- Sie Long Kek received the M.Sc. degree and the Ph.D. degree in mathematics from the Universiti Teknologi Malaysia, Jo... moreSie Long Kek received the M.Sc. degree and the Ph.D. degree in mathematics from the Universiti Teknologi Malaysia, Johor, Malaysia, in 2002 and 2011, respectively. He is currently a Senior Lecturer in the Department of Mathematics and Statistics, Universiti Tun Hussein Onn Malaysia. His research interests includes optimization and control, operation research and management science, and modelling and simulation.edit
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In this paper, an efficient computation approach is proposed for solving a general class of optimal control problems. In our approach, a simplified model-based optimal control model, which is adding the adjusted parameters into the model... more
In this paper, an efficient computation approach is proposed for solving a general class of optimal control problems. In our approach, a simplified model-based optimal control model, which is adding the adjusted parameters into the model used, is solved iteratively. In such a way, the differences between the real plant and the model used can be calculated, in turn, to update the optimal solution of the model used. During the computation procedure, the equivalent optimization problem is formulated, where the conjugate gradient algorithm is applied in solving the optimization problem. On this basis, the optimal solution of the modified model-based optimal control problem could be obtained repeatedly. Once the convergence is achieved, the iterative solution approximates to the correct optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, both linear and nonlinear examples are demonstrated to show the performance of the approach proposed. In conclusion, the efficiency of the approach proposed is highly presented.
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ABSTRACT Consider a discrete-time nonlinear system with random disturbances appearing in the real plant and the output channel where the randomly perturbed output is measurable. An iterative procedure based on the linear quadratic... more
ABSTRACT Consider a discrete-time nonlinear system with random disturbances appearing in the real plant and the output channel where the randomly perturbed output is measurable. An iterative procedure based on the linear quadratic Gaussian optimal control model is developed for solving the optimal control of this stochastic system. The optimal state estimate provided by Kalman filtering theory and the optimal control law obtained from the linear quadratic regulator problem are then integrated into the dynamic integrated system optimisation and parameter estimation algorithm. The iterative solutions of the optimal control problem for the model obtained converge to the solution of the original optimal control problem of the discrete-time nonlinear system, despite model-reality differences, when the convergence is achieved. An illustrative example is solved using the method proposed. The results obtained show the effectiveness of the algorithm proposed.
Research Interests: Mathematics, Applied Mathematics, Computer Science, System Identification, Optimal Control, and 15 moreModeling, Control, Algorithm, Parameter estimation, Kalman Filter, Kalman Filtering, Stochastic Control, Dynamical System, Nonlinear system, Linear Quadratic regulator, Electrical And Electronic Engineering, Non Linear Control, Iterative Algorithm, Discrete time, and Iterative Method
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In this paper, optimal control of an inverted pendulum on a cart system is studied. Since the nonlinear structure of the system is complex, and in the presence of random disturbances, optimization and control of the motion of the system... more
In this paper, optimal control of an inverted pendulum on a cart system is studied. Since the nonlinear structure of the system is complex, and in the presence of random disturbances, optimization and control of the motion of the system become more challenging. For handling this system, a discrete-time stochastic optimal control problem for the system is described, where the external force is considered as the control input. By defining a loss function, namely, the mean squared errors to be minimized, the stochastic approximation (SA) approach is applied to estimate the state dynamics. In addition, the Hamiltonian function is defined, and the first-order necessary conditions are derived. The gradient of the cost function is determined so that the SA approach is employed to update the control sequences. For illustration, considering the values of the related parameters in the system, the discrete-time stochastic optimal control problem is solved iteratively by using the SA algorithm. The simulation results show that the state estimation and the optimal control law design are well performed with the SA algorithm, and the motion of the inverted pendulum cart is addressed satisfactorily. In conclusion, the efficiency of the SA approach for solving the inverted pendulum on a cart system is verified.
