This thesis investigates the stability of a class of nonlinear, time-varying control systems using the Second Method of Liapunov. A recent investigation of this subject was done by Dr. Z. V. Rekasius when he presented the sufficient conditions for stability for a feedback system containing a single nonlinear, time-varying element whose input-output characteristic is contained in a finite sector. This stability criterion developed by Dr. Rekasius is used to extend the region of stability for a class of nonlinear, time-varying systems represented by the equations ẋ = A x + b f(σ,t) σ = cTx where x is an n-vector which represents the state of the system, A is an asymptotically stable n by n constant matrix, b and c are n-vectors, while U and f(,t) are the input-output, respectively, of the nonlinear, time-varying element. Liapunov’s second (direct) method is used in the iv stability analysis of this system. This method enables one to prove that a system is stable if a function V = V(x1,x2…xn,t) can be found which, together with its time derivative, satisfies the requirements of Liapunov' s stability theorems. A particular form of the Liapunov function, V, first proposed by Lure is assumed. By constraining the time derivative of the Liapunov function to have a particular form conditions for a stability criterion are developed and presented in the form of a theorem. The conditions of the theorem are designated as the Improved Criterion and the Integral Constraint. The Integral Constraint places restrictions on the input and output of the nonlinear, time-varying element while the Improved Criterion is used to calculate the maximum value of gain that the closed loop system may assume and still guarantee stability for the closed loop system. The method of this thesis can be used to find the sufficient conditions for stability and boundedness for closed loop systems containing a single nonlinear, time-varying element by a systematic approach. This approach is particularly useful since it applies the stability criterion developed for this class of systems in its most general form thus yielding the maximum gain predictable from the theorem.

Library of Congress Subject Headings

System analysis; Nonlinear theories

Publication Date


Document Type


Department, Program, or Center

Microelectronic Engineering (KGCOE)


Lee, Robert


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