SyNRAC: A Maple-Package for Solving Real Algebraic Constraints

Anai, Hirokazu; Yanami, Hitoshi

doi:10.1007/3-540-44860-8_86

Cited by 23 publications

(24 citation statements)

References 16 publications

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“…For the larger sized problems, we may use an efficient QE algorithm based on Sturm-Habicht sequence [1,10]. Some types of QE methods using Sturm-Habicht sequence are available in a maple package SyNRAC [4,19].…”

Section: Control Applicationmentioning

confidence: 99%

Sum of roots with positive real parts

Anai

Hara

Yokoyama

2005

Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation

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In this paper we present a method to compute or estimate the sum of roots with positive real parts (SORPRP) of a polynomial, which is related to a certain index of "average" stability in optimal control, without computing explicit numerical values of the roots. The method is based on symbolic and algebraic computations and enables us to deal with polynomials with parametric coefficients for their SORPRP. This leads to provide a novel systematic method to achieve optimal regulator design in control by combining the method with quantifier elimination. We also report some experimental result for a typical class of plants and confirm the effectiveness of the proposed method.

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Section: Control Applicationmentioning

confidence: 99%

Sum of roots with positive real parts

Anai

Hara

Yokoyama

2005

Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation

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“…Weispfenning's virtual substitution algorithm has been implemented in tools like SyNRAC [8], which is a toolbox in Maple for solving real algebraic constraints, Mathematica [9] and Reduce/Redlog [10]. We will use the implementation in Mathematica for the computational examples in Section IV.…”

Section: Quantifier Elimination Algorithmsmentioning

confidence: 99%

Verification of model predictive control laws using weispfenning's quantifier elimination by virtual substitution algorithm

Siaulys

Maciejowski

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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A method based on a quantifier elimination algorithm is suggested for obtaining explicit model predictive control (MPC) laws for linear time invariant systems with quadratic objective and polytopic constraints. The structure of the control problem considered allows Weispfenning's 'quantifier elimination by virtual substitution' algorithm to be used. This is applicable to first order formulas in which quantified variables appear at most quadratically. It has much better practical computational complexity than general quantifier elimination algorithms, such as cylindrical algebraic decomposition. We show how this explicit MPC solution, together with Weispfenning's algorithm, can be used to check recursive feasibility of the system, for both nominal and disturbed systems. Extension to cases beyond linear MPC using Weispfenning's algorithm is part of future work.

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“…QE, in short, converts a mathematical formula with quantifiers ∀, ∃ to the equivalent one without any quantifier. For the application of QE to the design of a control system, refer to the references [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

On the Hankel singular values for a parametric system

Kitamoto

Yamaguchi

2008

2008 SICE Annual Conference

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Given a system with a parameter k, its Hankel singular values, denoted by σ i (k) (i = 1, · · · , n), are naturally functions of k. In this paper, we show that σ i (k) can be expressed as a root of a bivariate polynomial f (x, k) with respect to x, and present an algorithm to compute the polynomial f (x, k). We then apply the expression of σ i (k) to examine the extrema and the asymptotic behaviors of σ i (k). We also show that the ratio σ i (k)/σ j (k) of two distinct Hankel singular values can also be expressed as a root of bivariate polynomial. This gives us a systematic method to examine various properties such as the extrema or the asymptotic behaviors of the ratio σ i (k)/σ j (k). Considering that the ratio σ i (k)/σ j (k) is quite important information for 'balanced model reduction', we can utilize the properties for a balanced model reduction of a parametric system.

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SyNRAC: A Maple-Package for Solving Real Algebraic Constraints

Abstract: In this paper we present a maple-package, named SyNRAC, for solving real algebraic constraints derived from various engineering problems. Our main tool is real quantifier elimination and we focus on its application to robust control design problems.

Cited by 23 publications

References 16 publications

Sum of roots with positive real parts

Sum of roots with positive real parts

Verification of model predictive control laws using weispfenning's quantifier elimination by virtual substitution algorithm

On the Hankel singular values for a parametric system

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