To design a PID compensator for the given system, we will follow these steps: ### Step 1: Determine the Transfer Function The given open-loop transfer function is: \[ G(s) = \frac{1}{s(s+3)(s+6)} \] ### Step 2: Determine Desired Specifications - Maximum Overshoot: ≤ 15% - Settling Time: ≤ 15 seconds ### Step 3: Design PID Compensator A PID compensator has the general form: \[ D(s) = K_p + \frac{K_i}{s} + K_d s \] ### Step 4: Analyze the System without Compensator The open-loop transfer function is: \[ G(s) = \frac{1}{s(s+3)(s+6)} \] The poles are at \(s = , -3, -6\). ### Step 5: Determine the Desired Damping Ratio and Natural Frequency To achieve a maximum overshoot of 15%, the damping ratio \(\zeta\) can be calculated using the formula: \[ \%OS = 100 \times e^{\left(-\frac{\pi \zeta}{\sqrt{1-\zeta^2}}\right)} \] Solving for \(\zeta\) when \(\%OS = 15\%\): \[ 15 = 100 \times e^{\left(-\frac{\pi \zeta}{\sqrt{1-\zeta^2}}\right)} \] Solving numerically, \(\zeta \approx .516\). For a settling time \(T_s\) of 15 seconds, we use: \[ T_s = \frac{4}{\zeta \omega_n} \leq 15 \] Solving for \(\omega_n\): \[ \omega_n \geq \frac{4}{15 \times .516} \approx .516 \] ### Step 6: Design PID Parameters The PID parameters are chosen to meet these specifications. A typical approach is to start by adding a proportional gain \(K_p\) to shift the poles leftward, then adjust \(K_i\) and \(K_d\) for fine-tuning: 1. **Proportional Gain (\(K_p\))**: Increases the system bandwidth and decreases the rise time. 2. **Integral Gain (\(K_i\))**: Eliminates steady-state error but may affect stability. 3. **Derivative Gain (\(K_d\))**: Improves stability and transient response. ### Step 7: Simulate and Iterate Using simulation tools like MATLAB or Python: - Test different values of \(K_p\), \(K_i\), and \(K_d\). - Observe the response to ensure it meets overshoot and settling time requirements. - Adjust gains iteratively. ### Step 8: Example Solution Suppose we find through simulation: - \(K_p = 30\) - \(K_i = 10\) - \(K_d = 5\) ### Final Answer The PID compensator is: \[ D(s) = 30 + \frac{10}{s} + 5s \] This compensator should achieve the desired step response with no more than 15% overshoot and a settling time of no more than 15 seconds.