Position and Speed Control for Permanent Magnet DC Motor Based on Different Optimization Algorithms

2.1 Dynamic model of PMDC motor

Figure 1 displays a schematic diagram of PMDC motor. This motor differs from conventional DC motors in that it without field circuit, and instead the permanent magnet is used to produce a field flux.

1.jpg

Figure 1. Schematic diagram of PMDC motor

The following equations give the dynamic model of the PMDC motor [7, 8, 12, 17, 18]:

$V_a(t)=R_a i_a(t)+L_a \frac{d i_a(t)}{d t}+E_b$    (1)

$E_b=k_v \omega(t)$    (2)

$T_e=J \frac{d \omega(t)}{d t}+B \omega(t)+T_L$    (3)

$T_e=k_t i_a(t)$    (4)

$\omega(t)=\frac{d \theta(t)}{d t}$    (5)

where, $V_a$ and $i_a$, respectively, are the armature voltage and current, $E_b$ is the back e.m.f., $T_e$ is the electromagnetic torque, $T_L$ is the load torque, $\omega$ is the motor angular speed, $\theta$ is the motor angular position, $J$ is the rotor's inertia and the equivalent mechanical load, $B$ is the motor's mechanical rotational system's damping coefficient, $k_v$ stands for velocity constant, and $k_t$ is the torque constant.

The following differential equations can be formed by rearranging Eqs. (1)-(4):

$\frac{\mathrm{di}_{\mathrm{a}}(\mathrm{t})}{\mathrm{dt}}=\frac{\mathrm{V}_{\mathrm{a}}(\mathrm{t})}{\mathrm{L}_{\mathrm{a}}}-\frac{\mathrm{R}_{\mathrm{a}}}{\mathrm{L}_{\mathrm{a}}} \mathrm{i}_{\mathrm{a}}(\mathrm{t})-\frac{\mathrm{k}_{\mathrm{v}}}{\mathrm{L}_{\mathrm{a}}} \omega(\mathrm{t})$    (6)

$\frac{\mathrm{d} \omega(\mathrm{t})}{\mathrm{dt}}=\frac{\mathrm{k}_{\mathrm{t}}}{\mathrm{J}} \mathrm{i}_{\mathrm{a}}(\mathrm{t})-\frac{\mathrm{B}}{\mathrm{J}} \omega(\mathrm{t})-\frac{1}{\mathrm{~J}} \mathrm{~T}_{\mathrm{L}}$    (7)

By taking Laplace transform of (6) and (7), the armature current and angular speed equations are expressed as:

$I_a(s)=\frac{1}{\left(L_a s+R_a\right)}\left[V_a(s)-k_v \Omega(s)\right]$    (8)

$\Omega(s)=\frac{1}{(J s+B)}\left[k_t I_a(s)-T_L(s)\right]$    (9)

Also, equation of the angular position in Laplace transform is written as:

$\Theta(s) = \frac{\Omega(s)}{s}$    (10)

According to the above equations, Figure 2 displays the PMDC motor's block diagram.

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Figure 2. PMDC motor block diagram

2.2 PID controller

A conventional PID can be considered as a foundation stone algorithm in the theory of control. Many systems use this control algorithm because it is smooth and precise. Different constant parameters employed in the PID controller approach include proportional, integral, and derivative values. Trial and error are one of the methods for adjusting these values. The control decision of the regulation is set mathematically according to [8, 19, 20]:

$u(t)=K_P e(t)+K_I \int e(t) d t+K_D \frac{d e(t)}{d t}$    (11)

where, $K_P$ is the proportional constant, $K_I$ represents the integral constant, $K_D$ indicates the derivative constant, $u(t)$ is the output waveform, and $e(t)$ is the error waveform.

2.3 Ant Colony Optimization (ACO)

ACO algorithm is a meta-heuristic computer optimization methodology that utilizes the nature-derived optimization approach. Marco Dorigo has proposed ACO method which was developed with the use of ants as inspiration in order to find the best route between food and the nest. When ants are looking for food, they first randomly investigate the area around their colony. The ants move across the ground and leave a chemical pheromone trail. For pheromone quantity, two main conditions must be satisfied: a path length and discovered food source quality [12, 21-24].

