A describing function approach to the design of robust limit-cycle controllers

Oliveira, Neusa Maria Franco; Kienitz, Karl Heinz; Misawa, E.A.

doi:10.1007/s11071-011-9983-8

Cited by 14 publications

(10 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This approach for analysis of nonlinear systems is easily found in several classic books [Ogata, 1970;Slotine & Li, 1991;Khalil, 1996;Glad & Ljung, 2000], consisting of an extension of linear techniques based on the concepts of the frequency response in order to verify effects of certain nonlinearities on a feedback dynamic system. The analysis by describing functions allows to predict stability and the possible existence of limit cycles [Ogata, 1970;Oliveira et al, 2012] and chaotic behavior, which can be interpreted as an interaction between limit cycles and equilibrium points [Genesio & Tesi, 1991;Neymeyr & Seelig, 1991;Genesio & Tesi, 1992;Savacı & Günel, 2006]. The transfer function of the linear part of Chua's circuit between the output x and the input u (see Fig.…”

Section: Stability Analysismentioning

confidence: 99%

The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

Medrano-T

Rocha

2014

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

Although Chua's circuit is one of the most studied nonlinear dynamical systems, its version with negative parameters remains practically untouched. This work reports an interesting and rich dynamic scenery that was hidden in this almost unexplored region. The study is focused on 2D parameter space and presents an analysis of stability based on describing functions. Numerical investigations present a gallery of period-adding cascades and a strong presence of basin boundary metamorphoses. The key to this new scenario is that for negative parameters, Chua's system does not satisfy the Shilnikov condition and it is shown that the homoclinic orbit organizes the parameter space completely different from as known. The obtained experimental results corroborate with the numerical and theoretical investigations.

show abstract

Section: Stability Analysismentioning

confidence: 99%

The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

Medrano-T

Rocha

2014

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

show abstract

“…Compared with the Lyapunov method and Poincare maps, the describing function (DF) method provides a relatively simple and efficient solution for the oscillation in the nonlinear control algorithm [14,15]. For the SM algorithm, the DF method and Routh Criterion were employed to predict the stability of oscillations in the uncertain SM system [16]. Its robustness could be tuned by the intersection angle between the Nyquist plot of the plant and the negative reciprocal DF of controller.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the oscillation in a second-order SM controller could be set by parameter tuning under the guidance of the DF method [12]. The traditional DF method is effective for these uncoupled nonlinearities in [12,16,17], which only depends on the amplitude of oscillations. However, the SM algorithm may depend on both the amplitude and frequency of oscillations with the increase of complexity of controllers, and it is hard for the traditional DF method to analyze the coupled nonlinearity in SM controllers.…”

Section: Introductionmentioning

confidence: 99%

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

Shen¹,

He²

2017

J. vibroeng.

View full text Add to dashboard Cite

The phenomena of oscillations in sliding mode control system usually cause potential damage or danger in engineering applications. This paper focuses on the estimation and adjustment of oscillation parameters (frequency and amplitude) using an improved describing function method and the characterization of the influence of the reduced-order sliding surface coefficients on the performance of the nonlinear system. The model of series connection of the zero-order holder and switching function is established in frequency domain. The improved describing function method could intuitively estimate the number, stability and parameters of oscillations for the coupled nonlinear system by its amplitude-phase characteristics and pole placement. The stability, parameters and attraction zones of three oscillations in a fourth-order plant are analyzed and simulated. The analytical results show that the reduced-order sliding mode controller possesses complex and colorful nonlinear behaviors. Besides, a coefficient tuning method is presented to eliminate the undesired oscillation which may lead to poor precision or break the stability. The simulations demonstrate that appropriate tuning on sliding surface may greatly improve the efficiency and robustness of the sliding mode control system.

show abstract

“…This approach is also a proper tool to characterize limit cycle behaviors by modeling nonlinear systems in the frequency domain. The robustness of the SM system has been improved by adjusting the intersection angle of curves of G(s) and −1/N (A) guided by the traditional DF approach [17]. The influences of a variety of switching functions on the limit cycle and its stability in SM systems were studied by the DF approach [18].…”

Section: Introductionmentioning

confidence: 99%

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

Shen

Qiu

2015

Nonlinear Dyn

View full text Add to dashboard Cite

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.

show abstract

A describing function approach to the design of robust limit-cycle controllers

Cited by 14 publications

References 5 publications

The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

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

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

Contact Info

Product

Resources

About