Topic #3 16.31 Feedback Control Frequency response methods Analysis e Synthes ● Performance Stability Copyright 2001 by Jonathan How
Topic #3 16.31 Feedback Control Frequency response methods • Analysis • Synthesis • Performance • Stability Copyright 2001 by Jonathan How. 1
Fall 2001 16.313-1 Introduction . Root locus methods have Advantages k Good indicator if transient response k Explicity shows location of all closed-loop poles Trade-offs in the design are fairly clear Disadvantages k Requires a transfer function model(poles and zeros) k Difficult to infer all performance metrics k Hard to determine response to steady-state(sinusoids Frequency response methods are a good complement to the root locus techniques Can infer performance and stability from the same plot Can use measured data rather than a transfer function model The design process can be independent of the system order Time delays are handled correctly Graphical techniques(analysis and synthesis) are quite simple
Fall 2001 16.31 3–1 Introduction • Root locus methods have: – Advantages: ∗ Good indicator if transient response; ∗ Explicity shows location of all closed-loop poles; ∗ Trade-offs in the design are fairly clear. – Disadvantages: ∗ Requires a transfer function model (poles and zeros); ∗ Difficult to infer all performance metrics; ∗ Hard to determine response to steady-state (sinusoids) • Frequency response methods are a good complement to the root locus techniques: – Can infer performance and stability from the same plot – Can use measured data rather than a transfer function model – The design process can be independent of the system order – Time delays are handled correctly – Graphical techniques (analysis and synthesis) are quite simple
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 dB
Fall 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
Fall 2001 16.313-3 Use logarithmic since if (s+1)(s+2) pg s+3)(s+4 log s+1|+log s+2-log s+3-log s+4 and each of these factors can be calculated separately and then added to get the total FRF Can also split the phase plot since (s+1)(s+2) arg (s+3)(s+4) arg(s +1)+arg(s+ 2) arg(s +3)-arg(s+4) The keypoint in the sketching of the plots is that good straightline approximations exist and can be used to obtain a good prediction of the system response
Fall 2001 16.31 3–3 • Use logarithmic since if log |G(s)| = (s + 1)(s + 2) (s + 3)(s + 4) = log |s + 1| + log |s + 2| − log |s + 3| − log |s + 4| and each of these factors can be calculated separately and then added to get the total FRF. • Can also split the phase plot since arg (s + 1)(s + 2) (s + 3)(s + 4) = arg(s + 1) + arg(s + 2) − arg(s + 3) − arg(s + 4) • The keypoint in the sketching of the plots is that good straightline approximations exist and can be used to obtain a good prediction of the system response
Fall 2001 16.313-4 Example ● Draw bode for S+ /10+ G Gw) j+1 /10+1 log|G(ju)}=log1+(u/1)2]12-log1+(a/102 ●A pproximation log1 +(w/ 211 logw/wil w>wi Two straightline approximations that intersect at w=w Error at wi obvious, but not huge and the straightline approxima tions are very easy to work with
Fall 2001 16.31 3–4 Example • Draw Bode for G(s) = s + 1 s/10 + 1 |G(jω)| = |jω + 1| |jω/10 + 1| log |G(jω)| = log[1 + (ω/1)2] 1/2 − log[1 + (ω/10)2] 1/2 • Approximation log[1 + (ω/ωi) 2 ] 1/2 ≈ 0 ω ωi log[ω/ωi] ω ωi Two straightline approximations that intersect at ω ≡ ωi • Error at ωi obvious, but not huge and the straightline approximations are very easy to work with. 10−2 10−1 100 101 102 100 101 102 Freq |G|