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Ang, K.H. and Chong, G.C.Y. and Li, Y. (2005) PID control system
analysis, design, and technology. IEEE Transactions on Control Systems
Technology 13(4):pp. 559-576.
http://eprints.gla.ac.uk/3817/
Deposited on: 13 November 2007
Glasgow ePrints Service
http://eprints.gla.ac.uk
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005 559
PID Control System Analysis, Design,
and Technology
Kiam Heong Ang, Gregory Chong, Student Member, IEEE, and Yun Li, Member, IEEE
Abstract—Designing and tuning a proportional-integral-deriva-
tive (PID) controller appears to be conceptually intuitive, but can
be hard in practice, if multiple (and often conflicting) objectives
such as short transient and high stability are to be achieved.
Usually, initial designs obtained by all means need to be adjusted
repeatedly through computer simulations until the closed-loop
system performs or compromises as desired. This stimulates
the development of “intelligent” tools that can assist engineers
to achieve the best overall PID control for the entire operating
envelope. This development has further led to the incorporation
of some advanced tuning algorithms into PID hardware modules.
Corresponding to these developments, this paper presents a
modern overview of functionalities and tuning methods in patents,
software packages and commercial hardware modules. It is seen
that many PID variants have been developed in order to improve
transient performance, but standardising and modularising PID
control are desired, although challenging. The inclusion of system
identification and “intelligent” techniques in software based PID
systems helps automate the entire design and tuning process to
a useful degree. This should also assist future development of
“plug-and-play” PID controllers that are widely applicable and
can be set up easily and operate optimally for enhanced produc-
tivity, improved quality and reduced maintenance requirements.
Index Terms—Patents, proportional-integral-derivative (PID)
control, PID hardware, PID software, PID tuning.
I. INTRODUCTION
W
ITH its three-term functionality covering treatment
to both transient and steady-state responses, propor-
tional-integral-derivative (PID) control offers the simplest and
yet most efficient solution to many real-world control problems.
Since the invention of PID control in 1910 (largely owning to
Elmer Sperry’s ship autopilot), and the Ziegler–Nichols’ (Z-N)
straightforward tuning methods in 1942 [34], the popularity
of PID control has grown tremendously. With advances in
digital technology, the science of automatic control now offers
a wide spectrum of choices for control schemes. However,
more than 90% of industrial controllers are still implemented
based around PID algorithms, particularly at lowest levels [5],
as no other controllers match the simplicity, clear functionality,
applicability, and ease of use offered by the PID controller
[32]. Its wide application has stimulated and sustained the
Manuscript received September 8, 2003; revised August 15, 2004. Manu-
script received in final form January 4, 2005. Recommended by Associate Ed-
itor D. W. Repperger. This work was supported in part by Universities U.K. and
in part by University of Glasgow Scholarships.
K. H. Ang is with Yokogawa Engineering Asia Pte Ltd., Singapore 469270,
Singapore (e-mail: KiamHeong.Ang@sg.yokogawa.com).
G. Chong and Y. Li are with the Intelligent Systems Group, Department of
Electronics and Electrical Engineering, University of Glasgow, Glasgow G12
8LT, U.K. (e-mail: g.chong@elec.gla.ac.uk; y.li@elec.gla.ac.uk).
Digital Object Identifier 10.1109/TCST.2005.847331
development of various PID tuning techniques, sophisticated
software packages, and hardware modules.
The success and longevity of PID controllers were character-
ized in a recent IFAC workshop, where over 90 papers dedicated
to PID research were presented [28]. With much of academic re-
search in this area maturing and entering the region of “dimin-
ishing returns,” the trend in present research and development
(R&D) of PID technology appears to be focused on the integra-
tion of available methods in the form of software so as to get the
best out of PID control [21]. A number of software-based tech-
niques have also been realized in hardware modules to perform
“on-demand tuning,” while the search still goes on to find the
next key technology for PID tuning [24].
This paper endeavours to provide an overview on modern
PID technology including PID software packages, commercial
PID hardware modules and patented PID tuning rules. To begin,
Section II highlights PID fundamentals and crucial issues. Sec-
tion III moves to focus on patented PID tuning rules. A survey
on available PID software packages is provided in Section IV.
In Section V, PID hardware and tuning methods used by process
control vendors are discussed. Finally, conclusions are drawn in
Section VI, where some differences between academic research
and industrial practice are highlighted.
II. T
HREE-TERM FUNCTIONALITY,DESIGN AND
TUNING
A. Three-Term Functionality and the Parallel Structure
A PID controller may be considered as an extreme form of
a phase lead-lag compensator with one pole at the origin and
the other at infinity. Similarly, its cousins, the PI and the PD
controllers, can also be regarded as extreme forms of phase-lag
and phase-lead compensators, respectively. A standard PID
controller is also known as the “three-term” controller, whose
transfer function is generally written in the “parallel form”
given by (1) or the “ideal form” given by (2)
(1)
(2)
where
is the proportional gain, the integral gain,
the derivative gain, the integral time constant and, the
derivative time constant. The “three-term” functionalities are
highlighted by the following.
