Acta Polytechnica Vol. 52 No. 5/2012

Research Trends for PID Controllers

Antonio Visioli

University of Brescia, Via Branze 38, I-25123 Brescia, Italy

Corresponding author: antonio.visioli@ing.unibs.it

Abstract
This paper analyses the most significant issues that have been recently been addressed by researchers in the field of
Proportional-Integral-Derivative (PID) controllers. In particular, the most recent techniques proposed for tuning and designing
PID-based control structures are briefly reviewed, together with methods for assessing their performance. Finally, fractional-order
and event-based PID controllers are presented among the most significant developments in the field.

Keywords: PID controllers, design, tuning, control structures.

1 Introduction
Despite the new results in control theory that have
been achieved by researchers year-by-year all over the
world, Proportional-Integral-Derivative (PID) controllers
are still the most widely-used controllers in industry.
This is because it is really very difficult to improve
their cost/benefit ratio, which is obviously a major
concern in industry. PID controllers are still a very
active field of research, because it is recognized that
they are often poorly tuned in industrial applications,
and the performance that is achieved can be improved
many times by using a more effective design technique
for the PID-based control system. A great impulse for
research on PID controllers was provided by a workshop
dedicated to them organized in Terrassa (Spain) in
2000, sponsored by the International Federation of
Automatic Control (IFAC). The success of the meeting is
witnessed not only by the number of participants but
also by the large number of papers and books published
in the last ten years specifically on this topic (see,
e.g. [3, 15, 23, 45, 46, 52, 55]). In order to further outline
the current state-of-the-art and the future perspectives
from an academic and also from an industrial viewpoint,
another IFAC conference dedicated to PID controllers
was held in Brescia (Italy) in March 2012. This paper
describes many significant results achieved recently in
the field of PID controllers, and reviews in particular
those related to tuning and designing PID-based control
structures, determining the stabilizing region of the
PID parameters, and performance assessment issues.
Finally, new control concepts applied to PID controllers,
namely fractional and event-based PID controllers, will
be highlighted.

2 Generalities
A PID controller is typically employed in a unity-feedback
control system like that shown in Figure 1, where P is

the process, y is the process variable, r is the set-point
signal and d is the load disturbance. In its basic form,
the PID controller can be described by the following
transfer function

C(s) = Kp
(

1 +
1
Tis

+ Tds
)
, (1)

where Kp is the proportional gain, Ti is the integral
time constant and Td is the derivative time constant.
Expression (1) is usually known as the ideal form. Other
forms (usually called series and parallel forms) can also
be employed [3]. In order to be effectively employed in
practical cases, additional functionalities also have to be
implemented. The most important functionalities can be
summarized as follows (details can be found in [52]).

• The derivative action has to be filtered in order to
make the controller proper and to filter the (high
frequency) measurement noise; in addition, the
derivative action is often applied directly to the
process variable instead of to the control error in
order to avoid the so-called derivative kick when
a step signal is applied to the set-point. The
derivative filter has to be taken into consideration in
the overall design of the controller [13, 17].

• The set-point value for the proportional action
can be weighted in order to obtain a two-degree-
of-freedom controller, i.e. in order to reduce the
overshoot in the set-point step response when
the controller is tuned in order to increase the
bandwidth of the system with the aim of increasing
the load disturbance rejection performance. In this
case a suitable choice of the value of the set-point
weight (or the application of a more sophisticated
techniques [47]) can yield a significant increment of
the control performance.

• Suitable techniques (see, e.g. [48]) should be imple-
mented properly in order to avoid the windup effect

144



Acta Polytechnica Vol. 52 No. 5/2012

e r u y 
d 

C(s) P(s) 

Figure 1: The unity feedback control scheme.

of the integral action which has a detrimental effect
when large set-point changes are applied.

3 Tuning and automatic tuning
techniques

Many tuning rules for PID controllers have been proposed
in the last century [23], and new rules have also been
proposed recently (for self-regulating, integral and
unstable processes). Among these new rules, it is worth
mentioning the SIMC tuning rules [9, 34] which have
been proven to provide very good results despite their
simplicity, and the AMIGO tuning rules, which are
capable of giving high performance for a wide range of
processes [2].

