ChE 182 Chemical Process Dynamics and Control

1. ChE 182Chemical Process Dynamics and Control Frequency Response Analysis

2. Frequency response analysis -how a process responds to a sinusoidal forcing input -how the response depends on the frequency of the input -used for stability analysis, control system design, and robustness analysis

3. Motivation for Frequency Response Analysis Problems with Routh Stability Criterion: • No measure of relative stability; does not indicate how much uncertainty can be tolerated before system becomes unstable • Only approximate results for processes with time-delay -These limitations are addressed by frequency response analysis

4. Frequency response analysis Two ways to perform frequency response analysis: • Conversion of input signal to Laplace domain and inversion to time domain • Substitution of s = jω in the transfer function Two ways to present frequency response information: • Bode plot • Nyquist plot

5. Method 1: First-order system Sinusoidal input:

6. Method 1: First-order system As t →∞: Long-time response or Frequency response

7. First-Order System Output signal amplitude: Amplitude ratio, AR: Normalized amplitude ratio, ARN:

8. First-Order System

9. Method 2: Stable Transfer Function

10. Method 2: Stable Transfer Function

11. Bode plot Plot of AR and φ as a function of ω • log-log plot of AR versus ω • semilog plot of φ versus ω

12. Nyquist plot Polar plot of G(jω) and ∠G(jω) • plot of Im(ω) versus Re(ω)