Data processing: generic control systems or specific application – Generic control system – apparatus or process – Optimization or adaptive control
Reexamination Certificate
1998-10-22
2001-11-13
Grant, William (Department: 2121)
Data processing: generic control systems or specific application
Generic control system, apparatus or process
Optimization or adaptive control
C700S037000, C700S045000, C706S900000
Reexamination Certificate
active
06317637
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates to the field of automatic control, and in particular a system and method for enhancing the performance of Proportional-Integral-Derivative (PID) controllers.
DESCRIPTION OF THE RELATED ART
The field of automatic control has proliferated in recent years. In response to advances in processor and memory technology, control algorithms are increasingly being implemented in software with the attendant benefits of increased flexibility and performance. Although the science of automatic control offers a wide spectrum of choices for control algorithms, the majority of industrial controllers in existence today are implemented according to the Proportional-Integral-Derivative (PID) algorithm. PID controllers became commercially available as early as the 1930s, and have been realized in various technologies including pneumatics, mechanics, and electronics [Process Dynamics and Control, Dale E. Seborg et al., ISBN 0-471-86389-0, 1989, page184]. The principles and operation of PID controllers are well-understood in the engineering community. Thus, manufacturers of control products have a strong incentive for providing products based on the PID algorithm.
FIG. 1
illustrates the structure of a PID-based control system which is well known in the prior art. The control system includes a PID controller and a process which is to be controlled. A process variable PV associated with the process is measured and compared to a set point value SP. An error signal e is computed as the difference of the set point and the process value, i.e.
e
(
t
)=
SP
(
t
)−
PV
(
t
) (I)
The error signal e is supplied as the input to the PID controller. The PID controller generates a control signal u
c
according to the relation
u
c
(
t
)
=
K
p
[
e
(
t
)
+
1
T
I
∫
0
t
e
(
τ
)
ⅆ
τ
+
T
D
e
.
(
t
)
]
,
(
2
)
which involves terms proportional to the error, the integral of the error, and the derivative of the error. The gain parameters K
p
, T
I
, and T
D
determine the behavior of the PID controller.
The control system of
FIG. 1
is configured to operate in an automatic control mode and a manual control mode. A switch is included to represent the choice of mode, although in practice the function of the switch may be implemented in software. In the automatic mode, the switch is disposed to couple the PID controller output u
c
to the process input u. In the manual mode, the switch is disposed to couple the output u
m
of the manual control interface (MCI) to the process input. The manual control interface receives a manual input m directly from the system user and converts the manual input into a manual control signal u
m
. The manual control interface allows the system user to directly control the dynamics of the process.
Modern PID controllers are typically implemented in software or digital hardware. Thus, the controller output u
c
is quite often computed according to the discrete-time relations
u
c
(
k
)
=
K
p
[
e
k
+
Δ
t
T
I
∑
j
=
0
k
(
e
j
+
e
j
-
1
2
)
-
T
D
e
k
-
e
k
-
1
Δ
t
]
,
(
4
)
where k is a discrete time index, &Dgr;t is the sample interval, SP
k
is current value of the set point, and PV
k
is the current value of the process variable. Equation (4) is sometimes referred to as the positional PID equation since it specifies the position of the variable u
c
. A PID algorithm based on equation (4) is thus referred to as a positional algorithm. A positional PID algorithm generally includes a storage location for the integrated error
IE
k
=
∑
j
=
0
k
(
e
j
+
e
j
-
1
2
)
Δ
t
.
(
5
)
In terms of the integrated error, the controller output may be expressed as
u
c
(
k
)
=
K
p
[
e
k
+
1
T
I
IE
k
-
T
D
e
k
-
e
k
-
1
Δ
t
]
.
(
6
)
Prior to the computation of the controller output u
c
(k), the integrated error is updated with the current error data as follows:
IE
k
=
IE
k
-
1
+
e
k
+
e
k
-
1
2
Δ
t
.
