A simple nonlinear PD controller for integrating processes

Research Article
A simple nonlinear PD controller for integrating processes
Chanchal Dey a
, Rajani K. Mudi b,n
, Dharmana Simhachalam b
a
Department of of Applied Physics, University of Calcutta, 92, A.P.C. Road, Calcutta 700009, West Bengal, India
b
Department of Instrumentation & Electronics Engineering, Jadavpur University, Sector III, Block LB/8, Salt-lake, Calcutta 700098, West Bengal, India
a r t i c l e i n f o
Article history:
Received 2 September 2012
Received in revised form
3 September 2013
Accepted 4 September 2013
Available online 2 October 2013
This paper was recommended for
publication by Prof. A.B. Rad
Keywords:
PID control
Nonlinear PD control
Integrating process
a b s t r a c t
Many industrial processes are found to be integrating in nature, for which widely used Ziegler–Nichols
tuned PID controllers usually fail to provide satisfactory performance due to excessive overshoot with
large settling time. Although, IMC (Internal Model Control) based PID controllers are capable to reduce
the overshoot, but little improvement is found in the load disturbance response. Here, we propose an
auto-tuning proportional-derivative controller (APD) where a nonlinear gain updating factor α con-
tinuously adjusts the proportional and derivative gains to achieve an overall improved performance
during set point change as well as load disturbance. The value of α is obtained by a simple relation based
on the instantaneous values of normalized error (eN) and change of error (ΔeN) of the controlled variable.
Performance of the proposed nonlinear PD controller (APD) is tested and compared with other PD and
PID tuning rules for pure integrating plus delay (IPD) and first-order integrating plus delay (FOIPD)
processes. Effectiveness of the proposed scheme is verified on a laboratory scale servo position control
system.
& 2013 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Proportional-integral-derivative (PID) controllers are widely
used in various industrial process control applications due to their
simplicity and effectiveness [1–6]. But, performances of model free
PID controllers are not usually satisfactory due to their oscillatory
responses and large settling time for integrating processes with
time delay [7–9]. For example, Fig. 1 shows such poor perfor-
mances of the well known Ziegler–Nichols tuned PID (ZNPID) [10]
controllers for IPD and FOIPD processes. In Fig. 1, initially a step set
point change is applied and when the process reaches its steady
state, an impulse load disturbance is introduced at the process
input. Here, overshoots are found to be more than 60%, which is
not acceptable in most of the applications [6]. On the other than
hand, model based PID control techniques, like IMC can provide
lower overshoot with faster settling on proper selection of tuning
parameters (close-loop time constant) [11–15]. Fig. 2 shows the
responses of model based IMC-PID controllers [11] for IPD and
FOIPD processes. It is found that lower overshoot with faster
settling is achieved in the set point response for IMC-PID com-
pared to ZNPID but no significant improvement is observed during
load rejection.
Due to the presence of integral action, PID controllers are likely
to produce oscillations for integrating plus dead time processes [1].
In general, proportional-derivative (PD) controllers, if properly
designed [6], are capable of providing reasonable performances
compared to PID controllers for integrating or zero-load processes
with delay [16]. Robots and manipulators are extensively used in
automation based manufacturing processes where any type of
overshoot and/or undershoot is highly undesirable [17] in position-
ing their arms. In spite of noise sensitivity, PD controllers help to
reduce the overshoot by introducing higher damping [18]. So, there
is a scope for designing improved PD controllers to achieve desired
performance for integrating processes with dead time. But, till
today, unlike PID controllers, probably there are less running
schemes for PD controllers [19].
For IPD processes, Chidambaram and Padma Sree [16] used
equating coefficient method to find the parameters of a PD
controller, which will be denoted here as CPPD. Kristiansson and
Lennartson [20] reported that the derivative action can signifi-
cantly improve the control performance compared to PI control
with equal stability margin for most of the plants including those
with noticeably large time delay. Authors in [21] proposed a
Lyapunov based approach to obtain PD parameters. Xu et al. [22]
designed a nonlinear PD controller with increased damping
corresponding to its linear counterpart. Its proportional and
derivative gains are modulated nonlinearly based on the instanta-
neous value of error (e) and the sign of change of error (Δe), and its
various tunable parameters are chosen heuristically maintaining
the stability of the system. Visioli [23] used genetic algorithm
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ISA Transactions
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2013.09.011
n
Corresponding author. Tel.: þ91 33 23352587; fax: þ91 33 23357254.
E-mail addresses: chanchaldey@yahoo.co.in, cdaphy@caluniv.ac.in (C. Dey),
rkmudi@yahoo.com, rkmudi@iee.jusl.ac.in (R.K. Mudi), chalamju10@gmail.com
(D. Simhachalam).
ISA Transactions 53 (2014) 162–172

based optimization scheme to minimize various integral errors
like ISE and ITSE and finally got PD settings (here termed as VPD)
for set point tracking. For FOIPD processes, Vítečková et al. [24]
and O'Dwyer [25] suggested PD controllers towards achieving
improved responses. There is a possibility to enhance the perfor-
mance of a PD controller by extending its integer order of the
derivative element to fractional order [26].