When a new technique is investigated, the following
points should be considered. First, the parameters related
to the additional functionalities mentioned in Section 2
should be explicitly taken into account. Other important
issues, e.g. the robustness of the controller to modelling
uncertainties, should also be considered [43]. Finally, the
availability of more and more advanced identification
techniques calls for procedures to select the most suitable
identification strategy for a given application and for
a general model-based design strategy [8, 18]. This is
clearly more relevant when automatic tuning techniques
are implemented. Obviously, each design methodology
should yield an optimal PID controller, i.e. it should
minimize some significant performance index (e.g. the
integrated absolute error when a set-point or load
disturbance step signal is applied) subject to constraints
usually represented by the robustness of the control
system of the control effort.

4 PID-based control structures
Many different PID-based control structures have been
considered as an effective means for obtaining high
performance while retaining simple implementation of the
control system. In this context it is worth highlighting (in
addition to the above-mentioned decoupling strategies)
structures and tuning techniques for dead time compensa-
tion schemes [22], for cascade [41] and ratio control
systems [50, 51] and methods for designing feedforward
control actions (both for the set-point following task
[28, 49, 53] and for the load disturbance rejection task

[11, 27, 44]). It should be stressed that the challenging
issue in a PID-based control system is to achieve a
suitable combination of the block scheme design and the
tuning of the controller parameters.

5 PID control for MIMO
processes

As it is recognized that many processes are multivariable
by their nature, the design of PID controllers for
application in multi-input-multi-output (MIMO) systems
is also a very interesting and relevant topic. In particular,
the tuning of decentralized or multivariable PID controllers
poses new challenges to be solved because of the coupling
effects in the process [35, 54]. Further, effective decoupling
strategies are needed where the best trade-off between
ease of implementation and obtained performance is
achieved [7, 19, 21, 29].

6 Stabilizing PID controllers
Great advances have been made in recent years on the
more theoretical issue of determining the complete set of
stabilizing PID controllers for a given process [33]. For
example, if a first-order-plus-dead-time (FOPDT) process

P(s) =
K

Ts + 1
e−Ls (2)

is considered, the stabilizing PID parameters can be
computed by solving fairly simple equations (numerically)
[32]. Similar procedures can also be employed for integral
processes [25]. It should be stressed that knowledge of
the set of stabilizing controllers provides information
related to the robustness and/or fragility of the controller,
and can also be employed for implementing advanced
tuning techniques.

7 Performance assessment and
retuning

In many practical cases, PID controllers are poorly
tuned because of lack of time and lack of skill of
the operator. As there are hundreds of control loops
in large plants, it is almost impossible for operators
to monitor each of them manually. It is therefore
important to have automatic tools that are first able
to assess the performance of a control system and, in
the event that it is not satisfactory, to suggest a way
to solve the problem (for example, if bad controller
tuning is detected, new appropriate controller parameter
values are determined). Many performance assessment
methodologies have been proposed in the literature and
have been applied successfully in industrial settings [14].
They are generally divided in two categories stochastic
performance monitoring in which the ability of the control

145



Acta Polytechnica Vol. 52 No. 5/2012

system to cope with stochastic disturbances is of main
concern (works that fall into this class mainly rely on the
concept of minimum variance control), and deterministic
performance monitoring in which performances related to
more traditional design specifications, e.g. set-point and
load rejection disturbance step response parameters,
are taken into account. Restricting the analysis to
the tuning assessment of PID controllers, methods for
determining the minimum variance PID controller have
been proposed in [16, 42]. Regarding deterministic
performance monitoring, a practical approach has been
proposed in [38, 39, 40]. It is based on a process
parameter estimation procedure which uses the set-point
step response (i.e. routine operation data). Its rationale
relies in the so-called “half rule”, which states that
the largest neglected (denominator) time constant is
distributed evenly to the effective dead time and the
smallest retained time constant. Then, the performance
that is obtained is evaluated by comparing the integrated
absolute error with the error that would have been
obtained by applying the SIMC tuning rule, which is
considered as a benchmark, i.e.

IAE = 2AL, (3)

where A is the amplitude of the set-point step signal.
In particular, for a FOPDT process (2), the gain is
estimated as

K = A
Ti

Kp
∫∞

0 e(t)dt
. (4)

Then, the sum of the time constant and the dead time
can be determined as

L + T =
limt→+∞

∫ t
0 eu(v)dv

A
(5)

where
eu(t) = Ku(t) −y(t). (6)

Because the apparent dead time L can be determined by
considering the time interval from the application of the
step signal to the set-point and the time instant when
the process output attains 2 % of the new set-point value
A (a suitable noise band can be employed in practical
cases to cope with measurement noise), the value of T
can easily be determined from 5.
With knowledge of the process parameters, the

performance can be assessed by considering the following
performance index (see (3))

J =
2AL∫∞

0 |e(t)|dt
, (7)

and, in the event that the performance is not satisfactory
(in principle, it should be J = 1, but from a practical
point of view J > 0.6 can be considered as acceptable),
the PID controller can be retuned by applying the SIMC
tuning rule (or some other suitable rule).

