Computation of multiple limit cycles in nonlinear control systems – a describing function approach
Abstract Limit cycles play an important role in nonlinear systems, provided that many control loops with common nonlinearities like relay, hysteresis, and saturation can present them. Thus, a proper description of this nonlinear phenomenon is highly desirable. A strategy for the linearized analysis...
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| Published in | Journal of aerospace technology and management Vol. 3; no. 1; pp. 21 - 28 |
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| Format | Journal Article |
| Language | English |
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Departamento de Ciência e Tecnologia Aeroespacial
01.01.2011
Instituto de Aeronáutica e Espaço (IAE) |
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| Abstract | Abstract Limit cycles play an important role in nonlinear systems, provided that many control loops with common nonlinearities like relay, hysteresis, and saturation can present them. Thus, a proper description of this nonlinear phenomenon is highly desirable. A strategy for the linearized analysis is the describing function method, which is a frequency domain approach that allows the limit cycle prediction and stability analysis. Some papers had discussed the method for the simplified analysis; however, they are concentrated in the prediction of only one limit cycle even for systems with multiple conditions. This paper proposes a systematic way of multiple limit cycle determination, as well as the stability analysis of each one. All theoretical/computational issues involved in the approach are also discussed. |
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| AbstractList | Limit cycles play an important role in nonlinear systems, provided that many control loops with common nonlinearities like relay, hysteresis, and saturation can present them. Thus, a proper description of this nonlinear phenomenon is highly desirable. A strategy for the linearized analysis is the describing function method, which is a frequency domain approach that allows the limit cycle prediction and stability analysis. Some papers had discussed the method for the simplified analysis; however, they are concentrated in the prediction of only one limit cycle even for systems with multiple conditions. This paper proposes a systematic way of multiple limit cycle determination, as well as the stability analysis of each one. All theoretical/computational issues involved in the approach are also discussed. Abstract Limit cycles play an important role in nonlinear systems, provided that many control loops with common nonlinearities like relay, hysteresis, and saturation can present them. Thus, a proper description of this nonlinear phenomenon is highly desirable. A strategy for the linearized analysis is the describing function method, which is a frequency domain approach that allows the limit cycle prediction and stability analysis. Some papers had discussed the method for the simplified analysis; however, they are concentrated in the prediction of only one limit cycle even for systems with multiple conditions. This paper proposes a systematic way of multiple limit cycle determination, as well as the stability analysis of each one. All theoretical/computational issues involved in the approach are also discussed. |
| Author | Brito, Alexandre Garro |
| AuthorAffiliation | Institute of Aeronautics and Space |
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| Keywords | Multimodal optimization Limit cycle Describing functions |
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| References | 1996; 1141 Nayfeh, A.H.; Mook, D. 1979 2009; 14 1990; 99 Gibson, J.E. 1964 Slotine, J.J.; Li, W. 1991 2000; 23 2002; 40 2008; 18 1997; 56 2001; 26 Leite Filho, W.; Bueno, M. 2003 1995; 18 Siljak, D. 1969 2005; 45 Gelb, A.; Velde, W. 1968 1994; 7 |
| References_xml | – volume: 99 start-page: 143 issue: 2 year: 1990 end-page: 155 article-title: A model of predator-prey dynamics as modified by the action of a parasite publication-title: Mathematical Biosciences – year: 1968 publication-title: Multiple-input describing functions and nonlinear system design contributor: fullname: Gelb, A.; Velde, W. – year: 1964 publication-title: Nonlinear automatic control contributor: fullname: Gibson, J.E. – volume: 14 start-page: 4041 issue: 12 year: 2009 end-page: 4056 article-title: Symbolic computation of limit cycles associated with hilbert's 16th problem publication-title: Communications in Nonlinear Science and Numerical Simulation – volume: 26 start-page: 3 year: 2001 end-page: 22 article-title: An eigenvalue method for calculation of stability and limit cycles in nonlinear systems publication-title: Nonlinear Dynamics – year: 2003 publication-title: Analysis of limit-cycle phenomenon caused by actuator's nonlinearity contributor: fullname: Leite Filho, W.; Bueno, M. – year: 1979 publication-title: Nonlinear oscillations contributor: fullname: Nayfeh, A.H.; Mook, D. – year: 1969 publication-title: Nonlinear systems - the parameter analysis and design contributor: fullname: Siljak, D. – year: 1991 publication-title: Applied nonlinear control contributor: fullname: Slotine, J.J.; Li, W. – volume: 40 start-page: 2197 issue: 11 year: 2002 end-page: 2205 article-title: Limit-cycle oscillation induced by nonlinear aerodynamic forces publication-title: American Institute of Aeronautics and Astronautics Journal – volume: 56 start-page: 3809 issue: 4 year: 1997 end-page: 3813 article-title: Number of limit cycles of the li´enard equation publication-title: Physical Review E – volume: 18 start-page: 755 issue: 3 year: 2008 end-page: 778 article-title: Bifurcation of multiple limit cycles for a rotor-active magnetic bearings system with time-varying stiffness publication-title: International Journal of Bifurcation and Chaos – volume: 14 start-page: 270 issue: 1 year: 2009 end-page: 283 article-title: Dynamics of three coupled limit cycle oscillators with application to artificial intelligence publication-title: Communications in Nonlinear Science and Numerical Simulation – volume: 18 start-page: 280 issue: 2 year: 1995 end-page: 286 article-title: Dynamics and control of limit cycling motions in boosting rockets publication-title: Journal of Guidance, Control and Dynamics – volume: 1141 start-page: 398 year: 1996 end-page: 407 article-title: Every niching method has its niche: fitness sharing and implicit sharing compared publication-title: Lecture Notes in Computer Science – volume: 45 start-page: 25 year: 2005 end-page: 30 article-title: On the implementation of the eigenvalue method for limit cycle determination in nonlinear systems publication-title: Nonlinear Systems – volume: 23 start-page: 611 issue: 4 year: 2000 end-page: 619 article-title: Multiple on-off valve control for a launch vehicle tank pressurization system publication-title: Journal of Guidance, Control and Dynamics – volume: 7 start-page: 65 issue: 6 year: 1994 end-page: 70 article-title: Multiple limit cycle for three dimensional lotka-volterra equations publication-title: Applied Mathmatics Letters |
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| Snippet | Abstract Limit cycles play an important role in nonlinear systems, provided that many control loops with common nonlinearities like relay, hysteresis, and... Limit cycles play an important role in nonlinear systems, provided that many control loops with common nonlinearities like relay, hysteresis, and saturation... |
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| SubjectTerms | Describing functions ENGINEERING, AEROSPACE Limit cycle Multimodal optimization |
| Title | Computation of multiple limit cycles in nonlinear control systems – a describing function approach |
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