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In this paper, a financial risk model, which is formulated from the risk management process of financial markets, is studied. By considering the presence of Gaussian white noise, the financial risk model is reformulated as a stochastic... more
In this paper, a financial risk model, which is formulated from the risk management process of financial markets, is studied. By considering the presence of Gaussian white noise, the financial risk model is reformulated as a stochastic optimal control problem. On this basis, two efficient computational approaches for state estimation, which are the extended Kalman filter (EKF) and unscented Kalman filter (UKF) approaches, are applied. Later, based on the state estimate given by the EKF and UKF approaches, a linear feedback control policy is designed from the stationary condition. For illustration, some parameter values and the initial conditions of the financial risk model are used for the simulation of the stochastic optimal control problem. From the results, it is noticed that the UKF algorithm provides a better state estimate with a smaller value of the sum of squared errors (SSE) as compared to the SSE given by the EKF algorithm. Thus, the estimated output trajectory has a high accuracy that is close to the real output. Moreover, the control effort assists in estimating the state dynamics at the minimum cost. In conclusion, the efficiency of the computational approaches for optimal control of the financial risk model has been well presented.
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In this paper, optimal control of an inverted pendulum on a cart system is studied. Since the nonlinear structure of the system is complex, and in the presence of random disturbances, optimization and control of the motion of the system... more
In this paper, optimal control of an inverted pendulum on a cart system is studied. Since the nonlinear structure of the system is complex, and in the presence of random disturbances, optimization and control of the motion of the system become more challenging. For handling this system, a discrete-time stochastic optimal control problem for the system is described, where the external force is considered as the control input. By defining a loss function, namely, the mean squared errors to be minimized, the stochastic approximation (SA) approach is applied to estimate the state dynamics. In addition, the Hamiltonian function is defined, and the first-order necessary conditions are derived. The gradient of the cost function is determined so that the SA approach is employed to update the control sequences. For illustration, considering the values of the related parameters in the system, the discrete-time stochastic optimal control problem is solved iteratively by using the SA algorithm....
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This paper proposes an optimized schedule for security staff using integer linear programming approach. It is important to improve the life quality of the security staff since the negative social life such as family problems, less social... more
This paper proposes an optimized schedule for security staff using integer linear programming approach. It is important to improve the life quality of the security staff since the negative social life such as family problems, less social support or even stress following from a poor work schedule. Therefore, this study aims to maximize the preference satisfaction of the security staff by allowing them to choose their preferred shift and day off while taking into consideration the restrictions of the university rules. The mathematical model of integer linear programming approach is developed and solved by using LPSolve IDE package. The result shows the overall preference satisfaction of the security staff towards work shift and days off is successfully maximized from 228.33 to 394.33. The comparison of the real schedule and the new proposed optimized schedule is made and all the constraints are successfully satisfied. The proposed schedule will be able to assist the university managem...
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Water is a basic human need. Water is essential for use in all stages of life, including domestic, and industrial services.Increasing water needs from time to time forced the authorities to increase the production of clean water for all... more
Water is a basic human need. Water is essential for use in all stages of life, including domestic, and industrial services.Increasing water needs from time to time forced the authorities to increase the production of clean water for all levels.This study was based on domestic water use in the district of Melaka Tengah. Forty respondents involved in this study. This study focuses on the use of water and charges imposed on each use of the water for a month.
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School children need to eat a well balance nutritious food that provides enough nutrients for development, preservation and restoration of the human body. Proper nutrient can prevent any undesirable diseases and infections. Recently,... more
School children need to eat a well balance nutritious food that provides enough nutrients for development, preservation and restoration of the human body. Proper nutrient can prevent any undesirable diseases and infections. Recently, medical discovery shows that by consuming well balanced nutritious food, it can help to prevent and diminish the risks of cancer and heart failure. Serving healthier meals is a major step towards achieving one of the objectives for this study but assembling a well balance and nutritious menu by hand is complex, ineffective and takes time. The objective of this paper is to develop a mathematical method for menu scheduling that satisfy the entire nutrient requirement for school children, reducing the processing time of optimal solution, minimize cost and also serve variety type of food every day. The data was obtained from the Ministry of Health Malaysian and also school authorities. Binary Programming along with optimization method was used to solve this...