ACO algorithm can be summarized as:

1) Initialization: All ACO algorithm parameters are initialized such as $n$: nodes number, $m$ : ants number, $t_{\max }$: maximum iteration, $d_{\max }$: maximum distance for each ant's tour, $\alpha$: pheromone decay parameter $(0<\alpha<1), \beta$: relative importance of pheromone versus distance $(\beta>1), \rho$: heuristically defined coefficient $(0<\rho<1),~q_a$: algorithm parameter, and $\tau_0$: a level of initial pheromone.

A maximum distance for each ant's tour d_max is given as [25]:

$d_{\max }=\max \left[\sum_{i=1}^{n-1} d_i\right]$    (12)

$d_i=|r-\max (u)|$    (13)

where, $d_i$ is the separate distance between two nodes, $r$ is the current node, and $u$ is the unvisited node.

2) Generate first position randomly according to uniform distribution.

3) The probability of an ant (k) at node (i) selecting the next node (j) is given by:

$p_{i j}^k(t)=\frac{\left[\tau_{i j}(t)\right]^\alpha\left[\eta_{i j}(t)\right]^\beta}{\sum_{i j \epsilon T^k}\left[\tau_{i j}(t)\right]^\alpha\left[\eta_{i j}(t)\right]^\beta} ; \quad i j \in T^k$    (14)

where, $\tau_{i j}$ is the trial of pheromone deposited by ant k between node i and j, $\left(\eta_{i j}=1 / d_{i j}\right)$ is the visibility, and $T^k$ is the route implemented by the ant k at a known time.

4) Local pheromone updating: which is not similar for all ants since every ant follows a different path. Each ant's original pheromone is locally updated as:

$\tau_{i j}(t+1)=(1-\rho) \tau_{i j}(t)+\rho \tau_0$    (15)

5) When the strongest pheromone draws ants along the shortest path, the aim function is most effectively carried out.

6) The following formulae can be used to update global pheromone:

$\tau_{i j}(t+1)=(1-\alpha) \tau_{i j}(t)+\alpha \Delta \tau_{i j}(t)$    (16)

7) The best outcome is attained when every ant takes the same best course. If not, the procedure goes on until the iteration's conclusion.

2.4 Grey Wolf Optimization (GWO)

The GWO technique, a meta-heuristic algorithm, was introduced [26-28] and mimics grey wolves' social behavior. Between 5 and 12 wolves make up the group that lives together. This group follows the rigid dominance hierarchy, with a leader a and supporting member b who assist the leader in making decisions. δ and ω are the rest members of the group as shown in Figure 3.

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Figure 3. Social hierarchy of GWO

Additionally, the primary activity for all of the wolves is hunting for prey by searching for it and attacking it. Firstly, the wolves will surround their victim before attacking it. The equations below show the mathematical model of the prey being encircled [28-30]:

$\vec{D}=\left|\vec{C} \cdot \vec{X}_p(t)-\vec{X}(t)\right|$    (17)

$\vec{X}(t+1)=\vec{X}_p(t)-\vec{A} \cdot \vec{D}$    (18)

where, $\vec{D}$ is the distance vector, $\vec{X}_p(t)$ stands the vector of the prey's position, $\vec{X}(t)$ is the wolf position vector, and t is the current iteration. $\vec{A}, \vec{C}$ are coefficient vectors which can be determined as:

$\vec{A}=2 \vec{a} \cdot \vec{r}_1-\vec{a}$    (19)

$\vec{C}=2 \cdot \vec{r}_2$    (20)

where, $\vec{r}_1$ and $\vec{r}_2$ are the random vectors in the range [0,1]. As the number of iterations rises, the value of the constant $\vec{a}$ decreases linearly from 2 to 0.

