29-05-2012, 11:58 AM
A Proposal for Optimal Tuning of Fractional Order Proportional Integral- Proportional Derivative (PIα-PDβ) Controllers
[attachment=23296]
Abstract
In this paper we propose a fractional PIα-PDβ controller
that tuned with integral performance criterion. The orders of
the integral and derivative parts, α and β, respectively, are
fractional. This controller is a generalization of a conventional
PI-PD controller. This expansion could provide much more
flexibility than a conventional PI-PD controller design. The
tuning method is based on the solution of an optimization
problem. For setting the six parameters of the fractional PIα-
PDβ controller, the proposed method is based on the minimum
integral squared error (ISE) criterion with a minimum control
effort. The integral criterion is calculated by using Hall-
Sartaurius method. Two cases of study are given; the results
demonstrate that gains and orders optimization leads to better
transient performance of the proposed fractional PIα-PDβ
control structure.
1. Introduction
Despite the significant developments of recent years
in control theory and technology, Proportional-Integral-
Derivative (PID) controller is the most industrially used
control algorithm. This can be explained by its
simplicity, low cost and ability to solve most of control
problems [1]. Difficulties are often faced, however,
while controlling plants with resonance, integral or
unstable transfer functions [2].
PI-PD controller proposed by S. Majhi and D.P.
Atherton [3] is a modified form of PID controller. The
PI-PD controller, which corresponds to PI control of the
plant transfer function changed by the PD feedback, can
produce improved control in several situations. His
implementation avoids the derivative kick problem
associated with derivative action in the forward path.
Further, the PD in the inner feedback loop can enable
placement of the open loop poles in appropriate
positions, thereby providing good control for open loop
system transfer functions having resonances, unstable or
integrating poles [4].
Fractional calculus is a mathematical topic with more
than 300 years old history but its application in physics
and engineering has been reported only in the recent
years. In the last decades, there are growing numbers of
applications of fractional calculus in different areas of
control engineering [5-7]. This fact is due to a better
understanding of the fractional calculus potentialities.
Many researchers have been interested in the use
and tuning of this type of controller and a lot of work is
now running in order to define new effective tuning
techniques and structure control for the fractional
controllers using classical control theory. For example,
in [8] the authors have showed that the fractional PIλDμ
controllers provide a more flexible tuning strategy that
can achieve control requirements which can never be
realized with classical PID controllers. In [9] a design
method of the fractional PIλDμ controller is derived
based on specified gain and phase margins with a
minimum integral squared error (ISE) criterion. A
proposition for the implementation of a fractional PIλ
controller for first-order systems with long time delay
has been given in [10]. In [11] the objective of the
design technique is to find out optimum settings for a
fractional PIλDμ controller in order to fulfill five
different design specifications for the closed-loop
system. In [12] the objective of the design of the
fractional PIλDμ controller is such that the feedback
control system fulfills different specifications regarding
robustness to plant uncertainties, load disturbances and
high frequency noise. Besides, the authors have
proposed an auto-tuning method for this kind of
controller.
In this work we propose the optimal parameters
tuning for a fractional order PIα-PDβ controller. The
purpose of this work is to illustrate how the fractional
order PIα-PDβ controller can improve the performance of
the control system. The parameters of the fractional
order PIα-PDβ are obtained by minimization of the
integral criterion. The presence of six optimized
parameters makes the task of designing PIα-PDβ more
challenging than conventional PI-PD controllers design,
also than other existing fractional controllers.
This paper is organized as follows. Section 2 we
978-1-4244-2728-4/09/$25.00 ©2009 IEEE
discuss the proposed fractional order PIα-PDβ control
structure, also we introduce the approximation of the
fractional PIα-PDβ controller by a rational function in a
limited frequency band of interest, the method of Hall-
Sartorius used for criterion calculus is presented. In
section 3, we introduce the basic ideas and the derived
formulations of the new conception strategy of the
fractional PIα-PDβ controller. In section 4, two cases of
study are presented for a position servo, to demonstrate
the advantages of the tuning method. Finally, section 5
draws the main conclusions and addresses some
perspectives of future developments.
2. Tools and Methods
2.1. Control Structure
The proposed fractional order PIα-PDβ controller is
performed in two steps: the first for the output of the
fractional order PIα controller and the second for the
output of the fractional order PDβ controller. By moving
the fractional order PDβ into an inner feedback loop, an
unstable or integrating process can be stabilized and then
can be controlled more effectively by the fractional order
PIα controller in the forward path. The final fractional
order PIα-PDβ controller combines these two individual
controllers together in an appropriate way. Therefore, the
control structure shown in Figure 1, which is known as a
fractional order PIα-PDβ control structure, is proposed.
