A Matlab tool for regions of attraction estimation via numerical algebraic geometry

Demenkov, Max

doi:10.1109/polyakhov.2015.7106722

Cited by 7 publications

(3 citation statements)

References 38 publications

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“…Buchberger's algorithm generalizes algorithms: Gaussian elimination for a system of linear algebraic equations and Euclid's algorithm for calculating the greatest common divisor of a set of one-dimensional polynomials. is algorithm was implemented on computers in symbolic computation programs using Gröbner bases for solving systems of polynomial equations [10][11][12].…”

Section: Gröbner Basesmentioning

confidence: 99%

The Investigation of Nonlinear Polynomial Control Systems

Чуканов

Chukanov

2021

Model. anal. inf. sist.

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The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

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Section: Gröbner Basesmentioning

confidence: 99%

The Investigation of Nonlinear Polynomial Control Systems

Чуканов

Chukanov

2021

Model. anal. inf. sist.

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“…There, we consider first-order optimality conditions of L in the test on Line 6, but switch to the Newton method on the first-order optimality conditions of (7), while memorising the current value. While minimising (7), we check the active set; when it does change, we revert to solving the convexification with the memorised value. Although this Figure 1: The motivation: the evolution of infeasibility (top row) and objective function (bottom row) when one switches from solving the convexification to the Newton method after a given number of steps on IEEE 30-bus test system (left), 118-bus test system (middle), and a snapshot of the Polish system (case2383wp; right).…”

Section: It Is Well-known That One Can Constructmentioning

confidence: 99%

“…Recently, Henrion and Korda [12] have shown that the domain of monotonicity of a polynomial system can be computed by solving an infinitedimensional linear program over the space of measures, whose value can be approximated by a certain hierarchy of convex semidefinite optimisation problems. See also the work of Valmórbida et al [29,28,27] in the context of partial differential equations, and elsewhere [7]. Dvijotham et al [9,10] showed that it can also be be cast as a certain non-convex semidefinite optimisation problem.…”

Section: Related Workmentioning

confidence: 99%

Hybrid Methods in Solving Alternating-Current Optimal Power Flows

Liddell,

Liu,

Marecek

et al. 2015

Preprint

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Many steady-state problems in power systems, including rectangular power-voltage formulations of optimal power flows in the alternatingcurrent model (ACOPF), can be cast as polynomial optimisation problems (POP). For a POP, one can derive strong convex relaxations, or rather hierarchies of ever stronger, but ever larger relaxations. We study means of switching from solving the convex relaxation to Newton method working on a non-convex augmented Lagrangian of the POP.

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