The DC motor represents a type of rotary electrical motor that can transform direct current electrical energy into mechanical energy. It is applied in a broad range of industrial, residential and commercial applications [1]. It comprises of a shunt field connected similarly to the armature. With various turns of a small gauge fence, the shunt field curve is constructed so that it has very high or low current flow in comparison with a series fence field. Thus, this kind of motor has good position control and excellent speed compared to others. It has therefore used in many implementations that necessitate at least five horsepowers.

In [2], the authors reported the transfer function of the DC motor speed model, which was of the form:

where K_b is the back electromotive force constant, J_m is the rotor inertia, L_a is the armature inductance, R_a is the armature resistance and B_m is the viscous friction coefficient. However, the values of the previous parameters are defined in Table 1, as reported in [2].

To improve the performance of the classical proportional-integral-derivative PID (controller), Podlubny et al. [3] proposed the so-called fractional-order PID controller (FoPID-controller) in 1997. They demonstrated that when his proposed controller is employed for controlling a system, a better responsiveness than traditional PID controller will be then yielded. Such controller has two extra parameters (δ and λ) added to the conventional PID-controller parameters (K_p, K_i, K_d) [4,5]. To find the optimal FoPID-controller, the best values of its parameters must be tuned [6]. Tuning the FoPID-controller parameters adds greater flexibility to the aimed design. For this purpose, a lot of optimization algorithms have been efficiently executed to find the optimal values of the PID-controller’s parameters [7, 8].

In this work, the bacterial foraging optimization (BFO) and the particle swarm optimization (PSO) algorithms are carried out to outline the best FoPID-controller parameters. The role of these algorithms is to minimize the objective function J that has the form [9, 10]:

where e_ss is the steady state error, T_s is the settling time, T_r is the rise time, M_p is the peak overshoot, and β is a scaling factor. It is relevant to note that the scaling factor β could be chosen by a designer as β = 0.5 [9, 10]. In particular, the execution of the PSO and BFO algorithms will provide an excellent overshoot, short settling time and short rise time to the closed-loop system of the DC motor speed model. These specifications can hence measure the powerful of the controlled system. The Laplacian operators s^±α yielded from the optimization process will be substituted by proper rational transfer functions of integer-order by implementing two schemes; the Outstaloup’s approach and the continued fraction expansion (CFE).

In PSO, a population of possible answers, referred to as particles, traverses the search space repeatedly. A potential solution to the optimization problem is represented by each particle. Particles move according to two parameters: the global best known position found by all particles in the population, and their personal best known position. Particles modify their positions at every iteration according to two primary principles: exploitation and exploration. Whereas exploitation entails traveling in the direction of the most well-established solutions, exploration is accomplished by permitting particles to travel toward uncharted areas of the search space. Each particle’s position and current velocity are the two primary factors that govern its motion. Every iteration updates a particle’s velocity based on its previous velocity, distance to its personal best location, and distance to the global best-known position. Particles can dynamically adjust their movement to balance exploitation and exploration thanks to this update equation (see [11] to get an overview of this algorithm).

In contrast, the optimization issue in BFO is formulated as a nutrient space, and the objective function is used to describe the nutrient concentration at various points within the space. The technique uses a population of synthetic bacteria, each of which stands for a possible fix for the optimization issue. At each iteration, the following three main mechanisms control how bacteria move through the nutrient space:

• • Chemotaxis: Similar to how actual bacteria travel toward higher food concentrations, bacteria gravitate toward areas with higher nutritional concentrations. The gradient of the nutrient content dictates the direction of movement, and the bacteria adapt by changing positions.

• • Reproduction: Reproduction is more likely in bacteria with higher fitness, or better solutions. Bacteria pass on traits from their parents to their offspring, however certain differences are added to increase population diversity.

• • Elimination-dispersal: Bacteria are eliminated from the population if they cannot locate enough nutrients or if they get stuck in local optima. New bacteria are also added to the colony in order to preserve its diversity and size.

The population of bacteria collectively searches the nutrient space through repetitions of these processes, becoming convergent towards optimal or nearly optimal solutions (see [12] for a summary of this algorithm).

