Stability, performance, and robustness using first order controllers.

摘要

This thesis presents a technique for determining the complete set (if one exists) of stabilizing first order controllers for a single-input single-output (SISO), linear, time invariant (LTI) plant P(s) of arbitrary order. In addition, techniques for reducing this set to only include controllers which satisfy certain robustness and performance specifications are presented. The method involves mapping the stability boundary in the frequency domain onto the controller parameter space. This divides the controller parameter space into regions, where every point in a region corresponds to a controller which will result in a closed loop system with a fixed number of unstable closed loop poles. Once this has been done, one need only determine if any of the regions results in no unstable closed loop poles. The results apply to a first order feedback controllers of the form C(s) = x1s+x2s+x3 . The stability set is shown to be computable explicitly for fixed x3 by solving two linear equations in x 1 and x2. The complete, three dimensional set is obtained by sweeping over the scalar parameter x 3. This technique can be applied to continuous time plants to solve the problem of simultaneously stabilizing multiple plants. An example showing how this method can be applied to robustly stabilize an interval family of plants is presented. The equivalent stability problem for discrete time plants can be solved after applying the Tchebyshev representation of a polynomial to the system and applying the same technique. The method is applied to solve two simultaneous stabilization problems in the discrete time domain: the determination of the maximum delay a plant can tolerate and the maximally deadbeat problem.; The guaranteed gain and phase margin problems can be formulated in terms of the above stability method. The solution is used to determine the complete set of stabilizing controllers which satisfy given robustness specifications in the form of gain and phase margin restrictions. In addition, a method using a family of target closed loop transfer functions can be used to choose a controller from within the resulting set to meet specified time domain performance requirements. Finally, a method of determining the complete set of first order controllers which satisfy a given H infinity norm condition is presented.