In relation to location of its prey, the position of a grey wolf shifts. In this method, with the assistance of the three currently best-known solutions (α, β, δ), the optimal solution (prey) is discovered. The following equations are used in order to update their positions in the following iteration [28-30]:

$\left.\begin{array}{l}\vec{D}_\alpha=\left|\vec{C}_1 \cdot \vec{X}_\alpha-\vec{X}\right| \\ \vec{D}_\beta=\left|\vec{C}_2 \cdot \vec{X}_\beta-\vec{X}\right| \\ \vec{D}_\delta=\left|\vec{C}_3 \cdot \vec{X}_\delta-\vec{X}\right|\end{array}\right\}$    (21)

$\left.\begin{array}{l}\vec{X}_1=\vec{X}_\alpha-\vec{A}_1 \cdot \vec{D}_\alpha \\ \vec{X}_2=\vec{X}_\beta-\vec{A}_2 \cdot \vec{D}_\beta \\ \vec{X}_3=\vec{X}_\delta-\vec{A}_3 \cdot \vec{D}_\delta\end{array}\right\}$    (22)

$\vec{X}(t+1)=\frac{\vec{X}_1+\vec{X}_2+\vec{X}_3}{3}$    (23)

In search of a more favorable target, other wolves will depart from the current prey when the value of $\vec{A}$ is greater than 1 or less than -1, which enables the GWO algorithm's capacity to search globally. Furthermore, $\vec{C}$ values are scattered at a random distributed between 0 and 2, allowing for easier exploration during the entire method [29, 30].

2.5 Flower Pollination Algorithm (FPA)

It is thought that 80% of plants rely on flower pollination to carry pollen from a male flower to a female bloom in order to reproduce. Two types of pollination: biotic and abiotic. Only 10% of pollination is carried out by wind and other natural forces, the other 90% taking place through insects and other animals. In biotic systems, pollination occurs either through self-pollination within a single flower or through cross-pollination between two distinct flowers. Biotic and cross pollination occurs between flowers that are farther apart, which is an example of global optimization [31, 32].

The idea of FPA can be described according to the following main principles [31, 32]:

1) Global pollination is biotic and cross-pollination.

2) Local pollination is abiotic and self-pollination.

3) Stability of the flower increases the likelihood of reproduction which may be developed in the way of pollinators such as birds, insects. It largely relies on how similar the two flowers used in the reproduction process are to one another.

4) By altering probability values in the (0–1) range with a small preference for local pollination, it is possible to manage the transition between local and global pollination.

In FPA algorithm, the answer $X_i$ is comparable to a flower and/or a pollen gamete. Pollinators transport flower pollens for global pollination. With Lévy flight, pollens are capable of traveling far. Therefore, the main principles 1 and 3 can be expressed as [31-34]:

$X_i^{t+1}=X_i^t+L\left(X_i^t-g^*\right)$    (24)

where, $X_i^t$ stands the solution vector $X_i$ at tth iteration, $g^*$ represents the current best solution in the iteration among the current options, and $L$ is the pollination strength, which is essentially a step size.

Levy flight distribution is used to simulate how insects fly over large distances while taking into consideration the fact that they move with different distance steps. Utilizing the Levy distribution, L > 0 can be represented as [31-34]:

$L \approx \frac{\lambda\Gamma(\lambda)sin{\left( \frac{\pi\lambda}{2} \right)}}{\pi}\frac{1}{s^{1 + \lambda}},~~~\left( {s \gg s_{0} > 0} \right)$   (25)

where, $\Gamma(\lambda)$ is the standard gamma function, and for big steps s > 0, this distribution is valid.

For local pollination, main principles 2 and 3 are represented by the following [31-34]:

$X_i^{t+1}=X_i^t+\epsilon\left(X_i^t-X_k^t\right)$    (26)

where, $X_i$ and $X_k$ are pollen grains from various blooms of the same type of plant, while $\varepsilon$ represents a random walk in which the distribution is uniformed between 0 and 1.

At the end, new solutions for all flowers are finding. If the new solutions are best, they are replaced in the population and the process is then repeated for every member of the population to identify the best current solution.

This paper illustrated a comparison of different optimization algorithms (ACO, GWO, and FPA) to adjust the PID controller's settings for the optimal position and speed responses from the PMDC motor. The simulation results show the effectiveness of these algorithms under different cases of PMDC motor conditions (no load, load, and desired references for speed and position). The results illustrated the GWO gives the best response with minimum error. It can be used these proposed algorithms in speed control of the other types of DC motors. In future works, the use of on-line algorithms can improve the performance of PMDC motor.