טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentNakhmani Arie
SubjectGeneralized Nyquist Criterion and Generalized Bode Diagram
for Analysis and Synthesis of Uncertain Control
Systems
DepartmentDepartment of Electrical Engineering
Supervisor Professor Emeritus Ezra Zeheb
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


Abstract

Our work is concerned with the parameterization of a simple, symmetric and bounded contour G and G-stability for systems with interval uncertainty. We utilize the generalized interpretation of amplitude and phase (on the contour G), and assume that if all the closed loop poles are contained inside the G-contour, then all stability and performance specifications are satisfied. We prove theorems for the analysis of linear time invariant (LTI) systems, and for clustering of open loop and closed loop poles inside a bounded contour G. In this work, we employ the positive and negative crossings notation for G-stability tests. This technique allows developing the theorems analytically.

Uncertainties in control systems models often have to be taken into account for analysis and/or design. Neglecting such uncertainties is often unjustified and is done only due to lack of methods to treat the uncertainties. We provide the tools for constructing the generalized Bode envelope for the given rational uncertain system, with interval coefficients. We prove generalized Mikhailov and Nyquist theorems. With the help of these theorems we can calculate the number of open loop and closed loop poles of the whole uncertain family inside the given G-contour.

The last part of this work is concerned with the problem of using our general framework in order to design the desired controller with the closed loop poles of the interval uncertain family distributed inside the given G-contour. We explain how to use the graphical and analytical design technique, using the notion of crossings. Also, we explicate how to find the set of G-stabilizing controllers for the whole uncertain family. The technique for controller design avoids the need to employ the phase or gain margin, and can assure the stability and transient response specifications at the same time for the whole family, defined by generalized Bode envelope.

The advantages of this present research are in the unified approach to continuous time and discrete time systems, and in the similarity of our techniques to classical control theory. The disadvantages of techniques presented here are in possible conservative results, because in most cases, the generalized Bode envelope includes more functions than are needed for an appropriate description of the uncertain family.

The future extension of the techniques considered in this work will include time-delay systems, MIMO systems, and certain non-linear systems. Also, more uncertainty types should be investigated.