Parameters Optimization Design Of Fractional Order PI~λD~μ Controllers For Time-Delay Systems

There are more than300years since the theory of fractional calculus founded. The fractional calculus is the theory of arbitrary order derivative and integral calculus, which is an expansion of traditional integral calculus theory. Many actual systems can be described by fractional-order differential equations more accurately than integer-order differential equations. By designing fractional-order controller or engineering fractional-order model, the systems can be controlled better effectively. At present, more and more scholars and engineers focus on the study of the fractional-order systems and have achieved certain results.In this dissertation parameters optimization design of fractional-order controller are discussed.Using a graphical stability criterion applicable to fractional-order time-delay systems, we optimize the fractional order of PIλDμ controllers for time-delay systems, discuss an analytical parameters tuning method of PIλ controllers with exact phase margin and robustness against the gain variations of the controlled plants, and research H∞design with fractional-order PDμ controllers. It is shown via simulation that the fractional-order controller can achieve better control effects than the integer-order controller. The main works of this dissertation are as follows:(1) We discusses the fractional-order optimization design and the phase and gain margins tuning of PIλDμ controllers for fractional-order time-delay systems. A graphical stability criterion for fractional-order time-delay systems is first applied to determine the stabilizing regions in integral-gain and derivative-gain space. Then, the stabilizing region is maximized with respect to fractional-order parameters λ and μ, to expect more various behaviors of the closed-loop systems. Furthermore, in the bigger stabilizing region, the phase margin and gain margin specifications are considered. Finally, an algorithm for achieving the given phase and gain margins is proposed. Illustrating examples are followed in each step to show the design procedures.(2) An analytical parameters tuning method of PIλ controllers is discussed with exact phase margin and robustness against the gain variations of the controlled plants. According to the definitions of phase margin and the robustness to the gain changes of the controlled plant at gain cross over frequency, three nonlinear equations containing the three tuning parameters of PIλ controller are obtained and an algorithm for analytically solving the nonlinear equations is proposed. The Ziegler-Nichols tuning rule for classical PI controller is used for the purpose of comparison. It is shown via simulation that the desired phase margin and robustness can be achieved by the proposed tuning rule of PIλ controllers. (3) This dissertation focuses on H∞performance design for fractional-order systems with time-delay, using fractional-order proportional-derivative (PDμ) controllers. First, the stabilizing parameters region in proportional-derivative space of PDμ controller, for a fixed derivative-order, is determined in terms of a graphical stability criterion applicable to fractional-order time-delay systems. Then, in the stabilizing region, the pairs of proportional and derivative gains of PDμ Controller are calculated, which satisfies the H∞-norm constraint of complementary sensitivity function and defines the H∞boundary curve for a range of frequencies. Finally, by changing the derivative-order of PDμ controller, we observe the relationship between the H∞curve and the derivative-order.