Chapter 6

TEMPERATURE CONTROL OF A DIFFUSIVE MEDIUM

USING THE CRONE APPROACH

Fady Christophy1,2, , Xavier Moreau1,†, Roy Abi Zeid Daou2

and Riad Assaf1,2,‡

1University of Bordeaux, Laboratory IMS, Talence, Bordeaux, France

2Lebanese German University, Faculty of Public Health, Biomedical Technologies department, Sahel Alma, Jounieh, Lebanon

Abstract

This chapter presents the design of the temperature control of a diffusive medium by using a unique robust controller for three different materials: aluminum, copper and iron. For the control-system design, the aluminum is selected as the material defining the nominal model. Then, the second generation CRONE control is used because the parametric uncertainty (due to the copper and the iron) leads to variations of open-loop gain. Finally, the responses in frequency-domain and in time-domain illustrate the influence of the position of the sensor versus the actuator on the stability degree robustness.

1. Introduction

The concept of fractional differentiation is an old one that was born at the end of the 17th century when L’Hospital and Leibniz exchanged on the meaning of half order derivatives

[Oldham, 1974]. Since that date, many mathematicians worked on this concept from a mathematical point of view. The most important contributions in this domain came in the 19th century with Riemann and Liouville [Miller, 1993] when they gave a coherent definition of a fractional derivation.

However, the applications on the fractional order systems started in the last quarter of the 20th century [Oustaloup, 1981]. In the last forty years, work on such systems propagated in

E-mail address: fady.chrystophy@u-bordeaux.fr.

† E-mail address: xavier.moreau@u-bordeaux.fr.

‡ E-mail address: riad.assaf@u-bordeaux.fr.

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several fields as the electrochemical processes [Kuhn, 2005] [Sabatier, 2006], dielectric polarization [Bohannan, 2000], induction machines [Benchellal, 2005] [Canat, 2005], viscoelastic materials [Moreau, 2002], suspension system [Abi Zeid Daou, 2010] [Abi Zeid Daou, 2011.a], automatic control [Oustaloup, 1983] [Oustaloup, 1995] [Lanusse, 1994] [Abi Zeid Daou, 2011.b], physiological movements as the muscle contraction [Sommacal, 2008], the respiratory process [Ionescu, 2011], thermal diffusion [Battaglia, 2001] [Agrawal, 2004] [Kusiak, 2005] [Melchior, 2007], and so on. In general, the development of diffusion phenomena equations naturally leads to fractional order systems [Podlubny, 2005].

The aim of this study is to design the temperature control of a diffusive medium by using an unique robust controller for three different materials (aluminum, copper and iron) constituting the medium. In more details, this chapter is divided as follows. In section 2, the material properties and the diffusive interface relations are recalled. Section 3 presents the SISO Crone control-system design used in the frequency-domain for the temperature control of the uncertain diffusive medium. Responses in frequency-domain and in time-domain are given to illustrate the influence of the position of the sensor versus the actuator on the stability degree robustness. At the end, a conclusion sums up all the results and some future works are proposed.

2. Modelling

Consider a homogeneous one-dimensional semi-infinite medium, of conductivity , of diffusivity and of initial temperature zero at every point (figure 1). It is subjected to a flux

density t (W/m2) on the outgoing normal surface n . This results in a temperature change, denoted T(x,t), function of time t and of abscissa x x 0 ; of the measuring temperature point inside the medium.

(t)

Semi-infinite medium

Figure 1. Illustration of the study area defining the process.

Initially, the goal is to achieve control of the temperature T(0,t) at x = 0 with a sensor placed in this position. For this we have a heating resistor bonded to a surface S of 1 cm2 at

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the end x = 0 of a bar (high thermal conductivity glue) for generating a command u(t) of flux f(t), or a flux density (t) = f(t)/S. The maximum flux Umax produced by the resistance is 12 W (1 A under 12 V over 1 cm2, thus 12 104 W/m2). The medium temperature is measured with a platinum sensor of type PT100 via a voltage amplifier [Sabatier, 2008].

