Research On Stability And Control Of Fractional-order Uncertain Systems

Due to the impacts of external disturbances and internal disturbances,system modeling usually needs analysis of the uncertainty of different parameters.Fractional-order uncertain systems have attracted wide attention because they well reflect many real-world physical systems.The robust stability and control on fractional-order uncertain systems is an important research direction.For different types of fractional-order uncertain systems,some methods for robust stability and stabilization have been proposed.However,it is difficult to analyze some types of complex systems by existing methods,such as fractional-order systems(FOS)with multi-parameters and FOS with uncertain order.Cylindrical Algebraic Decomposition(CAD)technique is an effective method for analyzing multi-parameters systems and has been widely used in interger-order systems.For FOS with multi-parameters,it is difficult to obtain robustness bounds of all parameters and the relationship between parameters with the existing methods.In addition,most existing studies consider FOS with order 0<?<1 and 1<?<2 separately due to the robust stability of FOS with order 0<?<1 is more difficult to analyze than the case of order 1<?<2.The results are conservative and there is no unified method to analyze fractional order uncertain systems with order 0<?<2.In this dissertation,for fractional-order linear systems with multi-parameters and fractional-order nonlinear systems with multi-parameters,the problems of robust stability and control are considered by using the CAD method.The CAD-based methods proposed in this dissertation solve the conservatism problem of the existing results.The results of this dissertation are complete and non-conservative,and suitable for fractional-order uncertain systems with order 0<?<2.The main contributions are as follows:First,for fractional-order linear system with multi-parameters,a method is established to obtain the stable parameter region.The system matrix of fractional-order uncertain systems is a parameter-dependent matrix.The existing methods need the simplification of the parameter-dependent matrix,so the results are conservative.This dissertation uses the CAD method to analyze parameters directly,and gives an algorithm for analyzing the robust stability of fractional-order linear systems with multi-parameters.The CAD-based method does not simplify the system matrix.The robustness bounds of all parameters can be obtained,including the relationship between parameters.Second,for fractional-order linear systems with uncertain order,a method is proposed to obtain the range of stable order.By coordinate substitution,the critical stable boundary of the complex plane is transformed into the critical hypersurface in the parameters space.The critical hypersurface decomposes the parameters space into several disconnected regions.Based on the CAD method,the stability of these regions can be tested and the range of stable order can be solved.The results can directly determine whether there is a coupling relationship between the uncertain order and the system parameters.Then,for the above two types of fractional-order uncertain linear systems,including fractional-order linear systems with multi-parameters and fractional-order linear systems with uncertain order,CAD-based parameterized controllers are designed.The problem of solving controller parameters is transformed into the problem of analyzing the stability of regions in the parameters space.By analyzing the robust stability of the control systems in the parameters space,the stable parameter region can be obtained.The constraint conditions between system parameters and controller parameters are given by the stable parameter region.Different control performances can be obtained by tuning control parameters in stable parameter region.The results provide the complete range of stable parameters for designing different controllers and show that the parameterized controller can effectively stabilize the fractional-order uncertain systems.Finally,the problem of Hopf bifurcation on fractional-order nonlinear systems with multi-parameters is considered.Fractional-order nonlinear systems with multi-parameters are more general than fractional-order nonlinear systems with single-parameter.The existing methods need to fix other parameters to analyze the jacobian matrix at the equilibrium point.The critical value of single bifurcation parameter is obtained by computing the real and imaginary parts of the eigenvalues.When the systems have multiple bifurcation parameters,it is difficult to obtain the range of critical value about multiple bifurcation parameters.In this dissertation,fractional-order nonlinear systems with multi-parameters and fractional-order nonlinear systems with uncertain order are considered.This dissertation transforms the characteristic polynomial of parameter-dependent jacobian matrix into the polynomial with bifurcation parameters,gives the range of the critical bifurcation parameters by combining the CAD technique and Hopf bifurcation conditions of fractional-order systems.Besides,for fractional-order nonlinear systems with multi-parameters,this dissertation proposes a parameterized controller for bifurcation control.The results include the relationship between control parameters and bifurcation parameters.The stable parameter region can be expanded and the critical bifurcation paramaters can be changed by tuing the parameterized controllers.