In this work, the system with forced term is considered. Two methods for adding the forced term to the differential dynamical systems for asymptotic stability are given. The first method is based on Lyapunov's direct method, and the second method is a new method based on stability analysis of Hopf bifurcations. The new method has a discriminant which is a rational expression consisting of the coefficients in the original system. Some further researches for adding forced term are also presented.
GUO Laigang
,
CHEN Yufu
. Study on stability of systems with forced term[J]. Journal of University of Chinese Academy of Sciences, 2017
, 34(3)
: 277
-288
.
DOI: 10.7523/j.issn.2095-6134.2017.03.002
[1] Leine R I. The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability[J]. Nonlinear Dynamics, 2010, 59(1/2): 173-182.
[2] B?la? M M, B?la? V E. World Knowledge for control applications by fuzzy-interpolative systems[J]. International Journal of Computers Communications & Control, 2008, 3(1):28-32.
[3] Bazoula A, Djouadi M S, Maaref H. Formation control of multi-robots via fuzzy logic technique[J]. International Journal of Computers Communications & Control, 2008, 3(2):179-184.
[4] Bellomo D, Naso D, Babuška R. Adaptive fuzzy control of a non-linear servo-drive: theory and experimental results[J]. Engineering Applications of Artificial Intelligence, 2008, 21(6): 846-857.
[5] Mohan B M, Sinha A. Analytical structure and stability analysis of a fuzzy PID controller[J]. Applied Soft Computing, 2008, 8(1): 749-758.
[6] Perry A G, Feng G, Liu Y F, et al. A design method for PI-like fuzzy logic controllers for DC-DC converter[J]. IEEE Transactions on Industrial Electronics, 2007, 54(5): 2 688-2 696.
[7] Vesselenyi T, Dzi?ac S, Dzi?ac I, et al. Fuzzy and neural controllers for a pneumatic actuator[J]. International Journal of Computers Communications & Control, 2007, 2(4): 375-387.
[8] Feng G. A survey on analysis and design of model-based fuzzy control systems[J]. IEEE Transactions on Fuzzy systems, 2006, 14(5): 676-697.
[9] Michels K, Klawonn F, Kruse R, et al. Fuzzy control: fundamentals, stability and design of fuzzy controllers[M]. New York: Springer, 2007.
[10] Precup R E, Preitl S, Rudas I J, et al. Design and experiments for a class of fuzzy controlled servo systems[J]. IEEE/ASME Transactions on Mechatronics, 2008, 13(1): 22-35.
[11] Tian E, Peng C. Delay-dependent stability analysis and synthesis of uncertain T-S fuzzy systems with time-varying delay[J]. Fuzzy Sets and Systems, 2006, 157(4): 544-559.
[12] Khalil H K, Grizzle J W. Nonlinear systems[M]. New Jersey: Prentice Hall, 1996.
[13] Haddad W M, Chellaboina V S. Nonlinear dynamical systems and control: a Lyapunov-based approach[M]. Princeton: Princeton University Press, 2008.
[14] Vidyasagar M. Nonlinear systems analysis[M]. Philadelphia: Society for Industrial & Applied Mathematics (SIAM), 2002.
[15] LaSalle J P. Some extensions of Liapunov's second method[J]. Circuit Theory, IRE Transactions on Circuit Theory, 1960, 7(4): 520-527.
[16] Yu P. Simplest Normal Forms of Hopf and Generalized Hopf Bifurcations[J]. International Journal of Bifurcation & Chaos, 2011, 9(10):1 917-1 939.
[17] Yu P. Computation of normal forms via a perturbation technique[J]. Journal of Sound and Vibration, 1998, 211(1): 19-38.
