An Intelligent Feedforward Controller Utilizing a Modified Gorilla Troops Optimization for Nonlinear Systems

Methods of linear control rely on the presence of a mathematical model of the system. Nevertheless, the majority of physical systems are nonlinear, and their mathematical models are unknown or incompletely understood, and time-dependent. Consequently, conventional techniques are limited in terms of effectiveness and stability [1]. To tackle this difficulty, the artificial neural network (ANN) has been recognized as one of the most effective tools for tracking highly nonlinear dynamic and complex processes due to the rapid advancement of artificial intelligence, which has proven more effective at managing these processes. Neural networks (NNs) can approximate a broad range of nonlinear functions with varying degrees of precision, which makes them useful for identifying and controlling nonlinear dynamic systems [2]. In this regard, the nonlinear autoregressive moving average (NARMA-L2) model is a type of artificial neural network, which was introduced by Narendra and Mukhopadhyay [3]. It is a simple, direct, yet efficient method for reproducing the dynamics of the nonlinear system. The NARMA-L2 model is capable of transforming nonlinear system dynamics into linear dynamics by eliminating the nonlinearities. Consequently, it is best suited for feedback linearization control [4].

There are a number of studies conducted on the NARMA-L2 model. For example, Gundogdu and Celikel [5] proposed an ANN-based NARMA-L2 controller for stepper motor control. They used a backpropagation algorithm (BPA) in order to find the optimal weights for the neural network. Rashad controlled the angular speed of a permanent magnet DC (PMDC) motor's rotor using a NARMA-L2 controller and the dynamic BPA to minimize the mean square of errors [6]. Moreover, El Hamidi et al. [7] proposed a hybrid neural network to establish a multilayer perceptron (MLP) based on NARMA-L2 as a model for controlling a nonlinear MIMO quadcopter system using the BPA to determine the best NN weights. Pedro, and Ekoru [8] used the NARMA-L2 control method for a nonlinear, servo-hydraulic, four-degrees-of-freedom active vehicle suspension. The Levenberg-Marquardt (LM) algorithm was used to train the weights of the network. However, the gradient-based methods, such as the BP and LM algorithms, are distinguished by their slow convergence rates, reliance on a suitable selection of inertial and learning factors, and proclivity to become trapped in local minima of the search space [9].

In designing the IFC structure, an effective inverse model of the system to be controlled must be created. To this end, the neural network is one of the most effective means of addressing IFC design requirements. NNs are well recognized to be universal approximators for a diverse variety of nonlinear functions [10, 11]. In the sequel, the inverse controller based on neural networks has been used a lot to control nonlinear systems. In this respect, most researchers have implemented MLP and RBF NNs within the feedforward control structure. For example, a bearingless induction motor was controlled by MLP NN with 22 hidden layer nodes using an IMC structure [10]. a comparatively large MLP network has been utilized to control unknown nonlinear systems using an IMC scheme [12]. In this method, the IMC structure consisted of two filters: set point and robustness filters. These filters have parameters that can be changed and should be chosen ahead of time, which may make this method less accurate. an arc furnace was controlled using two RBF networks in an IMC structure [13]. The first RBF was used to identify the system, and the system's inverse dynamic was learned using the second RBF. Nonetheless, this control strategy had to go through two stages of training: an identification stage to build a model of the forward plant and a second stage to synthesize the inverse controller.

As another type of neural network, wavelet neural networks (WNNs) are gaining increasing attention from researchers because WNNs combine the theory of wavelets and ANNs to create an integrative method that is more effective. Specifically, WNNs can learn and generalize like ANNs, and they can also navigate like the wavelet transform [14]. Due to these benefits, it has been proven that WNNs are better than other ANNs in terms of their ability to improve the mapping between inputs and outputs [15]. To this end, a comparative study in this paper will show that the WNN is better than both the MLP and the RBF. Despite the above-mentioned attractive features of the WNN, few researchers have implemented the WNN within an inverse controller. For instance, an IMC scheme using a WNN-based NARMA-L2 was used for controlling nonlinear systems [16]. However, this control method necessitates two NARMA-L2 structures, which increases the computational burden. In addition, Alwan [17] utilized a WNN-based NARMA-L2 with the Mexican hat function as the mother wavelet function for the IFC structure. The author utilized the BPA to determine the ideal NN weights.

