On multiple limit cycles in sliding-mode control systems via a generalized describing function approach

Abstract

This paper proposes a generalized describing function (DF) approach to analyze a class of frequency-dependent nonlinearities which are hard to deal with by traditional DF approaches. The proposed approach can intuitively predict the number, stability, and parameters of limit cycles by graphical illustrations of amplitude-phase characteristics. When applied to a 2-relative-degree sampling output feedback system, multiple limit cycles are found in the sliding-mode control. Both behaviors of complex limit cycles and influences of initial conditions on limit cycle oscillations are analyzed. Results show that each limit cycle has its own attraction region which can be estimated by the chattering amplitudes of the state variables, and initial conditions in different attraction regions may induce multiple limit cycles in the same control system. The undesired limit cycles can be eliminated by parameter tuning, such as control gains and sampling periods, to obtain a globally stable one. Simulations and hardware experiments further confirm the validity of the generalized DF approach.

Analysis of multiple oscillations in a sampling reduced-order sliding mode control in frequency domain

“…It also gives us a guidance to adjust the oscillation amplitude and frequency by analyzing the variation of ( ) when coefficients are tuned. Compared with the method in [27,28], the improved DF method is valid for the minimum and non-minimum phase system. In addition, the roots of ( ) can be calculated by the computer software.…”

“…The system could switch from a small limit cycle to a large one when a sufficiently large step signal applied to the input. A generalized DF approach was proposed to estimate multiple limit cycles in a SM system [27,28], and the relation among these limit cycles were revealed intuitively. Under the influence of the sampling interval, multiple limit cycles were found to coexist in a reduced-order SM system [22].…”

“…Under the influence of the sampling interval, multiple limit cycles were found to coexist in a reduced-order SM system [22]. However, the stability of multiple limit cycles in [22,[26][27][28] was analyzed by using the graphical DF approach which is only fit for the minimum phase system. Hence, the analysis of the stability of limit cycles needs further perfection.…”

Chattering-Free Single-Phase Robustness Sliding Mode Controller for Mismatched Uncertain Interconnected Systems with Unknown Time-Varying Delays

“…These have motivated the study of chattering removal problems and time-delay systems, leading to many useful results [1][2][3][4][5][6][7]. The chattering phenomenon causes reducing of control precision or in the worst-case, drive the system to its resonant [8]. In addition, time delay existence can induce degradation and/or instability in system performance [9,10].…”

Prediction of limit cycle oscillations in piecewise linear systems with multiple piecewise nonlinearities

“…By approximating the input and output signal as a unimodal sinusoidal form, DF of a PN is modeled as a phasor representation to enable the linear steady state frequency response analysis. The simplicity and accuracy of DF analysis has entailed many studies on the piecewise linear systems including the flight control systems involving the LCO analysis [7][8][9][10][11]. If, however, the frequency of LCO turns out to be relatively low compared to the system bandwidth of the linear part, or if there are more than two PNs in parallel, then the accuracy of the LCO parameter prediction significantly decreases [12].…”