Fractional-order Grey System And Novel Models Research

Grey theory is an important method for processing "small sample" and uncertain system with "poor information",which can be used for system analysis,prediction,control and decision by analyzing the known information.Compared with integer-order,the fractional-order calculus extends the order to the fraction or even the complex number.Although the complexity of the calculation and physical realization is higher,fractional calculus brings the flexibility of the application and the scope of problem solving for system analysis,which becomes an important tool for modeling complex physical processes.Considering the flexible and soft features for system description,fractional calculus is used in this paper,which could reveal the developing trend for the grey system,meanwhile provide more effective mathematical models and analytical methods for complex small sample and uncertain systems.On the basis of classical models,novel fractional-order grey system models,also called fractional-order grey models,will be constructed by better universality and higher modeling accuracy,which are essentially different with classical models in mathematical forms and modeling mechanisms.The research work in this thesis can be summed up as follows:(1)One class of grey integral model and grey stochastic model are proposed respectivelyThe grey integral model is presented in this thesis.The integral function is used to replace and describe the derivative function of the grey model with integral function.Its essence is to improve the background value of the continuous equation in the grey model.In addition,a class of stochastic GM(1,1)model is given by random coefficients for mining complex fluctuating data,which has more simple parameters characteristics and clear physical meaning,the optimization principle,solving process and simulation results of the stochastic grey model are also given.Meanwhile the extended forms of grey integral model and stochastic grey model in the fractional-order case are given.(2)Fractional-order linear time-invariant grey system and control problems are analyzed and simulatedThe fractional grey system equation is important for describing the fractional-order grey system,which is also the theoretical basis of the fractional-order grey model.The fractional grey linear time-invariant system is researched by fractional calculus and grey number in this thesis.Theoretical analysis and simulation experiments are given for fractional grey linear time-invariant system equation and its numerical solution.Meanwhile,the basic control problems of fractional-order grey system are discussed,for example controllability,observability and stability.The linear time-invariant fractional-order grey control is simulated by "kenel" and "grey degree" model,and H? control method for fractional-order grey model.(3)New discrete fractional-order accumulation grey models are establishedDiscrete grey models are established in this thesis by fractional-order accumulation calculus using Gamma function,including fractional accumulation GM(1,1)model,fractional accumulation GM(0,N)model and fractional-order accumulation interval grey model.Meanwhile,the optimal model of fractional-order accumulation GM(1,1)and its modified model based on Fourier series correction are given,and the actual load for thermal power plant is predicted by the fractional-order accumulative interval grey model.Based on the flexible characteristics of fractional-order accumulation operator,the new discrete models can effectively improve the reliability and accuracy of system modeling.Compared with the integer-order,it shows that the discrete fractional-order accumulation model can significantly improve the reliability and accuracy of system modeling.(4)New continuous fractional-order grey models are proposedIn this thesis,new continuous fractional-order grey models are proposed,which include GM(?m,1),GM(?1,1),GM(?2,1),multi-variable fractional-order grey model,fractional-order grey time-delay model,random model,integral model,scale model and et al.Examples are given for the optimization of GM(?1,1)and GM(?2,1)in detail,and the error feedback model are also discussed with higher universality and precision of modeling.The result shows its performance is greatly improved.In addition,the approximation and discretization of continuous fractional-order grey model are settled.The fractional-order diagram method is used for new continuous fractional-order grey models based on FOTF toolbox,it can also simplify the solving process of novel models to a certain extent as an important supplement to the numerical algorithm.(5)The implementation,analysis and simulation for grey circuit and fractional-order grey circuit are givenGrey circuit and fractional grey circuit are proposed for simulating uncertain grey number of the system in this thesis.Meanwhile,the model,principle,simulation analysis and experimental results of grey circuit and fractional grey circuit are given,for example the controllability,observability,stability of control problem in the grey linear circuit system,output and simulation analysis by virtual instruments;the output,resonance and stability of the fractional-order grey circuit.It is intuitive,convenient and accurate to to obtain grey system internal structure and get model time-frequency analysis results by Multisim virtual instrument simulation.The researches mentioned above can provide the circuit simulation and important theory references for fractional grey model theory and control system analysis.