EE 102 spring 2001-2002
Handout #26
Lecture 13
Feedback: static and dynamic
• feedback: overview, standard configuration, terms
• static linear case
• sensitivity
• static nonlinear case
• linearizing effect of feedback
• dynamic linear case
• Closed-loop, sensitivity, and loop transfer functions
13–1
Feedback: general
a portion of the output signal is ‘fed back’ to the input
standard block diagram:
u
e
A
y
F
• 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 system
in equations: y = Ae, e = u − F y
Feedback: static and dynamic
13–2
• feedback ‘loop’: e affects y, which affects e . . .
• overall system is called closed-loop system
• signals can be analog electrical (voltages, currents), mechanical, digital
electrical, . . .
• the − sign is a tradition only
feedback is very widely used
• in amplifiers
• in automatic control (flight control, hard disk & CD player mechanics)
• in communications (oscillators, phase-lock loop)
Feedback: static and dynamic
13–3
when 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 instability
Feedback: static and dynamic
13–4
Static linear case
static 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 are
numbers (‘gains’)
eliminate e from y = Ae, e = u − F y to get y = Gu where
A
G=
1 + AF
is 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 junction
Feedback: static and dynamic
13–5
observation: if L = AF is large (positive or negative!) then G ≈ 1/F and
is relatively independent of A
how close is G to 1/F ?
1/F − G
1
consider relative error :
=
(after some algebra)
1/F
1 + AF
1
1
=
S=
1 + AF
1+L
is called the sensitivity (and will come up many times)
for large loop gain, sensitivity ≈ 1/loop gain
thus:
for 20dB loop gain, G ≈ 1/F within about 10%
for 40dB loop gain, G ≈ 1/F within about 1%
etc.
Feedback: static and dynamic
13–6
Example: feedback amplifier
Av
v
vout
vin
R1
R2
described by: vout = Av, v = vin − (R1/(R1 + R2))vout
• vin is the input u; vout is 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 =
Feedback: static and dynamic
A
1 + AF
13–7
example: 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 smaller
variations in closed-loop gain
Feedback: static and dynamic
13–8
Sensitivity to small changes in A
how do small changes in the open-loop gain A affect closed-loop gain G?
∂
A
1
∂G
=
=
∂A ∂A 1 + AF
(1 + AF )2
so for small change δA, we have
1
δG ≈
δA
(1 + AF )2
express in terms of relative or fractional gain changes:
1
(δA/A) = S(δA/A)
(δG/G) ≈
1 + AF
hence the name ‘sensitivity’ for S
Feedback: static and dynamic
13–9
for small fractional changes in open-loop gain,
S≈
fractional change in closed-loop gain
fractional 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 fractional
changes in A yield much smaller fractional changes in G:
feedback has reduced the sensitivity of the gain G w.r.t. changes in the
gain A
Feedback: static and dynamic
13–10
we can relate (small) relative changes to changes in dB:
20
20
δ log X ≈
(δX/X)
δ(20 log10 X) =
log 10
log 10
(20/ log 10 ≈ 9, i.e., 10% relative change ≈ 0.9dB)
hence we have (for small changes in A),
δ(20 log10 G) ≈ S δ(20 log10 A)
thus (for small changes in open-loop gain),
dB change in closed-loop gain
S≈
dB change in open-loop gain
Example: ±2dB variation in A, with L ≈ 10, yields approximately ±0.2dB
variation in G
Feedback: static and dynamic
13–11
Summary:
for loop gain |L| 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 sensitivity
e.g., convert amplifier with gain 30 ± 2dB to one with gain 20 ± 0.7dB or
10 ± 0.2dB
Feedback: static and dynamic
13–12
Remarks:
• 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 feedback
components
• can also trade sensitivity for more gain, by setting AF ≈ −1
Feedback: static and dynamic
13–13
Nonlinear static feedback
We suppose now that the forward system is nonlinear static, i.e., A is a
function from R into R, e.g.,
1.5
1
A(e)
y
0.5
0
−0.5
−1
−1.5
−0.1
−0.05
0
e
0.05
0.1
0.15
very common for amplifiers, transducers, etc. to be at least a bit nonlinear
A is called the nonlinear transfer characteristic of the forward system
(never to be confused with transfer function!)
Feedback: static and dynamic
13–14
we’ll keep the feedback system F linear for now
u
e
A(·)
y
F
feedback system is described by y = A(e), e = u − F y
these are coupled nonlinear equations:
• maybe multiple solutions; maybe no solutions
• usually impossible to solve analytically
• can be solved graphically, or by computer
usually for each u ∈ R there is one solution y, so we can express the
closed-loop transfer characteristic as a function: y = G(u)
Feedback: static and dynamic
13–15
Example: open-loop characteristic A:
1.5
1
A(e)
y
0.5
0
−0.5
−1
−1.5
Feedback: static and dynamic
−0.1
−0.05
0
e
0.05
0.1
0.15
13–16
with feedback gain F = 0.2, yields closed-loop characteristic
1.5
1
y
0.5
G(u)
0
−0.5
−1
−1.5
−0.6
−0.4
−0.2
0
u
0.2
0.4
0.6
(you should check a few points!)
