W + ( ˆ
H1 + ˆ
H2) ˆ
X .
From this it is evident that if we choose
ˆ
H2 = − ˆ
H1,
then y will be a clean (noise-free) signal, equal to w. The following examples describe
real-world applications of this pattern.
Example 14.11:
A connection to the telephone network uses two wires (called
a twisted pair, consisting of tip and ring) to connect a telephone to a central
office. The central office may be, perhaps, 4 kilometers away. The two wires carry
voice signals to and from the customer premises, representing the voice signals as a
voltage difference across the two wires. Since two wires can only have one voltage
difference across them, the incoming voice signal and the outgoing voice signal
share the same twisted pair.
The central office needs to separate the voice signal from the local customer
premises (called the near-end signal) from the voice signal that comes from the
other end of the connection (called the far-end signal). The near end signal is
typically digitized (sampled and quantized), and a discrete-time representation of
the voice signal is transmitted over the network to the far end. The network itself
consists of circuits that can carry voice signals in one direction at a time. Thus, in
the network, four wires (or equivalent) are required for a telephone connection, one
wire pair for each direction.
As indicated in figure 14.4, the conversion from a two-wire to a four-wire connec-
tion is done by a device called a hybrid. A hybrid is a Wheatstone bridge, a circuit
that can separate two signals based on the electrical impedance looking into the lo-
cal twisted pair and the electrical impedance looking into the network. The design
of this circuit is a suitable topic for a text on electrical circuits.
A connection between subscribers A and B involves two hybrids, one in each sub-
scriber’s central office. The hybrid in B’s central office ideally will pass all of the
Lee & Varaiya, Signals and Systems
621
14.2. PARALLEL COMPOSITION
x = signal from A
twisted pair
Hybrid
Hybrid
at A's
at B's
CO
CO
A
B
echo
y = leakage of A's signal plus signal from B
x
H
H
2
1
w
w
2
1
y
w
echo canceller
echo path
Figure 14.4: A telephone central office converts the two-wire connection with a
customer telephone into a four-wire connection with the telephone network using
a device called a hybrid. An imperfect hybrid leaks, causing echo. An echo
canceller removes the leaked signal.
622
Lee & Varaiya, Signals and Systems
14. COMPOSITION AND FEEDBACK CONTROL
incoming signal x to B’s two-wire circuit, and none back into the network. How-
ever, the hybrid is not perfect, and some of the incoming signal x leaks through the
hybrid into the return path back to A. The signal y in the figure is the sum of the
signal from B and the leaked signal from A. A hears the leaked signal as an echo,
since it is A’s own signal, delayed by propagation through the telephone network.
If the telephone connection includes a satellite link, then the delay from one end
of the connection to the other is about 300ms. This is the time it takes for a radio
signal to propagate to a geosynchronous satellite and back. The echo traverses
this link twice: once going from A to B, and the second time coming back. Thus,
the echo is A’s own signal delayed by about 600ms. For voice signals, 600ms of
delay is enough to create a very annoying echo that can make it difficult to speak.
Humans have difficulty speaking when they hear their own voices 600ms later.
Consequently, the designers of the telephone network have put echo cancellers in
to prevent the echo from occuring.
Let ˆ
H1 be the transfer function of the hybrid leakage path. The echo canceller is
the filter ˆ
H2 placed in parallel composition with the hybrid, as shown in the figure.
The output w2 of this filter is added to the output w1 + w of the hybrid, so the signal
that actually goes back is y = w2 + w1 + w. If
ˆ
H2 = − ˆ
H1,
then y = w and the echo is cancelled perfectly. Moreover, note that as long as ˆ
H1 is
stable and causal, so will be the echo canceller ˆ
H2.
However, ˆ
H1 is not usually known in advance, and also it changes over time. So
either a fixed ˆ
H2 is designed to match a ‘typical’ ˆ
H1, or an adaptive echo canceller
is designed that estimates the characteristics of the echo path ( ˆ
H1) and changes ˆ
H2
accordingly. Adaptive echo cancellers are common in the telephone network today.
The following example combines cascade and parallel composition to achieve noise can-
cellation.
