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Latin American applied research

Print version ISSN 0327-0793

Lat. Am. appl. res. vol.42 no.2 Bahía Blanca Apr. 2012

 

ARTICLES

Generalization of internal model control loops using fractional calculus

L. A. D. Isfer†, G. S. da Silva†,‡, M. K. Lenzi† and E. K. Lenzi§

† Departamento de Engenharia Química, Universidade Federal do Paraná, Curitiba/PR, 81531-980, Brazil. lenzi@ufpr.br
‡ KLABIN Papeis, Fazenda Monte Alegre, Telêmaco Borba/PR, 84274-000, Brazil. gsales@klabin.com.br
§ Departamento de Física, Universidade Estadual de Maringá, Maringá/PR, 87020-090, Brazil. eklenzi@dfi.uem.br


Abstract - Process control represents an important tool for meeting product quality, process safety and environmental regulation. Different control strategies have been recently proposed in the literature; however, internal model control (IMC) has received great attention. Fractional calculus represents a fast growing field involving non-integer order derivatives. The aim of this work is the application of fractional calculus to develop generalized internal model control loops transfer functions, which is presented in two different approaches: firstly, the process model is considered perfect, i.e., equal to the internal model; secondly, the internal model is described by fractional transfer function. Simulation results showed that the proposed generalization could successfully control an industrial oven and a biochemical reactor described by fractional models, allowing better results when compared to integer order IMC.

Keywords - Process Control; Fractional Differential Equation; Caputo Representation; Internal Model Control; Generalization.


 

I. INTRODUCTION

Control systems play a key role in chemical and biochemical plants operation focusing on high production meeting product quality, process safety and environmental regulation, (Lenzi et al., 2005). Literature reports different conventional control algorithms, which have been successfully applied to the control of industrial systems (Seborg, 1999), being internal model control (IMC) originally reported by Garcia and Morari (1982). It must be highlighted that IMC is a model-based design technique and it also allows model uncertainty and trade-offs between performance and robustness to considered in a more systematic fashion (Zheng and Hoo, 2004). Literature also reports extensions of IMC to control of nonlinear systems (Nitsche et al., 2007), tuning rules (Vilanova, 2008), gain scheduled control (Xie and Eisaka, 2008), control of unstable systems (Wang et al., 2001), use of IMC in feedforward control loops (Mawire and McPherson, 2008) and applications with fuzzy logic (Duan et al., 2008), among others.

Fractional differential order equations represent a fast growing research field nowadays (Das, 2008). Different applications of fractional calculus have been reported, i.e., diffusion studies (Lenzi et al., 2006), rheology (Craiem et al., 2008), process identification (Isfer et al., 2010a), process control (Isfer et al., 2010b), electroanalytical chemistry (Oldham and Spanier, 2006), among others (Hilfer, 2000). Further details regarding the formalism of fractional calculus are beyond the scope of this work and can be found elsewhere (Oldham and Spanier, 2006). Fractional control has been successfully applied to mechanical (Pommier et al., 2002) and eletromechanical systems (Sabatier et al., 2004). The use of IMC control and fractional calculus was firstly reported by Valerio and Sa da Costa (2006). In their manuscript, IMC control was only used as an alternative tool for fractional PID controller tuning. To the best of our knowledge, applications of IMC control loops (Brosilow and Joseph, 2002) only use classical calculus, no studies of IMC control loops involving fractional calculus were found in the open literature.

The aim of this work is the application of fractional calculus to develop generalized internal model control loops transfer functions. The study is divided into two parts. In the first, the process model is considered perfect, i.e., equal to the internal model. In the second part, the internal model is described by fractional transfer function. Finally, the proposed generalization is applied to simulation studies in order to control an industrial oven and a biochemical reactor described by fractional models.

II. THEORETICAL FRAMEWORK

A fractional derivate can be obtained by several approaches. However, in this work, only the Caputo representation, presented bellow, will be considered.

(1)

where mbm+1; β ∈ ℜ; m ∈ ℵ.

The first advantage of this representation is the fact that initial conditions of the fractional differential equations can be expressed in terms of integer order derivatives, which usually have physical interpretation. Secondly, for this representation, the fractional derivate of a constant function is zero allowing the use of the classical deviation variables approach, simplifying the solution of the mathematical problem. The IMC loop considered for this work is presented by Fig. 1.

Fig. 1: Internal Model Control (IMC) closed loop.

The transfer functions of the controller and actuator can be grouped into only one term given by their product. By using block algebra, the following expressions used to describe the dynamic behavior of the controlled variable (Y) and manipulated variable (X) of the closed loop can be derived.