Fuzzy controllers have been successfully designed with
improved performances compared to their conventional counter-
parts [27–33]. In [28], the nonlinear gain modification scheme
following an operator's strategy continuously adjusts the output
scaling factor (considered to be the close-loop gain) of a fuzzy PD
controller with the help of 49 fuzzy If-Then rules, defined on the
current process states, i.e., e and Δe. Su et al. [29] developed a
hybrid fuzzy PD controller by combining two nonlinear tracking
differentiators to a conventional fuzzy PD controller. In spite of a
number of merits, there are many limitations while designing a
fuzzy controller, since, till now there is no standard methodology
for its various design steps. Moreover, no clear guidelines are
available for selecting appropriate values of its large number of
design parameters.
The above discussion and our literature survey reveal that
compared to the well established PI and PID control techniques,
less importance/attention has been given for the development of
conventional PD control. Hence, there is a good prospect for
further development of PD controllers with enhanced perfor-
mance. With this perspective in mind, and encouraging results
of [22,28], in this study, we are motivated to introduce a real time
nonlinear gain modification scheme for a PD controller. Due to lack
of a suitable auto-tuning scheme, here, we consider the most
widely accepted Ziegler–Nichols ultimate cycle based PID (i.e.,
ZNPID) tuning rules [10] by ignoring its integral part for the initial
setting of the proposed PD controller (APD). Note that, Ziegler–
Nichols rules were originally developed for the tuning of P, PI, and
PID controllers, but not for a PD controller. In the proposed APD,
the proportional and derivative gains are continuously adjusted
depending on the instantaneous process trend by introducing
a nonlinear gain updating parameter α. The basic idea behind this
real time gain adjustment mechanism is that when the process is
moving towards the set point, control action will be conservative
to avoid possible large overshoots and/or undershoots in the
subsequent operating phases, and when the process is moving
away from the set point, control action will be aggressive to bring
it back quickly to its desired value. It is to be mentioned that, our
proposed scheme is different from others [22,28] as far as its
design simplicity and practical implementation are concerned.
Nonlinear gain variation in [22] involves a number of heuristically
chosen tunable parameters along with an exponential function in
its gain update rules, whereas, in [28] the output scaling factor is
adjusted through a number of fuzzy conditional rules derived from
experts’ knowledge.
The performance of the proposed PD controller is tested and
compared with a large number of PID and PD tuning rules
reported over the last decade for IPD and FOIPD processes. In
addition, real time experimentation is also performed on a
laboratory scale DC servo position control system. Performance
analysis with respect to a number of performance indices reveals
that APD is capable of providing an overall improved performance
in comparison with PID settings given by AHPID [2], DMPID [3],
ZNPID [10], SLPPID [11], CPPID [16], VPID [23], PCPID [34], ACPID
[35], RRCPID [36], AMPID [37], HXCPID [38], RPID [50], and PD
settings by CPPD [16], VPD [23], and LLPPD [39] for the IPD process
under both set point change and load disturbance. Similarly,
enhanced performance is also observed for the FOIPD process in
comparison with PID tuning rules of DMPID [3], ZNPID [10], SPID
[12], AMPID [37], HXCPID [38], YPID [40], WCPID [41], RRCPID [42],
SLPID [44], KLPID [47], RPID [50] and PD settings of VVSPD [24],
DPD [25], EOMPD [45], and SJLPD [46]. Performance robustness
of the proposed APD is studied with þ25% perturbations in
process dead time and also in presence of measurement noise.
Its stability robustness is established from the gain margin (GM)
and phase margin (PM) values as well as through Kharitonov
polynomials [48]. The rest of the paper is divided into three sections.
In Section 2, we describe the various steps of the controller
design, its nonlinear gain variation mechanism for different operat-
ing points during transient phase, and its stability and robustness
issues. Section 3 presents simulation study with detailed perfor-
mance analysis as well as real time experimentation on a DC servo
position control system modeled as a FOIPD process. There is
a conclusion in Section 4.
2. The proposed controller
A simplified block diagram of the proposed APD is shown in
Fig. 3. In order to achieving a faster convergence of the system
with smaller overshoot and undershoot, both the proportional and
derivative gains are modified at each sampling instant based on
the instantaneous values of normalized error eN and change of
0 20 40 60 80 100 120 140 160 180 200
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Time(sec)
Response
IPD
FOIPD
Fig. 1. Responses for IPD ½GpðsÞ ¼ 0:0506eÀ 6s
=sŠ and FOIPD ½GpðsÞ ¼ eÀ 4s
=sð4sþ1ÞŠ
processes under ZNPID [10].
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
IPD
FOIPD
Fig. 2. Responses for IPD ½GpðsÞ ¼ 0:0506eÀ6s
=sŠ and FOIPD ½GpðsÞ ¼ eÀ 4s
=sð4sþ1ÞŠ
processes under IMC-PID [11].
-
r
Process
+
+
Normalization
Ne
-
y
Z
−1
+
e
e
(1 + )αpK
+
+NeΔ
-
×
Load
disturbance
+
+
+
+
Noise
Kd(1+γ )α
Z
−1
Fig. 3. Block diagram of the proposed auto-tuning PD controller (APD).
C. Dey et al. / ISA Transactions 53 (2014) 162–172 163

error ΔeN of the controlled variable. In this study, load disturbance
is applied at the process input and white noise is introduced at the
process output as shown in Fig. 3.