[attachment=23296]
Abstract
In this paper we propose a fractional PIα-PDβ controller
that tuned with integral performance criterion. The orders of
the integral and derivative parts, α and β, respectively, are
fractional. This controller is a generalization of a conventional
PI-PD controller. This expansion could provide much more
flexibility than a conventional PI-PD controller design. The
tuning method is based on the solution of an optimization
problem. For setting the six parameters of the fractional PIα-
PDβ controller, the proposed method is based on the minimum
integral squared error (ISE) criterion with a minimum control
effort. The integral criterion is calculated by using Hall-
Sartaurius method. Two cases of study are given; the results
demonstrate that gains and orders optimization leads to better
transient performance of the proposed fractional PIα-PDβ
control structure.
1. Introduction
Despite the significant developments of recent years
in control theory and technology, Proportional-Integral-
Derivative (PID) controller is the most industrially used
control algorithm. This can be explained by its
simplicity, low cost and ability to solve most of control
problems [1]. Difficulties are often faced, however,
while controlling plants with resonance, integral or
unstable transfer functions [2].
PI-PD controller proposed by S. Majhi and D.P.
Atherton [3] is a modified form of PID controller. The
PI-PD controller, which corresponds to PI control of the
plant transfer function changed by the PD feedback, can
produce improved control in several situations. His
implementation avoids the derivative kick problem
associated with derivative action in the forward path.
Further, the PD in the inner feedback loop can enable
placement of the open loop poles in appropriate
positions, thereby providing good control for open loop
system transfer functions having resonances, unstable or
integrating poles [4].
Fractional calculus is a mathematical topic with more
than 300 years old history but its application in physics
and engineering has been reported only in the recent
years. In the last decades, there are growing numbers of
applications of fractional calculus in different areas of
control engineering [5-7]. This fact is due to a better
understanding of the fractional calculus potentialities.
Many researchers have been interested in the use
and tuning of this type of controller and a lot of work is
now running in order to define new effective tuning
techniques and structure control for the fractional
controllers using classical control theory. For example,
in [8] the authors have showed that the fractional PIλDμ
controllers provide a more flexible tuning strategy that
can achieve control requirements which can never be
realized with classical PID controllers. In [9] a design
method of the fractional PIλDμ controller is derived
based on specified gain and phase margins with a
minimum integral squared error (ISE) criterion. A
proposition for the implementation of a fractional PIλ
controller for first-order systems with long time delay
has been given in [10]. In [11] the objective of the
design technique is to find out optimum settings for a
fractional PIλDμ controller in order to fulfill five
different design specifications for the closed-loop
system. In [12] the objective of the design of the
fractional PIλDμ controller is such that the feedback
control system fulfills different specifications regarding
robustness to plant uncertainties, load disturbances and
high frequency noise. Besides, the authors have
proposed an auto-tuning method for this kind of
controller.
In this work we propose the optimal parameters
tuning for a fractional order PIα-PDβ controller. The
purpose of this work is to illustrate how the fractional
order PIα-PDβ controller can improve the performance of
the control system. The parameters of the fractional
order PIα-PDβ are obtained by minimization of the
integral criterion. The presence of six optimized
parameters makes the task of designing PIα-PDβ more
challenging than conventional PI-PD controllers design,
also than other existing fractional controllers.
This paper is organized as follows. Section 2 we
978-1-4244-2728-4/09/$25.00 ©2009 IEEE
discuss the proposed fractional order PIα-PDβ control
structure, also we introduce the approximation of the
fractional PIα-PDβ controller by a rational function in a
limited frequency band of interest, the method of Hall-
Sartorius used for criterion calculus is presented. In
section 3, we introduce the basic ideas and the derived
formulations of the new conception strategy of the
fractional PIα-PDβ controller. In section 4, two cases of
study are presented for a position servo, to demonstrate
the advantages of the tuning method. Finally, section 5
draws the main conclusions and addresses some
perspectives of future developments.
2. Tools and Methods
2.1. Control Structure
The proposed fractional order PIα-PDβ controller is
performed in two steps: the first for the output of the
fractional order PIα controller and the second for the
output of the fractional order PDβ controller. By moving
the fractional order PDβ into an inner feedback loop, an
unstable or integrating process can be stabilized and then
can be controlled more effectively by the fractional order
PIα controller in the forward path. The final fractional
order PIα-PDβ controller combines these two individual
controllers together in an appropriate way. Therefore, the
control structure shown in Figure 1, which is known as a
fractional order PIα-PDβ control structure, is proposed.