(2a)
(2b)

The process model is usually divided into two terms, one concerning stable poles dynamics and the other containing unstable poles and zeros, resulting in Eq. (3).

(3)

According to Morari and Garcia (1982), one way to derive the IMC controller uses Eq. (4), which considers only the stable dynamics term and a tuning parameter filter. One can observe the model-based features of IMC by the use of part of the process transfer function. The filter parameters tc and r are used for the control loop tuning, in order to adjust the controller dynamic behavior.

(4)

where

In the first approach used for the IMC generalization proposed by this work, the process model is considered to be perfect. In this scenario, . If the process model has only stable dynamics, i.e., Gunstable =1, the IMC loop transfer function given by Eq. (2) is simplified to Eq. (5), i.e., for these hypothesis, the controlled variable behavior depends only on the filter. In order to generalize IMC loop transfer function two features of F(s) should be revisited. Firstly, a parameter can be introduced in Eq. (4). Secondly, parameters and r are allowed to be real numbers. These assumptions lead to Eq. (5), which describes the dynamic behavior of the controlled variable of generalized IMC loop transfer function for perfect model with no unstable dynamics.

(5)

Considering Yset-point(s) as unit step, i.e., Yset-point(s)=1/s, the inverse Laplace transform of Eq. (5), is given by Eq. (6), obtained by the use of the convolution theorem. This inverse transform can be found with the aid of Eq. (7) reported by Hartley and Lorenzo (1999), which represents a Laplace domain generalized transfer function with parameters a, v, k, r, that has inverse Laplace Transform given by Eq. (8).

(6)

where

(7)
(8)

In the second approach for IMC generalization, the control loop internal model is described by a fractional transfer function, as this type of transfer function usually provides a better description of the process behavior when compared to classical integer order transfer functions (Isfer et al., 2010a). The transfer functions given by Eq. (9) are commonly reported in literature to model the process dynamics. In this work, the internal model is described by the fractional two term transfer function presented bellow. It is also worth mentioning that this approach can be used for multi-term fractional order systems or even delayed systems.

(9)

Substituting the IMC generalized filter equation, i.e., Eq. (5) and Eq. (9) in Eq. (2), one can obtain the internal model control closed loop transfer function, which is given by Eq. (10).

(10)

This work focused on the particular case of r=1, which lead to Eq. (11),

(11)

The inverse Laplace transform of Eq. (11) is obtained with the aid of Mittag-Leffler function (Das, 2008) and is given by

(12)

where

III. RESULTS AND DISCUSSIONS

As mentioned before, the aim of this work is the generalization of IMC control loops using fractional calculus and its further application to control studies concerning previously identified fractional models of the CO2 concentration in a biochemical reactor (Isfer et al., 2010a) and the temperature behavior of an industrial oven (Isfer et al., 2010b) which are listed in Table 1.

Table 1. Transfer Functions considered for simulation.

A. Approach 01 - Perfect Model

IMC control of an oven

Firstly, the process model is considered perfect. From Eq. (5), it can be observed that for the assumptions here considered (model with no unstable dynamics), the dynamic behavior of the controlled variable of closed loop does not depend on the process model order. Consequently, as the aim of this work is the study of the fractional controller, an integer order process will be chosen and considered perfect. On the other hand, parameters r and cannot be randomly chosen. They must, instead, satisfy the physical realizability condition given by 1≤r, which is obtained from the transfer function that describes the dynamic behavior of the manipulated variable, given by Eq. (13), considering a integer order process.

(13)

The first set of results corresponds to simulation studies given by Eq. (6). Different values of parameters r, τC and were chosen taking into account realizability issues and are listed in Table 1. It can be also seen in Table 2, the ISE (integral of the square error) value of the controllers, which was evaluated for performance analysis. Case 0 is a benchmark of comparison, since it represents an integer order case. For simulation purposes, the convergence criterion is that absolute value of the series term is less than 10-50.

Table 2. IMC Parameters - Perfect Model - Oven Control.

Figure 2 presents the behavior of the controlled variable, focusing on the study of the influence of parameters and r. It can be observed that when is higher than 1, overshoot and oscillations are introduced in the behavior of the controlled variable, as one can note by comparison of Case II and Case III. Considering ISE values, Case I presents the best behavior, despite the overshoot present. It is worth mentioning the advantage of using the fractional filter if Case I is compared to Case 0.


Fig. 2: Controlled variable for perfect process model - influence of and r - oven control.

