This paper is concerned with consensus of a secondorder linear time-invariant multi-agent system in the situation that there exists a communication delay among the agents in the network.A proportional-integral consens...This paper is concerned with consensus of a secondorder linear time-invariant multi-agent system in the situation that there exists a communication delay among the agents in the network.A proportional-integral consensus protocol is designed by using delayed and memorized state information.Under the proportional-integral consensus protocol,the consensus problem of the multi-agent system is transformed into the problem of asymptotic stability of the corresponding linear time-invariant time-delay system.Note that the location of the eigenvalues of the corresponding characteristic function of the linear time-invariant time-delay system not only determines the stability of the system,but also plays a critical role in the dynamic performance of the system.In this paper,based on recent results on the distribution of roots of quasi-polynomials,several necessary conditions for Hurwitz stability for a class of quasi-polynomials are first derived.Then allowable regions of consensus protocol parameters are estimated.Some necessary and sufficient conditions for determining effective protocol parameters are provided.The designed protocol can achieve consensus and improve the dynamic performance of the second-order multi-agent system.Moreover,the effects of delays on consensus of systems of harmonic oscillators/double integrators under proportional-integral consensus protocols are investigated.Furthermore,some results on proportional-integral consensus are derived for a class of high-order linear time-invariant multi-agent systems.展开更多
This paper offeres an exact study on the robust stability of a kind of combined integrating control system, and the robust stability belongs to the analysis of a kind of quasi-polynomial with two independent time dela...This paper offeres an exact study on the robust stability of a kind of combined integrating control system, and the robust stability belongs to the analysis of a kind of quasi-polynomial with two independent time delays. The parameters of stable space under time delay uncertainty are fixed after Rekasius transformation, and then a new cluster treatment of characteristic roots (CTCR) procedure is adopted to determine the stable space. By this strategy we find that the unstable space is not continuous and both Karitonov vertices theory and Edge theory are unable to be extended to quasi-polynomial under time delay uncertainty.展开更多
The authors find by applying Mac Mahon's partition analysis that all magic labellings of the cube are of eight types,each generated by six basic elements.A combinatorial proof of this fact is given.The number of m...The authors find by applying Mac Mahon's partition analysis that all magic labellings of the cube are of eight types,each generated by six basic elements.A combinatorial proof of this fact is given.The number of magic labellings of the cube is thus reobtained as a polynomial in the magic sum of degree 5.Then the authors enumerate magic distinct labellings,the number of which turns out to be a quasi-polynomial of period 720720.The authors also find the symmetry group can be used to significantly simplify the computation.展开更多
Consider a generalized 3-body problem.The attraction force between any two bodies is proportional to the two"masses"and the b-th power of the mutual distance.Albouy and Fu have obtained the optimal upper bou...Consider a generalized 3-body problem.The attraction force between any two bodies is proportional to the two"masses"and the b-th power of the mutual distance.Albouy and Fu have obtained the optimal upper bound of the number of generalized Euler configurations for the cases b 1 and b=2,3.This paper obtains the optimal upper bound for the remaining real values of b in a systematic way.展开更多
基金supported in part by the National Natural Science Foundation of China (NSFC)(61703086, 61773106)the IAPI Fundamental Research Funds (2018ZCX27)
文摘This paper is concerned with consensus of a secondorder linear time-invariant multi-agent system in the situation that there exists a communication delay among the agents in the network.A proportional-integral consensus protocol is designed by using delayed and memorized state information.Under the proportional-integral consensus protocol,the consensus problem of the multi-agent system is transformed into the problem of asymptotic stability of the corresponding linear time-invariant time-delay system.Note that the location of the eigenvalues of the corresponding characteristic function of the linear time-invariant time-delay system not only determines the stability of the system,but also plays a critical role in the dynamic performance of the system.In this paper,based on recent results on the distribution of roots of quasi-polynomials,several necessary conditions for Hurwitz stability for a class of quasi-polynomials are first derived.Then allowable regions of consensus protocol parameters are estimated.Some necessary and sufficient conditions for determining effective protocol parameters are provided.The designed protocol can achieve consensus and improve the dynamic performance of the second-order multi-agent system.Moreover,the effects of delays on consensus of systems of harmonic oscillators/double integrators under proportional-integral consensus protocols are investigated.Furthermore,some results on proportional-integral consensus are derived for a class of high-order linear time-invariant multi-agent systems.
基金National Natural Science Foundation of China (No.60674088)
文摘This paper offeres an exact study on the robust stability of a kind of combined integrating control system, and the robust stability belongs to the analysis of a kind of quasi-polynomial with two independent time delays. The parameters of stable space under time delay uncertainty are fixed after Rekasius transformation, and then a new cluster treatment of characteristic roots (CTCR) procedure is adopted to determine the stable space. By this strategy we find that the unstable space is not continuous and both Karitonov vertices theory and Edge theory are unable to be extended to quasi-polynomial under time delay uncertainty.
基金supported by the National Natural Science Foundation of China under Grant No.12071311。
文摘The authors find by applying Mac Mahon's partition analysis that all magic labellings of the cube are of eight types,each generated by six basic elements.A combinatorial proof of this fact is given.The number of magic labellings of the cube is thus reobtained as a polynomial in the magic sum of degree 5.Then the authors enumerate magic distinct labellings,the number of which turns out to be a quasi-polynomial of period 720720.The authors also find the symmetry group can be used to significantly simplify the computation.
基金supported by National Natural Science Foundation of China(Grant Nos.10473025,10833001)
文摘Consider a generalized 3-body problem.The attraction force between any two bodies is proportional to the two"masses"and the b-th power of the mutual distance.Albouy and Fu have obtained the optimal upper bound of the number of generalized Euler configurations for the cases b 1 and b=2,3.This paper obtains the optimal upper bound for the remaining real values of b in a systematic way.
