# Stepwise Solution ## **Problem Restatement** Given the control system with transfer functions and feedback as shown, **find the values of \( K_p \) and \( K_t \)** such that: - Steady-state error for a **unit ramp** input is **< 5%** (\( e_{ss} < .05 \)) - **Damping ratio (\( \zeta \))** is **.5** - **Disturbance is zero** --- ## **Step 1: System Analysis and Error Calculation** ### **1.1. System Transfer Function** - **Forward path:** \( K_p \) (controller) and motor-load \( \frac{1}{15s+3} \), and integrator \( \frac{1}{s} \). - **Tachometer feedback:** \( K_t \). **Open-loop transfer function:** \[ G(s) = K_p \cdot \frac{1}{15s+3} \cdot \frac{1}{s} \] **Feedback:** \( K_t \) **Closed-loop transfer function:** \[ H(s) = \frac{K_p}{(15s+3)s + K_p + K_t} \] But let's use **steady-state error for unit ramp**. ### **1.2. Steady-State Error for a Unit Ramp** For a **Type 1** system (contains \( \frac{1}{s} \)), the steady-state error to a ramp (\( r(t) = t \)) is: \[ e_{ss} = \frac{1}{K_v} \] where \[ K_v = \lim_{s \to } s G_{open}(s) \] The open-loop transfer function seen by the reference input is: \[ G_{open}(s) = \frac{K_p}{(15s+3)s + K_p + K_t} \] But for steady-state error, use the **position error constant** for a ramp. First, find the transfer function from \( \theta_i(s) \) to \( \theta_o(s) \): \[ \text{Let} \quad G_c(s) = K_p, \quad G_p(s) = \frac{1}{15s+3} \cdot \frac{1}{s} \] \[ G_{open}(s) = K_p \cdot \frac{1}{15s+3} \cdot \frac{1}{s} \] Feedback is \( K_t \). So, the **characteristic equation** is: \[ 1 + G_{open}(s) \cdot K_t = 1 + \frac{K_p}{s(15s+3)} K_t \] But, for unity feedback with proportional plus tachometer feedback, the **effective open-loop transfer function** for error analysis is: \[ G_{open}(s) = \frac{K_p}{(15s+3)s + K_t} \] **Velocity error constant:** \[ K_v = \lim_{s \to } s \cdot G_{open}(s) = \lim_{s \to } s \cdot \frac{K_p}{(15s+3)s + K_t} = \frac{K_p}{K_t} \] **Steady-state error:** \[ e_{ss} = \frac{1}{K_v} = \frac{K_t}{K_p} \] **Constraint:** \[ e_{ss} < .05 \implies \frac{K_t}{K_p} < .05 \implies K_p > 20 K_t \] --- ## **Step 2: Damping Ratio and Parameter Selection** ### **2.1. Damping Ratio (\( \zeta \))** The **closed-loop characteristic equation** is: \[ (15s+3)s + K_p + K_t = \implies 15s^2 + 3s + K_t + K_p = \] This is a standard **second-order system**: \[ 15s^2 + 3s + (K_t + K_p) = \implies s^2 + \frac{3}{15}s + \frac{K_t + K_p}{15} = \] Compare with: \[ s^2 + 2\zeta\omega_n s + \omega_n^2 = \] So, - \( 2\zeta\omega_n = \frac{3}{15} = .2 \) - \( \omega_n^2 = \frac{K_t + K_p}{15} \) Given \( \zeta = .5 \): \[ 2 \times .5 \times \omega_n = .2 \implies \omega_n = .2 \] So, \[ \omega_n^2 = .04 = \frac{K_t + K_p}{15} \implies K_t + K_p = .6 \] --- ### **2.2. Solve for \( K_p \) and \( K_t \)** From above: \[ \begin{cases} K_p + K_t = .6 \\ K_p > 20 K_t \end{cases} \] Let \( K_t = x \), \( K_p = .6 - x \), and \( .6 - x > 20x \): \[ .6 - x > 20x \implies .6 > 21x \implies x < \frac{.6}{21} \approx .0286 \] So, \[ K_t < .0286, \quad K_p = .6 - K_t > .571 \] **Choose \( K_t = .02 \) (example), then:** \[ K_p = .6 - .02 = .58 \] **Check:** \[ K_p > 20 K_t \implies .58 > .4 \quad \text{(True)} \] **Steady-state error:** \[ e_{ss} = \frac{K_t}{K_p} = \frac{.02}{.58} \approx .0345 < .05 \] --- ## **Step 3: Final Answer** ### **Example Values:** \[ \boxed{K_p = .58, \quad K_t = .02} \] --- ## **Graph: Steady-State Error vs. \( K_t \) (for fixed \( K_p + K_t = .6 \))** ```python import numpy as np import matplotlib.pyplot as plt Kt = np.linspace(.01, .03, 100) Kp = .6 - Kt ess = Kt / Kp plt.plot(Kt, ess) plt.axhline(.05, color='red', linestyle='--', label='5% Error Bound') plt.xlabel('$K_t$') plt.ylabel('Steady-State Error $e_{ss}$') plt.title('Steady-State Error vs $K_t$ (for $K_p + K_t = .6$)') plt.grid(True) plt.legend() plt.show() ``` > **[Graph shows $e_{ss}$ remains below .05 for $K_t < .0286$]** --- ## **Summary Table** | Parameter | Value (Example) | |-----------|----------------| | \( K_p \) | .58 | | \( K_t \) | .02 | | \( e_{ss} \) | .0345 | | \( \zeta \) | .5 | --- ## **Conclusion** - **Choose \( K_p = .58 \), \( K_t = .02 \) (or any pair such that \( K_p + K_t = .6 \) and \( K_p > 20 K_t \)).** - This ensures **damping ratio .5** and **steady-state error < 5%** for a **unit ramp input**.