Figure 3 presents the behavior of the controlled variable, focusing on the study of the influence of parameter τC. Simulations considered the values of parameters and r of Case I. It can be observed that the higher the value of this parameter, the faster the controlled variable reaches the desired set-point, however, the shape of the dynamic behavior did not change.


Fig. 3: Controlled variable for perfect process model - influence of τc - oven control.

The dynamic behavior of the manipulated variable considering the process described by a fractional model is presented by Fig. 4 and by Fig. 5. It is important to notice that for the fractional controller, as the order of the transfer function denominator is greater than the order of the numerator, at t=0, the manipulated variable does not need to be greater than zero, differently when a integer order controller is used. It can be also observed from Figure 5 that for lower values of t, lower values of the manipulated variable are needed.


Fig. 4: Manipulated variable for perfect process model - oven control.


Fig. 5: Manipulated variable for perfect process model at initial values of time - oven control.

Consequently, it can be observed the use of fractional filter for IMC control provides better control of the oven system. Firstly, for the same value of τC, fractional values of r and provided a lower value of ISE. Secondly, fractional control provides lower values of the manipulated system, representing a more efficient system, as with lower values of the manipulated variable; the oven temperature control could be successfully achieved.

IMC control of a biochemical reactor

Regarding the biochemical reactor control, the controlled variable has the same behavior presented in Fig. 2, considering the same parameter values listed in Table 2. This happens because the models were derived using deviation variables and because of the perfect model hypothesis, the controlled variable does not depend on the process model. On the other hand, the manipulated variable presents different values which are presented in Fig. 6. It can be observed that the shape of the manipulated variable is now different when compared to the oven control, which probably happened due to the slower process dynamics. It is worth mentioning that fractional control Case I and integer control Case 0, present roughly the same behavior of the manipulated variable, but the fractional control lead to a faster controlled. In both oven a biochemical reactor control, the faster controller performance of the fractional system probably happened due to the memory effects (Lenzi et al., 2006) of the fractional derivates.


Fig. 6: Manipulated variable for perfect process model - biochemical reactor control.

B. Approach 02 - Fractional Internal Model

In the second part of this study, both internal model and f are considered as fractional transfer functions. Considering the process model and the internal model given by Eq. (10) and also for the particular case of r=1, Eq. (2b) is simplified to Eq. (14). Therefore, to address realizability issues, it is important to set parameter in such a way that a + 1 ≤ + 1. It is important to remember that from Eq. (12) realizability conditions need a ≤ + 1.

(14)

IMC control of an oven

Considering the transfer functions given in Table 1, Eq. (12) is rewritten as

(15)

Remembering that parameter r was set equal to 1, Table 3 presents the used values of parameter and τC used for simulation purposes, taking into account realizability conditions. Again, it is important to stress that no tuning rule was used, as the aim of this work is to evaluate the behavior of the control and the influence of the parameters.

Table 3. IMC Parameters - Fractional Internal Model - Oven Control.

From Figure 7, it can be observed that the oven can be successfully controlled using the generalized proposed IMC. It can be seen that higher values of τC lead to a faster controller loop, as ISE value is lowered. If τC is kept constant, higher values of lead to an increase in overshoot and introduce oscillation, while lower values of lead to an improvement in the control loop performance.


Fig. 7: Controlled variable for fractional internal model - oven control.

The dynamic behavior of the manipulated variable is presented in Fig. 8. It can be observed the same behavior for Approach 1, which is presented by Figure 4. Higher values of the manipulated variables are needed at the beginning of the control. Case I and Case V present roughly the same values, which are expected due to the controlled variable behavior.


Fig. 8: Manipulated variable for fractional internal model - oven control.

IV. CONCLUSIONS

The aim of this work is the application of fractional calculus to develop generalized internal model control loops transfer functions, which was achieved by two different approaches. In the first, the process model is considered perfect, i.e., equal to the internal model. In the second approach, the internal model is described by fractional transfer function, which usually provides better process description. Previously fractional identified models of an oven and a biochemical reactor were used for simulation purposes of the proposed approaches. The proposed approaches could successfully control the processes. It the first approach, results provided by the fractional filter were better when compared to integer order, more specifically, a reduction of 10% of ISE without the aid of tuning rules, for both oven and biochemical reactor control. In the second approach, comparison to integer order system was not possible due to the use of fractional models. On the other hand, control could also be achieved for the studied cases, by the use of the generalized IMC proposed by this work.

ACKNOWLEDGEMENTS
The authors are thanked to CNPq, Fundação Araucária and PRH24/ANP (Brazilian Agencies) for providing financial support for this project.

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Received: February 28, 2011.
Accepted: July 6, 2011.
Recommended by Subject Editor Jorge Solsona.

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