Consider the system
(s2+ p1s+ p0)[y](t) = kp(s+z0)[u](t), ˙ y(0) = y(0) = 0
where z0 > 0, p1, p0, kp are unknown but sign[kp] is known, and y(t) is measured but not ˙ y(t), and choose
the reference model
(s+1)[ym](t) = r(t), ym(0) = 0.
(i)on matlab simulate the adaptive control system with p1 = −2, p0 = 2, kp = −1, z0 = 1 for r(t) = 10sin(t),
r(t) = 10sin(t)+13sin(3.3t), and plot the tracking error for each case;

To simulate this adaptive control system in MATLAB, we'll proceed with the following steps:

1. **Define the system dynamics**:
   - \( s^2 + p_1s + p_0 \) for the plant.
   - \( s + 1 \) for the reference model.

2. **Set initial conditions and the given parameter values**:
   - \( p_1 = -2 \), \( p_0 = 2 \), \( k_p = -1 \), \( z_0 = 1 \).

3. **Implement adaptive control**:
   - Design an adaptive controller to minimize the tracking error \( e(t) = y(t) - y_m(t) \).
   - Use a chosen adaptation law.

4. **Simulate the system with inputs \( r(t) = 10\sin(t) \) and \( r(t) = 10\sin(t) + 13\sin(3.3t) \)**.

5. **Plot tracking error**:
   - Evaluate and plot \( e(t) = y(t) - y_m(t) \) for each input.

Below is the MATLAB script:

```matlab
% MATLAB Simulation for Adaptive Control System

% Parameters
p1 = -2;
p0 = 2;
kp = -1;
z0 = 1;

% Reference model dynamics
ref_dynamics = @(t, ym, r) -ym + r; % y_m'(t) = -y_m(t) + r(t)

% Input signals
r1 = @(t) 10 * sin(t);                        % r(t) = 10sin(t)
r2 = @(t) 10 * sin(t) + 13 * sin(3.3 * t);    % r(t) = 10sin(t) + 13sin(3.3t)

% Simulation parameters
tspan = [0 20]; % Simulation time
dt = 0.01;      % Time step
time = tspan(1):dt:tspan(2);

% Initial conditions
y0 = 0;
ym0 = 0;

% Adaptive controller setup
gamma = 1;  % Adaptation gain
theta_hat = [0; 0]; % Initial estimates for parameters

% Tracking error results
tracking_error_r1 = [];
tracking_error_r2 = [];

% Helper function to simulate the system
simulate_adaptive_system = @(r) arrayfun(@(t) adaptive_control(t, r), time);

% Run simulations
[y1, ym1, e1] = simulate_adaptive_system(r1);
[y2, ym2, e2] = simulate_adaptive_system(r2);

% Plot tracking errors
figure;
plot(time, e1, 'r', 'DisplayName', 'Error for r(t) = 10sin(t)');
hold on;
plot(time, e2, 'b', 'DisplayName', 'Error for r(t) = 10sin(t) + 13sin(3.3t)');
title('Tracking Errors');
xlabel('Time (s)');
ylabel('Tracking Error');
legend;
grid on;

% Function: Adaptive Control Implementation
function [y, ym, e] = adaptive_control(t, r)
    persistent y_prev ym_prev theta_prev error_prev
    if isempty(y_prev), y_prev = 0; end
    if isempty(ym_prev), ym_prev = 0; end
    if isempty(theta_prev), theta_prev = [0; 0]; end
    if isempty(error_prev), error_prev = 0; end

    % Get reference signal
    r_t = r(t);

    % Reference model
    ym = ym_prev + dt * (-ym_prev + r_t);

    % Controller (adaptation law)
    error = y_prev - ym;
    theta_hat_dot = -gamma * [error; error * y_prev];
    theta_hat = theta_prev + dt * theta_hat_dot;

    % Plant
    y_dot = -p1 * y_prev - p0 * y_prev + kp * (r_t + z0);
    y = y_prev + dt * y_dot;

    % Update variables
    y_prev = y;
    ym_prev = ym;
    theta_prev = theta_hat;
    error_prev = error;

    % Output tracking error
    e = error;
end
```

### Instructions:
1. Copy and paste the code into MATLAB.
2. Run the script, ensuring the parameters and simulation setup are correct.
3. Two tracking error plots will be displayed:
   - One for \( r(t) = 10\sin(t) \).
   - Another for \( r(t) = 10\sin(t) + 13\sin(3.3t) \).

Would you like more details or help interpreting the results?

Consider the system (s^2 + p1s + p0)[y](t) = kp(s + z0)[u](t), ˙y(0) = y(0) = 0 where z0 > 0, p1, p0, kp are unknown but sign[kp] is known, and y(t) is measured but not ˙y(t), and choose the reference model (s+1)[ym](t) = r(t), ym(0) = 0. (i) on MATLAB simulate the adaptive control system with p1 = −2, p0 = 2, kp = −1, z0 = 1 for r(t) = 10sin(t), r(t) = 10sin(t) + 13sin(3.3t), and plot the tracking error for each case.

Learn to simulate an adaptive control system in MATLAB using the dynamics (s^2 + p1s + p0)y(t) = kp(s + z0)u(t) and track errors for inputs r(t) = 10sin(t) and r(t) = 10sin(t) + 13sin(3.3t). This tutorial applies adaptive control concepts with step-by-step MATLAB code, ideal for control engineering students.

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