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This paper proposed a 7-level Cascaded H-Bridge Multilevel Inverter (CHBMI) with two diffenrent controller, ie, PID and Artificial Neural Network (ANN) controller to improve the output voltage performance and achieve a lower Total... more
This paper proposed a 7-level Cascaded H-Bridge Multilevel Inverter (CHBMI) with two diffenrent controller, ie, PID and Artificial Neural Network (ANN) controller to improve the output voltage performance and achieve a lower Total Harmonic Distortion (THD). A PWM generator is connected to the 7-level CHBMI to provide switching of the MOSFET. The reference signal waveform for the PWM generator is set to be sinusoidal to obtain an ideal AC output voltage waveform from the CHBMI. By tuning the PID controller as well as the self-learning abilities of the ANN controller, switching signals towards the CHBMI can be improved. Simulation results from the general CHBMI together with the proposed PID and ANN controller based 7-level CHBMI models will be compared and discussed to verifyl the proposed ANN controller based 7-level CHBMI achieved a lower output voltage THD value with a better sinusoidal output performance.
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This article examines an economic growth model that expresses the interaction between production, technology stock, and research and development (R&D) investments. The goal of this study is to maximize production. Considering the presence... more
This article examines an economic growth model that expresses the interaction between production, technology stock, and research and development (R&D) investments. The goal of this study is to maximize production. Considering the presence of Gaussian white noises, this model is reformulated as a stochastic optimal control problem, where the R&D investment rate is defined as the control input. We aim to explore the efficiency of Kalman filtering approaches for solving this problem. Here, the extended Kalman filter (EKF) and unscented Kalman filter (UKF) are applied for state estimation. The state equation linearization is made in the EKF, while the unscented transform is taken in the UKF for generating a set of sigma points. These approaches aim to estimate the state dynamics from different perspectives. With these state estimates, two different computational algorithms are proposed, the EKF for state-control (EKF4SC) and UKF for state-control (UKF4SC) algorithms. The optimal control...
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Direct torque control (DTC) is a method applied in induction motor (IM) drives to control the speed and torque of IM accurately and independently without feedback signal. However, in fast fourier transform (FFT) analysis, the total... more
Direct torque control (DTC) is a method applied in induction motor (IM) drives to control the speed and torque of IM accurately and independently without feedback signal. However, in fast fourier transform (FFT) analysis, the total harmonic distortion (THD) of the IM drives is high in DTC method with conventional inverter (CI). Therefore, the purpose of this study is to minimize the THD without affecting the drive’s performance. A DTC IM drive with multilevel inverter (MLI) is proposed in this study to reduce THD and preserve good speed and torque response of IM simultaneously. DTC IM drive with proposed MLI based THD minimization has several advantages over the DTC IM drive with CI, including higher generated output voltage with low distortion, operate under low switching frequency, and work with renewable energy. In order to prove the effectiveness of the proposed MLI based THD minimization in DTC IM drive, MATLAB Simulink is used to investigate the response of the IM drive and TH...
Even though interpolating bivariate data by Lagrange interpolation is straightforward, its repetitive calculations are quite boring and complicated if the number of data is large. Hence, there is a need to have a suitable tool in teaching... more
Even though interpolating bivariate data by Lagrange interpolation is straightforward, its repetitive calculations are quite boring and complicated if the number of data is large. Hence, there is a need to have a suitable tool in teaching and learning Numerical Methods for this topic. To simplify things, we have developed an Excel spreadsheet calculator to interpolate the bivariate data with 4 rows by 4 columns using Lagrange interpolation. The spreadsheet calculator can be used by educators and students who need its full solution. In addition, users only need to enter a dataset, two independent variables, and the values of the two independent variables which are not in the dataset to obtain a bivariate approximation solutions table in the respective target cells automatically. Besides, the excel command given helps in the teaching and learning process of this topic using Excel spreadsheet.