Feedback: static and dynamic
13–17
Observations: with feedback
• ‘gain’ is lower (note different horizontal scales)
• characteristic is more linear (for |y| < 1)
these phenomena are general . . .
closed-loop transfer characteristic function G satisfies
G(u) = y = A(e),
e = u − F G(u)
differentiate w.r.t. u:
de
G (u) = A (e) ,
du
Feedback: static and dynamic
de
= 1 − F G(u)
du
13–18
eliminate de/du to get
A(e)
G (u) =
1 + A(e)F
conclusions: for u s.t. |AF | 1,
• G ≈ 1/F (independent of u) i.e., G is nearly linear!
• slope of G is smaller than slope of A
(by factor 1 + AF )
Feedback: static and dynamic
13–19
Static Feedback Summary
• using feedback we can trade raw gain for lower sensitivity, greater
linearity
• benefits determined by S = 1/(1 + AF ):
sensitivity and nonlinearity are both reduced by S
• large loop gain L = AF (positive or negative) yields small S hence
benefits of feedback
Feedback: static and dynamic
13–20
Dynamic Feedback
we now assume:
• signals u, e, y are dynamic, i.e., change with time
• open-loop and feedback systems are convolution operators, with impulse
responses a and f , respectively
u
e
∗a
y
∗f
Feedback: static and dynamic
13–21
feedback equations are now:
y(t) =
t
0
a(τ )e(t − τ ) dτ,
e(t) = u(t) −
t
0
f (τ )y(t − τ ) dτ
• these are complicated (integral equations)
• it’s not so obvious what to do — current input u(t) affects future
output y(t̄), t̄ ≥ t
Feedback: static and dynamic
13–22
Feedback system: frequency domain
take Laplace transform of all signals:
Y (s) = A(s)E(s),
E(s) = U (s) − F (s)Y (s)
eliminate E(s) (just algebra!) to get
Y (s) = G(s)U (s),
G(s) =
A(s)
1 + A(s)F (s)
G is called the closed-loop transfer function
. . . exactly the same formula as in static case, but now A, F , G are
transfer functions
Feedback: static and dynamic
13–23
we define
• loop transfer function L = AF
• sensitivity transfer function S = 1/(1 + AF )
same formulas as static case!
for example, for small δA, we have
δA
δG
≈S
G
A
(but these are transfer functions here)
Feedback: static and dynamic
13–24
what’s new: L, S, G
• depend on frequency s
• are complex-valued
• can be stable or unstable
thus:
• “large” and “small” mean complex magnitude
• L (or G or S) can be large for some frequencies, small for others
• step response of G shows time response of the closed-loop system
Feedback: static and dynamic
13–25
Example
feedback system with
105
,
A(s) =
1 + s/100
F = 0.01
• open-loop gain is large at DC (105)
• open-loop bandwidth is around 100 rad/sec
• open-loop settling time is around 20msec
Feedback: static and dynamic
13–26
closed-loop transfer function is
G(s) =
1+
105
1+s/100
105
0.01 1+s/100
99.9
=
1 + s/(1.001·105)
• G is stable
• closed-loop DC gain is very nearly 1/F
• closed-loop bandwidth around 105 rad/sec
• closed-loop settling time is around 20µsec
. . . closed-loop system has lower gain, higher bandwidth, i.e., is faster
Feedback: static and dynamic
13–27
103
103
, so |L(jω)| = loop transfer function is L(s) =
1 + s/100
1 + (ω/100)2
4
10
3
|L(jω)|
10
2
10
1
10
0
10
−1
10
1
10
2
10
3
10
4
ω
10
5
10
6
10
• loop gain larger than one for ω < 105 or so
⇒ get benefits of feedback for ω < 105
• loop gain less than one for ω > 105 or so
⇒ don’t get benefits of feedback for ω > 105
Feedback: static and dynamic
13–28
sensitivity transfer function is S(s) =
1 + s/100
1001 + s/100
0
10
−1
|S(jω)|
10
−2
10
−3
10
−4
10
1
10
2
10
3
10
4
ω
10
5
10
6
10
• |S(jω)| 1 for ω < 104 (say)
• |S| ≈ 1 for ω > 105 or so
Feedback: static and dynamic
13–29
Thus, e.g., for small changes in A(0), A(j105)
δG(0) ≈ 10−3 δA(0) ,
G(0) A(0) Feedback: static and dynamic
δG(j105) δA(j105) G(j105) ≈ A(j105) 13–30
Example (with change of sign)
105
, F = 0.01
now consider system with A(s) = −
1 + s/100
(note minus sign!)
closed-loop transfer function is
100.1
G(s) =
1 − s/(0.999·105)
looks like G found above, but is unstable
• in static analysis, large loop gain ⇒ sign of feedback doesn’t much
matter
• dynamic analysis reveals the big difference a change of sign can make
Feedback: static and dynamic
13–31
Dynamic Feedback Summary
for LTI feedback systems,
• formulas same as static case, but now A, F , L, S are transfer functions
• hence are complex, depend on frequency s, and can be stable or
unstable
Feedback: static and dynamic
13–32