(
7
)
The set point SP, and the control parameters K
p
, T
I
and T
D
are controlled by the user. For example, a software based PID controller may provide a graphical user interface to facilitate user control of the set point, the control parameters, the manual/automatic mode switch, and manual input m.
One fundamental disadvantage associated with prior art positional PID algorithms is their tendency to produce large control discontinuities in response to parameter changes. For example, a change in the proportional gain KP will generally produce an abrupt change in the control output u
c
as may be observed by the magnitude of the partial derivative of the control output with respect to the proportional gain K
p
:
∂
[
u
c
(
k
)
]
∂
K
p
&RightBracketingBar;
pos
=
e
k
+
Δ
t
T
I
∑
j
=
0
k
(
e
j
+
e
j
-
1
2
)
-
T
D
e
k
-
e
k
-
1
Δ
t
.
(
8
)
These discontinuities in controller output due to parameter changes especially affect the behavior of gain scheduling PID controllers. Gain scheduling is characterized by using different sets of gains (K
p
, T
I
, T
D
) for different regions of operation of the controller. Often the gains are “scheduled” according the value of a variable such as the set point or process variable. For example, a gain schedule is illustrated by the following set of rules which control the gains to be used based on the value of the set point SP:
If 0
≦SP
≦30, use
K
P
=1
,T
I
=0.1,
T
D
=0.04;
If 30
<SP≦
70, use
K
P
=2,
T
I
=0.06,
T
D
=0.04;
If 70<SP≦100, use
K
P
=4,
T
I
=0.02
,T
D
=0.04.
Thus, the positional controller output exhibits a sudden discontinuity whenever the set point (i.e. scheduling parameter) migrates from one region to another.
PID algorithms which are less sensitive to control parameter changes may be obtained from an incremental form for the controller output as follows. Controller equation (4) holds for all values of the discrete index k. In particular, equation (4) implies that the previous controller output
u
c
(
k
-
1
)
=
K
p
[
e
k
-
1
+
Δ
t
T
I
∑
j
=
0
k
(
e
j
+
e
j
-
1
2
)
-
T
D
e
k
-
1
-
e
k
-
2
Δ
t
]
.
(
9
)
Taking the difference of equation (4) and equation (9), one obtains the so called incremental PID equation:
Δ
u
c
(
k
)
=
K
p
[
(
e
k
-
e
k
-
1
)
+
Δ
t
T
I
(
e
k
+
e
k
-
1
2
)
-
T
D
e
k
-
2
e
k
-
1
+
e
k
-
2
Δ
t
]
,
(
10
)
where the control increment &Dgr;u
c
(k) is the difference between the current controller output and the previous controller output, i.e. &Dgr;u
c
(k)=u
c
(k)−u
c
(k−1). Observe that the control increment &Dgr;u
c
(k) is computed in terms of the current error e
k
and the two previous error values e
k−1
and e
k−2
. Using the incremental PID equation (10), the current control output u
c
(k) may be written in a recursive form as follows:
u
c
(
k
)
=
u
c
(
k
-
1
)
+
K
p
[
(
e
k
-
e
k
-
1
)
+
Δ
t
T
I
(
e
k
+
e
k
-
1
2
)
-
T
D
e
k
-
2
e
k
-
1
+
e
k
-
2
Δ
t
]
.
(
11
)
A PID algorithm based on equation (10), or equivalently equation (11), is referred to as an incremental PID algorithm.
One disadvantage of the incremental algorithm is that it gives an incorrect response to changes in the controller parameters. For a Proportional or Proportional-Derivative controller (i.e. no integral action), the incremental algorithm will generate a control output which is obviously inconsistent with the positional algorithm. For example, consider a Proportional controller with output given by u=K
p
e. When the gain K
p
is changed, the subsequent incremental control
Conley Rose & Tayon PC
Grant William
Hartman Jr. Ronald D
Hood Jeffrey C.
National Instruments Corporation
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