Figure 8. Unit step responses of system with G2 (s) with PD-controller, and with the linear, nonlinear and optimal fuzzy fractional PDµ (µ = 1.15) controllers.

the optimal nonlinear controller produced better results than other fuzzy controllers, since the transient response (namely the overshoot, rise time, settling time, and steady-state error) and the error IAE are smaller.

Now, let us consider the FF-PDµ +Iλ-controller. In order to test the robustness of the fuzzy controller, we introduce a load disturbance of amplitude l = −2 after 8 seconds in system of Figure 1. We use the same (Kp, Kd) parameters of the linear FF-PDµ (µ = 1.15) controller and tuned the (Ki, λ) for a satisfactory control response. The final tuned parameters are (Ki, λ) = (10, 0.8). With Ke = 100, and using (21) we obtain Kf e, Ku, and Kf ie of the fuzzy controller.

Figure 9 shows the step and load responses of closed-loop system with FF-PDµ +Iλ controller, (µ, λ)=(1.15, 0.8), for the linear and nonlinear control surfaces. We observe the better response of the fuzzy controller to the reference and disturbance inputs with the nonlinear rule base (Table 2) compared to their linear counterpart (Table 1). If further improved is needed, we can fine tune it using the GA while minimizing the IAE criterion (25), as done in previous examples. Once more, we demonstrate the robustness and effectiveness of these controllers.

4.Optimal Fuzzy Fractional PID Controllers

In this section we design optimal fuzzy fractional PID controllers by using PSO algorithm [39, 31, 29, 30, 28].

Complimentary Contributor Copy

Figure 9. Unit-step and load responses of the fractional control system with G2(s) and with the linear and nonlinear FF-PDµ +Iλ controllers.

4.1.Approximations by Rational Functions

In this study we adopt discrete integer-order approximations to the fundamental element sα (α <) of a fractional-order control (FOC) strategy. The usual approach for obtaining discrete equivalents of continuous operators of type sα adopts the Euler, Tustin and AlAlaoui generating functions [10, 54, 14].

It is well known that rational-type approximations frequently converge faster than polynomial-type approximations and have a wider domain of convergence in the complex domain. Thus, by using the Euler operator w(z−1) = (1−z−1)/Tc, and performing a power series expansion of [w(z−1)]α = [(1− z−1 )/Tc]α gives the discretization formula corresponding to the Grunwald¨ -Letnikov definition (4):

where Tc is the sampling period and hα(k) is the impulse response sequence.

A rational-type approximation can be obtained through a Pade´ approximation to the impulse response sequence hα(k), yielding the discrete transfer function:

1 + a1z−1 + . . . + anz−n k=0

where m ≤ n and the coefficients ak and bk are determined by fitting the first m + n + 1 values of hα(k) into the impulse response h(k) of the desired approximation H (z−1).

Complimentary Contributor Copy

Thus, we obtain an approximation that matchs the desired impulse response hα(k) for the first m + n + 1 values of k [10]. Note that the above Pade´ approximation is obtained by considering the Euler operator but the determination process will be exactly the same for other types of discretization schemes [10].

4.2.Fuzzy Fractional Control System

Figure 10 presents the block diagram of the fuzzy control system to be analyzed, where G(s) is the process transfer function. The gains of the fuzzy fractional PDβ +I controller will be tuned through the application of a PSO algorithm in order to achieve a superior control performance of system of Figure 10. Note the presence of a saturation nonlinearity.

The generalized PIα Dβ controller is given here as [51]:

where the fractional-order operators are implemented by rational approximations of type (30).

In the system of Figure 10, we apply a fuzzy logic control (FLC) for the PDβ actions and the integral of the error is added to the output in order to find a fuzzy PDβ + I controller [6]. In fact, we can have a fuzzy PDβ +Iα controller, i.e. an added fractional integral action, as presented in [5]. This is the FLC of Figure 4 with an integer integral action in the feedforward path (α = 1). The block diagram of Figure 11 illustrates the configuration of the proposed fuzzy controller. In this controller, the control actions are the error e, the fractional derivative of e and the integral of e. The U represents the controller output. Also, the controller has four gains to be tuned, Ke, Kie, Kce corresponding to the inputs and Ku to the output.

