On the number of solutions of multivariable polynomial systems

Benallou, Abdelhanine; Mellichamp, Duncan A.; Seborg, Dale E.

doi:10.1109/tac.1983.1103214

Cited by 9 publications

(4 citation statements)

References 14 publications

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“…If it is not possible to reach form (20), i.e. the pivot procedure stops before the kth step, the solutions of system S x can be found by considering a smaller number of vectors * v v i and, consequently, by solving an equation of lower degree.…”

Section: Basic Proceduresmentioning

confidence: 99%

“…To explain the basic idea, let us first consider the case k ¼ m þ 2: Let us operate the pivot procedure in order to obtain a new matrix * V V satisfying (20), where the rows r 1 ; r 2 ; . .…”

Section: Extended Proceduresmentioning

confidence: 99%

“…Let us turn now to the general case, to see how the above reasoning can be extended to all situations in which m þ 15k4nðm À 1Þ þ 2: Without loss of generality, let us choose j ¼ 1 and let k be the smallest integer less than n such that k4m þ 1 þ kðm À 1Þ: Let us operate the pivot procedure in order to obtain a new matrix * V V satisfying (20) where the rows r 1 ; r 2 ; . .…”

Section: Extended Proceduresmentioning

confidence: 99%

“…This example is taken from References [20,21]. Let us consider the bilinear system of differential equations…”

Section: Examplementioning

confidence: 99%

See 3 more Smart Citations

Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach

Chesi

Garulli

Tesi

et al. 2003

Intl J Robust & Nonlinear

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This paper considers the problem of determining the solution set of polynomial systems, a well-known problem in control system analysis and design. A novel approach is developed as a viable alternative to the commonly employed algebraic geometry and homotopy methods. The first result of the paper shows that the solution set of the polynomial system belongs to the kernel of a suitable symmetric matrix. Such a matrix is obtained via the solution of a linear matrix inequality (LMI) involving the maximization of the minimum eigenvalue of an affine family of symmetric matrices. The second result concerns the computation of the solution set from the kernel of the obtained matrix. For polynomial systems of degree m in n variables, a basic procedure is available if the kernel dimension does not exceed m+1, while an extended procedure can be applied if the kernel dimension is less than n(m−1)+2. Finally, some application examples are illustrated to show the features of the approach and to make a brief comparison with polynomial resultant techniques

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Section: Basic Proceduresmentioning

confidence: 99%

Section: Extended Proceduresmentioning

confidence: 99%

Section: Extended Proceduresmentioning

confidence: 99%

“…This example is taken from References [20,21]. Let us consider the bilinear system of differential equations…”

Section: Examplementioning

confidence: 99%

See 2 more Smart Citations

Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach

Chesi

Garulli

Tesi

et al. 2003

Intl J Robust & Nonlinear

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show abstract

Modelling non‐linear devices exhibiting frequency‐dependent capacitances and inductances

Dervisoglu

Chua

1985

Circuit Theory & Apps

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The incremental capacitance and inductance of many electronic devices are frequency-dependent. This paper presents a systematic method for modelling such devices using only positive and/or negative linear time-imvariant capacitances and inductances.Modelling of a frequency-dependent capacitor or inductor is reduced to the realization of an odd rational function.It has been shown that any odd rational function can be realized as the driving-point function of a *L, * C one-port.It has also been shown that a linear higher-order capacitor or inductor of order 2 k + 1 can be realized by a *L, * C one-port which contains 2k + 2 elements.

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On pole assignment of linear systems by gain output feedback

Magni¹,

Champetier²

1987

Systems & Control Letters

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On the number of solutions of multivariable polynomial systems

Cited by 9 publications

References 14 publications

Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach

Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach

Modelling non‐linear devices exhibiting frequency‐dependent capacitances and inductances

On pole assignment of linear systems by gain output feedback

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