varying reference signals.
If realizability and good tracking of slowly varying reference signals are our
only speci cations, then is a good controller. The potential problem with this
K
controller can be seen by examining the whole closed-loop transfer matrix,
2
10
33 + 36
33 + 36
3
;
s
s
s
=
( + 3)2( + 4)
;
( + 3)2( + 4)
( + 3)2( + 4)
6
7
s
s
s
s
s
s
6
2
2
7
H
33 + 36
(33 + 36)(10 + )
(33 + 36)(10 + )
:
4
5
s
s
s
s
s
s
;
( + 3)2( + 4) ;( + 3)2( + 4)(10 ) s
( + 3)2( + 4)(10 )
s
s
s
s
;
s
s
s
;
s
The entries 22 and 23, which are the closed-loop transfer functions from the
H
H
sensor noise and reference input to , are unstable: for example, a reference input
u
with a very small peak can cause the actuator signal to have a very large peak, a
situation that is probably undesirable.
So with the controller , the I/O transfer function is quite benign, even
K
T
desirable, but the closed-loop system will probably have a very large actuator signal.
For a classical design approach, in which the requirement that the actuator signal
not be too large is not explicitly stated (i.e., it is side information), this example
provides a \paradox": the I/O transfer function is acceptable, but the controller is
not a reasonable design.
This example shows the importance of considering all of the closed-loop transfer
functions of interest in a control system, i.e., , and not just the I/O transfer
H
function . In our framework, there is no paradox to explain: the controller can
T
K
be seen to be unacceptable by examining the whole closed-loop transfer matrix ,
H
and not just .
T
This phenomenon of a controller yielding a stable I/O transfer function, but with
other closed-loop transfer functions unstable, is called internal instability. The qual-
i er internal stresses that the problem with the design cannot be seen by examining
the I/O transfer function alone.
Various arguments have been made to explain the \paradox" of this example,
i.e., why our controller is not an acceptable design. They include:
K
1. The unstable plant zero at = 10 is canceled by the controller pole at = 10.
s
s
Such unstable pole-zero cancellations between the plant and controller cannot
be allowed, because a slight perturbation of the plant zero, e.g., to = 9 99,
s
:
will cause the I/O transfer function to become unstable.
2. A state-space description of the closed-loop system will be unstable (it will
have an eigenvalue of 10), so for most initial conditions, the state will grow
larger and larger as time progresses. The unstable mode is unobservable from
p, which is why it does not appear as a pole in the I/O transfer function.
y
These are valid arguments, but in fact, they correspond to di erent \new", previ-
ously unstated, speci cations on our system. In addition to the speci cations of
7.2 INTERNAL STABILITY
153
realizability and stability of the I/O transfer function, (1) is a sensitivity speci ca-
tion, and (2) requires that other signals (the components of the state vector) should
not grow too large when the initial conditions are nonzero. Since these speci ca-
tions are probably necessary in any real control system, they should be explicitly
included in the speci cations.
7.2.2
The Desoer-Chan Definition
Desoer and Chan gave a de nition of internal stability of a closed-loop system
that rules out the problem with our example in section 7.2.1 and other similar
pathologies. The de nition is:
The closed-loop system with plant and controller is internally
De
nition
7.1:
P
K
stable if the four transfer matrices
u = (
yu ) 1
;
yu
(7.11)
H
K
I
;
P
K
P
1
u = (
yu ) 1
(7.12)
;
H
K
I
;
P
K
2
y = (
yu ) 1
;
yu
(7.13)
H
I
;
P
K
P
1
y = (
yu ) 1
;
(7.14)
H
I
;
P
K
2
are stable. In this case we say the controller stabilizes the plant .
K
P
These transfer matrices can be interpreted as follows. Suppose that 1 and 2
are an input-referred process noise and a sensor noise, respectively, as shown in
gure 7.3. Then the four transfer matrices (7.11{7.14) are the closed-loop transfer
matrices from these noises to and . So, roughly speaking, internal stability
u
y
requires that a small process or sensor noise does not result in a very large actuator
or sensor signal.
1
+
+ +
r
yu
2
P
+
r
q
q
y
u
K
Sensor and actuator noises used in the formal denition of
Figure
7.3
internal stability. stabilizes if the transfer matrices from 1 and 2 to
K
P
and are all stable.
u
y
The speci cation of internal stability can be made in our framework as follows.
We must include sensor and actuator-referred process noises in the exogenous input
154