In a second step, the goal is always to regulate the temperature T(0,t) at x = 0 but with a sensor placed at a position x > 0. An analysis of the influence of the position x of the sensor on the performance of the command is given.

2.1. Fractional Model at x = 0

At x = 0, the transfer function G(0,s) of the process is given by [Abi Zeid Daou, 2012]:

0 is an uncertain parameter, 0 0 min ; 0 max , not only because the parameters , et Cp (relation (2)) are given with an uncertainty (data values at 20°C), but also because the objective of the robust control for this study is to use the same control for different materials constituting the semi-infinite medium (aluminum, copper, iron, …).

In this study, we chose three materials: aluminum, copper and iron. Table 1 summarizes the materials’ characteristics necessary to calculate 0.

Table 1. Characteristics of the materials used

For these three materials, we get 0 0.735 ; 3.475 10 1 rad/s .

For the remainder of the control-system design, the aluminum is selected as a material defining the nominal model, hence 0nom = 1.76 10-1 rad/s. Thus, the uncertainties of 0 result in gain variations of G(0,j ) of a factor 2.17 (between the iron and copper we have: (3.475/0.735)0.5= 2.17). These gain variations cause gain variations in open loop, = 2.17, hence the use of the second generation CRONE control [Oustaloup, 1995].

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Figure 2 shows the Bode plots of G(0,j ) for the three materials.

Figure 2. Bode diagram of G(0,j ).

Moreover, the analytical expression of the step response T(0,t) for a flux f(t) of amplitude U0 is given by [Schneider, 1957] [Özisik, 1980] [Özisik, 1985] [Battaglia, 2008]:

Time (s)

Figure 3. Step responses of the process at x = 0 for a flux of amplitude U0 = 1 W.

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2.2. Fractional Model at x > 0

At x > 0, the transfer function G(x,s) of the process is expressed by [Assaf, 2012]:

Figure 4. Step responses of the process at x = 5mm for a flux of amplitude U0 = 1 W.

with

or, by introducing the nominal transfer function Gnom(0,s) established with the aluminum at x = 0,

Remark. The exponential function can be interpreted as a multiplicative structural uncertainty m(x):

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Moreover, the analytical expression of the step response T(x,t) for a flux f(t) of amplitude U0 is given by [Battaglia, 2008]:

where erfc(.) is the complementary error function.

Figure 4 shows for the three materials the step responses at x = 5mm for an input of amplitude U0 = 1 W.

2.3. Rational Model for Temporal Simulation x 0

If the numerical simulation of the fractional integrator I0.5(s) and of function E x, s cause

no problem in the frequency domain, it’s not the same for the time domain. In fact, apart from some analytical signals such as the Dirac pulse or unit step for which analytical expressions of time responses are well known, time simulation of the responses of the process model to any of the other inputs is more problematic. This is the case in particular when the process model is immersed in a control loop. It is then necessary to implement methods for approximating the fractional model to have a rational model.

Thus, the fractional integrator I0.5(s) is first approximated by a fractional integrator

with a < b and K0 a constant computed for the two integrators to have the same gain at midband m a b :

or by introducing the constant A = b/a > 1,

0.5

K0 m . (11)

A0.25

Then the fractional integrator bounded in frequency is approximated by a rational integrator frequency bounded [Oustaloup, 1995]:

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Finally, concerning the exponential function E x, s , we use the limited Taylor expansion of ez when z tends to zero:

Remark. z tends to zero is equivalent to tendering s to zero, and therefore to zero and t to infinity (time-frequency duality), or, considering the expression (5) of x, x tends to zero. Thus, this approximation is more accurate than the truncation order K is high, when x is small and it is with long time. In fact, this approximation is legitimate when << x or equivalently, when t >> x=1/ x.

In a first step, K is limited to 1, hence:

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