Example 14.12:
Consider a microphone in a noisy environment. For example,
a traffic helicopter might be used to deliver live traffic reports over the radio, but
the (considerable) background noise of the helicopter would be highly undesirable
Lee & Varaiya, Signals and Systems
623
14.3. FEEDBACK COMPOSITION
w
w 1
w 4
H 1
x
y
H 2
H 3
w
w
2
3
Figure 14.5: Traffic helicopter noise cancellation/equalization problem.
on the radio. Fortunately, the background noise can be cancelled. Referring to
Figure 14.5, suppose that w is the announcer’s voice, x is the engine noise, and
ˆ
H1 represents the acoustic path from the engine noise to the microphone. The
microphone picks up both the engine noise and the announcer’s voice, producing
the noisy signal w4. We can place a second microphone somewhere far enough from
the announcer so as to not pick up much of his or her voice. Since this microphone
is in a different place, say on the back of the announcer’s helmet, the acoustic path
is different, so we model that path with another transfer function ˆ
H2. To cancel the
noise, we design a filter ˆ
H3. This filter needs to equalize (invert) ˆ
H2 and cancel ˆ
H1.
That is, its ideal value is
ˆ
H3 = − ˆ
H1/ ˆ
H2.
Of course, as with the equalization scenario, we have to ensure that this filter re-
mains stable. Once again, in practice, it is necessary to make the filter adaptive.
14.3
Feedback composition
Consider the feedback composition in figure 14.6. It is a composition of two systems
with transfer functions ˆ
H1 and ˆ
H2. We assume that these systems are causal and that ˆ
H1
and ˆ
H2 are proper rational polynomials in z or s. The regions of convergence of these
two transfer functions are those suitable for causal systems (the region outside the largest
624
Lee & Varaiya, Signals and Systems
14. COMPOSITION AND FEEDBACK CONTROL
x
e
w
y
H 1
H 2
!
H
H =
1 H 2
1 + H 1 H 2
Figure 14.6: Negative feedback composition of two LTI systems with transfer func-
tions H1 and H2.
circle passing through a pole, for discrete time, and the region to the right of the pole with
the largest real part, for continuous-time).
In terms of Laplace or Z transforms, the signals in the figure are related by
ˆ
Y = ˆ
H ˆ ˆ
2H1E ,
and
Ê = ˆ
X − ˆ
Y .
Notice that, by convention, the feedback term is subtracted, as indicated by the minus
sign adjacent to the adder (for this reason, this composition is called negative feedback).
Combining these two equations to elimintate Ê, we get
ˆ
Y = ˆ
H ˆ
1H2( ˆ
X − ˆ
Y ),
which we can solve for the transfer function of the composition,
ˆ
Y
ˆ
H ˆ
H
ˆ
1
2
H =
=
.
(14.1)
ˆ
X
1 + ˆ
H ˆ
1H2
This is often called the closed-loop transfer function, to contrast it with the open-loop
transfer function, which is simply ˆ
H ˆ
1H2. We will assume that this resulting system is
causal, and that the region of convergence of this transfer function is therefore determined
by the roots of the denominator polynomial, 1 + ˆ
H ˆ
1H2.
Lee & Varaiya, Signals and Systems
625
14.3. FEEDBACK COMPOSITION
The closed-loop transfer function is valid as long as the denominator 1 + ˆ
H ˆ
1H2 is not
identically zero (that is, it is not zero for all s or z in C – it may be zero some s or z in C).
This is sufficient for the feedback loop to be well-formed, although in general, this fact
is not trivial to demonstrate (Exercise 8 considers the easier case where ˆ
H ˆ
1H2 is causal
and strictly proper, in which case the system ˆ
H ˆ
1H2 has state-determined output). We will
assume henceforth, without comment, that the denominator is not identically zero.
Feedback composition is useful for stabilizing unstable systems. In the case of cascade
and parallel composition, a pole of the composite must be a pole of one of the components.
The only way to remove or alter a pole of the components is to cancel it with a zero. For
this reason, cascade and parallel composition are not effective for stabilizing unstable
systems. Any error in the specification of the unstable pole location results in a failed
cancellation, which results in an unstable composition.
In contrast, the poles of the feedback composition are the roots of the denominator 1 +
ˆ
H ˆ
1H2, which are generally quite different from the poles of ˆ
H1 and ˆ
H2. This leads to the
following important conclusion:
The poles of a feedback composition can be different from the poles of its component
subsystems. Consequently, unstable system can be effectively and robustly stabilized by
feedback.
The stabilization is robust in that small changes in the pole or zero locations do not result
in the composition going unstable. We will be able to quantify this robustness.
14.3.1
Proportional controllers
In control applications, one of the two systems being composed, say ˆ
H2, is called the plant.
