Analysis Of Elementary Fractional Transfer Functions And Applications

The concept of fractional calculus was proposed 300 years ago.It is the theory of any order differential and integral calculus.In control science,fractional calculus equation can be used to describe the fractional order systems and the dynamic and steady state performances of the system are analyzed.The control effect of the system is achieved by designing the fractional order controllers or establishing the fractional order models.In this dissertation,we first introduce the background and development of fractional calculus and the properties of the first kind and second kind of elementary fractional transfer functions are discussed.Then the second kind of elementary fractional transfer functions with an additional zero or pole are investigated,by frequency domain method,and the effect of adding a zero or pole to the second kind of elementary fractional transfer functions on the response performances in time domain are analyzed,indirectly.On this basis,using the fractional order controllers,the compensation design for the first kind and second kind of elementary fractional transfer functions are discussed and the specific design examples are given.Finally,the simulation results show that the compensated fractional order system can achieve a better control effect.The main works of this dissertation are as follows:(1)We discuss the range of parameter values for the stability of the first kind and second kind of elementary fractional transfer functions,analyze the relationship between different time response and fractional order,and the two kind of elementary fractional transfer functions in the frequency domain are studied.(2)The effects of adding a zero or pole to second kind of elementary fractional transfer functions on the system performance are studied.According to the definition of the phase margin and cut-off frequency,the relationship between phase margin and cut-off frequency,respectively,of fractional order system with an additional zero or pole and the adjusting parameters is analyzed,and the response characteristics,in time domain,of fractional order system with an additional zero or pole are analyzed,indirectly,according to the frequency domain characteristics.Finally,simulation studies are given to verify the performance changes after adding a zero or pole.(3)Using fractional order controllers,the first kind and second kind of elementary fractional transfer functions are compensated.The compensation for the first kind of elementary fractional transfer functions by fractional order PI~? controller and the compensation for the second kind of elementary fractional transfer functions via fractional order PD~? controller are considered.It is seen that the compensation is equivalent to adding a fractional zero in the open-loop elementary fractional transfer functions.Similar to the discussion of adding a zero or pole to the second kind of elementary fractional transfer-functions,the response characteristics in time domain of the compensated fractional order system are analyzed indirectly in frequency domain.Finally,simulation verifications are followed.The main contribution of this dissertation is that by adding a zero or pole to the elementary fractional transfer functions,the response performances,in time domain,of the resulted transfer functions are analyzed indirectly in frequency domain.It provides a simple and practical method for the analysis and design of the fractional order system.Also the compensation design of two kinds of elementary fractional transfer functions using fractional order controller gives a concrete application example,further validates the feasibility of the method.