2.1. Design of the proposed controller APD
Discrete form of a PD controller at kth sampling instant can be
described as:
uðkÞ ¼ Kp eðkÞþ
Td
Δt
ΔeðkÞ

ð1Þ
or
uðkÞ ¼ KpeðkÞþKdΔeðkÞ; ð2Þ
when
Kd ¼ Kp
Td
Δt

Here, Kp is the proportional gain, Td is the derivative time, Kd is
the derivative gain, and Δt is the sampling interval. Error e(k) and
change of error Δe(k) at kth sampling instant are defined by
eðkÞ ¼ rÀyðkÞ; and ΔeðkÞ ¼ eðkÞÀeðkÀ1Þ: ð3Þ
In Eq. (3), r is the set point and y(k) is the process output at kth
instant. It has already been mentioned that, due to absence of any
simple auto-tuning scheme for the initial setting of the propor-
tional and derivative gains of a PD controller, we have chosen the
widely practiced Ziegler–Nichols PID tuning rules [10] by dropping
its integral term. Therefore, the values of Kp and Td of the initial PD
controller are obtained by the following relations:
Kp ¼ 0:6Ku; ð4Þ
and
Td ¼ 0:125Tu: ð5Þ
where Ku is the ultimate gain and Tu is the ultimate period
obtained by relay-feedback test [51]. These initial values of Kp
and Kd (Eq. (2)) get modified continuously by the nonlinear gain
updating factor α, which is defined by
αðkÞ ¼ eNðkÞ Â ΔeNðkÞ; ð6Þ
where eNðkÞ ¼ eðkÞ=jrj and ΔeNðkÞ ¼ eNðkÞÀeNðkÀ1Þ.
In Eq. (6), eN(k) and ΔeN(k) are considered as the normalized
values of e(k) and Δe(k), respectively. Eq. (6) indicates that the
instantaneous value of α(k) may vary between À1 and þ1, since
the usual range of eN(k) or ΔeN(k) is [À1, 1] for an acceptable close-
loop performance where the peak overshoot remains within 100%.
Observe that the gain updating parameter α changes with the
instantaneous values of eN and ΔeN, hence, it contains the
information regarding the current position and direction of move-
ment of the process in the response trajectory. We utilize this
intelligence property of α to realize the desired gain variation
strategy mentioned earlier, i.e., control action will be made
conservative or aggressive depending on the movement of the
process towards or away from its set point, respectively. Keeping
in mind such a real time gain adjustment in APD, we propose the
following update rules:
Ka
pðkÞ ¼ Kpð1þαðkÞÞ; ð7Þ
Ka
dðkÞ ¼ Kdð1þγjαðkÞjÞ: ð8Þ
Here, Ka
p and Ka
d are the time varying nonlinear proportional
and derivative gains. γ is a positive constant for providing an
appropriate variation of damping to get the desired close-loop
response. In steady state condition αðkÞ ¼ 0, hence Ka
pðkÞ ¼ Kp and
Ka
dðkÞ ¼ Kd. During transient phases, Ka
p may be higher or lower
than Kp depending on the sign of α but Ka
d will always be higher
than Kd. Such nonlinear gain variations are expected to provide
necessary damping required for achieving an enhanced control
performance. Detailed discussion about this online gain modifica-
tion mechanism at different operating points and its influence on
the close-loop performance is provided in the following Section 2.2.
Thus, the proposed APD can be expressed as:
ua
ðkÞ ¼ Ka
pðkÞ eðkÞþKa
dðkÞ ΔeðkÞ: ð9Þ
For proper tuning of the controller, suitable value of γ is to be
selected, which may be done either using operator's knowledge or
by trial depending on the process dynamics. In addition, for
specific performance based applications where some constraints
are imposed on the system behavior, the value of γ may be
obtained through some optimization technique, so that the result-
ing system response meets those specified performance indices.
Through an extensive simulation study on a large number of
integrating (IPD and FOIPD) processes, we suggest the following
simple empirical relation for γ, which gives an overall satisfactory
performance for all such cases.
γ ¼ 2 Â Ku Â Tu Â K: ð10Þ
In Eq. (10), K is the open-loop gain of the related process.
Observe that γ depends on the dynamics of the process under
control, which may be characterized by its critical point (Ku, Tu).
Moreover, γ does not increase the design complexity as all the
parameters, i.e., Ku, Tu, and K are obtained from the relay-feedback
test [51], which is also used for setting the initial parameters of the
proposed APD. We use the same relation of γ, i.e., Eq. (10), for both
the IPD and FOIPD processes in our simulation as well as in
experimental study on a DC servo system. In (Section 3) to come,
we will observe that considerable deviations of γ from its respec-
tive nominal values bring only a little change in the close-loop
performance of IPD as well as FOIPD processes.
2.2. Online gain modification
From the dynamic proportional and derivative gain expres-
sions, as given by Eqs. (7) and (8), it is evident that α makes real
time variations in Ka
p and Ka
d. The gain update rules (Eqs. (7) and
(8)) are chosen in such a way that the proposed APD will be able to
provide an improved performance under both set point change
and load disturbance. Note that, unlike the linear control surface of
PD controller with static gains, control surface of APD as shown in
Fig. 4 is highly nonlinear in nature. Now, we explain how the
proportional and derivative gains of APD are modified by the
proposed scheme for providing appropriate control action under
different operating conditions for achieving the desired perfor-
mance. For a better understanding, we refer Fig. 5, which represents
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
u
a
ΔeN
eN
Fig. 4. Control surface of APD.
