A method and an algorithm for obtaining the Stable Oscillatory Regimes Parameters of the Nonlinear Systems, with two time constants and Relay with Delay and Hysteresis NUŢU VASILE, MOLDOVEANU CRISTIAN-EMIL, ŞOMOIAG PAMFIL Department of Mechanical Engineering, Military Technical Academy No.81-83, George Cosbuc, Sector 5, Bucharest, ROMANIA vnutu@yahoo.com , mcrristi@gmail.com , pamfils@yahoo.com Abstract: - The object of this paper is the oscillatory stable regime of a particular nonlinear system. This particular nonlinear system includes a relay, and its linear part is characterized by a transfer function with two time constants. In the beginning, the paper shows a method that can be used in the calculus of the parameters for the limit stable cycle, which is appropriated for these nonlinear systems. After that, the method is particularized for two different commutation lows of the relay nonlinearity: relay with hysteresis and relay with delay time. The different commutation laws induced, for the same linear parts, different shapes and parameters for the oscillatory stable regimes proper to assembly nonlinear system. These differences are presented in a case study. Key-Words: - nonlinear systems, oscillatory regime, relay, delay time, hysteresis, calculus method, algorithm. 1 Introduction The systems that include linear and nonlinear parts, in assembly, are nonlinear systems. The figure 1 shows a typical structure for these nonlinear systems. N H ε y r + _ ) ( s H L u Fig.1. A typical nonlinear system The previous structure uses the feedback principle, with a unitary reaction. The controlled process (or plant) is described by a linear transfer function, , which also includes the linear parts of the actuator. The regulator functionality and the nonlinear actuator characteristic are described by a nonlinear function, . The error signal, ) ( s H L N H ε , is the difference between the system input (or reference), r , and system output, y . Based on the error signal, the nonlinear parts generates the command , that are the input of the plant. u Following, we will consider that the kind of the nonlinearity is a real relay, with delay time and hysteresis. This is the model of certain actuators, frequently included in many industrial and civilian applications, even in advanced equipments: liquid alimentation parts, air or gas conditioning devices. The main function of theses nonlinear systems is to maintain the values of any system parameters, in specified ranges [1]. Usually, these systems work with a constant input values. According to the system input level, the response of those systems bring to a punctually stationary state or to an oscillatory stable regime. The oscillatory stable regime is exclusively induced by the system nonlinearity. The characteristics of oscillatory stable regime, if it appears, can be analytically expressed [2], [3], [4], [8], and in other situations can be obtained only by simulations. The relay model is proper for the actuators that work in maximal regime. An ideal relay model of the system nonlinearity can induce a very fast oscillatory regime, which increases the actuators stress and wearing. Nevertheless, in many automatic systems, is not preferred the ideal relay work for the nonlinear part. The relay model can be deliberately depreciated, using delays or hysteresis characteristics, in order to increase the systems oscillatory regime periods. In this way, the commutations frequency of the actuators can be reduced, the actuators wearing decreases and the WSEAS TRANSACTIONS on SYSTEMS and CONTROL Nutu Vasile, Moldoveanu Cristian-Emil, Somoiag Pamfil ISSN: 1991-8763 413 Issue 8, Volume 4, August 2009