A Graphical Tuning Of PI~λD~μ Controllers For Fractional Order Systems With Time-delay

Fractional order calculus is an old field in mathematic study fields and an expansion of traditional integral calculus theory. It is more than300years since the theory of fractional order calculus was founded, and so far, a great achievement has been attained in the study of this theory, which provides a new theory foundation for the application of fractional order calculus in other subjects.In this paper, the theoretical foundation of fractional order calculus, the structure of fractional order control systems, the feature of time-delay systems and the research of the control for time-delay systems are discussed. Based on the discussion, using a graphical stability criterion applicable to fractional-order time-delay systems, the stabilizing region and tuning of PI^D" controller for fractional order time-delay systems are studied, the entire region of the controller parameters that meet the requirement was given, which makes the design of the controllers easier and more flexible. The main works of this dissertation are as follows:(1) For fractional order time-delay systems, the PIλ controller is designed, and the stabilizing region is determined according to the graphical stability criterion. The influence of λ on the size of the stabilizing region is discussed. After that the stabilizing region is maximized analytically with respect to parameter λ. The stability margin is considered in the stabilizing region to get better dynamic performance, and the illustrative examples are given finally.(2) For PD" controller of fractional order time-delay systems, the parameter stabilizing region is studied. For first-order plant with time-delay, the influence of parameter μ on the shape and size of the stabilizing region is discussed. The stability margin and relative stability degree are considered, respectively in stabilizing region. The simulation indicates that the parameters in stability margin and relative stability degree curves could ensure a better performance.(3) The PIλDμ controller is designed for fractional order processes with time-delay. Then fix one of the three parameters:kpk1and kd, the stabilizing region of the other two parameters is discussed. Further, the stabilizing regions could be drawn in the3-dimensional spaces by gridding the fixed parameter. The stability margin and relative stability degree are considered, respectively in stabilizing region, and examples are given to show the design procedure.The innovation of this dissertation is applying a graphical stability criterion to fractional order time-delay systems. Using the criterion, the parameter stabilizing region of the fractional order PID controller and the design of the system performance are studied, which provide a simple and useful means for fractional order time-delay systems’analyze and design. Otherwise, the stabilizing region’s optimization with respect to parameterλ is discussed when PIλ controller is designed, which has a instructive meaning for the choose of parameter...