C. Dey et al. / ISA Transactions 53 (2014) 162–172164

a typical close-loop response of an under-damped second-order
process due to set point change and load disturbance.
(i) During the operating stage, when the process is moving fast
towards the set point (e.g., point like A or C in Fig. 5), there is a
possibility of larger overshoot or undershoot in the subsequent
operating phase. To avoid such situations, a considerable amount
of damping should be present in the control action. This may be
possible either by increasing the derivative action or by decreas-
ing the proportional action separately, or by making such
changes simultaneously. In this case, since e and Δe are of
opposite sign, α will be negative, as a result, Ka
p oKp (Eq. (7)).
At the same time Eq. (8) indicates that Ka
d 4Kd. A combined
effect of the increased damping and reduced proportional gain
will make the speed of response slow, which is expected to
reduce the overshoot or undershoot. Thus, the proposed gain
variation mechanism fulfills the requirement of appropriate
control action for achieving an improved transient response.
(ii) Opposite to the previous operating stage, when the process is
far from the set point and moving away very fast from it (e.g.,
point B in Fig. 5), both the proportional and derivative gains
should be large enough to restore the controlled variable
quickly to its desired value. Under such situations, both
e and Δe have large values with the same sign, thereby making
α large and positive according to Eq. (6). Such a large positive
value of α makes Ka
p 4Kp and Ka
d 4Kd respectively according to
Eqs. (7) and (8). Therefore, ua
4u (i.e., control action becomes
more aggressive) according to Eqs. (2) and (9). Similar situa-
tion is also observed during load disturbance. Immediately
after a load disturbance, e may be small, but Δe will be
sufficiently large (e.g., point D in Fig. 5) and they are of the
same sign, as a result α becomes positive. Therefore, according
to Eqs. (7) and (8) both the proportional and derivative gains
of APD will be higher than those of conventional PD controller.
These higher gains will make APD to generate the required
strong control action for restoring the process quickly to its
desired value. Thus, APD is capable of providing the required
variation in control action to improve the process recovery.
From the above discussion, it appears that our simple gain
modification scheme always attempts to generate an appropriate
control action towards achieving an enhanced performance both
under set point change and load disturbance. Next, we mention
how the proposed scheme is tested for good stability margins as
well as robustness against parametric variations.
2.3. Stability and robustness
The proposed APD reduces to a simple nonlinear controller due
to nonlinear gain variation through α. We know that the stability
analysis for nonlinear control systems is not straight forward.
Here, we study the relative stability for APD by calculating the
stability margins along with their corner frequencies for two
boundary values of α, i.e., at its maximum (αmax) and minimum
(αmin) values. Under close-loop operation of the process, gain
margin (GM) and phase margin (PM) values are calculated at
αmax and αmin. For both the IPD and FOIPD processes, APD will be
found to provide good stability margins in terms of GM and PM,
which will be justified in the result section. In addition to the
relative stability margins, stability robustness of the close-loop
system for a given process is tested by Kharitonov's interval
polynomials [48] at the boundary values of α (αmax and αmin) along
with 725% simultaneous perturbations in process parameters.
Stability robustness is ensured as all the roots of Kharitonov's
polynomials will be found to be negative.
3. Results
In this section, we present the detailed performance analysis of
our proposed APD through simulation as well as real time
implementation. For simulation study, we consider two well
known integrating process models—pure integrating process with
delay (IPD) [12,23,49] and first-order integrating process with
delay (FOIPD) [43,44]. For the IPD process, performance of the
proposed APD is compared with the reported PID tuning relations
of AHPID [2], DMPID [3], ZNPID [10], SLPPID [11], CPPID [16], VPID
[23], PCPID [34], ACPID [35], RRCPID [36], AMPID [37], HXCPID
[38], and RPID [50], and PD settings of CPPD [16], VPD [23], and
LLPPD [39]. Similarly, for the FOIPD process, performance of APD is
compared with PID tuning rules reported in DMPID [3], ZNPID
[10], SPID [12], AMPID [37], HXCPID [38], YPID [40], WCPID [41],
RRCPID [42], SLPID [44], KLPID [47], and RPID [50], and PD settings
in VVSPD [24], DPD [25], EOMPD [45], and SJLPD [46]. For an in-
depth comparison, in addition to the response characteristics,
several performance indices, such as percentage overshoot (%OS),
rise time (tr), settling time (ts), integral absolute error (IAE) and
integral time absolute error (ITAE) are calculated for each setting.
Here, ts is calculated following 2% criterion. To verify the robust-
ness of the proposed APD, performance indices are also evaluated
at the nominal as well as 25% increased value of process dead time.
Experimental verification of the proposed APD along with all the
above reported PID and PD settings is made on a laboratory scale
DC servo position control system [52] identified to be a FOIPD
model under both set point change and load disturbance. To verify
the noise sensitivity of the proposed controller, white noise (with
mean¼0 and variance¼0.1) is added to the measured controlled
variable during simulation as well as experimental verification. For
all the simulation and experimental studies, the only additional
tuning parameter γ is estimated using Eq. (10). During perfor-
mance study, first we apply a step set point change followed by an
impulse load disturbance as shown in Fig. 3.