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In this paper, the fourth version of Richardson’s extrapolation Excel spreadsheet calculator for computing differentiations numerically has been upgraded. In this version, a graphical user interface (GUI) form was developed to capture... more
In this paper, the fourth version of Richardson’s extrapolation Excel spreadsheet calculator for computing differentiations numerically has been upgraded. In this version, a graphical user interface (GUI) form was developed to capture users’ inputs so that users will not confuse the input and output in Excel spreadsheet. It makes this version of GUI Excel spreadsheet solver more user friendly, attractive and interactive. A summative evaluation of this Richardson’s extrapolation GUI Excel spreadsheet solver has been conducted by involving 20 postgraduate students by using questionnaire. The findings showed that the majority of the students agreed that the Richardson’s extrapolation graphical user interface (GUI) spreadsheet solver provides an interesting and interactive learning environment.
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Solving systems of ordinary differential equations (ODEs) by using the fourth-order Runge-Kutta (RK4) method in classroom or in examinations is quite tedious, tiring and boring since it involves many iterative calculations. Hence, there... more
Solving systems of ordinary differential equations (ODEs) by using the fourth-order Runge-Kutta (RK4) method in classroom or in examinations is quite tedious, tiring and boring since it involves many iterative calculations. Hence, there is a need to design a suitable tool in teaching and learning the numerical methods involved, especially those for solving systems of ODEs. Here, we present a new approach to solving systems of ODEs by the RK4 method through the use of an EXCEL spreadsheet to tackle these drawbacks. In doing so, we employ the concept of relative row, relative column and fixed column in the spreadsheet to obtain the solution of systems of ODEs by the RK4 method. With the appropriate differential function given by the user, it is found that the way suggested here is faster than applying a scientific calculator and the solution obtained is significantly more accurate. Besides, the concept presented here can be extended to solve systems of ODEs up to n equations using the...
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There are two common used methods to find the minimum completion time for a project scheduling. These methods are Critical Path Method (CPM) and Program Evaluation Review Technique (PERT). In CPM, a network diagram, which is Activity on... more
There are two common used methods to find the minimum completion time for a project scheduling. These methods are Critical Path Method (CPM) and Program Evaluation Review Technique (PERT). In CPM, a network diagram, which is Activity on Node (AON), is drawn and the slack time for every activity is calculated such that the project’s critical path could be found. It is important that the critical path can suggest the shortest possible completion time. On the other hand, PERT concerns on uncertainty and risk in a project. It has three time estimates, which are optimistic, pessimistic and most likely, and all the time estimates mentioned follows the beta distribution. Besides, the probability in completing the project within certain duration is calculated by using the standard normal distribution. As the risk cannot be avoided in a project, it is important to keep track on any changes and to minimize the completion time for a project. Both of the methods are used to calculate the shorte...
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In this paper, an efficient computation approach is proposed for solving a general class of optimal control problems. In our approach, a simplified model-based optimal control model, which is adding the adjusted parameters into the model... more
In this paper, an efficient computation approach is proposed for solving a general class of optimal control problems. In our approach, a simplified model-based optimal control model, which is adding the adjusted parameters into the model used, is solved iteratively. In such a way, the differences between the real plant and the model used can be calculated, in turn, to update the optimal solution of the model used. During the computation procedure, the equivalent optimization problem is formulated, where the conjugate gradient algorithm is applied in solving the optimization problem. On this basis, the optimal solution of the modified model-based optimal control problem could be obtained repeatedly. Once the convergence is achieved, the iterative solution approximates to the correct optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, both linear and nonlinear examples are demonstrated to show the performance of the approa...
