Robust Fault-tolerant Control For Uncertain Singular Fractional Order Systems

With the development of technology industry,the scale and complexity of the system is increasing,so the fault-tolerant control technology of the complex system has become a hot research topic in the field of control science.Compared with the integer order system,the fractional order system can describe the real system more accurately.The singular fractional order system is the extension of the general system,and it contains not only the static information but also the dynamic constraints,so the description of the singular fractional order system is more accurate than the traditional description.In recent years,many scholars have studied the fault-tolerant control of the singular integer order systems and fractional order systems,and have made a lot of achievements.Therefore,this paper mainly studies the fault-tolerant control of uncertain singular fractional order systems.The main research works are as follows:1.This part introduces the development history of fractional calculus,its purpose and significance,fractional systems theory,fault-tolerant control problem,singular fractional systems theory and their research status.Next,we introduce some basic knowledge of fractional calculus theory,and give three common functions of fractional calculus:the definitions and basic properties of Mittag Leffler function,Gamma function and Beta function.Then,the definitions of three fractional derivatives of Grünwald-Letnikov type,Riemann-Liouville type and Caputo type and their basic properties are given.Finally,the basic definitions and theorems of the admissibility of the singular fractional order systems are given,which play key roles in considering the admissibility of uncertain singular fractional systems.2.The problem of observation fault-tolerant control for uncertain singular fractional order systems with the order 0<α<1 and 1<α<2 is studied by using linear matrix inequality(LMI)methods.First,the system model to be studied in this chapter is introduced.Then,the control input uf(t)of the system with actuator failure is described and the observer is designed.For the observer based augmentation system,the problem of robust fault-tolerant control with actuator failure with order 0<α<1 and 1<α<2 are proposed,respectively.By applying Petersen Lemma,the sufficient conditions for the solvability of robust fault-tolerant control problems are given in the form of linear matrix inequalities.3.The issue of H∞ fault-tolerant control for uncertain singular fractional order systems with perturbation is studied.Firstly,the uncertain singular fractional order system with disturbance is given and the state feedback controller is designed.Then,the sufficient conditions of robust H∞ fault-tolerant control for uncertain singular fractional order systems with the order 0<α<1 and 1<α<2 are studied,and new LMIs formula are proposed.By using the signal-preserving method of the system,the system is decomposed into factors by introducing a polynomial which does not affect the stability of the system,the H∞ performance of the system is improved,and the conclusion also makes the system admissible.Compared with the existing methods,this method not only keeps the system robust stability,but also has better control performance.Finally,four numerical examples are given to illustrate the validity of the theorem and ensure the validity of bounded norm.4.Fault-tolerant control for actuator fault of uncertain generalized fractional order T-S fuzzy system is studied.Firstly,the model of T-S fuzzy uncertain singular fractional order system is given,and the state feedback controller is designed for the T-S fuzzy system.Then,for the problem of actuator failures of the system,by using the full rank decomposition method,the sufficient criteria and the solutions form of the gain matrix of state feedback controller for uncertain singular fractional order T-S fuzzy systems with the order 0<α<1 and 1<α<2 are proved,respectively.Finally,the feasibility analysis of the conclusions in this chapter is carried out by using the linear matrix inequality toolbox in MATLAB,and the feasible solutions are obtained.The simulation of the system is drawn by using the Simulink Toolbox to verify the correctness and effectiveness of the conclusions.