Fractional calculus is the calculus so that the order of its integration or differentiation could be real or complex [13, 14]. The primary operation of the non-integer calculus is the fractional-order differential operator Datα that has the form:

where a and t denote respectively the lower and the upper bounds of the above operator, α is the fractional-order value, and ℜ(α) is the real part of α. In what follows, we recall some basic definitions that illustrate the Riemann-Liouville operator, followed by the Caputo operator.

Definition 3.1

Let f be an integrable piecewise continuous function on any finite subinterval of t ∈ (0,+∞), then the Riemann-Liouville fractional integral operator of f(t) of order α is defined as [15]:

where Γ(·) is the Gamma function, t > 0 and 0 < α ≤ 1.

Definition 3.2

Let α ∈ ℝ⁺ and m ∈ ℕ such that m – 1 < α < m, then the Caputo fractional derivative operator of order α is defined by [15]:

It is well-known that the frequency response of dynamical systems is a commonly scheme for realizing the designed controllers. From this point of view, the Laplace transform was extended to involve the fractional calculus.

Definition 3.3

By assuming that the initial state equals zero, the Caputo fractional-order derivative operator has Laplace transform of the form [15]:

Definition 3.4

By assuming that the initial state equals zero, the Riemann-Liouville fractional integral operator has Laplace transform of the form [15]:

FoPID-controller is typically implemented to enhance the systems’ performance in many industrial applications. It provides to the closed-loop system extra degrees of freedom via its integration component λ and its differentiator component δ [16,17]. The output signal of this controller m(t) might be expressed as:

where e(t) is the error signal and K_p, K_i, K_d, λ, δ are real constants. Obviously, if λ = δ = 1, then the conventional PID controller is yielded [18, 19]. Now, if one takes the Laplace transform to Eq. (7), then we generate the following integro-differential equation of fractional-order [7]:

Clearly, the parameters (K_p, K_i, K_d, λ, δ) need to be adjusted by applying an optimization algorithm [20, 21]. In this regard, we execute the BFO and PSO algorithms, and then we must consider the Laplacian operators (s^λ and s^δ) generated in Eq. (8). To deal with such operators within a limited frequency band, we should approximate them to equivalent rational transfer functions of integer-order [22]. There exist several common approximations that could be utilized for this purpose, such as the Oustaloup, least square method, Carlson, Continued Fractional Expansion (CFE), AbdelAty et al., El-Khazali, Chareff and Matsuda approximations [22–26]. Figure 1 represents the closed-loop system of the DC motor speed model.

In Figure 1, the output from the plant is monitored and the feedback will be sent to the controller by which it can be compared with the system input to determine deviations from the expected output. This would allow the controller to make any necessary adjustments and regulations. Next, we describe briefly two approximations of these operators; the Oustaloup and the CFE approximations.

4.1 Oustaloup’s Approach

The Oustaloup approximation is one of the well-known methods that can be implemented to yield certain odd-order rational transfer functions. The bandwidth during which time this approach is taken into consideration can be customized to produce, within a predefined frequency band, a good fitting to the fractional-order values s^±α, where 0 < α < 1. In this connection, the rational function, which can be utilized for approximating s^α over the frequency range of interest (ω_b.ω_h), can be expressed as [22]:

where the poles, zeros and the gain can be computed respectively according to following formulas:

Because of the frequencies geometric distribution, the unity gain geometric frequency ω_u can be computed by using the following relation:

where the approach at hand relies on the lower frequency range (ω_b, ω_h) and the order filer N, with observing that the transfer function (9) is always of order n = 2N + 1.

4.2 The CFE Approach

This scheme is considered the principal mathematical method that can be used to approximate the Laplacian operator by suitable rational transfer functions of integer-order. This scheme is proposed in accordance with the following relation [27]:

where 0 < α < 1 and n ∈ ℕ.