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In this paper, an unconstrained quadratic programming problem with uncertain parameters is discussed. For this purpose, the basic idea of optimizing the unconstrained quadratic programming problem is introduced. The solution method of... more
In this paper, an unconstrained quadratic programming problem with uncertain parameters is discussed. For this purpose, the basic idea of optimizing the unconstrained quadratic programming problem is introduced. The solution method of solving linear equations could be applied to obtain the optimal solution for this kind of problem. Later, the theoretical work on the optimization of the unconstrained quadratic programming problem is presented. By this, the model parameters, which are unknown values, are considered. In this uncertain situation, it is assumed that these parameters are normally distributed; then, the simulation on these uncertain parameters are performed, so the quadratic programming problem without constraints could be solved iteratively by using the gradient-based optimization approach. For illustration, an example of this problem is studied. The computation procedure is expressed, and the result obtained shows the optimal solution in the uncertain environment. In con...
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ABSTRACT An iterative algorithm, which is called the integrated optimal control and parameter estimation algorithm, is developed for solving a discrete time nonlinear stochastic control problem. It is based on the integration of the... more
ABSTRACT An iterative algorithm, which is called the integrated optimal control and parameter estimation algorithm, is developed for solving a discrete time nonlinear stochastic control problem. It is based on the integration of the principle of model-reality differences and Kalman filtering theory, where the dynamic integrated system optimization and parameter estimation algorithm are used interactively. In this approach, the weighted least-square output residual is included in the cost function by appropriately monitoring the weighted matrix. An improved linear quadratic Gaussian optimal control model, rather than the original optimal control problem, is solved. Subsequently, the model optimum is updated using the adjusted parameters induced by the differences between the real plant and the model used. These updated solutions converge to the true optimum, despite model-reality differences. For illustration, the optimal control of a nonlinear continuous stirred tank reactor problem is considered and solved by using the method proposed.
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Motivated by the works of a Richardson’s Extrapolation spreadsheet calculator for differentiation, we have developed the Euler’s spreadsheet calculator using VBA programming to solve ordinary differential equations (ODEs). Users simply... more
Motivated by the works of a Richardson’s Extrapolation spreadsheet calculator for differentiation, we have developed the Euler’s spreadsheet calculator using VBA programming to solve ordinary differential equations (ODEs). Users simply need to enter the independent and dependent variables used, a starting value and ending value for the independent variable, an initial value for the dependent variable, the step size, the ODE and exact function for the ODE. Lastly click the APPLY button which is associated with the VBA programming written to solve the ODEs by the Euler’s method, and finally its full solution is automatically calculated and displayed. Hopefully, this Euler’s ODEs spreadsheet calculator can help educators to prepare their marking scheme easily and assist students in checking their answers
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The non-linear constant increment of power demands due to loads caused a complexity in the operation of the power system network and might also cause insecurity without adequate control in the system with large power flows. A successful... more
The non-linear constant increment of power demands due to loads caused a complexity in the operation of the power system network and might also cause insecurity without adequate control in the system with large power flows. A successful alternative energy source gives new challenges when connected to the power grid system. It is however that with the presence of environmental conditions, there is a constant fluctuation of generated power from renewable energy sources. This can be explained when wind power is used as a source of injection into an electric grid, where the power quality will be affected due to the fluctuating condition of the nature of the wind and comparatively new types of its generators panel. Power system control is introduced in this matter using a controller known as FACTS (Flexible AC Transmission System). FACTS controllers such as STATCOM (Static Synchronous Compensator) and SSSC (Static Synchronous Series Compensator) can function to be a terminal voltage regu...
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Tay (2006) has proposed solving methods using the Casio fx-570MS calculator to overcome the tediousness of doing recursive calculations. Here, we present another alternative, that is solving a non-linear system using Newton's method... more
Tay (2006) has proposed solving methods using the Casio fx-570MS calculator to overcome the tediousness of doing recursive calculations. Here, we present another alternative, that is solving a non-linear system using Newton's method in Microsoft Office Excel. For this, we just make use of the MULT function to do matrix multiplication and MINVERSE function to do the matrix inverse operation. The concept presented here can be developed into a solver where the user just needs to input the initial vector X, the corresponding formula of Jacobian matrix J(x,y) and non-linear system vector F(x,y). The full solutions will be displayed automatically. The solver is easy and user friendly for students and educator who needs its full solutions quickly.
