Let's break down and **solve** the problem step by step, then I'll show you how to **graph the in MATLAB**. --- ## **1. State-S Representation** Given transfer function: \[ \frac{Y(s)}{U(s)} =frac{1s^2} \] This is a **double integrator**. Let: - \( x_1(t) = y(t) \) (output/attitude) - \( x_2(t) = \dot{y}(t) \) (angular velocity) So, \[ \dot{x}_1 = x_2 \] \[ \dot{x}_2 = u(t) \] --- **2. State Feedback Law** Given feedback: \[ u(t) = -2x_2(t) - x_1(t) \] Substitute into state equations: \[ \dot{x}_1 = x_2 \] \[ \dot{x}_2 = -2x_2 - x_1 \] Or, in matrix form: \[ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} & 1 \\ -1 & -2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \] --- ## **3. Characteristic Equation and Roots** The system matrix \( A \) is: \[ A = \begin{bmatrix} & 1 \\ -1 & -2 \end{bmatrix} \] The characteristic equation: \[ \det(sI - A) = \left| \begin{matrix} s & -1 \\ 1 & s+2 \end{matrix} \right| = s(s+2) + 1 = s^2 + 2s + 1 = (s+1)^2 \] So the **roots** are: \[ s = -1, -1 \] (Double pole at -1) --- ## **4. Initial Conditions** \[ x_1() = ,\quad x_2() = 1 \] --- ## **5. MATLAB Code for Response** Let's simulate and plot \( x_1(t) \) and \( x_2(t) \): ```matlab % State-space matrices A = [ 1; -1 -2]; B = [; ]; % No external input, closed-loop C = eye(2); % To get both x1 and x2 D = [; ]; % Initial conditions x = [; 1]; % Time vector t = linspace(, 10, 100); % Simulate response [y, t, x] = initial(ss(A, B, C, D), x, t); % Plot figure; plot(t, x(:,1), 'b-', 'LineWidth', 2); hold on; plot(t, x(:,2), 'r--', 'LineWidth', 2); legend('x_1(t) = y(t) (Attitude)', 'x_2(t) = \dot{y}(t) (Rate)'); xlabel('Time (s)'); ylabel('States'); title('Rocket Attitude State Response'); grid on; ``` --- ## **6. Explanation of the Code** - **A**: system matrix from state-feedback closed loop. - **x**: initial states. - **initial**: MATLAB function to simulate the response of a state-space model. - **Plot**: Shows both attitude and rate over time. --- ### **Summary Table** | Step | Action | Result | |------|---------------------|-------------------------------| | 1 | State-space form | \(A = \begin{bmatrix} & 1\\-1 & -2\end{bmatrix}\) | | 2 | Characteristic Eq. | \((s + 1)^2\) | | 3 | Roots | \(s = -1, -1\) | | 4 | Initial conditions | \(x_1()=,\ x_2()=1\) | | 5 | MATLAB plot | See code above | --- If you run this MATLAB code, you'll see the **system response** (attitude and rate) starting from your specified initial conditions! If you need further explanation or have questions about the state-space math or MATLAB code, just ask.