Math Problem Statement

Consider the system (s2+ p1s+ p0)y = kp(s+z0)u, ˙ 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 = 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;

Solution

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

  1. Define the system dynamics:

    • s2+p1s+p0s^2 + p_1s + p_0 for the plant.
    • s+1s + 1 for the reference model.
  2. Set initial conditions and the given parameter values:

    • p1=2p_1 = -2, p0=2p_0 = 2, kp=1k_p = -1, z0=1z_0 = 1.
  3. Implement adaptive control:

    • Design an adaptive controller to minimize the tracking error e(t)=y(t)ym(t)e(t) = y(t) - y_m(t).
    • Use a chosen adaptation law.
  4. Simulate the system with inputs r(t)=10sin(t)r(t) = 10\sin(t) and r(t)=10sin(t)+13sin(3.3t)r(t) = 10\sin(t) + 13\sin(3.3t).

  5. Plot tracking error:

    • Evaluate and plot e(t)=y(t)ym(t)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)=10sin(t)r(t) = 10\sin(t).
    • Another for r(t)=10sin(t)+13sin(3.3t)r(t) = 10\sin(t) + 13\sin(3.3t).

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Math Problem Analysis

Mathematical Concepts

Control Systems
Adaptive Control
Differential Equations
Signal Processing

Formulas

Plant dynamics: (s^2 + p1s + p0)[y](t) = kp(s + z0)[u](t)
Reference model: (s + 1)[ym](t) = r(t)
Tracking error: e(t) = y(t) - ym(t)

Theorems

Lyapunov Stability Theory (for adaptive control convergence)

Suitable Grade Level

Undergraduate - Control Engineering