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Two-valued PID controller

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A discrete-time proportional integral derivative (PID) controller, the manipulated variable of which takes two values, i.e., ON and OFF, is proposed and analyzed. Oversampling technology is employed in the controller design. As to the structure of the controller, a cascade type and a built-in type are proposed. An experimental example is presented
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 1, FEBRUARY 1998 183
Two-Valued PID Controller
Akihiko Yoneya, Takehiro Kondo, Yoshihiro Hashimoto,
and Yoshitaka Togari
AbstractA discrete-time proportional integral derivative (PID) con-
troller, the manipulated variable of which takes two values, i.e., ON and
OFF, is proposed and analyzed. Oversampling technology is employed in
the controller design. As to the structure of the controller, a cascade type
and a built-in type are proposed. An experimental example is presented.
Index TermsON–OFF control, oversampling technology, proportional
control.
I. INTRODUCTION
In many control applications, there are restrictions that make it
difficult to construct an ideal control system. The manner in which
these restrictions are dealt with, when considering the controller
implementation, is important, especially when the design of an ideal
control system is the desired result. This letter deals with a control
system restriction by a manipulated variable that can take only two
values, with the application of the temperature control of an electric
furnace as an example.
To maintain temperature control of an electric furnace, the electric
power must be continuously adjusted. There are two methods of
adjusting alternating current; one is the
ONOFF type, in which
the alternating current is fed with the unit of the current cycle by
using zero-cross switching, and the other employs a silicon controlled
rectifier (SCR), the firing timing of which is adjusted according to the
command for power. When this latter method is used, the consuming
current may contain many superharmonic components, which often
causes some trouble. On the other hand, zero-cross switching does not
produce as much superharmonic current. However, when the cycle-
count method, which utilizes zero-cross switching, is used, some
additional time lag may be involved in the controller, thus, possibly
weakening control performance.
Recently, oversampling technology, which uses two-value signals,
has been used, especially in the audio field [2]. For example, a high-
resolution D/A conversion is achieved by feeding the output of a
single-bit D/A converter, which operates at a higher rate than an
analog low-pass filter, to obtain a high-resolution output.
This oversampling technology can also be used in the temperature
control of an electric furnace by replacing the single-bit D/A converter
with a zero-cross switch, which feeds or interrupts electric current
for an alternating current cycle. This letter proposes some methods
of achieving a control system with a two-value manipulated variable
by using oversampling technology. Although electric furnace control
is the application considered in this letter, the proposed approach can
also be used in many other control applications.
II. S
CHEME
The control system considered in this letter has a plant with a
continuous-value output and a discrete-time controller featuring a
two-value manipulated variable. Although the control scheme should
be developed based on the control system’s features, a PID-based
Manuscript received May 6, 1996; revised April 28, 1997.
The authors are with Nagoya Institute of Technology, Nagoya 466, Japan
(e-mail: yoneya@system.nitech.ac.jp).
Publisher Item Identifier S 0278-0046(98)00903-4.
Fig. 1. General structure.
control is employed in this letter, because of the following: 1) the
PID control is well studied [1] and familiar to many engineers; 2) it
is easy to implement in a control system; and 3) the PID controller
has a suitably high performance level for many cases. The design of
the controller is fundamentally based on the continuous-time transfer
function. In spite of the fact that the controller to be designed is
one of discrete time, only the lower frequency component of the
manipulated variable would be considered in the controller design.
This is because the manipulated variable is a two-value signal with
many large high-frequency components, the instantaneous value of
which is not significant in itself.
In this letter, the above-mentioned control system is constructed
using the principle of the
modulator, which plays a main role
in the oversampling D/A converter and converts a continuous-value
signal to a two-value one. The dashed-framed part of Fig. 1 shows a
diagram of a first-order
modulator which has a dynamic element
in its feedback path. Whereas
is specified as unity in almost all
cases for A/D and D/A converters, it is assumed that
is proper,
stable, and of a relative order of zero.
Analyzing this
modulator by the describing function method,
replacing the quantizer with an approximately equivalent proportional
element, the gain of which is
becomes and the
transfer function from
to becomes
(1)
Using the substitution
, this equation can be approximated
as
where if . Hence, if
is the discretization of with a certain sampling period, the
dynamic characteristics from
to are close to that of in
the lower frequency domain. Roughly speaking, the
modulator
converts a continuous-value signal to a discrete one with the dynamics
of
.