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Least squares method, which is a statistical method with minimum sum squares of errors (SSE), is used for curve fitting and parameter estimation. In general, the Gauss-Newton (GN) and the Levenberg-Marquardt (LM) methods are the popular... more
Least squares method, which is a statistical method with minimum sum squares of errors (SSE), is used for curve fitting and parameter estimation. In general, the Gauss-Newton (GN) and the Levenberg-Marquardt (LM) methods are the popular least squares method. In this paper, a nonlinear least squares problem and the LM method are discussed. In our study, the derivation of the LM algorithm is briefly described and the relevant necessary condition is satisfied. During the calculation procedure, the optimal solution, which is the optimal parameter estimate, is obtained once the convergence is achieved. For illustration, the related models for an exponential distribution with two unknown parameters, and for the average monthly high temperature with four unknown parameters are constructed. Their respective unknown parameters are estimated by applying the LM method. Besides, the best model selection is suggested to represent the dataset of the concentration of a blood sample. Moreover, a nu...
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In this paper, we have improved the limitations of our previous Richardson's extrapolation spreadsheet calculator for computing differentiations numerically. These limitations are the value of D(0,O) keyed in by users using 3-point... more
In this paper, we have improved the limitations of our previous Richardson's extrapolation spreadsheet calculator for computing differentiations numerically. These limitations are the value of D(0,O) keyed in by users using 3-point central difference formula, and the fact that the previous spreadsheet calculator can only calculate the approximate definite differentiation up to level 4 x 4. If the function to be differentiated is complicated, calculating D(0,O) using 3-point central difference formula can be tedious as parentheses should be put in a proper order when writing the calculation command. Otherwise, the calculation command may lead to a wrong answer. In this improved Richardson's extrapolation spreadsheet calculator, we redesigned the Richardson's extrapoIation spreadsheet calculator, where users are only required to give the value of x, the function to be differentiated f(x), and the step size h value without writing the command to obtain D(0,O). Consequently,...
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Motivated by the work of a spreadsheet solution of a system of ordinary differential equations (ODEs) using the fourth-order Runge-Kutta (RK4) method, a RK4 spreadsheet calculator for solving a system of two first-order ODEs was developed... more
Motivated by the work of a spreadsheet solution of a system of ordinary differential equations (ODEs) using the fourth-order Runge-Kutta (RK4) method, a RK4 spreadsheet calculator for solving a system of two first-order ODEs was developed using VBA programming. The main feature of this spreadsheet calculator is to provide a user-friendly interface input form for users to insert the required information instead of the standard cells in Excel. Users are prompted step by step to give relevant information beginning from providing independent and dependent variables used in the system of ODEs. Secondly, they are required to give the interval for the independent variable, initial values for the dependent variables, a step size h and desired accuracy for computation. Thirdly, they have to enter the system of two first-order ODEs and the exact functions if it exists. The ODES and the exact function can be typed in Mathematical form. After Solve button is clicked, its calculation is automati...
This paper discusses the data-driven regression modelling using first-order linear ordinary differential equation (ODE). First, we consider a set of actual data and calculate the numerical derivative. Then, a general equation for the... more
This paper discusses the data-driven regression modelling using first-order linear ordinary differential equation (ODE). First, we consider a set of actual data and calculate the numerical derivative. Then, a general equation for the first-order linear ODE is introduced. There are two parameters, namely the regression parameters, in the equation, and their value will be determined in regression modelling. After this, a loss function is defined as the sum of squared errors to minimize the differences between estimated and actual data. A set of necessary conditions is derived, and the regression parameters are analytically determined. Based on these optimal parameter estimates, the solution of the first-order linear ODE, which matches the actual data trend, shall be obtained. Finally, two financial examples, the sales volume of Proton cars and the housing index, are illustrated. Simulation results show that an appropriate first-order ODE model for these examples can be suggested. From our study, the practicality of using the first-order linear ODE for regression modelling is significantly demonstrated.