^{1}

^{2,3,4}

^{1}

^{2}

^{3}

^{4}

This paper addresses a new approach for modeling of versatile controllers in industrial automation and process control systems such as pneumatic controllers. Some fractional order dynamical models are developed for pressure and pneumatic systems with bellows-nozzle-flapper configuration. In the light of fractional calculus, a fractional order derivative-derivative (FrDD) controller and integral-derivative (FrID) are remodeled. Numerical simulations illustrate the application of the obtained theoretical results in simple examples.

Fractional calculus is a powerful mathematical tool with a long history, but its application to engineering and modeling of physical systems has attracted much attention only in recent years [

Pneumatic controllers are essential parts of the industrial automation systems containing several diaphragms and bellows [

Recently in [

In this paper, in order to consider some real and nonideal features of a pneumatic structure, using a four-parameter fractional derivative Zener model for viscoelastic materials [

This paper is organized as follows. Section

In this section, a brief background of fractional calculus is presented. The definition of the fractional integral is the extension of Cauchy formula for evaluating the integration. The

The Laplace transform of RL fractional derivative of order

Another definition for fractional derivative has been introduced by Caputo:

In contrast to RL operator, the Laplace transform of the Caputo operator needs the integer order derivative of the function as the initial conditions, as shown by the following relation:

For more information about fractional calculus see, for example, [

Many industrial processes and pneumatic controllers involve the flow of a gas or air through connected pipelines and pressure vessels. Thus, it is logical to introduce a parameter for characterizing their specifications. Some quantities in pipelines and pressure vessels can be viewed as resistance (

It can be shown that during the change of state of a polytropic process, from isothermal to adiabatic state, the capacitance is constant and can be obtained as [

A conventional apparatus in the industrial pneumatic control systems is the nozzle-flapper configuration. The nozzle-flapper amplifier converts displacement into a pressure signal. Since typical industrial process control systems require large output power to operate large pneumatic actuating valves, the power amplification of the nozzle-flapper amplifier is usually insufficient. To overcome this problem, a pneumatic relay can be connected to the nozzle-flapper. Therefore, a complete pneumatic amplifier is composed of two stages: nozzle-flapper as the first and pneumatic relay as the second amplifier stages. A schematic diagram of such a configuration is depicted in Figure

Schematic diagram of a force-distance type of pneumatic proportional controller.

Assuming that the relationship between the variation in the nozzle back pressure

A simple model of bleed-type relay.

Following [

For the flapper, since there are two small movements (

Block diagram for the fractional order nozzle-flapper-relay.

The transfer function between the control pressure

As can be seen, this controller configuration yields a fractional order proper transfer function. In the following sections, by inserting a restrictor in the feedback path, a fractional order derivative and integral performance are achieved.

It should be emphasized that in using transfer function (by the definition) all initial conditions must be set to zero. Since we have utilized the Caputo’s fractional derivative, all initial conditions are those which have been considered in the integer order cases. In other words, from (

Consider the pneumatic controller configuration shown in Figure

Pneumatic controller with a restriction in the feedback path.

Assuming a small positive step change in the error signal

Thus, a simplified block diagram for the controller which is called fractional order derivative controller can be developed as depicted in Figure

A simplified block diagram for fractional order PD controller.

The transfer function of the control pressure with respect to the error signal is

Note that if

Similar to the previous discussions in Section

Pneumatic controller with a restriction in the feedback path which depicts an integral performance.

The bellows denoted by I is connected to the control pressure source without any restriction, though a restriction can be implemented for it. The bellows denoted by II is connected to the control pressure source through a restriction denoted by

A simplified block diagram for fractional order PI controller.

In this block diagram we denote the fractional order model for the restrictor-bellows II with orders

Using Mason’s formula, one can develop the transfer function for the block diagram sketched in Figure

Notice that if

Using a similar method, one can develop a fractional PID controller for a nozzle-flapper-relay configuration.

In this section some simulations are given for the developed theory in the previous sections.

Consider the following values for a nozzle-flapper amplifier discussed in Section

The step responses are depicted in Figure

Step response of the nozzle-amplifier-relay (

The values of

Now consider the nozzle-flapper-relay studied in Section

Moreover the step changes in the error signal yield a differentiation performance for the apparatus designed in Figure

Step response of the fractional order derivative nozzle-flapper-relay for (

As the final example consider the nozzle-flapper-relay studied in Section

Step response of the fractional order derivative nozzle-flapper-relay for (

We have employed the Simulink package to simulate these numerical examples. All standard blocks have been taken from its library. However, the fractional order derivative blocks have been generated via Oustaloup approximation frequency technique which is a routine and reliable method in simulating the fractional order control systems [

In this paper in the light of fractional calculus, some new models have been developed for three configurations of nozzle-flapper-relay amplifier. Indeed using a typical form of the four-parameter fractional derivative Zener model for viscoelastic materials, a fractional order model has been provided for the diaphragm of pneumatic relays and the spring property of the bellows. After developing block diagrams for each configuration, a fractional order derivative and integral controller with two freedoms in orders have been obtained. The fractional orders that are imposed in the models can be used as other adjustable parameters of the controllers which inherently depend upon the material of the bellows and diaphragms. Some numerical simulations are proposed to clarify the obtained results.

The authors declare that there is no conflict of interests regarding the publication of this paper.