With the aim of finding an integer-order approximation of s^α, one can substitute s instead of z in (14). This substitution allows the n^th-order approximation of s^α to be expressed around the center frequency ω₀ = 1rad/sec as [27]:

where 0 < α_i< 1, i = 0, 1, 2, · · ·, 5. It should be mentioned that the values of the coefficients of α_i are taken from reference [28], for i = 0, 1, · · ·, 5.

A lot of research workers have examined the FoPID-controller design using several popular optimization techniques. The type of approach used to replace the Laplacian operators plays a key role in the effectiveness of such controller [22, 29]. As we mentioned before, we intend in this work to tune the five parameters of the FoPID-controller by using the BFO and PSO algorithms. The performance of the DC motor speed model will be then optimized by evaluating the unit-step response. The enhancement of the system’s performance in time domain is identical to a minimization problem [30]. In particular, for a suitable design of the FoPID-controller (i.e. finding the best parameters of (8)), the objective function J reported in (2) should be minimized by using the BFO and PSO algorithms. The two generated fractional-order operators (s^λ and s^δ) will be then approximated using the Oustaloup and the CFE approximations. As a result, four FoPID-controllers C_i(s) would be then yielded. These controllers would, in their turn, imply four closed-loop systems H_i(s), where i = 1, 2, 3, 4. Such closed-loop systems will finally compete to each other to outline which controller will be the best.

To this aim, we first execute the PSO algorithm via CFE and Oustaloup approaches. Accordingly, we obtain respectively the following results

• • The FoPID-controller via CFE approach:

  
  

  (16)C1(s)=48+0.31s0.177+2.6s0.166.

  • - The two Laplacian operators (s^0.177 and s^0.166) are approximated using the CFE approach as follows:

    
    

    (17)s0.177=2.2541s5+46.1835s4+1.6220e         +002s3+1.4413e+002s2         +31.4520s+1s5+31.4520s4+1.4413e+002s3         +1.6220e+002s2+46.1835s         +2.2541,(18)s0.166=2.1419s5+44.4013s4+1.5719e         +002s3+1.4070e+002s2         +30.9712s+1s5+30.9712s4+1.4070e+002s3         +1.5719e+002s2+44.4013s         +2.1419.

  • - The closed-loop system H₁(s) is then given by:

    
    

    (19)H1(s)=155s10+7814s9+1.3e005s8         +7.9e005s7+2.3e006s6+3.2e006s5         +7.9e005s7+2e006s4+7.6e005s3         +1.2e005s2+7154s+136.80.0008s13+0.12s12+8.3s11         +417s10+1.1e004s9+1.5e005s8         +8.5e005s7+2.4e006s6+3e006s5         +2.3e006s4+7.7e005s3+1.2e005s2         +7157s+136.8.

• • The FoPID-controller via Oustaloup approach:

  
  

  (20)C2(s)=87+1.36s0.165+7.34212s0.247.

  • - The two Laplacian operators (s^0.165 and s^0.247) are approximated using the Oustaloup approach as follows:

    
    

    (21)s0.165=2.138s5+86.88s4+482.7s3         +414.6s2+55.07s+1s5+55.07s4+414.6s3         +482.7s2+86.88s+2.138,(22)s0.247=3.119s5+117.5s4+605.4s3         +482.2s2+59.39s+1s5+59.39s4+482.2s3         +605.4s2+117.5s+3.119.

  • - The closed-loop system H₂(s) is then given by

    
    

    (23)H2(s)=302.5s10+2.9e004s9         +8.7e005s8+9.2e006s7         +3.9e007s6+6.2e007s5         +4e007s4+1e007s3         +9.9e005s2+3.4e004s         +368.30.0008s13+0.15s12+13.1s11         +909s10+4.3e004s9         +1e006s8+9.7e006s7         +3.9e007s6+6.2e007s5         +4.1e007s4+1e007s3         +9.9e005s2+3.4e004s         +368.

Now, in a similar manner to the previous discussion, we execute here the BFO algorithm via CFE and Oustaloup approaches. As a result, we obtain respectively the following results:

• • The FoPID-controller via CFE approach:

  
  

  (24)C3(s)=18.26+16.56s0.5342+13.59s0.748.

