Chapter 11

FUZZY FRACTIO NAL PID CO NTRO LLERS:

ANALYSIS, SYNTH ESIS AND IMPLEMENTATION

Ramiro S. Barbosa and Isabel S. Jesus†

GECAD − Knowledge Engineering and Decision Support Research Center Institute of Engineering / Polytechnic of Porto (ISEP/IPP)

Dept. of Electrical Engineering, Porto, Portugal

Abstract

Fuzzy logic controllers (FLC) have developed greatly in recent years. They offer easy and robust solutions to complex problems, allowing human reasoning to be applied to the control of systems. This chapter introduces the concepts of fractional calculus in FLC. The resulting fuzzy fractional PID controllers are investigated in terms of their structures and respective digital implementations. In the first part of chapter, a simple and effective tuning methodology is proposed and compared with traditional approaches. The methodology for tuning the fuzzy controllers is based on the prior knowledge of integer/fractional-order control strategy, making the procedure adequate to replace an existent controller in order to improve the system control performance. In the second part of chapter, are devised optimal fuzzy fractional PID controllers by using a particle swarm optimization algorithm. A comparative study between fuzzy integer and fractional PID controllers is presented. The feasibility and effectiveness of the proposed controllers are tested on several benchmark systems that are representative of industrial processes. The simulation results show the better performance of fractional controllers over the conventional PID or fuzzy PID controllers.

PACS: 05.45-a, 52.35.Mw, 96.50.Fm

Keywords: fuzzy control, PID controller, fractional PID control, fractional calculus, fuzzy fractional PID control, optimization

1.Introduction

In recent years, fractional-order (FO) PID controllers have been a fruitful field of research [3, 50, 51]. However, no effective and simple tuning rules still exist for these con-

E-mail address: rsb@isep.ipp.pt

†E-mail address: isj@isep.ipp.pt

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trollers as those given for the integer PID controllers [2]. The FO-PID controller involves an integrator/differentiator of real, irrational or, even, of complex order [11, 50]. It has been demonstrated the good performance of this type of controller, in comparison with the conventional PID algorithms. The FO-PID extends the capabilities of the classical counterpart and, thus, have a wider domain of application, such as in suspension systems, robotics, signal processing, control and diffusion [44, 50, 51, 33]. On the other hand, the FLC have also been successfully applied in the control of many physical systems, particularly those with uncertainty, unmodelled, disturbed and/or nonlinear dynamics [36, 37, 18, 27].

Fuzzy control emerged on the foundations of Zadeh’s fuzzy set theory [43, 12, 22]. This kind of control is based on the ability of a human being to find solutions for particular problematic situations. It is well know from our experience, that humans have the ability to simultaneously process a large amount of information and make effective decisions, although neither input information nor consequent actions are precisely defined. Through multivalent fuzzy logic, linguistic expressions in antecedent and consequent parts of IF-THEN rules describing the operator’s actions can be efficaciously converted into a fully-structured control algorithm. The fuzzy method offer a systematic procedure to design controllers for many kind of systems, that often leads to a better performance than that of the conventional PID controller. It is a methodology of intelligent control that mimics human thinking and reacting by using a multivalent fuzzy logic and elements of artificial intelligence. Many applications using FLC can be found in [40, 36, 48, 53, 27].

It has been proved that the use of the fuzzy fractional controllers improved the results for many kind of systems, since it gives additional flexibility to the design. In this line of thought, many applications using this type of controllers were developed in the last few years [6, 3, 28, 4, 5, 17, 16, 47, 29, 30]. In [6, 3], the authors proposed the use of fuzzy fractional PD and PID controllers in the control of integer plants. They proved their effectiveness and superior robustness when compared with conventional integer or fuzzy PID controllers. In [4, 5] the fuzzy FO-PID controllers are tested on several fractional-order plants showing their superior control system performance. In [29, 30] are devised optimal fuzzy FO-PID controlers for nonlinear control systems. The above studies are devoted to discrete fractional-order PID controllers and their digital implementation. Other studies using continuous fuzzy FO-PID controllers are found in [18, 17, 16, 47].