The control action U is generally a nonlinear function of error E, fractional change of error CE, and integral of error IE:

h i

U (k) = [f (E, CE) + IE] Ku = f Kee (k) , KceDβ e (k) + KieIe (k) Ku (32)

where Dβ is the discrete fractional derivative implemented as rational approximation (30) using the Euler scheme (29); the integral of error is calculated by rectangular integration:

To further illustrate the performance of the fuzzy PDβ +I a saturation nonlinearity is included in the closed-loop system of Figure 10, and inserted in series with the output of the fuzzy controller. The saturation element is defined as:

where u and n are respectively the input and the output of the saturation block and sign(u) is the signum function.

Complimentary Contributor Copy

Figure 10. Block diagram of the fuzzy control system.

Figure 11. Fuzzy PDβ + I controller.

Here we give an emphasis of the proposed FLC presented in Figure 11. The basic structure for FLC is illustrated in Figure 2. The fuzzy rule base, which reflects the collected knowledge about how a particular control problem must be treated, is one of the main components of a fuzzy controller. The other parts of the controller perform make up the tasks necessary for the controller to be efficient. The rule base for the fuzzy PDβ +I controller is presented in Table 2. The membership functions for the premises and consequents of the rules are shown in Figure 12a). With two inputs and one output the input-output mapping of the fuzzy logic controller is described by a nonlinear surface, presented in Figure 12b).

The fuzzy controller will be adjusted by changing the parameter values of Ke, Kce, Kie and Ku. The fuzzy inference mechanism operates by using the product to combine the conjunctions in the premise of the rules and in the representation of the fuzzy implication. For the defuzzification process we use the centroid method.

4.3.Particle Swarm Optimization

In this study we use a particle swarm optimization (PSO) algorithm for the tuning of a fractional order controller, applied to a system. The PSO is one of the latest evolutionary techniques developed by Dr. Eberhart and Dr. Kennedy in 1995, inspired by social behaviour of bird flocking or fish schooling. The PSO scheme optimizes searching by virtue of the swarm intelligence produced by the cooperation and competition among the particles of a species [20, 35, 21]. The social system is discussed through the collective behaviours of simple individuals interacting with their environment and each other. Examples of this are the bird flock or fish school. Some applications of PSO are found in the field of nonlinear dynamical systems, data analysis, electrical engineering, function optimization, artificial neural network training, fuzzy control and many others in real world applications [20, 21, 49, 55, 28].

It is important to mention that a reliable execution and analysis of a PSO usually re-

Complimentary Contributor Copy

Figure 12. Memberships functions for E, CE and v (a) Nonlinear control surface (b).

quires a large number of simulations to provide that stochastic effects have been properly considered. In the PSO algorithm, all particles represent a potential solution to a problem, which is performed by adjusting their position taking into account both personal and group experiences [21, 28, 52]. In each iteration, the velocity (v) is actualized by expression (35a). The new particle position (cpk+1 ) is found by adding their actual position with the new velocity, as shown in (35b).

where ω is the inertia weight, c1 and c2 are the individual and sociality weights coefficients for modelling attractive forces from the local and global best, respectively; r1 and r2 are aleatory numbers between [0, 1], lbp is the local best position, cp is the current position, and gbp is the global best position [28]. The inertia term, forces the particle to move in the same direction as before by adjusting the old velocity. The cognitive term (personal best), forces the particle to go back to the previous best position. On the other hand, the social learning term (global best), forces the particle to move to the best previous position of its neighbors. The PSO optimizes an objective function by iteratively improving a swarm of solutions, called particles, based on special management of memory. Each particle is modified by referring to the memory of individual swarm’s best information. Due to the collective intelligence of these particles, the swarm is able to repeatedly improve its best observed solution and converging to an optimum.

4.4.Objective Function for Optimization

The optimization fitness function corresponds to the minimization of the integral time absolute error (ITAE) criterion, that measure the response error as defined as [29]:

Complimentary Contributor Copy

where (Ke, Kce, Kie, Ku) are the PDβ +I controller parameters to be optimized. Other criteria are possible like the integral time square error (ITSE), integral absolute error (IAE), integral square error (ISE), among others [31, 34, 17, 16].

