To design a PID compensator for the given system with desired performance specifications, follow these steps: ### System Details The transfer function \( G(s) \) is given by: \[ G(s) = \frac{1}{s(s+3)(s+6)} \] ### Desired Specifications - Maximum overshoot: No more than 15% - Settling time (\( t_s \)): No more than 15 seconds ### Step-by-Step Solution #### Step 1: Determine the Closed-Loop Transfer Function The closed-loop transfer function \( T(s) \) for a unity feedback system is: \[ T(s) = \frac{G(s)C(s)}{1 + G(s)C(s)} \] Where \( C(s) \) is the compensator, which we need to design. #### Step 2: Calculate Desired Damping Ratio and Natural Frequency 1. **Maximum Overshoot (\( M_p \))**: The maximum overshoot for a second-order system is given by: \[ M_p = e^{\left(-\frac{\zeta \pi}{\sqrt{1-\zeta^2}}\right)} \] Solving for \(\zeta\) when \( M_p = .15 \): \[ .15 = e^{\left(-\frac{\zeta \pi}{\sqrt{1-\zeta^2}}\right)} \] Solving gives \(\zeta \approx .516\). 2. **Settling Time (\( t_s \))**: The settling time for a second-order system is: \[ t_s = \frac{4}{\zeta \omega_n} \] Given \( t_s \leq 15 \) seconds, solve for \(\omega_n\): \[ 15 \geq \frac{4}{.516 \omega_n} \implies \omega_n \geq \frac{4}{15 \times .516} \approx .517 \] #### Step 3: Design the PID Compensator A PID compensator is of the form: \[ C(s) = K_p + \frac{K_i}{s} + K_d s \] #### Step 4: Adjust System Poles 1. **Desired poles**: With \(\zeta = .516\) and \(\omega_n = .517\), the desired characteristic polynomial is: \[ s^2 + 2\zeta\omega_n s + \omega_n^2 = s^2 + .532s + .267 \] 2. **Place the poles**: Modify the open-loop transfer function by selecting \( K_p, K_i, \) and \( K_d \) to achieve these pole locations. #### Step 5: Use Root Locus or Frequency Response Method - **Root Locus**: Adjust the PID parameters to shift the root locus to pass through the desired pole locations. - **Frequency Response**: Use Bode plots to adjust phase margin and gain to meet the desired specifications. #### Step 6: Validate Design Simulate the closed-loop system with the designed PID controller to ensure specifications are met: - Check the overshoot. - Verify the settling time. #### Final Answer Select \( K_p, K_i, \) and \( K_d \) such that the closed-loop system meets the desired overshoot and settling time. Use simulation tools like MATLAB to find precise values. This step-by-step process will guide you through designing a PID compensator that meets the specified performance criteria. Adjust \( K_p, K_i, \) and \( K_d \) iteratively to achieve the desired response.