• The proportional term—providing an overall control ac-
tion proportional to the error signal through the all-pass
gain factor.
1063-6536/$20.00 © 2005 IEEE
560 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005
• The integral term—reducing steady-state errors through
low-frequency compensation by an integrator.
• The derivative term—improving transient response
through high-frequency compensation by a differentiator.
The individual effects of these three terms on the closed-loop
performance are summarized in Table I. Note that this table
serves as a first guide for stable open-loop plants only. For op-
timum performance,
, (or ) and (or ) are mu-
tually dependent in tuning.
The message that increasing the derivative gain,
,
will lead to improved stability is commonly conveyed from
academia to industry. However, practitioners have often found
that the derivative term can behave against such anticipation
particularly when there exists a transport delay [23], [28].
Frustration in tuning
has hence made many practitioners
switch off or even exclude the derivative term. This matter has
now reached the point that requires clarification, which will be
discussed in Section II-E.
B. Series Structure
A PID controller may also be realized in the “series form”
if both zeros are real, i.e., if
. In this case, (2) can
be implemented as a cascade of a PD and a PI controller in the
form [23]
(3)
where
(4)
C. Effect of the Integral Term on Stability
Refer to (2) or (3) for
and 0. It can be seen that,
adding an integral term to a pure proportional term will increase
the gain by a factor of
(5)
and will increase the phase-lag at the same time since
(6)
Hence, both stability gain margin (GM) and phase margin (PM)
will be reduced, i.e., the closed-loop system will become more
oscillatory or potentially unstable.
D. Integrator Windup and Remedies
If an actuator that realizes the control action has an effective
range limit, then the integrator may saturate and future correc-
tion will be ignored until the saturation is offset. This causes
low-frequency oscillations and may lead to instability. A usual
measure taken to counteract this effect is “anti-windup” [4], [8],
[29]. This is realized by inner negative feedback of some ex-
cess amount of the integral action to the integrator such that
TABLE I
E
FFECTS OF
INDEPENDENT P, I ,
AND DT
UNING
saturation will be taken out. Nearly all software packages and
hardware modules have implemented some form of integrator
anti-windup protection.
As most modern PID controllers are implemented in digital
processors, they can accommodate more mathematical func-
tions and modifications to the standard three terms shown in (1)
to (3). A simple and most widely adopted anti-windup scheme
can be realized in software or firmware by modifying the inte-
gral action to
(7)
where
represents the saturated control action and is a
correcting factor. It is found that the range of [0.1,1.0] for
results in extremely good performance if PID coefficients are
tuned reasonably [23].
It is also reported that, in the “series form,” the PI part may be
implemented to counter actuator saturation without the need for
a separate anti-windup action, as shown in Fig. 1 [4], [29]. When
there is no saturation, the feedforward-path transfer is unity and
the overall transfer from
to is the same as the last
factor in (3).
E. Effect of the Derivative Term on Stability
Generally, derivative action is valuable as it provides useful
phase lead to offset phase lag caused by integration. It is also
particularly helpful in shortening the period of the loop and
thereby hastening its recovery from disturbances. It can have
a more dramatic effect on the behavior of second-order plants
that have no significant dead-time than first-order plants [29].
However, the derivative term is often misunderstood and mis-
used. For example, it has been widely perceived in the control
community that adding a derivative term will improve stability.
It will be shown here that this perception is not always valid.
In general, adding a derivative term to a pure proportional term
will reduce phase lags by
(8)
which alone tends to increase the PM. In the meantime, however,
the gain will be increased by a factor of
(9)
and, hence, the overall stability may be improved or degraded.
ANG et al.: PID CONTROL SYSTEM TECHNOLOGY 561
Fig. 1. Anti-windup PI part of a “series form.”
To prove that adding a differentiator could actually destabilise
the closed-loop system, consider without loss of generality a
common first-order lag plus delay plant as described by
(10)
where
is the process gain; is the process time-constant;
and
is the process dead-time or transport delay. Suppose that
it is controlled by a proportional controller with gain
and
now a derivative term is added. This results in a combined PD
controller as given by
(11)
The overall open-loop feedforward-path transfer function be-
comes
(12)
with gain becoming
(13)
where the inequality has been obtained because
is monotonic with . This
implies that the gain is not less than 0 dB if
and
or and
(14)
In these cases, the 0 dB gain crossover frequency
is at infinite,
where the phase
(15)
Hence, by Bode or Nyquist criterion, there exist no stability mar-
gins and the closed-loop system will be unstable.
This phenomenon could have contributed to the difficulties
in the design of a full PID controller and also to the reason that
80% of PID controllers in use have the derivative part omitted
or switched off [21]. This means that the functionality and po-
tential of a PID controller is not fully exploited. Nonetheless,
it is shown that the use of a derivative term can increase sta-
bility robustness and can help maximize integral gain so as to
Fig. 2. Increasing derivative gain could decrease stability margins and
destabilise the closed-loop system.
