On the Existence of Oscillatory Modes in a Nonlinear System with Hystereses

On the Existence of a Periodic Mode in a Nonlinear System

“…The paper [1] dealt with the nonlinear control system mÿ(t) + βẏ(t) + ky(t) = αl −1 E T (t) ∆ − y(t) − mg, αγ −1Ṫ (t) + T (t) = u…”

“…Note that, given a specific continuous function T (t), a single-valued branch of E(T ) is selected, which is a continuous function ranging in the interval [E 1 , E 2 ] (see [1]).…”

“…Efficiently verifiable conditions on the coefficients and initial values of the state variables in system (1), (2) and on the parameters of the controller (3) ensuring the onset of oscillatory motions [4, pp. 10] in the closed-loop system were obtained in [1], where the following definition of oscillatory motion (oscillatory mode) was used. A solution of system (1), (2), that is, a pair of functions (y(t), T (t)) satisfying the system and the initial conditions with the control u given by ( 3) is called an oscillatory mode if there exist positive constants (mode parameters) t * 1 , t * 2 , y, and y (where 0 < t * 1 < t * 2 and 0 < y < y < ∆) such that the following conditions are satisfied: 1.…”

“…In the present paper, based on the results in [1], we obtain sufficient conditions for the existence of a periodic mode in the closed-loop system (1)-(3).…”

“…For each t such that y(t) = y, there exists a ξ ∈ [t + t * 1 , t + t * 2 ] such that y(ξ) = y and y(τ ) = y for all τ ∈ (t, t + t * 1 ). An analysis of the results in [1] shows that the problem of finding conditions on the parameters of the closed-loop system (1)-(3) guaranteeing the existence of oscillatory modes turns out to be very difficult. However, from the viewpoint of applications, the problem of determining conditions for the existence of a periodic mode in the closed-loop system is more important.…”