Fractional Order PI~λD~μ Controller Design Based On Sensitivity Constraint

In the past, we use the integer order differential equation to describe the processes in nature, but with the development of science and technology, we find that many phenomena, in the nature, can not be described accurately using traditional integer order differential equations. In fact, the reality of the world, most of them are of fractional order in nature. So, fractional model should become mathematical model for describing the phenomenon of nature. Fractional order calculus for what we can see, can feel, can control the thing has a great influence in the natural world.In this dissertation, firstly, the development and background of the theory of fractional order calculus are introduced. Then, taking the THJSK-1 type tank experiment platform as the engineering background, tank control system, tank model, the feature of time-delay systems, and the fractional order operator approximation are discussed. Based on the discussion, using the relationship of the bound of sensitivity transfer function and the gain-margin and the phase-margin of the system, the stabilizing region and tuning of PIλ、 PIλDμ controllers for time-delay systems are studied, the entire region of the controller parameters that meet the requirement is given, and the superiority of fractional order controller is verified. The main works of this dissertation are as follows:(1) According to the tank experiment device and working principle, we establish mathematical models of single-tank, and test the correctness and effectiveness of the models.(2) For the time-delay systems, according to the the relationship of the bound of sensitivity transfer function and the gain-margin and the phase-margin of the system,the parameter tuning of PIλ、 PIλDμ controllers are investigated. Comparing with the traditional integer order, the simulations and the level closed-loop control experiment, shows that the fractional order controller can ensure a better dynamic performance.The innovation of this dissertation is using the relationship of the bound of sensitivity transfer function and the gain-margin and the phase-margin of the system, which gives the information on the relative stability of the system. Using an algebraic derivation, the parameters of PIλ、PIλD1μ controllers are tuned by plotting the curve of sensitivity bound in the stabilizing parameter region of the controller. Simulation example and experiment prove that the designed PIλ controller can achieve better dynamic performances and robustness, showing the effectiveness of the method.