For an integral processes, the procedure is similar.
The process model is

P(s) =
K

s(Ts + 1)
e−Ls (8)

and the sum of the lags and of the dead time of the
process can be determined as

L + T =
limt→+∞

∫ t
0 eu(v)dv

A
, (9)

where
eu(t) := K

∫ t
0
u(v)dv −y(t) (10)

The process gain K can be determined as

K = A
Ti

Kp
∫∞

0
∫ t

0 e(v)dvdt
. (11)

If the set-point following task is of concern and a PID
controller is employed, the performance index becomes

J =
3.45AL∫∞

0 |e(t)|dt
. (12)

It is worth noting that this methodology can also be
applied, with appropriate modifications, when a load
disturbance step response is available.

8 Fractional-order PID
controllers

A topic which continues to be the subject of many
investigations is fractional-order PID (FOPID) controllers,
which can be considered a generalization of standard
integer-order PID controllers, where the orders of
integration and differentiation are not necessary integer
[20, 36]. The typical formulation of a FOPID controller
is

C(s) = Kp
(

1 +
1

Tisλ
+ Tdsµ

)
, (13)

where λ and µ are the noninteger orders of the integral
and derivative terms respectively. An alternative form
(which includes the filter of the derivative term) is

C(s) = Kp
Tis

λ + 1
Tisλ

Tds
µ + 1

Td
N
s + 1

. (14)

FOPID controllers have the great advantages of providing
more flexibility in their design, as the user can also
tune also the order of integration and differentiation
in addition to the proportional gain and the integral
and derivative time constants. This implies that the
frequency response of the open-loop system can be
shaped with more degrees-of-freedom, thus allowing the
user to meet more control requirements. For example,
the iso-damping property can be pursued, namely the

146



Acta Polytechnica Vol. 52 No. 5/2012

capability of the control system to achieve the same
phase margin (i.e.that is, the same overshoot in the
set-point step response) independently from (moderate)
variations of the process gain [4]. Further, the minimum
integrated absolute error when a step signal is applied to
the set-point or to the load disturbance can be decreased
with respect to standard PID controllers [24, 26].

However, FOPID controllers are more difficult to im-
plement (the fractional controller has to be approximated
by a usually high-order integer-order controller) and, in
spite of the theoretical results that have been recently
achieved, there is still much work to be done before they
are ready for widespread use in industrial settings. In
particular, the effectiveness of the tuning rules that have
been devised needs to be fully demonstrated and the
substitution of classical PID controllers with FOPID
controllers in control structures (see Section 4) still has
to be addressed, as has the presence of the additional
functionalities mentioned above (e.g. set-point weight,
anti-windup, feedforward action, etc.) which makes
standard industrial controllers successful in practical
applications.

9 Event-based PID controllers
The recent introduction of wireless transmitters and
wireless actuators in the process industry has motivated
new interest in PID modifications that allow effective
control using nonperiodic information updates. Indeed,
the underlying assumption in process control has always
been that control is executed on a periodic basis, and
that a new measurement value is available each execution.
However, it is well known that in some processes a small
stationary control error or smooth oscillations of the
process output around the set-point may not constitute
hard design constraints, but a reduction in the information
exchanged between the agents that take part in the
control loop (sensors, controllers, actuators) can be one of
the tightest requirements. In fact, when wireless sensors
and actuators are involved, reduction of the information
flow implies a decrement of computing operations and
transmissions, and thus longer lifetime of batteries. With
these demands, one of the most convenient strategies is
to use event-based sampling and control approaches. In
particular, in order to minimize power consumption,
a wireless transmitter may transmit a new measurement
only if the measurement has changed by a significant
amount or, in general, when a logical condition becomes
true. In the process control field, the logical condition is
usually a composition of Boolean operations, where the
variables are the signals (or a function of them such
as an estimation, the derivative, the integration, etc.)
that the sensor receives from the process or the control
action produced by a controller [1, 31, 37]. In general, in
event-based control strategies, the controller can be
divided into four logical blocks as shown in Figure 2: the
sensor unit (SU), the control unit (CU), the actuator

unit (AU) and a governor (G). The units and their tasks
can be described as follows:

• The sensor unit is composed of the sensor and its
on-board intelligence. Its task is to measure the
process output and to calculate the error between
the measured signal and a constant set-point value
received from the governor.

