Implementation of Legendre Neural Network to Solve Time-Varying Singular Bilinear Systems

: Bilinear singular systems can be used in the investigation of different types of engineering systems. In the past decade, considerable attention has been paid to analyzing and synthesizing singular bilinear systems. Their importance lies in their real world application such as economic, ecologi-cal, and socioeconomic processes. They are also applied in several biological processes, such as population dynamics of biological species, water balance, temperature regulation in the human body, carbon dioxide control in lungs, blood pressure, immune system, cardiac regulation, etc. Bilinear singular systems naturally represent different physical processes such as the fundamental law of mass action, the DC motor, the induction motor drives, the mechan-ical brake systems, aerial combat between two aircraft, the missile intercept problem, modeling and control of small furnaces and hydraulic rotary multi-motor systems. The current research work discusses the Legendre Neural Network’s implementation to evaluate time-varying singular bilinear systems for finding the exact solution. The results were obtained from two methods namely the RK-Butcher algorithm and the Runge Kutta Arithmetic Mean (RKAM) method. Compared with the results attained from Legendre Neural Network Method for time-varyingsingularbilinear systems, the output proved to be accurate. As such, this research article established that the proposed Legendre Neural Network could be easily implemented in MATLAB. One can obtain the solution for any length of time from this method in time-varying singular bilinear systems.


Introduction
Differential Equations (DEs) are algebraic relations that exist between functions and their derivatives. These DEs are the backbone of any sort of physical system. Partial differential equations (PDE) or ordinary differential equations (ODE) are the basis upon which most of the chemistry, physics, math, engineering etc., are modeled. In most cases, it is not simple to get an analytical solution for DEs. Therefore, researchers started considering new and dynamic numerical methods to approximate their solutions.
Numerical methods have few limitations, for instance, high computational cost. However, they are widely used for resolving DEs, and they evolved since the first differential equation was derived. Finite differences, finite elements, finite volumes and spectral methods are some of the conventional methods available for spatial discretization of Partial Differential Equations (PDEs) [1]. In this case of discretizing Ordinary Differential Equations (ODEs), some of the following conventional methods are applied i.e., the Euler Method, the Runge-Kutta Method, the RK-Gill Method [2] and the RK-Butcher Algorithm [3].
Artificial Intelligence (AI) experienced rapid development in recent years due to the researchers shifted their attention towards neural network methods [4]. Artificial Neural Networks (ANN) are applied in a wide of domains such as control systems [5], image processing techniques [6] and pattern recognition [7] since they produce promising output. With a proven track record, neural network methods, especially neural network function approximation capabilities, are applied to solve DEs through neural network models.
Legendre Neural Network was leveraged in the study conducted by Mall et al. [8], in which a novel method was proposed as a solution for ODE. To solve two DEs such as Linear Coefficients Delay Differential-Algebraic Equations and Singularly Perturbed DE [9], Legendre Neural Network was proposed by Liu et al. [10]. Yang et al. [11] used Legendre Neural Networkbased algorithm for elliptical partial DEs. In the research conducted by Chen et al. [12], the researchers used Block Trigonometric Exponential Neural Network to find a probable solution for Continuous-Time Model. A new algorithm was proposed by Toni Schneidereit et al. based on Artificial Neural Network to resolve ODs [13].
In the current research paper, the author proposes a novel approach to resolve timevarying singular bilinear systems with the help of the highly accurate Legendre Neural Network method [14].

Legendre Neural Networks
There are two components present in single layers Legendre Neural Network such as input node and output node [15]. Its functional expansion depends on Legendre polynomials. Legendre polynomials constitute a set of orthogonal polynomials which are obtained as a solution for Legendre differential equations. Legendre polynomials are simply denoted L n (u) in which n is the order of polynomial whereas u lies between −1 and 1. Legendre polynomials are a group of orthogonal polynomials and attained to resolve Legendre differential equations. Fig. 1 shows the structure of the Legendre Neural Network. Having its functional expansion based on Legendre polynomial P n (x), Legendre Neural Network for a single layer has one input and one output. The mathematical model for Legendre Neural Network for N nodes of a polynomial P n (x) is as follows Here, the network's input value is denoted by x, the output is denoted by y A , the weight of the input node of j th hidden node is denoted via w j , b j corresponds to the threshold for j th hidden node, and finally, the weight vector of the j th hidden node is denoted by α j . To simplify the Eq. (1), let us take w j = 1 and b j = 0, then the model in the Eq. (1) becomes As per the universal approximation theorem, Singularly Perturbed Differential Equations (SPDEs) represent its analytical solution, whereas y A (x) represents its approximate solution Here, the intervals are discretized, which denotes the boundary points. The weight α j can be solved as given herewith.
This can be described simply as follows H matrix is the first left term in Eq. (4) that corresponds to the neural network's output matrix after the linear L ∈ operator and Bf is the first proper Eq. (4). To mitigate the error between proper solution y (x) and approximate solution y A (x), the optimization should be done by using extreme Machine Learning (ML) algorithm [16] as given as follows.

Time-Varying Singular Bilinear Systems
Here, the first-order time-varying singular system is considered.
In this equation x (0) = x 0 , K corresponds to n × n singular matrix, whereas n × n matrix is denoted by A and n × r matrix is denoted by B. The n-state vectors are denoted by x (t), while the r-input vector corresponds to u (t).
Based on the above discussion, the time-varying singular bilinear system is rewritten in the following form.
The rewritten form of Eq. (9) is given below.
Here, E (t) ∈ R n×n denotes the singular matrix whereas the state corresponds to x (t) ∈ R n , control.
It is challenging to solve a time-varying singular bilinear system compared to its counterpart i.e., time-invariant singular bilinear system [17]. So, various researchers attempted different transformation methods to get rid of this challenge. The current research study leveraged Legendre Neural Network to find a highly accurate solution for a time-varying singular bilinear system [18].

Simulation Example
In this research work, the author considered a time-varying singular bilinear system as proposed earlier [19,20].
If Eq. (5) is solved, then the exact solution for x(t) is as shown below Legendre Neural Network was used to further assess the discrete solution for Eq. (10), and in this stage, 0.25 is considered the step size (t). The results attained from different methods such as the RK-Butcher algorithm and the Runge Kutta Arithmetic Mean Method (RKAM) were compared with that of the solution attained from Legendre Neural Network. Tabs. 1 and 2 show the results and the analytic solution determined using Eq. (12). These tables further shows the error between analytic solution and discrete solution.

Conclusions
Legendre Neural Network obtained highly accurate discrete solutions compared to other methods such as the RK-Butcher algorithm and the Runge Kutta Arithmetic Mean Method (RKAM). It can be observed from Tabs. 1-2 that Legendre Neural Network Method attained only minimal absolute error in contrast to RKAM and RK-Butcher Algorithms because these algorithms produced a considerable error. To conclude, the current study results established that Legendre Neural Network is a promising candidate to evaluate time-varying singular bilinear systems.