed thread in the link below. You can comment or request more details if the solution described in the answer is unclear.

https://www.mathworks.com/matlabcentral/answers/437018-how-can-i-download-from-mathwork-site