• The control unit implements the control algorithm,
which determines the control action by taking into
account the last received sampled error and sends it
to the actuator unit.

• The actuator unit receives the control action signal
from the control unit and applies it to the actuator.
The governor, which, in practice, can be implemented
together with one of the previous two blocks, receives
the desired set-point value from a user interface or
from a hierarchically higher controller, and sends it
to the sensor unit.

These blocks can be implemented in a unique machine
or in two or more physical entities. In this last case,
the data has to be sent from one to each of the others
in a network. It is clear that communication between
two entities implies more effort than exchanging the
data into a single machine, especially when they are
battery-powered. For this reason, it is recommended to
use event-triggered data exchanging for all the signals
sent between two machines and normal time-driven
sampling for the data elaborated by a unique machine.
This implies that the control system has to be designed
to deal effectively with an asynchronous sampling rate,
and this opens new challenges both from a theoretical
viewpoint and from a practical viewpoint, as the timing
of the events influences the system performance and
limit cycles may arise. Further, in addition to the PID
gains, there are in general other parameters (threshold
values) employed in the control algorithm that have
to be tuned, thus making the overall control design
more complex. For reasons such as these, in recent
years event-based sampling and control techniques have
received special attention from several research groups,
and various effective methodologies have been proposed.
Among the various solutions, it is worth mentioning the
PIDplus technique, which has already been applied
successfully in industry [6]. It involves restructuring the
PID controller to reflect the reset contribution for the
expected process response since the last measurement
update. The so-called symmetric send-on-delta (SSOD)
PI controller has recently been presented [5]. The
approach that is employed is based on quantization of
the sampled signal by a quantity multiple of a given
parameter ∆, so that the relationship between the input
and output of the event-generator block is symmetric
with respect to the origin. In this context, necessary and
sufficient conditions on the controller parameters for the
existence of equilibrium points without limit cycles

147



Acta Polytechnica Vol. 52 No. 5/2012

CUSU AU

G

Plant

Figure 2: Scheme of a generic event based control
strategy. The dashed arrows indicate the possibility of
event-triggered data transmission.

can be determined, and this can be exploited to devise
effective tuning rules. It is also shown that the choice of
parameter ∆ does not influence the stability of the
system, and it can therefore be selected just in order to
handle the trade-off between reducing the desired number
of events and reducing the steady-state error.

10 CACSD tools
The availability of more and more sophisticated and high
performance software tools has favored the realization
of more and more effective Computed Aided Control
System Design (CACSD) tools specifically designed
for PID-based control systems [10, 12, 30]. CACSD
tools surely contributes to the rapid dissemination of
new methodologies through researchers and, most of
all, makes their applicability much easier in industry,
as theoretical issues can be made transparent to the
user. Moreover, the user can often better understand
the design procedure and the physical meaning of the
design parameters. It is therefore recognized that each
new proposed design methodology should also be made
available in a suitable software tool.

11 Conclusions
Despite the long history of PID controller research
and application, there are still many open issues and
challenges related to many different aspects of the overall
industrial control system design. Indeed, the need to
keep improving the performance of control systems while
keeping them as simple as possible, together with the
need to satisfy technological requirements, calls for new
methodologies to be applied in this field. A (surely
not exhaustive) selection of relevant topics that are
currently the subject of investigation has been presented

here, highlighting in particular the requirements that
newly-devised methodologies should satisfy.

Acknowledgements

The author thanks Manuel Beschi, Jesús Chacón,
Sebastián Dormido, Fabrizio Padula, Aurelio Piazzi, José
Sánchez and Massimiliano Veronesi for collaborating with
him in various research fields on PID controllers.

References

[1] K.E. Årzèn. A simple event-based PID controller.
In Proceedings of 14th World Congress of IFAC.
Bejining, China, 1999.

[2] K. J. Åström, T. Hägglund. Revisiting the
Ziegler–Nichols step response method for PID
control. Journal of Process Control 14:635–650,
2004.

