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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

Abstract—A 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 Terms—ON–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

ON–OFF 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

EFERENCES

[1] K. J.

˚

Astr

¨

om, C. C. Hang, P. Persson, and W. K. Ho, “Toward intelligent

PID control,” Automatica, vol. 28, no. 1, pp. 1–9, Jan. 1992.

[2] J. C. Candy and G. C. Temes, Oversampling Delta-Sigma Data Convert-

ers: Theory, Design, and Simulation. New York: IEEE Press, 1991.