This is a physical system that is given to us to control. Its transfer function is determined
by its physics. The second system being composed, say ˆ
H1, is the controller. We design
this system to get the plant to do what we want. The following example illustrates a
simple strategy called a proportional controller or P controller.
Example 14.13: For this example we take as the plant the simplified continuous-
time helicopter model of example 12.2,
1
˙
y(t) =
w(t).
M
626
Lee & Varaiya, Signals and Systems
14. COMPOSITION AND FEEDBACK CONTROL
Here y(t) is the angular velocity at time t and w(t) is the torque. M is the moment
of inertia.
We have renamed the input w (instead of x) because we wish to control the heli-
copter, and the control input signal will not be the torque. Instead, let’s define the
input x to be the desired angular velocity. So, to get the helicopter to not rotate, we
provide input x(t) = 0.
Let us call the impulse response of the plant h2, to conform with the notation in
Figure 14.6; it is given by
∀ t ∈ R, h2(t) = u(t)/M,
where u is the unit step. The transfer function is
ˆ
H2(s) = 1/(Ms),
with
RoC(h) = {s ∈ C | Re(s) > 0}.
ˆ
H2 has a pole at s = 0, so this is an unstable system.
As a compensator we can simply place a gain K in a negative feedback composition,
as shown in Figure 14.7. The intuition is as follows. Suppose we wish to keep the
helicopter from rotating. That is, we would like the output angular velocity to be
zero, y(t) = 0. Then we should apply an input of zero, x(t) = 0. However, the
plant is unstable, so even with a zero input, the output diverges (even the smallest
non-zero initial condition or the smallest input disturbance will cause it to diverge).
With the feedback arrangement in Figure 14.7, if the output angular velocity rises
above zero, then the input is modified downwards (the feedback is negative), which
will result in a negative torque being applied to the plant, which will counter the
rising velocity. If the output angular velocity drops below zero, then the torque will
be modified upwards, which again will tend to counter the dropping velocity. The
output velocity will stabilize at zero.
To get the helicopter to rotate, for example to execute a turn, we simply apply a
non-zero input. The feedback system will again compensate so that the helicopter
will rotate at the angular velocity specified by the input.
The signal e is the difference between the input x, which is the desired angular
velocity, and the output y, which is the actual angular velocity. It is called the error
signal. Intuitively, this signal is zero when everything is as desired, when the output
angular velocity matches the input.
Lee & Varaiya, Signals and Systems
627
14.3. FEEDBACK COMPOSITION
error
torque
x
e
w
y
H
1
1= K
H =
2
desired
Ms
!
angular
angular
velocity
velocity
K / M
H = s + K / M
Figure 14.7: A negative feedback proportional controller with gain K.
K = 0
!
!2
2
K =
K = +2
K = !2
K = !
root locus
Figure 14.8: Root locus of the helicopter P controller.
A compensator like that in example 14.13 and Figure 14.7 is called a proportional controller or P controller. The input w to the plant is proportional to the error e. The
objective of the control system is to have the output y of the plant track the input x as
closely as possible. I.e., the error e needs to be small. We can use (14.1) to find the
transfer function of the closed-loop system.
Example 14.14: Continuing with the helicopter of example 14.13, the closed loop
system transfer function is
K ˆ
H(s)
K/M
ˆ
G(s) =
=
.
(14.2)
1 + K ˆ
H(s)
s + K/M
628
Lee & Varaiya, Signals and Systems
14. COMPOSITION AND FEEDBACK CONTROL
which has a pole at s = −K/M. If K > 0, the closed loop system is stable, and
if K < 0, it is unstable. Thus, we have considerable freedom to choose K. How
should we choose its value?
As K increases from 0 to ∞, the pole at at s = −K/M moves left from 0 to −∞. As
K decreases from 0 to −∞, the pole moves to the right from 0 to ∞. The locus of the
pole as K varies is called the root locus, since the pole is a root of the denominator
polynomial.
Figure 14.8 shows the root locus as a thick gray line, on which are marked the
locations of the pole for K = 0, ±2, ±∞. Since there is only one pole, the root locus
comprises only one ‘branch’. In general the root locus has as many branches as
the number of poles, with each branch showing by the movement of one pole as K
varies.
