Theses & Dissertations
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PublicationStabilisation of fractional order dynamical control systems based on backstepping method(Universiti Malaysia Perlis (UniMAP), 2018)Nowadays, boundary control of integer order partial differential equations (IPDEs) has become an important research area. This is due to the increasing demand on highprecision control of many mechanical systems. In general, the universal phenomenon can be modeled more accurately using fractional order partial differential equations (FPDEs). Therefore, there has been a growing interest in investigating the solution and properties of FPDEs. Compared with the study on control of IPDEs, the results on control of FPDEs are very limited and need to be explored. This thesis focuses on the development of systematic procedures based on backstepping method for stabilisation of FPDE and fractional order partial integro differential equation (FPIDE) systems. Stabilisation for linear FPDE and FPIDE systems is achieved by using two approaches. The first approach uses a finite dimensional backstepping method through the design of coordinate transformations. These transformations have the form of recursive relationships with infinite number of iterations. From numerical simulation, the result showed that the kernel converges to a bounded but possibly discontinuous function. The second approach uses the infinite dimensional backstepping method. In this approach, an integral transformation maps the FPIDE system to a suitably selected Mittag-Leffler stable target system. The kernel is defined by the solution of the kernel hyperbolic partial integro differential equation (PIDE). From numerical simulation, the result showed that the kernel is not only bounded but twice continuously differentiable function. In addition, the infinite dimensional backstepping method is used to design observer for state estimation of linear FPIDE with Dirichlet and Neumann boundary conditions. Two setups are provided. The first setup is anti-collocated when sensor and actuator are placed at the opposite ends. The second setup is collocated when sensor and actuator are placed at the same end. The results showed that both approaches yield satisfactory performance in dealing with unstable linear FPDE and FPIDE systems. Moreover, the semi-discretised backstepping method is introduced to find the boundary controller of nonlinear FPDE system. For this purpose, three cases of nonlinear FPDE system are considered, which are nonlinear FPDE system with space fractional derivative, nonlinear FPDE system with time fractional derivative and nonlinear FPDE system with space and time fractional derivatives. All cases of nonlinear FPDE system are transformed into equivalent stable closed loop and the analytic forms of feedback control laws are designed. For the first case, the convergence of the closed loop system is guaranteed by LaSalle's invariance principle while for the other two cases the convergence is guaranteed by Mittag-Leffler stability. From numerical simulation, the results showed that the proposed semi-discretised backstepping method is powerful to stabilise the nonlinear FPDE system. However, the symbolic calculation of the virtual control becomes more expensive when the value of the discretisation step size approaches to zero.
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PublicationSimplex and interior point methods for solving budgetary allocation linear programming problem in food industry(Universiti Malaysia Perlis (UniMAP), 2018)Mathematical optimization models have essential criteria for optimizing various industrial linear problems mainly involve the profit maximization or cost reduction. There are various significant mathematical models employed for solving budgetary linear programming in industries such as the Simplex and Interior Point Methods. In this study, two mathematical optimization models are developed namely Simplex Method (SM) and Interior Point Method (IPM) to solve linear programming problem. The objective function of linear programming model is to maximize the profit in food manufacturing company. By referring to the constraints that determined by the company in the aspect of production quantities, planned amount of profits and revenues, both models are developed to satisfy all constraints in order to obtain optimal nwnber of product quantities to be produced monthly. The products that been referred mainly come from six type of products namely, Red Beans (K₁), Green Beans (K₂), Chick Peas (K₃), Hamos (K₄), White Beans (K₅) and Large Beans (K₆) from LANA Company for Food Ltd for a period of 10 months in the year 2014. The results indicated that the IPM method produced optimal profit values than its counterpart method and thus it's a promising optimization model for linear programming problem in industries.
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PublicationShooting-euler method for two-point fuzzy boundry value problems(Universiti Malaysia Perlis (UniMAP), 2016)Differential equation plays an important role in modeling of real world problems. However, many real world problems involved are very difficult to obtain clear and exact model. Therefore, differential equation with fuzzy set is essential to provide better understanding. In this study, a two-point fuzzy boundary value problem (FBVP) is considered and solved numerically by using the shooting method. First, the FBVP is interpreted under the concept of generalized differentiability and then followed by reducing it into parametric forms, which fmally obtained four different cases. For each case, the shooting method is used to construct the general procedure of obtaining solutions. Then, Euler method is used to approximate the fmal solutions. A numerical example is provided in order to show the application of the proposed procedures. All numerical simulations are executed by using MA TLAB. From numerical results, it is clear that the approximate solutions obtained by using the proposed procedures are slightly different compared to analytical solutions obtained by using Adomian decomposition method (ADM). Since the values close to each other, thus it can be concluded that the proposed procedure is applicable to solve any two-point FBVP under the concept of generalized differentiability.
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PublicationReliability simulation of scheduled preventive maintenance for surface-mount technology machine(Universiti Malaysia Perlis (UniMAP), 2016)This thesis presents an improved maintenance approach for the surface-mount technology; particularly the AX-50 I machine's scheduled of preventive maintenance by using reliability simulation. This is to predict the failure of the machine and estimate the probability that the machine will not fail over some operational time in order to maintain its functional from sudden breakdown, especially during critical time. The main aim of this study is to propose a better schedule of preventive maintenance than current schedule which is once a month based on the machine's reliability. Data source is collected from July 2014 until June 2015 for 24 hours (2 shifts) of working day from Monday to Sunday. The mathematical model used in this study is based on reliability and model simulation was developed by using Arena software. In conclusion, the best scheduled preventive maintenance for this machine is once for every two weeks. This frequency of scheduled preventive maintenance totally reduces the machine's breakdown as it will extend machine's re1iability and its lifespan thus prolong machine's time to fail.
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PublicationDifferential transformation method (DTM) for solving SIS and SI epidemic models(Universiti Malaysia Perlis (UniMAP), 2015)Differential equations play an important role in modeling real world problem. One of the importance of differential equations is seen in the SIS and SIR epidemic models. Many researches have been proposed to solve SIS and SIR models numerically. However, there is no study about the application of Differential Transformation Method (DTM) for solving SIS and SI epidemic models. In this study, DTM is proposed to solve the SIS and SI epidemic models for constant population. Firstly, the theoretical background of DTM is studied and followed by constructing the solutions of SIS and SI epidemic models. Furthermore, the convergence analysis of DTM is proven by proposing two theorems. Finally, numerical computations are made and compare with the exact solution. From the numerical result, the solution produced by DTM approaches the exact solution which agreed with the proposed theorems. In summary, DTM is an alternative technique to be considered in solving many practical problems involving differential equations.
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