Theses & Dissertations

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https://hdl.handle.net/20.500.14170/2002

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Browsing Theses & Dissertations by Author "Ibtisam Kamil Hanan"
  • Publication
    Stabilisation of fractional order dynamical control systems based on backstepping method
    (Universiti Malaysia Perlis (UniMAP), 2018)
    Ibtisam Kamil Hanan
    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.