3.1. Integrating process with delay (IPD)
Integrating processes are frequently encountered in process
industries. Chien and Freuhauf [49] suggested that a number of
chemical processes can be approximated by a pure integral plus
time delay (IPD) model, defined by
Gp ¼
KeÀ θs
s
: ð11Þ
Here, we consider the well known IPD process model with open
loop gain K¼0.0506 and dead time θ¼6 s [12,23,49]. Performance
of APD for this IPD process is investigated along with other
model based PID tuning rules proposed in AHPID [2], SLPPID [11],
0 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Time(sec)
Response
A C D
e
Δe
A
B
C
D
B
Fig. 5. Typical close-loop response of an under-damped second-order process.

CPPID [16], VPID [23], PCPID [34], ACPID [35], RRCPID [36], AMPID
[37], HXCPID [38], and RPID [50], and PD settings in CPPD [16], VPD
[23], and LLPPD [39]. In addition, model free PID tuning rules of
DMPID [3] and ZNPID [10] are also considered for performance
comparison. Close-loop responses with the nominal value of dead
time (i.e., θ¼6 s) are shown in Fig. 6 for different controllers. The
related performance indices are given in Table 1a. It is found that
most of the model based and model free PID controllers fail to
provide satisfactory performance under set point change and load
variation simultaneously. In case of VPID [23], the process shows
highly oscillatory response and for AHPID [2] it completely diverges.
In comparison with VPD [23] and CPPD [16], APD offers lower
overshoot with faster settling during set point change and at the
same time process recovery is found to be improved under load
variation as shown in Fig. 7(a). In case of LLPPD [39] due to over
damped response no overshoot is found but load rejection is quite
poor compared to APD.
Performance robustness of the proposed APD is tested with
25% increased process dead time, i.e., θ¼7.5 s, and the respective
performance indices are listed in Table 1b. From Table 1b it is
found that under VPID [23], CPPID [16], and SLPPID [11] the
process fails to settle within the simulation period and AMPID
[37] shows a large overshoot in the set point response as well as
poor load regulation. On the other hand, our proposed APD shows
consistently improved performance compared to VPD [23] and
CPPD [16]. Thus, from the process responses as shown in
Figs. 6 and 7 and performance indices of Tables 1a and 1b it is
found that the proposed APD shows an overall improved perfor-
mance compared to others tuning relations. Moreover, response
characteristics of APD shown in Fig. 7(c) reveal that the proposed
scheme can overcome the effects of measurement noise.
Fig. 7(b) shows the variation of proportional and derivative
gains of the proposed APD with the gain updating factor α for
θ¼6 s, which shows a considerable change in the derivative gain
Ka
d during transient phases of the close-loop operation. To check
the sensitivity of the tuning parameter γ, performance indices are
evaluated with 725% perturbations in its initial value (γ¼12.49)
for both the nominal (i.e., θ¼6 s) and increased (i.e., θ¼7.5 s)
values of dead time (Tables 1a and 1b). No considerable deviation
is observed in the performance of APD due to such variations of γ,
which justiﬁes the robustness of our proposed scheme. Stability
analysis of APD is provided in Table 1c in terms of GM and PM
values with their corner frequencies. Stability robustness is
observed (Table 1c) from the Kharitonov polynomials for two
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
2
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
-100
-50
0
50
100
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
CPPID [16]
PCPID [34] SLPPID [11]
RRCPID [36]
HXCPID [38]
VPD [23]
RPID [50]
ZNPID [10] VPID [23]
AHPID [2]
ACPID [35] DMPID [3]
AMPID [37]
CPPD [16] LLPPD [39]
Fig. 6. Responses for the IPD process GpðsÞ ¼ 0:0506eÀ 6s
=s.

extreme values of α (αmax and αmin) along with 725% simulta-
neous perturbations in process parameters (K and θ).
3.2. First-order integrating process with delay (FOIPD)
First-order integrating process with delay can be described by
Gp ¼
KeÀ θs
sðτ sþ1Þ
: ð12Þ
Here, we consider K¼1, θ¼4 s, and τ¼4 s [43,44]. Performance
of APD is compared with a number of model free and model based
PID tuning relations given by DMPID [3], ZNPID [10], SPID [12],
AMPID [37], HXCPID [38], YPID [40], WCPID [41], RRCPID [42],
SLPID [44], KLPID [47], and RPID [50], and model based PD settings
of VVSPD [24], DPD [25], EOMPD [45], and SJLPD [46]. Responses
of (12) with θ¼4 s are shown in Fig. 8 and the performance indices
for θ¼4 s and 5 s (with þ25% perturbation) are given in Tables 2a
and 2b respectively. Fig. 8 shows that the process either goes to
the verge of instability for RRCPID [42] or completely diverges
under HXCPID [38], SLPID [44], and RPID [50]. In comparison with
VVSPD [24] and DPD [25], our proposed APD exhibits lower
overshoot with faster recovery as depicted in Fig. 9(a). Fig. 9
(c) justiﬁes the robustness of APD against measurement noise.
Fig. 9(b) shows the variations of proportional and derivative gains
of APD with gain updating factor α. Under both set point change
Table 1a
Performance indices for the IPD process GpðsÞ ¼ 0:0506eÀ 6s
=s.