In the first part of chapter, are combined the features of fuzzy controllers with those of fractional controllers of PID-type. The resulting fuzzy FO-PID controller is investigated in terms of its digital implementation and robustness. The combined advantages of the two controllers results in a better controller with superior robustness and wider domain of application. The tuning methodology of these controllers is based on the prior knowledge of integer or fractional-order control. First, a fractional-order controller is built and tuned (or used one already implemented). Then, we replace it with a linear fuzzy fractional controller displaying exactly the same step response. After, we make the controller nonlinear and fine tune it in order to get better control of the system. The fuzzy fractional controller will give, at least, the same performance of its linear counterpart.

In the second part of chapter, are designed optimal fuzzy FO-PID controllers by using particle swarm optimization (PSO) algorithm. PSO is one of the latest evolutionary techniques inspired by social bahavior of bird flocking or fish schooling [20, 21, 35]. A fractional nonlinear control system with saturation in actuator is analyzed in terms of step

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responses, variation of controller parameters and time domain specifications as the fractional order of the fuzzy controller is changed. The influence of the order of the discrete rational approximations in control system performance is also studied.

The chapter is organized as follows. Section 2 presents the fundamentals of continuous and discrete fractional-order PID controllers. Section 3 outlines a tuning methodology for the design of fuzzy FO-PID controllers. Two decomposed structures for fuzzy FO-PID controllers are presented. In section 4, are devised optimal fuzzy FO-PID controllers using the PSO algorithm. The proposed fuzzy FO-PID controllers are tested and assessed their effectiveness and robustness in the control of integer and fractional-order plants commonly found in industry. Finally, section 5 draws the main conclusions.

2.Basics of Fractional PID Controllers

The fractional-order controller of PID-type, usually named PIλ Dµ controller, may be

where λ and µ are the orders of the fractional integrator and differentiator, respectively. The parameters Kp, Ki and Kd are correspondingly the proportional, integral and derivative gains of the controller. Clearly, taking (λ, µ) ≡ {(1, 1) , (1, 0) , (0, 1) , (0, 0)} we get the classical {PID, PI, PD, P}-controllers, respectively. Other PID controllers are possible, namely: PDβ controller, PIα controller, PIDβ controller, and so on. The PIλDµ-controller is more flexible and gives the possibility of adjusting more carefully the dynamical proprieties of a control system [51, 6].

where D( ) (≡ 0Dtα) denotes the differential operator of integration and differentiation (differintegral) to a fractional-order α = {−λ, µ} <.

The two most commonly used definitions for the differintegral are the Riemann-

where n − 1 < α < n, n is an integer, f (t) is the applied function, and (x) represents the Gamma function of x.

The Grunwald¨ -Letnikov definition is (α <):

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262 Ramiro S. Barbosa and Isabel S. Jesus

where h is the time increment and [v] means the integer part of v.

The ”memory” effect of these operators is demonstrated by above equations, where the convolution integral in (3) and the infinite series in (4), reveal the unlimited memory of these operators, ideal for modeling hereditary and memory properties in physical systems and materials.

The fractional integrals and derivatives can also be defined in the s-domain. Consider-

where L represents the Laplace operator and F (s) = L {f (t)}. Definition (5) gives an easily interpretation of the fractional-order operator in the frequency domain. In fact, the open-loop Bode diagrams of amplitude and phase of sα have correspondingly a slope of 20α dB/dec and a constant phase positionated at απ/2 rad over the entire frequency domain [46, 7, 15].

From a control (and signal) processing perspective, approach (4) seems to be the most useful and intuitive, particularly for a discrete-time implementation [10, 39]. Thus, using (4), a discrete fractional PIλDµ control equation can be obtained from (2) as (h ≈ T , T is the sampling period):

Replacing (7) in (6), the discrete PIλDµ controller is then expressed as:

The control equation (8) shows that the current value of control signal u (k) depends on all previous values of error e (k), making the computation too heavy as time increases and so unsuitable for a practical implementation of these algorithms. This fact illustrates the global character (i.e., unlimited memory) of the fractional-order operators. For practical implementation of fractional integral and derivative (7) we often apply the ”short memory principle” [50], resulting in expression:

where v = 0 for k < L/T or v = k − L/T for k > L/T ; L is the memory length and

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