The performance of the NARMA-L2 controller is directly proportional to the precision of the system's model estimation. The BPA is the most commonly employed learning technique in the literature on NARMA-L2. However, the gradient descent-based BPA has some problems, like a slow rate of convergence and a tendency to get stuck at local minimums in the search space. In order to avoid the limitations mentioned above with gradient-based methods, evolutionary algorithms have gained a lot of attention for their ability to find the global solution to a problem. Therefore, this work focuses on a novel swarm intelligence algorithm known as the Gorilla Troops Optimizer (GTO). Abdollahzadeh et al. [18] originally proposed this algorithm in 2021. GTO was inspired by the way gorillas live as a group and how intelligent they are in social situations. Research so far shows that GTO does a great job on benchmark function optimization [19]. In addition, in order to increase the exploration ability of the original GTO algorithm by adopting more optimal solutions to the optimization process, an enhanced version, termed the modified gorilla troops optimization (MGTO) algorithm, is proposed in this study to optimize the controller's parameters. The simulation results from this study demonstrate that the proposed algorithm is an effective method for optimizing the WNN-based NARMA-L2 network structure.

The principal contributions of this study are summarized in several points. First, a WNN-based NARMA-L2 is proposed to control nonlinear systems using an inverse feedforward controller. The advantage of using a NARMAL2 model is that the proposed method can be used to control unknown nonlinear systems. In particular, only a single NARMA-L2 is needed to learn the controlled plant's forward dynamics, after which the control algorithm can be applied directly without additional training. Second, a modified version of the GTO algorithm called MGTO is proposed for training the WNN to optimize its weights. Finally, utilizing different neural network techniques (MLP and RBF) and the most prevalent activation functions of WNN (Mexican Hat, Gaussian, and Morlet) in the IFC scheme, the proposed control approach demonstrated its superiority via a comparative study. The rest of the paper is organized as follows: Section 2 describes the fundamental principles of the WNN-based NARMA-L2 network. The GTO and its modified version are elaborated upon in Section 3. In Section 4, several performance evaluations and comparative tests are provided to demonstrate the efficacy of the proposed IFC scheme. Finally, Section 5 provides a brief conclusion to the study.

Artificial gorilla troops optimization is a newly proposed nature-inspired and gradient-free optimization algorithm that mimics the group lifestyle of gorillas [18]. The gorilla lives in a group known as a troop, which is comprised of an adult male gorilla known as the silverback, multiple adult female gorillas, and their progeny. A silverback gorilla is an adult gorilla named after the distinctive hair on its back during puberty and it lives for more than 12 years. Furthermore, the silverback is the troop's leader. It takes all decisions, mediates conflicts, decides group movements, guides gorillas to sources of food, and ensures the safety of the group. Younger male gorillas aged 8 to 12 years are known as blackbacks because they lack the silvery-colored back hairs of their older counterparts. They are part of the Silverback team and serve as the team's backup defense. Generally, both male and female gorillas prefer to leave the group where they were born and join another one. Conversely, mature male gorillas may separate from the original group and establish their own troops by the attraction of migrating females. Nonetheless, some male gorillas prefer to reside with the original troop and follow the silverback. If the silverback perishes, these males may engage in a violent struggle for group dominance and mate with adult females [19].

3.1 Mathematical model of artificial gorilla troops optimization algorithm

The GTO algorithm's specific mathematical model is developed utilizing the previously mentioned approach of gorilla group behavior in nature. Similar to other intelligent algorithms, GTO is comprised of three major parts: initialization, global exploration, and local exploitation. Each of these is explained in more detail below [19].