Controller %OS tr ts IAE ITAE
Model based PID tuning methods
VPID [23] (2001) 118.20 9.81 143.75 37.36 2899.0
CPPID [16] (2003) 61.26 10.54 88.75 24.14 1545.0
AHPID [2] (2004) Diverges completely
PCPID [34] (2005) 34.47 12.27 83.56 25.6 1597.0
ACPID [35] (2007) 27.51 21.99 144.21 39.45 3020.0
SLPPID [11] (2008) 2.85 18.29 59.03 14.67 1508.0
RPID [50] (2008) 40.46 19.56 133.27 39.73 3053.0
RRCPID [36] (2009) 0.00 64.12 64.12 29.33 1963.0
AMPID [37] (2010) 56.96 11.34 62.70 24.92 1478.0
HXCPID [38] (2011) 7.05 63.08 148.34 41.40 2749.0
Model free PID tuning methods
ZNPID [10] (1942) 67.63 11.23 90.15 30.8 1919.0
DMPID [3] (2009) 20.96 11.80 107.64 24.71 1642.0
Model based PD tuning methods
VPD [23] (2001) 16.67 11.64 61.48 16.41 988.4
CPPD [16] (2003) 13.77 12.05 58.62 15.99 946.4
LLPPD [39] (2006) 0.00 53.72 53.72 23.80 1440.0
Proposed auto-tuning PD method
APD γ ¼ 12:49 4.01 12.86 41.87 15.06 859.6
γþ25% ¼ 15:61 3.28 13.27 41.87 15.09 863.1
γÀ25% ¼ 9:37 4.73 12.86 41.87 15.04 856.5
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
-2
0
2
4
Time(sec)
Controlaction
0 50 100 150 200 250
-0.03
-0.02
-0.01
0
0.01
0 50 100 150 200 250
3
3.05
3.1
0 50 100 150 200 250
9
10
11
12
13
Time(sec)
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
-15
-10
-5
0
5
10
15
Time(sec)
Controlaction
APD
APD
α
Kd
a
Kp
a
APD
APD
Fig. 7. (a) Response and the corresponding control action for the IPD process GpðsÞ ¼ 0:0506eÀ 6s
=s. (b) Variation of α, Ka
p, and Ka
d under APD for the IPD process
GpðsÞ ¼ 0:0506eÀ 6s
=s. (c) Response and the corresponding control action with measurement noise for the IPD process GpðsÞ ¼ 0:0506eÀ 6s
=s.
Table 1b
Performance indices for the IPD process with 25% increased dead time
GpðsÞ ¼ 0:0506eÀ 7:5s
=s.
VPID [23] (2001) Unstable
CPPID [16] (2003) 103.80 11.83 Not settled 52.37 4445.0
AHPID [2] (2004) Completely diverges
PCPID [34] (2005) 67.14 13.55 80.25 30.11 1889.0
ACPID [35] (2007) 35.08 21.73 145.65 43.23 3296.0
SLPPID [11] (2008) 36.12 17.43 Not settled 52.51 6895.0
RPID [50] (2008) 50.22 23.19 121.98 44.94 3489
RRCPID [36] (2009) 0.00 74.66 74.66 30.81 2226.0
AMPID [37] (2010) 97.96 12.69 107.36 36.05 2560.0
HXCPID [38] (2011) 7.46 60.77 147.75 42.04 2853.0
Model free PID tuning methods
ZNPID [10] (1942) 115.28 12.69 98.32 43.51 2943.0
DMPID [3] (2009) 49.15 13.54 110.80 29.58 2001.0
VPD [23] (2001) 42.35 13.12 109.08 28.07 2056.0
CPPD [16] (2003) 38.94 13.55 107.36 26.36 1877.0
LLPPD [39] (2006) 0.00 37.65 37.65 23.93 1432.0
APD γ ¼ 12:49 27.14 13.55 66.48 22.68 1534.0
γþ25% ¼ 15:61 26.19 13.98 66.48 22.62 1537.0
γÀ25% ¼ 9:37 28.10 13.98 66.48 22.74 1531.0
Table 1c
Stability and robustness analysis for the IPD process GpðsÞ ¼ 0:0506eÀ 6s
=s.
APD (αmax) APD (αmin) Kharitonov's Polynomials
with αmax and αmin and
725% variation in K and θ
GM (dB) 4.68 (Inf rad/s) 3.83 (Inf rad/s) s2
þ0.51sþ0.15¼0
s2
þ1.49sþ0.14¼0
PM (deg) 64.9 (0.19 rad/s) 68.7 (0.20 rad/s) s2
þ1.28sþ0.14¼0
s2
þ0.59sþ0.15¼0

and load disturbance, APD shows an overall improved perfor-
mance than other PID and PD settings as revealed by Tables 2a and
2b. Its robustness is observed with þ25% perturbation in dead
time as well as 725% variation in γ from its nominal value 16.51
(Tables 2a and 2b). GM and PM along with their corner frequencies
are listed in Table 2c, which represents good stability margins.
Kharitonov's polynomials of Table 2c also confirm its stability
robustness under maximum and minimum values of α as well as
725% simultaneous variations in K, τ, and θ.
