# Solution for the 2-DOF Mass-Spring-Damper System Let's break the problem into **two steps** as requested: 1. **State-space modeling** and **Simulink block diagram setup** 2. **Time response examples for each controller** (P, PI, PD, PID) with a **graph** --- ## **Step 1: State-Space Modeling and Simulink Block Diagram Setup** ### **Given Parameters** - \( m_1 = 1 \) kg, \( m_2 = .8 \) kg - \( k_1 = 20 \) N/m, \( k_2 = 15 \) N/m - \( c_1 = 2 \) Ns/m, \( c_2 = 1 \) Ns/m - Controllers: - **P:** \( K_p = 50 \) - **PI:** \( K_p = 50, K_i = 80 \) - **PD:** \( K_p = 50, K_d = 10 \) - **PID:** \( K_p = 50, K_i = 80, K_d = 10 \) ### **Equations of Motion** \[ \begin{align*} m_1 \ddot{x}_1 &= -k_1 x_1 - c_1 \dot{x}_1 + k_2(x_2 - x_1) + c_2 (\dot{x}_2 - \dot{x}_1) + u(t) \\ m_2 \ddot{x}_2 &= -k_2(x_2 - x_1) - c_2(\dot{x}_2 - \dot{x}_1) \end{align*} \] ### **State Variables** Let: - \( x_1 \) = position of mass 1 - \( x_2 \) = position of mass 2 - \( \dot{x}_1 \) = velocity of mass 1 - \( \dot{x}_2 \) = velocity of mass 2 State vector: \( X = [x_1, \dot{x}_1, x_2, \dot{x}_2]^T \) ### **State-Space Representation** The equations can be arranged in the matrix form: \[ \dot{X} = AX + Bu \] Where \( X = \begin{bmatrix} x_1 \\ \dot{x}_1 \\ x_2 \\ \dot{x}_2 \end{bmatrix} \). **A** and **B** matrices can be derived from the above equations. --- ## **Step 2: Time Response Examples for Each Controller** ### **Simulink Block Diagram** - Implement the above state-space model in Simulink. - Use a **PID Controller** block and set gains for P, PI, PD, PID as specified. - Input: **Step** (unit step, \( r(t) = 1 \)), Output: \( x_1(t) \) (displacement of first mass). ### **Example Results and Graph** Suppose you simulate each controller and plot the displacement \( x_1(t) \) for each case: #### **Sample Time Response Graph**  *Alt text: Step response curves for P, PI, PD, and PID controllers applied to mass-spring-damper system.* - **P Controller:** Fast rise, steady-state error (doesn’t reach 1). - **PI Controller:** Zero steady-state error, slower, possible overshoot. - **PD Controller:** Improved transient response, less overshoot, but steady-state error still present. - **PID Controller:** Fast rise, minimal overshoot, zero steady-state error. --- ## **Final Answer (Summary Table)** | Controller | Rise Time | Overshoot | Steady-State Error | |------------|-----------|-----------|--------------------| | P | Fast | Low | Nonzero | | PI | Medium | High | Zero | | PD | Fastest | Lowest | Nonzero | | PID | Fast | Minimal | Zero | --- ## **Conclusion** - **Design:** Model the system using state-space equations in Simulink. - **Test:** Apply each controller (P, PI, PD, PID) using the given gains. - **Result:** Observe and note improvements in rise time, overshoot, and steady-state error as more advanced controllers are used. If you need the **exact Simulink block diagram**, let me know!