4.5.Case Studies

We analyze the closed-loop system of Figure 10 with a fuzzy fractional PDβ + I controller (Figure 11). In all the experiments, the fractional order derivative Dβ in scheme of Figure 11 is implemented by using a 4th order Pade´ discrete rational transfer function (m = n = 4) of type (30). It is used a sampling period of Tc = 0.1 s. The PDβ +I controller is tuned through the minimization of the ITAE (36) using a PSO. We use δ = 15.0. We establish the following values for the PSO parameters: population number P N = 25, c1 = c2 = 1, ω = 0.9 and a maximum number of iterations IM ax = 50. For guarantee that stochastic effects are properly considered, the experiments consist on executing the PSO 10 times, and we get the best result, i.e. the simulation that leads to the smaller J, and consequently a combination of controller gains, that leads to a better transient and steady state responses.

4.5.1.Example 1. High-Order Transfer Function

In the first case, we compare a fuzzy fractional PDβ +I controller which leads to the lower error (β = 0.9), with a fuzzy integer PID controller (β = 1). Figure 13a) shows the unit step responses of both controllers. The plant system G3(s) used is represented by a high−order transfer function [1]:

The controller parameters, corresponding to the minimization of the ITAE index, lead to the values for the fuzzy integer PID controller: {Ke, Kce, Kie, Ku} ≡ { 1.0705, 3.8194, 0.4206, 4.4784 }, with J = 2.3763, and for the fuzzy fractional PDβ +I controller to the following values: {Ke, Kce, Kie, Ku} ≡ {0.7193, 0.3551, 0.2702, 3.2029}, with J = 0.8409. These values lead us to conclude that the fuzzy fractional order controller produced better results than the integer one, since the transient response (namely, the settling time, rise time and overshoot) and the error J are smaller, as can be seen in Figure 13a). Figure 13b) shows the ITAE error as function of β. The graph shows that lowest error is produced for β = 0.9. We also see that the fractional controller is always better than its integer version (considering 0 < β ≤ 1).

Figure 14 illustrates the variation of FLC parameters (Ke, Kce, Kie, Ku) as function of the order’s derivative β, while Figure 15 shows the variation of the transient response parameters, namely the settling time ts, rise time tr , peak time tp and overshoot ov(%) versus β, for the closed-loop step response, when minimizing the ITAE index. The variation of FLC parameters and the transient response parameters reveal a smooth variation with β.

Complimentary Contributor Copy

Figure 13. Step responses of the closed-loop system with a fuzzy PD+I and PDβ +I (β = 0.9) controllers (a) Error J versus β for G3(s) (b).

4.5.2.Example 2. Non-Minimum Phase System

In a second experiment, we consider a fuzzy PDβ +I controller which leads for lower error to β = 0.8, applied to a process G4(s) represented by a non−minimum system given by the transfer function (38) [1].

Once again, we consider for comparison the corresponding integer version (β = 1). Figure 16a) shows the unit step responses of both controllers.

The controller parameters, corresponding to the minimization of the ITAE index, lead to the values for the fuzzy integer controller: {Ke, Kce, Kie, Ku} ≡ { 0.9953, 1.3058, 0.4360, 0.1991 }, with J = 70.9016, and for the fuzzy fractional controller: {Ke, Kce, Kie, Ku}≡ {0.0873, 0.0569, 0.0417, 2.5740}, with J = 35.4189. These values lead us to remain the previously conclusions drawn for G3(s), namely that the fuzzy fractional order controller produced better results than the integer one, since the transient response (in particular the settling time, rise time and overshoot) and the error J are smaller. Figure 16b) shows the ITAE error as function of β. The graph shows a lowest error for β = 0.8. Also, the fractional controller (for 0 < β ≤ 1) is always better than the conventional fuzzy PID controller.

Figure 17 illustrates the variation of FLC parameters (Ke, Kce, Kie, Ku) as function of the order’s derivative β, while Figure 18 shows the variation of the transient response parameters, namely ts, tr , tp and ov(%) versus β, for the closed-loop step response, when minimizing the ITAE index. The variation of FLC parameters and the transient response parameters reveal a smooth variation with β.

Complimentary Contributor Copy

Figure 14. The PDβ +I parameters (Ke, Kce, Kie, Ku) versus β for G3(s).

