The Gain And Phase Margins Calculation And Controllers Design

Robust control is wide used in the space aircraft and industry systems design to ensure the stability and robustness of control systems. In the past two decades, many researches provided rich reslts in the robust control theory and its applications. The disturbance, model error and parameter perturbation in the contol loops are the major resource to effect the stability of control system. In the frequence domain, the gain and phase margins measure the robustness of closed-loop systems. At present, the gain and phase margins are only suitable for stable single-loop processes. The stability margin calculation and the controller design for unstable processes or multi-loop processes still need to be taken into account in the further study.This paper develops a frequency domain method to solve the gain and phase margins calculation and controller design problems. Based on the frequency response analysis, complex variable analysis, iterative drawing method and optimization theory are employed to solve the characteristic equations, which enables the accuracy calculation of the stability margin and control system design. The main results and contributions are listed as follows:(1) The combined gain and phase margin is proposed for stable and unstable system.The limitations of the conventional or separate gain and phase margins are shown by case study. Then, the combined gain and phase margin is introduced to overcome the limitations and suit both stable and unstable systems. The calculation method is proposed to solve the characteristic equation with a virtual compensator. With the combined gain and phase margin, its applications in stabilized parameters range of PD controller calculation and PID controller design are demonstrated.(2) A lead/lag compensator design problem is considered for a class of unstable delay processes based on a new set of gain and phase margin specifications.Due to the nature of the unstable system, both upper and lower gain margins are required to measure the true stability robustness with regard to gain change, which leads to three stability margins should be taken into account. The stability condition is developed by the Nyquist theorem and the three stability margins conditions. In addition to a phase margin, such a combined margin problem leads to a set of nonlinear and coupled equations which have no analytical solution. Two types of Nyquist curve is determined and an effective graphical method is developed such that the solution is determined from the intersections of the curves constructed by a transformed set of nonlinear equations. The tuning procedure is presented and examples are given for its illustration and comparison.(3) Tuning of multi-loop PI controllers based on gain and phase margin specifications is proposed.A single-iteration strategy is proposed for the design of a multi-loop PI controller to achieve the desired gain and phase margins for two-input and two-output (TITO) processes. To handle loop interactions, a TITO system is converted into two equivalent single loops with uncertainties drawn from interactions. The maximum uncertainty is estimated for the initial controller design in one loop and single-input and single-output (SISO) controller design is applied based on Nyquist band. This controller is substituted to other equivalent loop for design, and finally, the first loop controller is refined on knowledge of other loop controller. For SISO controller tuning, a new method is presented to determine the achievable gain and phase margins as well as the relevant controller parameters. Examples are given for illustration and comparison.(4) Exact computation of loop gain margins of multivariable feedback system based on vector function optimization is presented.A frequency domain approach is proposed to accurately computing loop gain margins for multivariable feedback systems. With the help of vector mapping method, the stability boundary calculation problem is converted to some constrained optimization. Then, the optimization problem is transformed from complex field to real field, and is solved numerically by the Lagrange multiplier and Newton-Raphson iteration algorithm. Meanwhile, we utilize norm inequality to estimate the frequency range for the stabilizing gain computation. The proposed approach can determine all the stabilizing boundaries and provide exact gain margins in comparison with conservativeness of the results reported before.(5) Loop gain and phase margins calculation for two-input two-output systems in frequency domainA frequency domain approach is proposed to accurately computing loop gain and phase margins for TITO systems. With the help of vector analysis and analytic geometry methods, the stability boundaries computation problem is converted to find the intersection points of some geometry curves. The frequency ranges are estimated by the matrix norm inequalities method to reduce computation burden. All the stability boundaries are calculated in the frequency ranges, and the gain and phase margins are determined naturally in the stability regions.