Algebraic loop in MRAC discrete model

Hi Yovel,

I do understand your frustration not having this problem resolved. Could you provide details about what you want your code to do, I demonstrated examples in my previous code to help you understand the concept of implementing predictor or an observer in the control loop to reduce the error resulting from the delay because they can estimate the system's current state based on past information, compensating for delays. Predictors on one hand use past inputs and outputs to forecast the system's behavior, while observers on the other hand estimate the system's internal state. These tools help in making real-time adjustments, reducing errors caused by delays and enhancing system performance. When you wrote a lengthy code with no comments incorporated, it made my job little difficult to analyze your code but I still tried my best to analyze your code and provided my comments to improve your code. I do appreciate your efforts implementing an adaptive control algorithm, trying to calculate control inputs based on the error between a reference model and a plant model, and updating adaptive controller parameters to improve the control performance. To reduce the error in control loop, I will help you understand by providing example snippet code which will help you to implement this concept in your code to resolve the issue. So, I implemented an adaptive control algorithm in Matlab. The algorithm calculates control inputs based on the error between a reference model and a plant model. It then updates adaptive controller parameters to enhance control performance. Pay close attention to part of the snippet code below, “Calculate control input based on error”

% Adaptive Control Algorithm Implementation

% Define plant model parameters

A = [0.5, 0.2; 0.1, 0.3]; B = [1; 0]; C = [1, 0]; D = 0;

% Define reference model parameters

Am = [0.4, 0.1; 0.05, 0.35]; Bm = [1; 0]; Cm = [1, 0]; Dm = 0;

% Initialize adaptive controller parameters

theta = zeros(4, 1); % [theta1; theta2; theta3; theta4]

% Simulation loop

for k = 1:100

    % Calculate control input based on error

    e = ym(k) - y(k); % Calculate error between plant output y and reference model output ym

    u(k) = -theta(1)*e - theta(2)*y(k) - theta(3)*u(k-1); % Calculate control input u

    % Update adaptive controller parameters

    theta_dot = 0.1 * [e; y(k); u(k-1); u(k)] * e'; % Update rule for theta
    theta = theta + theta_dot; % Update theta parameters
end

You probably already know what is the above code snippet doing but I still want you to understand about the simulation loop where the code calculates the control input 'u' based on the error 'e' between the plant output 'y' and the reference model output 'ym'. The control input is updated using a linear combination of the error and previous control inputs.

Hopefully, now, following these steps and suggestions should help you answer your question, “ Is it possible to add something to the controller \ to the system to reduce the error resulting from the delay?”

Please let me know if you have any further questions.