3.1.1 Initialization phase

Assume that N gorillas exist in a space of the D-dimension. The i-th gorilla's position in space can be characterized as X_i= (x_i,1, x_i,2, …, x_{i, D}), i=1, 2, …, N. Accordingly, the gorilla population initialization process can be defined as follows:

$X_{N \times D}=\operatorname{rand}(N, D) \times(u b-l b)+l b$          (11)

where, N represents the population size and D represents the number of variables, and ub and lb are the search space's upper and lower limits, respectively.

3.1.2 Exploration phase

When gorillas leave their original group, they will explore a variety of natural surroundings that they may or may not have encountered previously. All gorillas have assumed candidate solutions in the GTO algorithm, and the silverback is considered to be the best solution in each optimization process. As a way to accurately simulate this kind of natural migration behavior, the gorilla's position update equation for the exploration stage consisted of three distinct methods, which include migrating to unknown locations, migrating around familiar places, and migrating to other groups, as seen in Eq. (12):

$G X(t+1)$$=\left\{\begin{array}{lr}(u b-l b) \times r_2+l b, & r_1<p \\ \left(r_3-c\right) \times X_r(t)+L \times Z \times X(t), & r_1 \geq 0.5 \\ X(t)-L \times\left(\begin{array}{c}L \times\left(X(t)-G X_r(t)\right)+ \\ r_4 \times\left(X(t)+G X_r(t)\right)\end{array}\right) & , r_1<0.5\end{array}\right.$          (12)

where, t denotes the current iteration, GX(t+1) represents the search agents' candidate positions in the next iteration, and X(t) denotes the individual gorilla's current position vector. Moreover, r₁, r₂, r₃, and r₄ represent random values between [0, 1], p is a parameter with a range from 0 to 1 that must be set prior to optimization to determine the probability of selecting the migration mechanism to a previously unknown position, X_r(t) and GX_r(t) are two locations randomly chosen within the current gorilla population, and Z represents a problem-dimensional row vector with randomly generated values for each element in [-C, C]. Additionally, the parameter C is computed using Eq. (13):

$C=\left(\cos \cos \left(2 \times r_5\right)+1\right) \times\left(1-\frac{I t}{\text { MaxIt }}\right)$          (13)

where, cos denotes the cosine function, r₅ denotes a random value between 0 and 1 and it is updated with each iteration, It is the current iteration value, and MaxIt denotes the maximum number of iterations. Finally, the parameter L in Eq. (12) can be calculated as follows:

$L=C \times l$          (14)

where, l represents a random value between -1 and 1.

After the exploration phase is complete, all newly generated candidate solutions GX(t+1) have their fitness values assessed. If GX is superior to X, which is shown by F(GX) < F(X), where F(.) represents the fitness value for a particular problem, it will be kept and will take the place of the original solution X(t). Additionally, silverback X_silverback is chosen as the best solution for this period.

3.1.3 Exploitation phase

At the time of the troop's formation, the silverback is strong and healthy, whereas the other male gorillas are still juveniles. They abide by all of Silverback's decisions in search of a variety of food resources and satisfy him with unwavering loyalty. Unquestionably, the silverback ages and eventually dies, while younger blackbacks in the troop may engage in violent competition with the other males in the group for mating with adult females and taking over the group. As aforementioned the exploitation phase of GTO models two behaviors: having followed the silverback and competition for adult female gorillas. Simultaneously, the parameter W is established to regulate the switching between them.

If $C \geq W$, the first technique of silverback tracking is selected. The following is the mathematical formula:

$G X(t+1)=L \times M \times\left(X(t)-X_{\text {silverback }}\right)+X(t)$          (15)

where, X_silverback denotes the best solution found thus far and X(t) represents the gorilla position vector. Additionally, L is calculated using Eq. (14) and the parameter M can be calculated using the formula below:

$M=\left(\mid \frac{1}{N} \sum_{i=1}^N\right.$ $\left.\left.G X_i(t)\right|^{2^L}\right)^{\frac{1}{2^L}}$          (16)

where, N is the size of the population and GX_i(t) represents each gorilla's position vector in the current iteration.

