On the input-output stability of time-varying nonlinear feedback systems--Part II: Conditions involving circles in the frequency plane and sector nonlinearities

The object of this paper is to outline a stability theory based on functional methods. Part I of the paper was devoted to a general feedback configuration. Part II is devoted to a feedback system consisting of two elements, one of which is linear time-invariant, and the other nonlinear. An attempt is made to unify several stability conditions, including Popov's condition, into a single principle. This principle is based on the concepts of conicity and positivity, and provides a link with the notions of gain and phase shift of the linear theory. Part II draws on the (generalized) notion of a "sector non-linearity." A nonlinearity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is said to be INSIDE THE SECTOR <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{α,β}</tex> if it satisfies an inequality of the type <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(Nx-)ₜ, (Nx-)ₜ≤0</tex> . If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is memoryless and is characterized by a graph in the plane, then this simply means that the graph lies inside a sector of the plane. However, the preceding definition extends the concept to include nonlinearities with memory. There are two main results. The first result, the CIRCLE THEOREM, asserts in part that: If the nonlinearity is inside a sector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{α, β}</tex> , and if the frequency response of the linear element avoids a "critical region" in the complex plane, then the closed loop is bounded; if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α > 0</tex> then the critical region is a disk whose center is halfway between the points <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/α</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/β</tex> , and whose diameter is greater than the distance between these points. The second result is a method for taking into account the detailed properties of the nonlinearity to get improved stability conditions. This method involves the removal of a "multiplier" from the linear element. The frequency response of the linear element is modified by the removal, and, in effect, the size of the critical region is reduced. Several conditions, including Popov's condition, are derived by this method, under various restrictions on the nonlinearity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> ; the following cases are treated: (i) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is instantaneously inside a sector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{α, β}</tex> . (ii) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> satisfies (i) and is memoryless and time-invariant. (iii) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> satisfies (ii) and has a restricted slope.