  • - The two Laplacian operators (s^0.534 and s^0.748) are approximated using the CFE approach as follows:

    
    

    (25)s0.534=13.24s5+1.93e+002s4         +5.27e+002s3         +3.68e+002s2+59.42s+1s5+59.42s4+3.68e+002s3         +5.27e+002s2+1.931e         ++002s13.24,(26)s0.748=50.02s5+6.08e+002s4         +1.44e+003s3         +8.65e+002s2+1.14e         +002s+1s5+66.11s4+4.26e         +002s3+6.29e+002s2         +2.37e+002s+16.91.

  • - The closed-loop system H₃(s) is given by

    
    

    (27)H3(s)=9420s10+2.9e005s9+3.3e006s8         +1.6e007s7+4.1e007s6         +5.5e007s5+4.1e007s4         +1.6e007s3+3.4e006s2         +3.2e005s+1.1e0040.013s13+2.2s12+166.4s11         +1.5e004s10+3.8e005s9+3.8e006s8         +1.7e007s7+4.2e007s6+5.6e007s5         +4.1e007s4+1.6e007s3+3.4e006s2         +3.2e005s+1.1e004.

• • The FoPID-controller via Oustaloup approach:

  
  

  (28)C4(s)=9.92+15.81s0.831+20.81s0.390.

  • - The two Laplacian operators (s^0.831 and s^0.390) are approximated using the Oustaloup approach as follows:

    
    

    (29)s0.831=45.88s5+1010s4+3039s3         +1414s2+101.7s+1s5+101.7s4+1414s3         +3039s2+1010s+45.88,(30)s0.390=6.028s5+199.1s4+899s3         +627.7s2+67.75s+1s5+67.75s4+627.7s3         +899s2+199.1s+6.028.

  • - The closed-loop system H₄(s) is given by (31)

    
    

    (31)H4(s)=7971s10+4.6e005s9+8.4e006s8         +5.6e007s7+1.6e008s6+2e008s5         +1.3e008s4+4.1e007s3+5.9e006s2         +3.2e005s+56990.016s13+3.01s12+250.3s11         +1.9e004s10+6.6e005s9+9.9e006s8         +6.1e007s7+1.6e008s6+2.1e008s5         +1.3e008s4+4.1e007s3+5.9e006s2         +3.2e005s+5699.

In the following content, we aim to carry out a numerical competition between all H_i(s) to outline which FoPID-controller is the best, where i = 1, 2, 3, 4. For this purpose, we present a graphical comparison in Figure 2 between the closed-loop systems H_i(s), which are generated with the help of applying the FoPID-controllers (C_i(s), i = 1, 2, 3, 4).

To highlight the variations between all design methods, some numerical results related to the step response specifications of the closed-loop transfer functions H_i(s), i = 1, 2, 3, 4, are exhibited in Table 2.

In Figure 2 and Table 2, one might choose the FoPID-controller generated by PSO algorithm via the CFE approach C₁(s) as the best controller among the others. Although it provides to the closed-loop system a slightly high overshoot, but it at the same time provides a good rise time and an excellent settling time. However, another might see that the FoPID-controller generated by BFO algorithm via the Oustaloup approach C₄(s) is the best. It provides to the closed-loop system a good settling time and an excellent rise time.

In this work, four optimal FoPID-controllers have been designed for the DC motor speed model. The PSO and BFO algorithms have been implemented to tune the parameters of such controllers. The CFE and Outstaloup’s approaches have been used to approximate the Laplacian operators in the form of integer-order transfer functions. The proposed controllers have been compared to each others, and as an results, it has been shown that FoPID-controller generated by PSO algorithm via the CFE approach is the best one. Overall, it can be said that the FoPID-PSO-CFE method can efficiently manage the nonlinear dynamics present in DC motors, resulting in faster settling times, smoother operation, and less overshoot. Based on available data, FoPID-PSO-CFE controllers can be an effective tool for improving DC motor speed control systems’ performance and efficiency in practical applications like industrial processes or mechatronic systems.

Table 1. The values of the model’s parameters.

Table 2. Numerical comparisons between step response specifications of the closed-loop systems.