Note that in principle, the same transfer function as the closed-loop transfer function can
be achieved by a cascade composition. But as in example 14.1, the resulting system is
not robust, in that even the smallest change in the pole location of the plant can cause
the system to go unstable (see problem 6). The feedback system, however, is robust, as
shown in the following example
Example 14.15: Continuing with the P controller for the helicopter, suppose that
our model of the plant is not perfect, and its actual transfer function is
1
ˆ
H2(s) =
,
M(s − ε)
for some small value of ε > 0. In that case, the closed loop transfer function is
K/M
ˆ
H(s) =
,
s − ε + K/M
which has a pole at s = ε − K/M. So the feedback system remains stable so long as
ε < K/M.
Lee & Varaiya, Signals and Systems
629
14.3. FEEDBACK COMPOSITION
In practice, when designing feedback controllers, we first quantify our uncertainty about
the plant, and then determine the controller parameters so that under all possible plant
transfer functions, the closed-loop system is stable.
Example 14.16: Continuing the helicopter example, we might say that ε < 0.5. In
that case, if we choose K so that K/M > 0.5, we would guarantee stability for all
values of ε < 0.5. We then say that the proportional feedback controller is robust
for all plants with ε < 0.5.
We still have a large number of choices for K. How do we select one? To understand
the implications of different choices of K we need to study the behavior of the output y
(or the error signal e) for different choices of K. In the following examples we use the
closed-loop transfer function to analyze the response of a proportional controller system
to various inputs. The first example studies the response to a step function input.
Example 14.17:
Continuing the helicopter example, suppose that the input is a
step function, ∀t, x(t) = au(t) where a is a constant and u is the unit step. This
input declares that at time t = 0, we wish for the helicopter to begin rotating with
angular velocity a. The closed-loop transfer function is given by (14.2), and the
Laplace transform of x is ˆ
X (s) = a/s, from table 13.3, so the Laplace transform of
the output is
K/M
ˆ
Y (s) = ˆ
G(s) ˆ
X (s) =
· a
s + K/M s
Carrying out the partial fraction expansion, this becomes
−a
a
ˆ
Y (s) =
+
.
s + K/M
s
We can use this to find the inverse Laplace transform,
∀t, y(t) = −ae−Kt/Mu(t) + au(t).
The second term is the steady-state response yss, which in this case equals the
input. So the first term is the tracking error ytr, which goes to zero faster for
larger K. Hence for step inputs, the larger the gain K, the smaller the tracking error.
630
Lee & Varaiya, Signals and Systems
14. COMPOSITION AND FEEDBACK CONTROL
In the previous example, we find that the error goes to zero when the input is a step
function. Moreover, the error goes to zero faster if the gain K is larger than if it is smaller.
In the following example, we find that if the input is sinusoidal, then larger gain K results
in an ability to track higher frequency inputs.
Example 14.18: Suppose the input to the P controller helicopter system is a sinu-
soid of amplitude A and frequency ω0,
∀ t ∈ R, x(t) = A(cosω0t)u(t).
We know that the response can be decomposed as y = ytr + yss. The transient re-
sponse ytr is due to the pole at s = −K/M, and so it is of the form
∀ t ∈ R, ytr(t) = Re−Kt/M,
for some constant R. The steady-state response is determined by the frequency
response at ω0. The frequency response is
∀
K/M
ω ∈ R,
H(ω) = ˆ
H(iω) =
,
iω + K/M
with magnitude and phase given by
|
K/M
H(ω)| =
, ∠H(ω) = −tan−1 ωM .
[ω2 + (K/M)2]1/2
K
So the steady-state response is
∀t, yss(t) = |H(ω0)|Acos(ω0t + ∠H(ω0)).
Thus the steady-state output is a sinusoid of the same frequency as the input but
with a smaller amplitude (unless ω0 = 0). The larger ω0 is, the smaller the output
amplitude. Hence, the ability of the closed-loop system to track a sinusoidal input
decreases as the frequency of the sinusoidal input increases. However, increasing
K reduces this effect. Thus, larger gain in the feedback loop improves its ability to
track higher frequency sinusoidal inputs.
In addition to the reduction in amplitude, the output has a phase difference. Again,
if ω0 = 0, there is no phase error, because tan−1(0) = 0. As ω0 increases, the phase
lag increases (the phase angle decreases). Once again, however, increasing the gain
K reduces the effect.
Lee & Varaiya, Signals and Systems
631
14.3. FEEDBACK COMPOSITION
The previous two examples suggest that large gain in the feedback loop is always better.
For a step function input, it causes the transient er