Figure 1. Fuzzy fractional PID controlled system.

Note that (9) is given in the form of a FIR filter [24]. Other discrete-time approximations in the form of IIR filters are also possible [54, 8, 9, 10, 13, 14, 45]. In section 4 is presented a method to obtain digital rational approximations of fractional-order derivatives and integrals.

3.Tuning Methodology for Fuzzy Fractional PID Controllers

In this section, we will apply fuzzy logic control (FLC) for the design of fuzzy FO-PID controlled systems [36, 48]. Despite of variety of possible fuzzy controller structures, the controller is usually arranged in cascade with the system being controlled. This type of arrangement is shown in Figure 1 and will be used in this study. Other fuzzy schemes may be adopted such as fuzzy supervisory control and adaptive fuzzy control [48, 40, 26].

The main idea here is to explore the fact that the FLC, under certain conditions, is equivalent to a PID controller [43, 37, 26] and to extrapolate this fact for the case of fractionalorder controllers. In a certain sense, the fuzzy PID controllers are a special case of the more general fuzzy FO-PID controllers, in which are involved two extra tuning parameters: the fractional orders (λ, µ) of controller equation (6).

The basic form of a fuzzy controller is illustrated in Figure 2 [48], consisting of a fuzzy rule base, an inference mechanism, and fuzzification and defuzzification interfaces. In general, the mapping between the inputs and the outputs of a fuzzy system is nonlinear [23, 26]. However, it is possible to construct a rule base with a linear input-output mapping [43, 26]. For that, the following conditions must be fulfilled:

•Use triangular input sets that cross at the membership value 0.5

•The rule base must be complete AND combination (cartesian product) of all input families

•Use the algebraic product (*) for the AND connective

•Use output singletons, positioned at the sum of the peak positions of the input sets

•Use sum-accumulation and centre of gravity for singletons (COGS) defuzzification

It seems reasonable to start with the design of a conventional integer/fractional PID controller and from there to proceed to a fuzzy control design. In this way, the linear fuzzy

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Figure 2. Structure of a fuzzy controller.

controller may be used in a design procedure based on integer/fractional PID control, as follows [26, 6, 3]:

1.Build and tune an integer/fractional PID controller

2.Replace it with an equivalent linear fuzzy controller

3.Make the fuzzy controller nonlinear

4.Fine-tune it

With the above procedure, the design of fuzzy FO-PID controllers will be greatly simplified, particularly if the controller was already implemented and it is desirable to enhance its performance. Moreover, this new type of controllers extends the potentialities of both fuzzy and fractional controllers and performs, at least, as well as its linear counterpart [26, 6, 3].

3.1.Fuzzy Fractional PD Controller

The time-domain equation of a fractional PDµ-controller is given by (Ki = 0 in (2)):

The corresponding discrete-time fractional PDµ -controller is:

Figure 3 illustrates the block diagram of the fuzzy fractional PDµ (FF-PDµ ) controller. As can be seen, the controller acts on the error, e(k), and on the fractional change of error,

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e E

Dµ Kfe

Figure 3. Fuzzy fractional PDµ-controller.

Dµe(k). The control signal is U . The controller has three tuning gains, Ke and Kf e, corresponding to the inputs and Ku to the output. The following relations are thus verified:

where E and F E represent the terms of error and fractional change of error, respectively. The control signal U is generally a nonlinear function of E and FE:

Comparing (16) with (12), it yields the relation between the gains of the conventional and fuzzy PDµ controllers:

KeKu = Kp

The linear FF-PDµ -controller provides all the advantages of the conventional fractional PDµ-controller.

For an equivalent linear FF-PDµ -controller, the conclusion universe should be the sum of the premise universes and the input-output mapping should be linear. Table 1 lists a linear rule base for the FF-PDµ controller composed of four rules. There are only two fuzzy labels (Negative and Positive) used for the fuzzy input variables and three fuzzy labels (Negative, Zero and Positive) for the fuzzy output variable. This rule base should satisfy above mentioned conditions in order to provide a linear mapping.