If $C<W$, it indicates that the competition for adult female gorillas’ mechanism has been selected and in this case, the gorillas' position can be updated as follows:

$G X(t+1)=$$X_{\text {silverhack }}-\left(X_{\text {silverhack }} \times Q-X(t) \times Q\right) \times A$          (17)

$Q=2 \times r_6-1$          (18)

$A=\varphi \times E$          (19)

$E=\left\{N_1, \quad r_7 \geq 0.5 N_2, \quad r_7<0.5\right.$          (20)

In Eq. (17), X(t) represents the current position and Q represents the impact force, which is calculated by Eq. (18). In addition, the coefficient A utilized to simulate the level of violence in the competition is calculated using Eq. (19), in which φ is a constant value, and the values of E are assigned by Eq. (20), where r₆ of Eq. (18) and r₇ of Eq. (20) contain random values ranging from 0 to 1. Finally, if r₇ > 0.5, E is identified as a 1-by-D array of random numbers with normal distribution, where D is the dimension of space. Otherwise, if r₇ < 0.5, E would be equal to a normal distribution-compliant stochastic number.

At the conclusion of the exploitation procedure, the fitness values of the newly created candidate GX(t+1) solution are also computed. If F(GX) < F(X), the solution GX will be maintained and utilized in the next optimization, whereas the best solution among all individuals is denoted by the silverback X_silverback.

At the conclusion of the exploitation procedure, the fitness values of the newly created candidate GX(t+1) solution are also computed. If F(GX) < F(X), the solution GX will be maintained and utilized in the next optimization, whereas the best solution among all individuals is denoted by the silverback X_silverback.

3.2 Modified artificial gorilla troops optimization algorithm (MGTO)

Even though the original algorithm attempts to incorporate both exploration and exploitation operations, it has been demonstrated that the GTO tends to get stuck into sub-optimal regions of the search space. This problem is specifically attributable to the original GTO's insufficient exploitation ability. Moreover, equality between exploration and exploitation processes may degrade the algorithm's performance in instances where more exploration operators are required and vice versa [25]. Consequently, training the proposed WNN-based NARMA-L2 in this work resulted in insufficient precision. As a solution to this issue, two modifications have been made in this study to increase the exploration capability of the original algorithm by adopting more optimal solutions to the optimization process. The proposed algorithm was referred to as the Modified artificial gorilla troops optimization algorithm (MGTO). Specifically, as the first modification, the worst n solutions are replaced with new candidate solutions derived from the best solution obtained thus far. In this study, n was set to 20 solutions, and the candidate solutions have been generated using the following formula:

$G X(i, j)=$ Silverback $(j)+\left(G X\left(X_{r 1}, j\right)\right.$$\left.-G X\left(X_{r 2}, j\right)\right) * \mu_{i, j}$          (21)

where, i and j represent the position and dimension of GX, respectively, Silverback is the best solution found so far, X_r1 and X_r2 are two different random integers chosen from 1 to the maximum number of solutions and must be distinct from the index of the current solution, and μ_i,j represents a random value selected from [-1, 1]. As a second modification, a random solution is generated and its objective function is assessed at each iteration. If this newly generated solution's objective function is worse than the worst solution, the worst solution is replaced with the best solution's position. Instead of that, the worst solution is substituted with the newly generated solution. In order to generate the candidate solution, the following formula was used:

$G X_{\text {worst }}^i= \begin{cases}X_{\text {silverback }}^i & \text { if } J_{\Lambda}^i \geq J_{\text {worst }}^i \\ \Lambda^i & \text { if } J_{\Lambda}^i<J_{\text {worst }}^i\end{cases}$          (22)

where, $\Lambda^i$ represents a random solution value selected from [-1, 1], $J_{\Lambda}^i$ represents the objective function of the random solution and $J_{\text {worst }}^i$ represents the objective function of the worst solution. Notably, these two modifications significantly enhanced the original algorithm's searching capability without the need for additional parameters of control to implement the suggested steps. These characteristics qualify the proposed MGTO to achieve superior optimization results compared to other comparable methods, as illustrated in Section 4.3.

Finally, the steps of the proposed MGTO algorithm are presented in the pseudo-code Algorithm and in Figure 4.

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Figure 4. Flowchart of the MGTO