The power spectrum of the quantization error
(Fig. 1) is highly
dependent on
and , and experimental study is required to
estimate the effect of the quantization error on the manipulated
variable.
III. S
TRUCTURES OF CONTROLLER
The PID control scheme used here is a derivative advanced type
with incomplete derivative term shown as
(2)
where the derivative gain is
and , , and are
the Laplace transforms of the continuous-time manipulated variable
0278–0046/98$10.00 1998 IEEE
184 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 1, FEBRUARY 1998
Fig. 2. Effective resolution.
, the continuous-time control error , and the continuous-time
plant output
, respectively.
The controller structure with which this letter is concerned is
shown in Fig. 1. It is assumed that all the elements operate with
the same clock, whereas the oversampling A/D and D/A converters
use multirate clocks. This assumption also means that the control
calculation is performed every clock period.
A. Cascade-Type Controller
The controller structure is based on the following elements:
, where denotes time discretization. In this controller,
the continuous-value manipulated variable is first calculated with
a conventional operational element, after which the continuous
manipulated variable is converted to a two-value manipulated
variable with the
modulator, so that the lower frequency
component of the two-value manipulated variable may follow that of
the continuous-value one. We call this a cascade-type controller.
B. Built-In-Type Controller
The built-in-type controller structure consists of the following
elements:
. The modulator
acts as a part of the proportional integral (PI) operation. We call
this a built-in-type controller. A new parameter
( )is
introduced, so that the loop transfer function in the
modulator
may have some lag, otherwise, the
modulator does not work
well. The value of
affects the windup characteristics of the
controller, but does not influence the controller’s dynamics, as long
as no saturations occur. This type of controller is expected to have
less time lag attributable to the
modulator than the cascade
type, because a part of the control operation is performed when
a continuous-value signal is converted to a two-value one in the
built-in-type controller.
C. Performance
The aim of this letter is to design a controller, the manipulated
variable of which takes two values and which works like one with
a continuous-value manipulated variable. Therefore, one of the most
important things is the signal-to-noise (S/N) ratio of the manipulated
variable over a defined bandwidth, namely, an effective resolution.
Fig. 3. Step responses of built-in-type and cycle-count control systems.
Fig. 2 shows the relationship between the bandwidth and the
effective resolution when the sinusoidal signal frequency is 0.1 Hz
and the sampling period is 0.02 s and where the amplitude of
the sinusoidal signal is taken as 80% of the maximum realizable
without saturation. The PID parameters used for this example are
s, and s.
For any type of controller, the effective resolution becomes higher
as the bandwidth becomes lower with the rate of
20 dB/s, since a
first-order
modulator is used. Fig. 2 also shows that effective
resolution depends on the controller type and
. In this example,
both the cascade controller and the built-in-type controller with
have a higher resolution than the built-in-type controller
with
. This is because the latter controller has which is
a low-cut filter with a cutoff frequency of 1.6 Hz, and lower frequency
components in
are not fed back well within the modulator.
IV. E
XPERIMENTAL EXAMPLE
An experimental example is shown to illustrate the effectiveness
of the proposed controller. The plant is an alternating current electric
resistive furnace, and the controlled variable is the temperature in the
furnace. The manipulated variable is the electric power feed, which is
switched on or off for each current cycle. The power supply frequency
is 60 Hz, and the control period is the same.
A built-in-type controller with
is used, and the PID
parameters are set at
C , s, s, and
, so that good control responses may be obtained. Fig. 3
shows a control response when the reference temperature changes
from 400
C to 600 C. It is shown that the plant is well controlled
and the proposed controller is practical.
This figure also shows a control response with the cycle-count
controller, the maximum count of which is 30 and the PID parameters
of which are the same as the proposed controller. In this case, the
controlled variable does not converge to the reference input, because
the control system is unstable due to the time lag which accompanies
the cycle-count method.
R
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[1] K. J.
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Astr
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[2] J. C. Candy and G. C. Temes, Oversampling Delta-Sigma Data Convert-
ers: Theory, Design, and Simulation. New York: IEEE Press, 1991.
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