[3] K. J. Åström, T. Hägglund. Advanced PID Control.
ISA Press, Research Triangle Park, USA, 2006.

[4] R. S. Barbosa, J. A. Tenreiro Machado, I. M.
Ferreira. Tuning of PID controllers based on
Bode’s ideal transfer function. Nonlinear Dynamics
38:305–321, 2004.

[5] M. Beschi, S. Dormido, J. Sanchez, A. Visioli.
Characterization of symmetric send-on-delta PI
controllers. Journal of Process Control, 2012.

[6] T. L. Blevins. PID advances in industrial control.
In Proceedings IFAC Conference on Advances in
PID Control. Brescia, I, 2012.

[7] J. Garrido, F. Vázquez, F. Morilla. Centralized
multivariable control by simplified decoupling.
Journal of Process Control 22:1044–1062, 2012.

[8] E. Grassi, K. Tsakalis. PID controller tuning by
frequency loop-shaping: Application to diffusion
furnace temperature control. IEEE Transactions on
Control Systems Technologies 8:842–847, 2000.

[9] C. Grimholt, S. Skogestad. Optimal PI-control and
verification of the SIMC tuning rule. In Proceedings
IFAC Conference on Advances in PID Control.
Brescia, I, 2012.

[10] J. L. Guzman, K. J. Åström, S. Dormido, et al.
Interactive learning modules for PID control. IEEE
Control Systems Magazine 28:118–134, 2008.

[11] J. L. Guzmán, T. Hägglund. Simple tuning rules
for feedforward compensators. Journal of Process
Control 21:92–102, 2011.

148



Acta Polytechnica Vol. 52 No. 5/2012

[12] J. L. Guzmán, P. García, T. Hägglund, et al.
Interactive tool for analysis of time-delay systems
with dead-time compensators. Control Engineering
Practice 16:824–835, 2008.

[13] T. Hägglund. Signal filtering in PID control. In
Proceedings IFAC Conference on Advances in PID
Control. Brescia, I, 2012.

[14] M. Jelali. An overview of control performance
assessment technology and industrial applications.
Control Engineering Practice 14:441–466, 2006.

[15] M. A. Johnson, M. H. Moradi (eds.). PID control -
New Identification and Design Methods. Springer,
London, UK, 2005.

[16] B.-S. Ko, T. F. Edgar. PID control performance
assessment: the single-loop case. AIChE Journal
50:1211–1218, 2004.

[17] A. Leva, M. Maggio. A systematic way to extend
ideal pid tuning rules to the real structure. Journal
of Process Control 21:130–136, 2011.

[18] A. Leva, S. Negro, A. V. Papadopoulos. PI/PID
autotuning with contextual model parametrisation.
Journal of Process Control 20:452–463, 2010.

[19] T. Liu, W. Zhang, F. Gao. Analytical decoupling
control strategy using a unity feedback control
structure for MIMO processes with time delays.
Journal of Process Control 17:173–186, 2007.

[20] C. A. Monje, B. M. Vinagre, V. Feliu, Y. Q. Chen.
Tuning and auto-tuning of fractional order controllers
for industry applications. Control Engineering
Practice 16:798–812, 2008.

[21] P. Nordfeldti, T. Hägglund. Decoupler and PID
controller design of TITO systems. Journal of
Process Control 16:923–936, 2006.

[22] J. E. Normey-Rico, E. F. Camacho. Dead-time
compensators: A survey. Control Engineering
Practice 16:407–428, 2008.

[23] A. O’Dwyer. Handbook of PI and PID Tuning
Rules. Imperial College Press, 2006.

[24] F. Padula, A. Visioli. Tuning rules for optimal PID
and fractional-order PID controllers. Journal of
Process Control 21:69–81, 2011.

[25] F. Padula, A. Visioli. On the stabilizing PID
controllers for integral processes. IEEE Transactions
on Automatic Control 57:494–499, 2012.

[26] F. Padula, A. Visioli. Optimal tuning rules for
proportional-integral-derivative and fractional-
order proportional-integral-derivative controllers
for integral and unstable processes. IET Control
Theory and Applications 6:776–786, 2012.

[27] M. Petersson, K.E. Årzèn, T. Hägglund. A compari-
son of two feedforward control structure assessment
methods. International Journal of Adaptive Control
and Signal Processing 17:609–624, 2003.

