针对一类多项式形式的Hopf分岔系统,提出了一种鲁棒稳定的控制器设计方法.使用该方法设计控制器时不需要求解出系统在分岔点处的分岔参数值,只需要估算出分岔参数的上下界,然后设计一个参数化的控制器,并通过Hurwitz判据和柱形代数剖分...针对一类多项式形式的Hopf分岔系统,提出了一种鲁棒稳定的控制器设计方法.使用该方法设计控制器时不需要求解出系统在分岔点处的分岔参数值,只需要估算出分岔参数的上下界,然后设计一个参数化的控制器,并通过Hurwitz判据和柱形代数剖分技术求解出满足上下界条件的控制器参数区域,最后在得到的这个区域内确定出满足鲁棒稳定的控制器参数值.该方法设计的控制器是由包含系统状态的多项式构成,形式简单,具有通用性,且添加控制器后不会改变原系统平衡点的位置.本文首先以Lorenz系统为例说明了控制器的推导和设计过程,然后以van der Pol振荡系统为例,进行了工程应用.通过对这两个系统的控制器设计和仿真,说明了文中提出的控制器设计方法能够有效地应用于这类Hopf分岔系统的鲁棒稳定控制,并且具有通用性.展开更多
The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems c...The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.展开更多
An algorithm is given for computing in a very efficient way the topology of two real algebraic plane curves defined implicitly.The authors preform a symbolic pre-processing that allows us later to execute all numerica...An algorithm is given for computing in a very efficient way the topology of two real algebraic plane curves defined implicitly.The authors preform a symbolic pre-processing that allows us later to execute all numerical computations in an accurate way.展开更多
Making use of the discriminant sequence for polynomials, WR algorithm, Wu' s elimination and a partial cylindrical algebraic decomposition, we present here a practical algorithm for automated inequality discoverin...Making use of the discriminant sequence for polynomials, WR algorithm, Wu' s elimination and a partial cylindrical algebraic decomposition, we present here a practical algorithm for automated inequality discovering which can discover new inequalities automatically without requiring to put forward any conjectures beforehand. That is complete for an extensive class of inequality-type theorems. Also this algorithm is applied to the classification of the real physical solutions of geometric constraint problems. Many inequalities with various backgrounds have been discovered or rediscovered by our program, DISCOVERER, which implements the algorithm in Maple.展开更多
In this paper, we study the relationship between iterated resultant and multivariate discriminant. We show that, for generic form f（xn） with even degree d, if the polynomial is squarefreed after each iteration, the ...In this paper, we study the relationship between iterated resultant and multivariate discriminant. We show that, for generic form f（xn） with even degree d, if the polynomial is squarefreed after each iteration, the multivariate discriminant A（f） is a factor of the squarefreed iterated resulrant. In fact, we find a factor Hp（f, [x1 , xn]） of the squarefreed iterated resultant, and prove that the multivariate discriminant A（f） is a factor of Hp（f,[x1,... ,xn]）. Moreover, we conjecture that Hp（f, [x1,..., xn]） =△（f） holds for generic form f, and show that it is true for generic trivariate form f（x, y, z）.展开更多
文摘针对一类多项式形式的Hopf分岔系统,提出了一种鲁棒稳定的控制器设计方法.使用该方法设计控制器时不需要求解出系统在分岔点处的分岔参数值,只需要估算出分岔参数的上下界,然后设计一个参数化的控制器,并通过Hurwitz判据和柱形代数剖分技术求解出满足上下界条件的控制器参数区域,最后在得到的这个区域内确定出满足鲁棒稳定的控制器参数值.该方法设计的控制器是由包含系统状态的多项式构成,形式简单,具有通用性,且添加控制器后不会改变原系统平衡点的位置.本文首先以Lorenz系统为例说明了控制器的推导和设计过程,然后以van der Pol振荡系统为例,进行了工程应用.通过对这两个系统的控制器设计和仿真,说明了文中提出的控制器设计方法能够有效地应用于这类Hopf分岔系统的鲁棒稳定控制,并且具有通用性.
基金supported by National Natural Science Foundation of China (Grant Nos. 10271022, 60373093,60533060)the Natural Science Foundation of Zhejiang Province (Grant No. Y7080068)the Foundation of Department of Education of Zhejiang Province (Grant Nos. 20070628 and Y200802999)
文摘The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.
文摘An algorithm is given for computing in a very efficient way the topology of two real algebraic plane curves defined implicitly.The authors preform a symbolic pre-processing that allows us later to execute all numerical computations in an accurate way.
文摘Making use of the discriminant sequence for polynomials, WR algorithm, Wu' s elimination and a partial cylindrical algebraic decomposition, we present here a practical algorithm for automated inequality discovering which can discover new inequalities automatically without requiring to put forward any conjectures beforehand. That is complete for an extensive class of inequality-type theorems. Also this algorithm is applied to the classification of the real physical solutions of geometric constraint problems. Many inequalities with various backgrounds have been discovered or rediscovered by our program, DISCOVERER, which implements the algorithm in Maple.
文摘In this paper, we study the relationship between iterated resultant and multivariate discriminant. We show that, for generic form f（xn） with even degree d, if the polynomial is squarefreed after each iteration, the multivariate discriminant A（f） is a factor of the squarefreed iterated resulrant. In fact, we find a factor Hp（f, [x1 , xn]） of the squarefreed iterated resultant, and prove that the multivariate discriminant A（f） is a factor of Hp（f,[x1,... ,xn]）. Moreover, we conjecture that Hp（f, [x1,..., xn]） =△（f） holds for generic form f, and show that it is true for generic trivariate form f（x, y, z）.
