S. Boyd EE102Lecture 11Feedback: static analysis• feedback: overview, standard configuration, terms• static linear case• sensitivity• static nonlinear case• linearizing effect of feedback11–1Feedback: generala portion of the output signal is ‘fed back’ to the inputstandard block diagram:PSfrag replacementsuyeAF• u is the input signal; y is the output signal; e is called the error signal• A is called the forward or open-loop system or plant• F is called the feedback systemin equations: y = Ae, e = u − F yFeedback: static analysis 11–2• feedback ‘loop’: e affects y, which affects e . . .• overall system is called closed-loop system• signals can be analog electrical (voltages, currents), mechanical, digitalelectrical, . . .• the − sign is a tradition onlyfeedback is very widely used• in amplifiers• in automatic control (flight control, hard disk & CD player mechanics)• in communications (oscillators, phase-lock loop)Feedback: static analysis 11–3when properly designed, feedback systems are• less sensitive to component variation• less sensitive to some interferences and noises• more linear• faster(when compared to similar open-loop systems)we will also see some disadvantages, e.g.• smaller gain• possibility of instabilityFeedback: static analysis 11–4Other feedback configurationswe will also see other feedback configurations, e.g.PSfrag replacementsuryePCwhich is often used in automatic controlfor now we stick to the ‘standard configuration’ (p.11–2)Feedback: static analysis 11–5sometimes the ‘feedback loop’ is not clear (e.g., in amplifier circuits)PSfrag replacementsVDDIdVinVoutRlRsDSGhere we haveVout= Rlf(VGS), VGS= Vin− (Rs/Rl)Vout,where Id= f(VGS)Feedback: static analysis 11–6Static linear casestatic case: signals do not vary with time, i.e., signals u, e, y are(constant) real numbers(dynamic analysis of feedback is very important — we’ll do it later)suppose forward and feedback systems are linear, i.e., A and F arenumbers (‘gains’)eliminate e from y = Ae, e = u − F y to get y = Gu whereG =A1 + AFis called the closed-loop system gain (A is called open-loop system gain)L = AF is called the loop gain — it is the gain around the feedback loop,cut at the summing junctionFeedback: static analysis 11–7observation: if L = AF is large (positive or negative!) then G ≈ 1/F andis relatively independent of Ahow close is G to 1/F ?consider relative error :1/F − G1/F=11 + AF(after some algebra)S =11 + AF=11 + Lis called the sensitivity (and will come up many times)for large loop gain, sensitivity ≈ 1/loop gainthus:for 20dB loop gain, G ≈ 1/F within about 10%for 40dB loop gain, G ≈ 1/F within about 1%etc.Feedback: static analysis 11–8Example: feedback amplifierPSfrag replacementsvinvAvvoutR1R2described by: vout= Av, v = vin− (R1/(R1+ R2))vout• vinis the input u; voutis the output y• v is the ‘error signal’ e• open-loop gain is A• feedback gain is F = R1/(R1+ R2)vout= Gvin, where closed-loop gain is G =A1 + AFFeedback: static analysis 11–9example: for F = 0.1 and A ≥ 100, G ≈ 10 within 10%as A varies from, say, 100 to 1000 (20dB variation),G varies about 10% (around 1dB variation)in this example, large variations in open-loop gain lead to much smallervariations in closed-loop gainFeedback: static analysis 11–10Sensitivity to small changes in Ahow do small changes in the open-loop gain A affect closed-loop gain G?∂G∂A=∂∂AA1 + AF=1(1 + AF )2so for small change δA, we haveδG ≈1(1 + AF )2δAexpress in terms of relative or fractional gain changes:(δG/G) ≈11 + AF(δA/A) = S(δA/A)hence the name ‘sensitivity’ for SFeedback: static analysis 11–11for small fractional changes in open-loop gain,S ≈fractional change in closed-loop gainfractional change in open-loop gain(so ‘sensitivity ratio’ is perhaps a better term for S)for large loop gain (positive or negative), |S| ¿ 1, so small fractionalchanges in A yield much smaller fractional changes in G:feedback has reduced the sensitivity of the gain G w.r.t. changes in thegain AFeedback: static analysis 11–12we can relate (small) relative changes to changes in dB:δ(20 log10X) =20log 10δ log X ≈20log 10(δX/X)(20/ log 10 ≈ 9, i.e., 10% relative change ≈ 0.9dB)hence we have (for small changes in A),δ(20 log10G) ≈ S δ(20 log10A)thus (for small changes in open-loop gain),S ≈dB change in closed-loop gaindB change in open-loop gainExample: ±2dB variation in A, with L ≈ 10, yields approximately ±0.2dBvariation in GFeedback: static analysis 11–13Summary:for loop gain |L| À 1,• gain is reduced by about |L|• sensitivity of gain w.r.t. A is reduced by about |L|thus, feedback allows us to trade gain for reduced sensitivitye.g., convert amplifier with gain 30 ± 2dB to one with gain 20 ± 0.7dB or10 ± 0.2dBFeedback: static analysis 11–14Remarks:• feedback critical with vacuum tube amplifiers(gains varied substantially with age . . . )• get benefits for ‘negative’ (AF > 0) or ‘positive’ (AF < 0) feedback —makes little difference in static case• sensitivity w.r.t. F is not small — need accurate, reliable feedbackcomponents• can also trade sensitivity for more gain, by setting AF ≈ −1Feedback: static analysis 11–15Nonlinear static feedbackWe suppose now that the forward system is nonlinear static, i.e., A is afunction from R into R, e.g.,−0.1 −0.05 0 0.05 0.1 0.15−1.5−1−0.500.511.5PSfrag replacementseyA(e)very common for amplifiers, transducers, etc. to be at least a bit nonlinearA is called the nonlinear transfer characteristic of the forward system(never to be confused with transfer function!)Feedback: static analysis 11–16we’ll keep the feedback system F linear for nowPSfrag replacementsuyeA(·)Ffeedback system is described by y = A(e), e = u − F ythese are coupled nonlinear equations:• maybe multiple solutions; maybe no solutions• usually impossible to solve analytically• can be solved graphically, or by computerusually for each u ∈ R there is one solution y, so we can express theclosed-loop transfer characteristic as a function: y = G(u)Feedback: static analysis 11–17Example: open-loop characteristic A:−0.1 −0.05 0 0.05 0.1 0.15−1.5−1−0.500.511.5PSfrag replacementseyA(e)Feedback: static analysis 11–18with feedback gain F = 0.2, yields closed-loop characteristic−0.6 −0.4 −0.2 0 0.2 0.4 0.6−1.5−1−0.500.511.5PSfrag