3.3. Real time implementation
Servo position control system is a typical example of integrat-
ing process. Here, performance of the proposed APD is verified on
a DC servo position control system. The schematic block diagram
of the position control system is shown in Fig. 10(a) and its
experimental setup is shown in Fig. 10(b). The hardware setup is
a Quanser make DC Motor Control Trainer (DCMCT) [52] and it has
been identified as a FOIPD model. A small delay of 0.01 s is
introduced by the Simulink delay block in the forward path of
the process loop. This DC servo motor is a high quality 18-Watt
motor of Maxon brand. This is a graphite brush DC motor with low
inertia rotor. It has zero cogging and very low unloaded running
friction. The transfer function of this servo motor (as provided by
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
-1000
-500
0
500
1000
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
1.5
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
-1000
-500
0
500
1000
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
0
0.5
1
Time(sec)
Response
0 50 100 150 200 250
-50
0
50
Time(sec)
Response
WCPID [41]
KLPID [47] RRCPID [42]
AMPID [37]
DPD [25]
RPID [50]
ZNPID [10] YPID [40]
SPID [12]
SLPID [44] DMPID [3]
HXCPID [38]VVSPD [24]
EOMPD [45] SJLPD [46]
1.5
Fig. 8. Responses for the FOIPD process GpðsÞ ¼ eÀ 4s
=sð4sþ1Þ.
Table 2a
Performance indices for the FOIPD process GpðsÞ ¼ eÀ 4s
=sð4sþ1Þ.
WCPID [41] (2002) 22.19 17.56 57.89 29.70 2561.0
SPID [12] (2003) 34.47 15.27 101.09 35.32 3017.0
KLPID [47] (2006) 34.88 17.57 87.27 39.64 3390.0
RRCPID [42] (2007) 30.26 12.96 Not settled 56.03 6836.0
SLPID [44] (2008) Completely diverges
RPID [50] (2008) Completely diverges
AMPID [37] (2010) 58.83 11.81 69.41 28.51 2122.0
HXCPID [38] (2011) Completely diverges
Model free tuning PID tuning methods
ZNPID [10] (1942) 69.74 12.96 87.27 38.0 3083.0
YPID [40] (1999) 67.63 17.57 Not settled 62.78 6304.0
DMPID [3] (2009) 25.06 14.11 127.59 31.30 2654.0
VVSPD [24] (2000) 16.46 12.96 49.83 18.01 1263.0
DPD [25] (2001) 26.29 11.81 67.68 20.52 1564.0
SJLPD [46] (2006) 0.00 36.0 36.0 14.94 1164.0
EOMPD [45] (2009) 1.73 28.51 41.76 24.74 1882.0
APD γ ¼ 16:51 6.64 15.84 37.73 17.91 1237.0
γ þ25% ¼ 20:64 5.82 15.84 38.31 17.80 1226.0
γÀ25% ¼ 12:39 7.05 15.84 38.31 18.03 1248.0

Quanser [52]) is
GpðsÞ ¼
19:9eÀ0:01s
sð0:09sþ1Þ
: ð13Þ
Quanser-Q8 DAQ board interfaces the DCMCT with the PC
through USB port. With the help of QuaRC soft-ware based on
Matlab–Simulink we implement the proposed auto-tuning PD con-
troller. Similarly, other reported PID and PD controllers are also
implemented for their performance evaluation on DCMCT. Real Time
Workshop (RTW) and Real Time Windows Target (RTWT) generate C
code using Microsoft Cþ þ
Professional from the QuaRC block
diagram, and the Quanser-Q8 board acts as the intermediary for
two way data flow from the physical servo system to and from the
QuaRC model. A high resolution encoder is used for position sensing
of the DC motor. Performances of the proposed APD and other PID/
PD tuning rules (DMPID [3], ZNPID [10], SPID [12], VVSPD [24], DPD
[25], AMPID [37], HXCPID [38], YPID [40], WCPID [41], RRCPID [42],
SLPID [44], EOMPD [45], SJLPD [46], KLPID [47], and RPID [50]) are
tested on the DCMCT. Responses of the DC servo motor for different
controllers other than APD during set-point tracking and load
variation for the nominal value of dead time (θ¼0.01 s) are shown
in Fig. 11. We observe that even with this nominal value of dead time
VVSPD [24], AMPID [37], HXCPID [38], RRCPID [42], SLPID [44], SJLPD
[46], and RPID [50] provide unstable performances. The close-loop
response and its corresponding control action for the proposed APD
is shown in Fig. 12(a). Performance of APD has also been tested with
measurement noise as shown in Fig. 12(b). Fig. 13 shows only stable
performances of different controllers with 25% higher value of
dead time, i.e., θ¼0.0125 s. Fig. 14(a) and (b), respectively, shows
the responses of APD without and with measurement noise. Thus,
results obtained from both nominal and increased values of dead
time (Figs. 11–14) reveal that the proposed APD exhibits an overall
improved performance compared to other PID and PD tuning rules as
well as robustness against measurement noise.
To summarize, from the simulation as well as real time experi-
mentation, it is observed that for both IPD and FOIPD processes, the
proposed APD shows consistently improved overall performance
under set point change and load disturbance. APD is also found to
provide good stability margins and performance robustness. Stability
robustness of the proposed scheme is also verified at the boundary
values of the gain modifying parameter α (i.e., αmax and αmin) along
with 725% simultaneous perturbations in process parameters.
4. Conclusion
We proposed a real time gain modification scheme through
nonlinear parameterization of a PD auto-tuner. The proportional
and derivative gains of the proposed auto-tuning PD controller
(APD) have been modified in each instant by a nonlinear gain
updating factor α defined on the instantaneous process states.