Fig. 3. Time-domain effect of an increasing gain on the closed-loop
performance.
achieve the best performance [7]. However, care must be taken,
as it is difficult to tune the differentiator properly. An example is
given in Figs. 2 and 3 for plant (10) with
10, 1 s and
0.1 s, which is initially controlled by a PI controller with
0.644 and 1.03 s It can be seen that if a differen-
tiator is added with
0.0303 s, both the GM and the PM
will be maximized while the transient response improves to the
best. However, if
is increased further to 0.1 s, the GM and
transient response will deteriorate. The closed-loop system can
even be destabilised if the derivative gain is increased to 20%
of the proportional gain. Hence, the derivative term should be
tuned and used properly.
F. Remedies on Singular Derivative Action
A pure differentiator is not “casual.” It does not restrict
high-frequency gains, as shown in (9) and demonstrated in
Fig. 2. Hence, it will results in a theoretically infinite high
control signal when a step change of the reference or distur-
bance occurs. To combat this, most PID software packages
and hardware modules perform some forms of filtering on the
differentiator.
562 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005
1) Averaging Through a Linear Low-Pass Filter: A
common remedy is to cascade the differentiator with a low-pass
filter, i.e., to modify it to
(16)
Most industrial PID hardware provides a
setting from 1 to
33 and the majority falls between 8 and 16 [72]. A second-order
Butterworth filter is recommended in [17] for further attenuation
of the high-frequency gains.
2) Modified Structure: The issue of improving transient per-
formance has recently become such a crucial one that atten-
tion of the fundamental unity negative feedback structure has
been proposed in the R&D of PID control [4]. In cascade con-
trol applications, the inner-loop often needs to be less sensitive
to set-point changes than the outer-loop. For the inner-loop, a
variant to the standard PID structure may be adopted, which
uses the process variable (PV) instead of the error signal, for
the derivative term [40], i.e.
(17)
where
is the PV, and is the reference
signal or set-point. It is also proposed that, in order to further
reduce sensitivity to set-point changes, the proportional term
may also be changed to act upon the PV, instead of the error
signal, i.e., [40]
(18)
Structure (17) is sometimes referred to as “Type B” (or PI-D)
control and structure (18) as “Type C” (or I-PD) control, while
structures (1) to (3) as “Type A” PID control. Note that, Types B
and C alter the foundations of conventional feedback control and
can make the PID schemes more difficult to analyze with stan-
dard techniques on stability and robustness, etc. For set-point
tracking applications, however, one alternative to using Type B
or C is perhaps a set-point filter that has a critically-damped
dynamics so as to achieve soft-start and smooth control [13].
Nevertheless, the ideal, parallel, series and modified forms of
PID structures can all be found in present software packages
and hardware modules. Readers may refer to Techmation’s Ap-
plications Manual [72] for a list documenting the structures em-
ployed in some of the industrial PID controllers.
3) Removal of Singular Action Through a Nonlinear Median
Filter: Another method is to use a median filter, which is
nonlinear and widely applied in image processing. It compares
several neighboring data points around the current one and
selects their median for a “nonsingular” action. This way,
unusual or unwanted spikes resulting from a step command
or disturbance, for example, will be filtered out completely.
Pseudocode of a three-point median filter is illustrated in Fig. 4
[23]. The main benefit of this method is that no extra parameter
is needed, though it is not very suitable for use in under-damped
processes.
Fig. 4. Three-point median filter to eliminate singular derivative action.
G. Tuning Objectives and Existing Methods
Preselection of a controller structure can pose a challenge in
applying PID control. As vendors often recommend their own
designs of controller structures, their tuning rules for a specific
controller structure does not necessarily perform well with other
structures. One solution seen is to provide support for individual
structures in software. Readers may refer to [16] and [22] for de-
tailed discussions on the use of various PID structures. Nonethe-
less, controller parameters are tuned such that the closed-loop
control system would be stable and would meet given objec-
tives associated with the following:
• stability robustness;
• set-point following and tracking performance at transient,
including rise-time, overshoot, and settling time;
• regulation performance at steady-state, including load dis-
turbance rejection;
• robustness against plant modeling uncertainty;
• noise attenuation and robustness against environmental
uncertainty.
With given objectives, tuning methods for PID controllers can
be grouped according to their nature and usage, as follow [4],
[13], [23].
• Analytical methods—PID parameters are calculated from
analytical or algebraic relations between a plant model
and an objective (such as internal model control (IMC) or
lambda tuning). These can lead to an easy-to-use formula
and can be suitable for use with online tuning, but the
objective needs to be in an analytical form and the model
must be accurate.
• Heuristic methods—These are evolved from practical ex-
perience in manual tuning (such as the Z-N tuning rule)
and from artificial intelligence (including expert systems,
fuzzy logic and neural networks). Again, these can serve
in the form of a formula or a rule base for online use, often
with tradeoff design objectives.
• Frequency response methods—Frequency characteristics
of the controlled process are used to tune the PID con-
troller (such as loop-shaping). These are often offline and
academic methods, where the main concern of design is
stability robustness.