[28] A. Piazzi, A. Visioli. A noncausal approach for PID
control. Journal of Process Control 16:831–843,
2006.

[29] S. Piccagli, A. Visioli. An optimal feedforward
control design for the set-point following of mimo
processes. Journal of Process Control 19:978–984,
2009.

[30] PID controller laboratory, www.pidlab.com. http:
//www.pidlab.com.

[31] J. Sánchez, A. Visioli, S. Dormido. A two-degree-of-
freedom PI controller based on events. Journal of
Process Control 21:639–651, 2011.

[32] G. J. Silva, A. Datta, S. P. Bhattacharyya. New
results on the synthesis of PID controllers. IEEE
Transactions on Automatic Control 47:241–252,
2002.

[33] G. J. Silva, A. Datta, S. P. Bhattacharyya. PID
Controllers for Time-Delay Systems. Birkhauser,
Boston, USA, 2005.

[34] S. Skogestad. Simple analytic rules for model
reduction and PID controller tuning. Journal of
Process Control 13:291–309, 2003.

[35] S. Tavakoli, I. Griffin, P. J. Fleming. Tuning
of decentralized PI (PID) controllers for TITO
processes. Control Engineering Practice 14:1069–
1080, 2006.

[36] D. Valerio, J. Sa da Costa. Introduction to the
single-input, single-output fractional control. IET
Control Theory and Applications pp. 1033–1057,
2011.

[37] V. Vasyutynskyy, K. Kabitzsh. Event-based control:
Overview and generic model. In Proceedings of
8th IEEE International Workshop on Factory
Communication Systems. 2010.

[38] M. Veronesi, A. Visioli. Performance assessment
and retuning of PID controllers. Industrial and
Engineering Chemistry Research 48:2616–2623,
2009.

[39] M. Veronesi, A. Visioli. An industrial application of
a performance assessment and retuning technique
for PI controllers. ISA Transactions 49:244–248,
2010.

[40] M. Veronesi, A. Visioli. Performance assessment and
retuning of PID controllers for integral processes.
Journal of Process Control 20:261–269, 2010.

149



Acta Polytechnica Vol. 52 No. 5/2012

[41] M. Veronesi, A. Visioli. A simultaneous closed-loop
automatic tuning method for cascade controllers.
IET Control Theory and Applications 5:263–270,
2011.

[42] M. Veronesi, A. Visioli. Global minimum-variance
PID control. In Proceedings IFAC World Congress,
pp. 7891–7896. Milan, I, 2012.

[43] R. Vilanova. IMC based robust PID design: Tuning
guidelines and automatic tuning. Journal of Process
Control 18:61–70, 2008.

[44] R. Vilanova, O. Arrieta, P. Ponsa. IMC based
feedforward controller framework for disturbance
attenuation on uncertain systems. ISA Transactions
48:439–448, 2009.

[45] R. Vilanova, A. Visioli (eds.). PID Control in
the Third Millennium: Lessons Learned and New
Approaches. Springer, London, UK, 2012.

[46] A. Visiol, Q.-C. Zhong. Control of Integral Processes
with Dead Time. Springer, London, UK, 2010.

[47] A. Visioli. Fuzzy logic based set-point weight tuning
of PID controllers. IEEE Transactions on Systems,
Man, and Cybernetics - Part A 29:587–592, 1999.

[48] A. Visioli. Modified anti-windup scheme for PID
controllers. IEE Proceedings - Control Theory and
Applications 150(1):49–54, 2003.

[49] A. Visioli. A new design for a PID plus feedforward
controller. Journal of Process Control 14:455–461,
2004.

[50] A. Visioli. Design and tuning of a ratio controller.
Control Engineering Practice 13:485–497, 2005.

[51] A. Visioli. A new ratio control architecture. Industrial
and Engineering Chemistry Research 44:4617–4624,
2005.

[52] A. Visioli. Practical PID control. Springer, London,
UK, 2006.

[53] A. Visioli, A. Piazzi. Improving set-point follow-
ing performance of industrial controllers with a
fast dynamic inversion algorithm. Industrial and
Engineering Chemistry Research 42:1357–1362,
2003.

[54] Q.-G. Wang, Z. Ye, W. J. Cai, C. C. Hang. PID
Control for Multivariable Processes. Springer,
London, UK, 2008.

[55] C. C. Yu. Autotuning of PID Controllers: A Relay
Feedback Approach. Springer, London, UK, 2007.

150