Scaling the input gains may be necessary to preserve the linearity of the fuzzy controller. However, that should be made without affecting the tuning [6, 3]. For example, considering (16), and a scale factor M (M > 0):

This scaling has some advantages, as it will avoid saturation and will provide a simpler design, since the universes ranges of inputs and outputs are normalized to a prescribed interval, say percentage of full scale [−100, 100].

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Table 1. Rule base for the FF-PDµ controller.

Rule 1 If E is N and F E is N then u is N

Rule 2 If E is N and F E is P then u is Z

Rule 3 If E is P and F E is N then u is Z

Rule 4 If E is P and F E is P then u is P

3.2.Fuzzy Fractional PID Controller

The inclusion of an integral action is necessary whenever the closed-loop system exhibits a steady-state error. The fuzzy fractional PDµ +Iλ (FF-PDµ +Iλ) controller combine the fractional-order integral action with a fuzzy PDµ-controller, as illustrated in Figure 4.

The control signal U is generally a nonlinear function of error E, fractional change of error FE, and fractional integral of error FIE:

U = (f (E, F E) + F IE) Ku = f (Kee (k) , Kf eDµe (k)) + Kf ieD−λe (k) Ku

Comparing (20) with the discrete fractional PIλDµ-controller (6), it yields the relation between the gains of the conventional and fuzzy fractional PID controllers:

KeKu = Kp

Kf ieKu = Ki

The linear FF-PDµ+Iλ controller provides all the advantages of the conventional fractional PIλDµ-controller.

3.3.Illustrative Examples

In this section we apply the proposed fuzzy FO-PID controllers in the control of integer and fractional-order plants. In all experiments, the simulation parameters are: absolute memory computation of approximation (7), scale factor M = 0.1, and sampling period T = 0.05 s.

3.3.1.Example 1: Integer-Order Plant

Consider the normalized transfer function of a double integrator plant, which serves as a model of many dynamic systems:

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Let us design an equivalent linear FF-PDµ -controller. By configuring the fuzzy inference system (FIS) and selecting three scaling factors, we obtain a FF-PDµ -controller that reproduces the exact control performance as the fractional PDµ-controller. We first fix Ke = 100, since the error universe is chosen to be percentage of full scale [−100, 100], and the maximum error to a unit step is 1. The values of Kf e and Ku are obtained using expressions (17). Figure 5 shows the input families and the linear control surface obtained by using the rule base of Table 1 while satisfying the conditions of linear mapping; E is the error, F E is the fractional derivative of error and u is the output of the fuzzy PDµ controller. Note that this result represents the step 2 – replace the conventional controller with an equivalent linear fuzzy controller – of the design procedure.

In order to enhance the performance of the control system we now proceed to step 3 of the design – make the fuzzy controller nonlinear. Thus, after verifying that the linear FF-PDµ -controller is properly designed, we may adjust the FIS settings such as its style, membership functions and rule base to obtain a desired nonlinear control surface. In this work, we choose to change the fuzzy rule base, as illustrated in Table 2. This rule base is commonly used in fuzzy control systems and consists of 49 rules with 7 linguistic terms (NL – Negative large, NM – Negative medium, NS – Negative small, ZR - Zero, PS – Positive small, PM – Positive medium and PL – Positive large). The membership functions for the premises and consequents of the rules are shown in Figure 6a). With two inputs and one output the input-output mapping of the fuzzy logic controller is described by the nonlinear surface of Figure 6b). For the defuzzication process we use the centroid method.

Figure 7 shows the step responses of both linear and nonlinear fuzzy PDµ (µ = 0.7) controllers. We verify that the nonlinear controller improves the system control performance, namely the overshoot and settling time.

For comparison purposes, we also compute the integral of the absolute error (IAE):

Z t

0

For t = 20 s we get IAE (linear) = 2.84 and IAE (nonlinear) = 2.00.

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Figure 5. Linear surface (a) with the corresponding input families (b).

Table 2. Fuzzy control rules.

