Fall 2001 16.313-2 Frequency response Function Given a system with a transfer function G(s), we call the G(jw) ∈0,∞) the frequency response function(FRF) GGjw)=G(jw)l arg G ljw The frf can be used to find the steady-state response of a system to a sinusoidal input. If t)→(G(s)→y(t) and e(t)=sin 2t, G(2j)=0.3, arg G(2j)=800, then the steady-state output is y(t)=0.3sin(2t-80°) The FRF clearly shows the magnitude(and phase) of the response of a system to sinusoidal input e a variety of ways to display this 1. Polar(Nyquist) plot-Re vs. Im of G w) in complex plane Hard to visualize, not useful for synthesis, but gives definitive tests for stability and is the basis of the robustness analysis 2. Nichols Plot-GGjw) vs. arg Gw), which is very handy for systems with lightly damped poles 3. Bode Plot-Log G(jw and arg G(jw) vs Log frequency Simplest tool for visualization and synthesis Typically plot 20log G which is given the symbol dBFall 2001 16.31 3–2 Frequency response Function • Given a system with a transfer function G(s), we call the G(jω), ω ∈ [0, ∞) the frequency response function (FRF) G(jω) = |G(jω)| arg G(jω) – The FRF can be used to find the steady-state response of a system to a sinusoidal input. If e(t) → G(s) → y(t) and e(t) = sin 2t, |G(2j)| = 0.3, arg G(2j) = 80◦ , then the steady-state output is y(t)=0.3 sin(2t − 80◦ ) ⇒ The FRF clearly shows the magnitude (and phase) of the response of a system to sinusoidal input • A variety of ways to display this: 1. Polar (Nyquist) plot – Re vs. Im of G(jω) in complex plane. – Hard to visualize, not useful for synthesis, but gives definitive tests for stability and is the basis of the robustness analysis. 2. Nichols Plot – |G(jω)| vs. arg G(jω), which is very handy for systems with lightly damped poles. 3. Bode Plot – Log |G(jω)| and arg G(jω) vs. Log frequency. – Simplest tool for visualization and synthesis – Typically plot 20log |G| which is given the symbol dB