Fig. 9. (a) Response and the corresponding control action for the FOIPD process GpðsÞ ¼ eÀ4s
=sð4sþ1Þ. (b) Variation of α, Ka
p, and Ka
d under APD for the FOIPD process
GpðsÞ ¼ eÀ 4s
=sð4sþ1Þ. (c) Response and the corresponding control action with measurement noise for the FOIPD process GpðsÞ ¼ eÀ 4s
=sð4sþ1Þ.
Table 2c
Stability and robustness analysis for the FOIPD process GpðsÞ ¼ eÀ 4s
=sð4sþ1Þ.
APD (αmax) APD (αmin) Kharitonov's polynomials
with αmax and αmin and
725% variation in K, τ, and θ
GM (dB) 9.07 (0.51 rad/s) 9.17 (0.52 rad/s) s3
þ0.61s2
þ0.57sþ0.04¼0
s3
þ0.52s2
þ0.17sþ0.01¼0
PM (deg) 53.1 (0.18 rad/s) 54.2 (0.17 rad/s) s3
þ0.52s2
þ0.57sþ0.01¼0
s3
þ0.61s2
þ0.17sþ0.04¼0
Table 2b
Performance indices for the FOIPD process with 25% increased dead time
GpðsÞ ¼ eÀ 5s
=sð4sþ1Þ.
WCPID [41] (2002) 33.11 17.57 48.68 32.61 2834.0
SPID [12] (2003) 46.27 15.84 106.28 38.56 3307.0
KLPID [47] (2006) 42.32 17.57 83.81 42.78 3674.0
RRCPID [42] (2007) Completely diverges
SLPID [44] (2008) Completely diverges
RPID [50] (2008) Completely diverges
AMPID [37] (2010) 87.43 12.38 125.29 43.10 3908.0
HXCPID [38] (2011) Completely diverges
Model free tuning PID tuning methods
ZNPID [10] (1942) 91.52 13.54 97.06 47.14 4025.0
YPID [40] (1999) 78.68 18.15 Not settled 73.18 7531.0
DMPID [3] (2009) 39.39 14.69 127.02 34.65 2957.0
VVSPD [24] (2000) 34.47 13.54 100.52 27.88 2341.0
DPD [25] (2001) 49.47 12.38 Not settled 43.83 4378.0
SJLPD [46] (2006) 10.32 11.81 42.34 15.74 1170.0
EOMPD [45] (2009) 4.59 24.48 45.22 26.41 2049.0
APD γ ¼ 16:51 18.92 15.26 52.71 23.14 1782.0
γ þ25% ¼ 20:64 18.51 15.26 52.13 22.98 1769.0
γÀ25% ¼ 12:39γ 19.74 15.26 52.71 23.31 1796.0

DCMCT
Hardware
Quanser DAQ
Board (Q8)
PC with Maltab and RTW
and RTWT
PC Inertial Load
Encoder DAQ D/A Converter Amplifier DC Motor
PCI Link
Motor
Load
Encoder
Fig. 10. (a) Schematic diagram of DCMCT. (b) Experimental setup of DC servo rig (Quanser DCMCT).
0 1 2 3 4 5
0
5
10
Time(sec)
Response
0 1 2 3 4 5
0
5
10
Time(sec)
Response
0 1 2 3 4 5
0
2
4
6
8
Time(sec)
Response
0 1 2 3 4 5
0
2
4
6
8
Time(sec)
Response
0 1 2 3 4 5
0
2
4
6
8
Time(sec)
Response
0 1 2 3 4 5
0
5
10
Time(sec)
Response
0 1 2 3 4 5
-10
-5
0
5
10
Time(sec)
Response
0 1 2 3 4 5
0
5
10
Time(sec)
Response
WCPID [41] SPID [12]
KLPID [47] DMPID [3]
DPD [25] EOMPD [45]
Fig. 11. Responses with nominal value of dead time for DCMCT.
Fig. 12. (a) Response and the corresponding control action for the proposed APD with nominal value of dead time for DCMCT. (b) Response and the corresponding control
action for the proposed APD under measurement noise with nominal value of dead time for DCMCT.

An empirical relation has also been suggested for the selection of
the single additional tuning parameter γ. The proposed gain
modification scheme can be easily incorporated in conventional
control loops. Performance analysis for integrating processes with
varying dead time under both set point change and load distur-
bance has revealed that the proposed APD is capable of providing
an improved overall performance compared to other tuning rules
reported in the literature. Performance robustness of the close-
loop system under APD has been observed with considerable
variations in process dead time as well as with measurement
noise. Stability robustness has also been established by applying
725% simultaneous perturbations in process parameters.
Further works may be done for setting more appropriate
parameters of the initial PD controller, and finding more suitable
value of γ for a given process. Similar gain adjustment schemes
may be tried for other real world problems, which exhibit inverse
response. All these possibilities are under investigation.
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Time(sec)
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-5
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Controlaction
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
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Time(sec)
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-20
0
20
Time(sec)
Controlaction
APD
APDAPD
APD
Time(sec)
Fig. 14. (a) Response and the corresponding control action for the proposed APD with 25% increased value of dead time for DCMCT. (b) Response and the corresponding
control action for the proposed APD under measurement noise with 25% increased value of dead time for DCMCT.

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