The system performance can be further improved. For that, we go to step 4 (and last) of the design procedure – fine tune the controller. The nonlinear fuzzy controller will be adjusted by changing the parameter values of Ke, Kf e, and Ku. In this study we propose the use of a genetic algorithm (GA) to fine tune the gains of the controller. A GA is a search technique used in computer science to find approximate solutions in optimization and search problems. GA are a particular class of evolutionary algorithms that use techniques inspired by evolutionary biology such as inheritance, mutation, natural selection, and crossover, established by the Darwin’s theory of evolution [19, 38, 41, 32]. The advantage of GA is in their parallelism. GA is travelling in a search space using more individuals than other methods. However, GA also have disadvantages, namely the computational cost, because many times these algorithms are slower than other methodologies.

The GA fitness function corresponds to the minimization of the integral absolute error (IAE) criterion (as defined in (24)):

where (Ke, Kf e, Ku) are the fuzzy fractional PDµ controller parameters to be optimized.

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Figure 6. Memberships functions for E, FE and u (a) Nonlinear control surface (b).

In Figure 7, the comparison of the unit-step responses of the closed-loop system with G1(s) controlled by the linear and nonlinear PDµ -controllers, and with the optimal nonlinear FF-PDµ (µ = 0.7) controller is given. We establish the following values for the GA parameters: population size P = 20, crossover probability C = 0.8, mutation probability M = 0.05 and number of generations Ng = 50. The interval of the FLC parameters used in the GA optimization are defined around 10% of the nominal parameters obtained with the linear controller. In this case IAE (optimal) = 0.075. As can be seen, the optimal nonlinear controller improves significantly the control system performance, in terms of overshoot, rise time, and settling time, when compared with other fuzzy controllers. In this way, the fuzzy fractional controller provides greater flexibility than the integer/fractional controller and can be used to better adjust the dynamical properties of a control system.

3.3.2.Example 2: Fractional-Order Plant

Many real dynamical processes are modeled by fractional-order transfer functions [50, 44, 25]. Here we consider the fractional-order plant model given in [51]:

An integer-order PD controller and a fractional-order PDµ -controller were designed in [51]:

Figure 8 shows the unit-step responses of the closed-loop fractional-order system with the conventional PD-controller and with the PDµ-controller for G2(s). The comparison shows that for satisfactory feedback control of the fractional-order system is better to use a fractional-order controller instead of a classical integer-order controller. Note, however,

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Figure 7. Unit step responses of system with G1 (s) and with the linear, nonlinear and optimum fuzzy fractional PDµ (µ = 0.7) controllers.

that the control system presents a steady-state error, since no integral action is employed. In this case, for t = 6 s: IAE (PD) = 0.86 and IAE(PDµ) = 0.55.

Let us now design an equivalent linear FF-PDµ controller. As in previous example, we fix Ke = 100. The values of Kf e and Ku are obtained using expressions (17). The input families and the linear control surface are illustrated in Figure 5. This result represents the step 2 of the design procedure. In order to enhance the performance of the control system we proceed to step 3 of the methodology.

For obtaining the desired surface control, we apply again the nonlinear fuzzy rule base of Table 2. The corresponding membership functions and surface control are shown in Figures 6.

In Figure 8, the comparison of the unit-step responses of the closed-loop system with G2(s) controlled by the linear and nonlinear FF-PDµ (µ = 1.15) controllers is given. We verify that the nonlinear controller improved the overshoot, settling time, and steady-state error, when compared with the linear fuzzy controller. In this case IAE (nonlinear) = 0.40, which is smaller than the linear counterpart.

To further improve the system performance we fine tune the controller (step 4, and last, of the design procedure). The fuzzy controller parameters (Ke, Kf e, Ku) are tuned using a GA while minimizing the IAE index (25). Figure 8 shows the unit-step responses of the closed-loop system with G2(s) controlled with the optimum nonlinear FF-PDµ (µ = 1.15) controller. The GA parameters are: population size P =20, crossover probability C = 0.8, mutation probability M = 0.05 and number of generations Ng = 25. The interval of the FLC parameters used in the GA optimization are defined around 10% of the nominal parameters obtained with the linear controller. In this case IAE (optimal) = 0.037. Clearly,

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