Part A | 10.3
258 Part A Fundamentals
H.z/ D
1 z
1
Z
I
1
Ä
ka
s 2 .s C a/
(10.135)
is the inverse Laplace transform operator, and Z is the
z-transform operator.
ZfI
1
Œka=.s
2
.s C a//g can be computed by first
evaluating the inverse Laplace transform via partial
fraction techniques and then performing z-transform on
the time functions via standard z-transform tables
I
1
a
s 2 .s C a/
D I
1
A
s
C
B
s 2 C
C
s C a
D I
1
1
s 2
1=a
s
C
1=a
s C a
D t
1
a
C
1
a
e
at
) H.z/ D k
z 1
z
Tz
.z 1/
2
z=a
z 1
C
z=a
z e aT
D k
T
z e
aT
.z 1/
.1e
aT
/
a
.z 1/ .z e aT /
:
(10.136)
Consider ˛ D 27, T D 0:1, and we have
H.z/ D
k.0:0655z C 0:027 83/
.z 1/.z 0:0672/
(10.137)
Similar to the continuous-time analog, the closed-loop
discrete-time transfer function, H cl .z/, can be expressed
as
H cl .z/ D
H.z/
1 C H.z/
) H cl .z/
D .k .0:0655z C 0:027 83//
z
2
C .0:0655k 1:0672/ z
C0:027 83k C 0:0672
1 :
(10.138)
Unlike the continuous-time stability consideration
where its closed-loop poles must lie on the left-hand
side of the complex s-plane, the closed-loop poles of
a closed-loop discrete-time system must lie within the
unit circle of the complex z-plane. At k D 20 the closedloop discrete poles are located at 0:12 ˙ 0:78i, and are
within the unit circle. At k D 200 the closed-loop discrete poles are located at 0:58, 4:9, and are outside
the unit circle. Thus, given a discrete-time controller,
increasing its gain causes instability whereas the system stability is independent of the controller gain if
a continuous-time controller is used. Similarly, when
the sampling interval is increased to 0.5, the closed-loop
poles become 8:17 and 0:09, and thus the closedloop system becomes unstable.
Digital PID Controller
To implement a PID controller on a computer, its
derivative and integral terms must be approximated
with difference equations as signals only exist at a regular sampling time interval.
Proportional Control. The proportional term in
a continuous-time system is u.t/ D k P .y d .t/ y.t//. On
a digital computer, this term is modified as
u.k/ D k P .y d .k/ y.k// ;
(10.139)
where k denotes the sampled time instants.
Integral Control. The integral term in a continuoustime system is given as
u.t/ D
k P
T I
t
Z
0
e../d :
(10.140)
The integral term is commonly approximated using the
first-order backward difference method because of its
numerical stability property
P
I.t/ D
k P
T I
e.t/ )
I.k/ I.k 1/
T
D
k P
T I
e.k/
) I.k/ D I.k 1/ C
Tk P
T I
e.k/ :
(10.141)
Here k D 0; 1; 2; : : :, and I.1/ D 0.
Derivative Control. The derivative (differential) term
in a continuous-time system is given as
u.t/ D k P T D P
e.t/ :
(10.142)
With consideration that P
y d .t/ D 0, the derivative term is
approximated using the backward difference method as
follows
u.t/ D k P T D P
e.t/ D Dk P T D P
y.t/
) u.k/ D Dk P T D
 y.k/ y.k 1/
T
Ã
: (10.143)
Example 10.6 AUV Forward Speed Digital PID Control
Consider again controlling the forward speed of the
AUV with a digital PID controller. Figure 10.36 shows
the block diagram of the closed-loop discrete-time system generated using MATLAB/Simulink.
In the block, a continuous-time PID block is used,
and its controller parameters are K p D 17:26, T i D 2:6
258 Part A Fundamentals
H.z/ D
1 z
1
Z
I
1
Ä
ka
s 2 .s C a/
(10.135)
is the inverse Laplace transform operator, and Z is the
z-transform operator.
ZfI
1
Œka=.s
2
.s C a//g can be computed by first
evaluating the inverse Laplace transform via partial
fraction techniques and then performing z-transform on
the time functions via standard z-transform tables
I
1
a
s 2 .s C a/
D I
1
A
s
C
B
s 2 C
C
s C a
D I
1
1
s 2
1=a
s
C
1=a
s C a
D t
1
a
C
1
a
e
at
) H.z/ D k
z 1
z
Tz
.z 1/
2
z=a
z 1
C
z=a
z e aT
D k
T
z e
aT
.z 1/
.1e
aT
/
a
.z 1/ .z e aT /
:
(10.136)
Consider ˛ D 27, T D 0:1, and we have
H.z/ D
k.0:0655z C 0:027 83/
.z 1/.z 0:0672/
(10.137)
Similar to the continuous-time analog, the closed-loop
discrete-time transfer function, H cl .z/, can be expressed
as
H cl .z/ D
H.z/
1 C H.z/
) H cl .z/
D .k .0:0655z C 0:027 83//
z
2
C .0:0655k 1:0672/ z
C0:027 83k C 0:0672
1 :
(10.138)
Unlike the continuous-time stability consideration
where its closed-loop poles must lie on the left-hand
side of the complex s-plane, the closed-loop poles of
a closed-loop discrete-time system must lie within the
unit circle of the complex z-plane. At k D 20 the closedloop discrete poles are located at 0:12 ˙ 0:78i, and are
within the unit circle. At k D 200 the closed-loop discrete poles are located at 0:58, 4:9, and are outside
the unit circle. Thus, given a discrete-time controller,
increasing its gain causes instability whereas the system stability is independent of the controller gain if
a continuous-time controller is used. Similarly, when
the sampling interval is increased to 0.5, the closed-loop
poles become 8:17 and 0:09, and thus the closedloop system becomes unstable.
Digital PID Controller
To implement a PID controller on a computer, its
derivative and integral terms must be approximated
with difference equations as signals only exist at a regular sampling time interval.
Proportional Control. The proportional term in
a continuous-time system is u.t/ D k P .y d .t/ y.t//. On
a digital computer, this term is modified as
u.k/ D k P .y d .k/ y.k// ;
(10.139)
where k denotes the sampled time instants.
Integral Control. The integral term in a continuoustime system is given as
u.t/ D
k P
T I
t
Z
0
e../d :
(10.140)
The integral term is commonly approximated using the
first-order backward difference method because of its
numerical stability property
P
I.t/ D
k P
T I
e.t/ )
I.k/ I.k 1/
T
D
k P
T I
e.k/
) I.k/ D I.k 1/ C
Tk P
T I
e.k/ :
(10.141)
Here k D 0; 1; 2; : : :, and I.1/ D 0.
Derivative Control. The derivative (differential) term
in a continuous-time system is given as
u.t/ D k P T D P
e.t/ :
(10.142)
With consideration that P
y d .t/ D 0, the derivative term is
approximated using the backward difference method as
follows
u.t/ D k P T D P
e.t/ D Dk P T D P
y.t/
) u.k/ D Dk P T D
 y.k/ y.k 1/
T
Ã
: (10.143)
Example 10.6 AUV Forward Speed Digital PID Control
Consider again controlling the forward speed of the
AUV with a digital PID controller. Figure 10.36 shows
the block diagram of the closed-loop discrete-time system generated using MATLAB/Simulink.
In the block, a continuous-time PID block is used,
and its controller parameters are K p D 17:26, T i D 2:6