Spectral abscissa(SA)is defined as the real part of the rightmost characteristic root(s)of a dynamical system,and it can be regarded as the decaying rate of the system,the smaller the better from the viewpoint of fast...Spectral abscissa(SA)is defined as the real part of the rightmost characteristic root(s)of a dynamical system,and it can be regarded as the decaying rate of the system,the smaller the better from the viewpoint of fast stabilization.Based on the Puiseux series expansion of complex-valued functions,this paper shows that the SA can be minimized within a given delay interval at values where the characteristic equation has repeated roots with multiplicity 2 or 3.Four sufficient conditions in terms of the partial derivatives of the characteristic function are established for testing whether the SA is minimized or not,and they can be tested directly and easily.展开更多
Stability is usually in the sense of Lyapunov′s asymptotical stability,thus the solutions starting from points close to a stable equilibrium may have a very long transient.In the applications of time-delayed feedback...Stability is usually in the sense of Lyapunov′s asymptotical stability,thus the solutions starting from points close to a stable equilibrium may have a very long transient.In the applications of time-delayed feedback controls,it is important not only to determine the stable regions in the gain plane or gain space,but also to find out the abscissa that can be used as an index of stability.Based on the D-subdivision method,this paper proposes a simple algorithm for finding and labeling the stable regions in feedback gain plane with abscissa.The labeled sub-regions with smaller abscissa are better in applications.The main results are presented for the controlled pendulum or inverted pendulum under a delayed feedback,and are illustrated with two case studies.展开更多
For the nonpositive Hermite-Fejér interpolation based on the Laguerre abscissas, a pointwise two-sided estimate of the degree of approximation in the aleatoric interval [0, A] is first established.
A necessary and sufficient condition of regularity of (0, 1,… , m - 2, m) interpolation on the zeros of the Laguerre polynomials Ln(α) (x) (α≥ -1) in a manageable form is established. Meanwhile, the explicit repre...A necessary and sufficient condition of regularity of (0, 1,… , m - 2, m) interpolation on the zeros of the Laguerre polynomials Ln(α) (x) (α≥ -1) in a manageable form is established. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, it is shown that, if the problem of (0, 1,…, m - 2, m) interpolation has an infinity of solutions, then the general form of the solutions is f0(x) + Cf1(x) with an arbitrary constant C.展开更多
In this paper,an approach of square coordinate transformation is proposed to approximate the spectral abscissa for continuous-time switched linear systems.By applying elementary transformations iteratively,a series of...In this paper,an approach of square coordinate transformation is proposed to approximate the spectral abscissa for continuous-time switched linear systems.By applying elementary transformations iteratively,a series of minimums of least μ1 matrix set measures are obtained,which are utilized to approximate the spectral abscissa of the switched system.The approach is developed into tractable numerical algorithms that provide upper bound estimates of the spectral abscissa.Numerical simulations show the effectiveness of the proposed method.展开更多
This article investigates the convergence and growth of multiple Dirichlet series. The Valiron formula of Dirichlet series is extended to n-tuple Dirichlet series and an equivalence relation between the order of n-tup...This article investigates the convergence and growth of multiple Dirichlet series. The Valiron formula of Dirichlet series is extended to n-tuple Dirichlet series and an equivalence relation between the order of n-tuple Dirichlet series and its coefficients and exponents is obtained.展开更多
基金Project supported by the National Natural Science Foundation of China(No.12072370)。
文摘Spectral abscissa(SA)is defined as the real part of the rightmost characteristic root(s)of a dynamical system,and it can be regarded as the decaying rate of the system,the smaller the better from the viewpoint of fast stabilization.Based on the Puiseux series expansion of complex-valued functions,this paper shows that the SA can be minimized within a given delay interval at values where the characteristic equation has repeated roots with multiplicity 2 or 3.Four sufficient conditions in terms of the partial derivatives of the characteristic function are established for testing whether the SA is minimized or not,and they can be tested directly and easily.
基金supported by the National Natural Science Foundation of China (No.11372354)
文摘Stability is usually in the sense of Lyapunov′s asymptotical stability,thus the solutions starting from points close to a stable equilibrium may have a very long transient.In the applications of time-delayed feedback controls,it is important not only to determine the stable regions in the gain plane or gain space,but also to find out the abscissa that can be used as an index of stability.Based on the D-subdivision method,this paper proposes a simple algorithm for finding and labeling the stable regions in feedback gain plane with abscissa.The labeled sub-regions with smaller abscissa are better in applications.The main results are presented for the controlled pendulum or inverted pendulum under a delayed feedback,and are illustrated with two case studies.
基金Supported by Science and Research Fund Item of Education Department of Zhejiang Province(20050408).
文摘For the nonpositive Hermite-Fejér interpolation based on the Laguerre abscissas, a pointwise two-sided estimate of the degree of approximation in the aleatoric interval [0, A] is first established.
文摘A necessary and sufficient condition of regularity of (0, 1,… , m - 2, m) interpolation on the zeros of the Laguerre polynomials Ln(α) (x) (α≥ -1) in a manageable form is established. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, it is shown that, if the problem of (0, 1,…, m - 2, m) interpolation has an infinity of solutions, then the general form of the solutions is f0(x) + Cf1(x) with an arbitrary constant C.
基金supported by the National Key Basic Research Program(973 Plan)under Grant No.2014CB845302the National Natural Science Foundation of China under Grant No.61273121
文摘In this paper,an approach of square coordinate transformation is proposed to approximate the spectral abscissa for continuous-time switched linear systems.By applying elementary transformations iteratively,a series of minimums of least μ1 matrix set measures are obtained,which are utilized to approximate the spectral abscissa of the switched system.The approach is developed into tractable numerical algorithms that provide upper bound estimates of the spectral abscissa.Numerical simulations show the effectiveness of the proposed method.
基金Supported by the National Science Foundation of China(10771011)the National Key Basic Research Project of China(2005CB321902)
文摘This article investigates the convergence and growth of multiple Dirichlet series. The Valiron formula of Dirichlet series is extended to n-tuple Dirichlet series and an equivalence relation between the order of n-tuple Dirichlet series and its coefficients and exponents is obtained.