4.5.3.Example 3. Second-Order Transfer Function With Time Delay

In a third experiment, we consider a fuzzy PDβ +I controller which leads for lower error to β = 0.9, applied to a process G5(s) represented by a second-order transfer function with a time delay (39).

Once more time, we consider for comparison the corresponding integer version (β = 1). Figure 19 a) shows the unit step responses of both controllers.

The controller parameters, corresponding to the minimization of the ITAE index, lead to the values for the fuzzy integer controller: {Ke, Kce, Kie, Ku} ≡ {1.1616, 1.0076, 0.1993, 1.3628}, with J = 8.3228, and for the fuzzy fractional controller: {Ke, Kce, Kie, Ku}≡ {0.6600, 0.7851, 0.1171, 2.7525}, with J = 5.8728. These values lead us to remain the previously conclusions drawn for G3(s) and G4(s), namely that the fuzzy fractional order controller produced better results than the integer ones, since the transient response (in particular the settling time, rise time and overshoot) and the error J are smaller. Figure 19b) shows the ITAE error as function of β. The graph shows that for this process we have a lower error for β = 0.9.

Figure 20 illustrates the variation of FLC parameters (Ke, Kce, Kie, Ku) as function of the order’s derivative β, while Figure 21 shows the variation of the transient response parameters, namely ts, tr , tp and ov(%) versus β, for the closed-loop step response, when minimizing the ITAE index. The variation of FLC parameters and the transient response parameters reveal a smooth variation with β.

In conclusion, with the fuzzy fractional PDβ +I controller we get the best controller tuning, superior to the performance revealed by the integer-order scheme. Moreover, we prove the effectiveness of this control structure when used in systems with time delay. In fact, sys-

Complimentary Contributor Copy

Figure 15. Parameters ts, tr , tp, ov(%) versus β of the step responses of the closed-loop system with G3(s) and with a fuzzy PDβ +I controller.

Figure 16. Step responses of the closed-loop system with fuzzy PD+I and PDβ +I (β = 0.8) controllers (a) Error J versus β for G4(s) (b).

tems with time delay are more difficult to be controlled with the classical methodologies, however the proposed algorithm reveals that is very effective in the control of this type of systems.

4.6.Influence of Order of Rational Approximations

Here we analyze the influence of the order of rational approximations used in the simulations for different values of m = n. In the experiments, the controller performance is evaluated for several values of order m = n = {1, 2, ..., 10}. For this study we use the

Complimentary Contributor Copy

Figure 17. The PDβ +I parameters (Ke, Kce, Kie, Ku) versus β for G4(s).

Figure 18. Parameters ts, tr , tp, ov(%) versus β of the step responses of the closed-loop system with G4(s) and with a fuzzy PDβ +I controller.

second-order transfer function with time delay G5(s).

Figure 22 shows the unit step responses of PDβ +I controller for various orders of approximations m, and for the best case of 10 experiments, i.e. the simulation that leads to the smaller J. We observe that the responses are very similar varying slightly with order m.

Complimentary Contributor Copy

Figure 19. Step responses of the closed-loop system with fuzzy PD+I and PDβ +I (β = 0.9) controllers (a) Error J versus β for G5(s) (b).

Figure 20. The PDβ +I parameters (Ke, Kce, Kie, Ku) versus β for G5(s).

For this process we have a lower error for β between 0.8 and 0.9 as function of m. These facts can also be seen in Figure 23 that shows the ITAE error as function of β and m. From the graph, it is clear that the best approximations are found for 2 6 m 6 5.

Figure 24 illustrates the variation of FLC parameters (Ke, Kce, Kie, Ku) as function of order’s derivative β and order of approximation m, while Figure 25 shows the variation of the transient response specifications, namely ts, tr , tp and ov(%) versus β and m, for the closed-loop step response. It is observed that the FLC parameters and the transient response specifications varies with β and m. For instance, the system overshoot is lower for relatively high values of β and wide range of order’s 2 6 m 6 10, as seen in Figure 25.

Complimentary Contributor Copy

Figure 21. Parameters ts, tr , tp, ov(%) versus β of the step responses of the closed-loop system with G5(s) and with a fuzzy PDβ +I controller.

Figure 22. Step responses of the closed-loop system with fuzzy PDβ +I controller versus order m for G5(s).

The same analysis can be done for other parameters.

Complimentary Contributor Copy