The Investigation of Nonlinear Polynomial Control Systems

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gr¨obner basis method is used to assess the stability of a dynamical system. A description of the Gr¨obner 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 Gr¨obner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gr¨obner 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 Gr¨obner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gr¨obner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gr¨obner basis. The application of the Gr¨obner 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 Gr¨obner 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 Gr¨obner bases is considered.

e paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. e apparatus of the Gröbner basis method is used to assess the stability of a dynamical system. A description of the Gröbner 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 Gröbner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. e use of the Gröbner basis for nding 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. e 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 Gröbner basis method is given. An example of nding the critical points of a nonlinear polynomial system using the Gröbner basis method and the Wolfram Mathematica application so ware is given. e Wolfram Mathematica program uses the function of determining the reduced Gröbner basis. e application of the Gröbner basis method for estimating the a raction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector eld 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. e use of Gröbner bases in the gradient method for nding the Lyapunov function of a nonlinear dynamical system is considered. e coordination of input-output signals of the system based on the construction of Gröbner bases is considered. Introduction e most of the dynamical systems in technology and nature are nonlinear dynamical systems. e canonical relations of a nonlinear system can be approximated by polynomials of the components of the state and control vectors. Stability testing using the method of Lyapunov functions is widely applied to nonlinear systems.
ere are several methods in the literature to identify candidates for Lyapunov functions [1]: • decomposition of the sum of squares [2]; • using the Gröbner basis to select parameters [3]; • use of homotopy operators for decomposition of the vector eld of states of the system [4,5]; • the assumption that the derivative of the Lyapunov function is negative de nite, and then obtain by integration and check the positive de niteness (gradient method) [6].
Gröbner bases are used to solve problems in the theory of nonlinear systems. Some of the applications of the Gröbner basis can be named: estimation of equilibrium states of a nonlinear system; nding the critical points of a given nonlinear system with the Lyapunov function; coordination of input-output signals of the system.
Gröbner bases facilitate the solution of a system of multidimensional polynomial equations in the same way as the Gaussian elimination algorithm makes it possible to solve a system of linear algebraic equations. In lexical ordering the Gröbner basis has a triangular structure, reminiscent of the triangular structure in the Gaussian elimination method. e theory of control of dynamic objects can be divided into two subgroups [7]: (1) systems in which the principle of superposition operates, and linear control methods can be used; (2) systems in which the superposition principle does not work, and it is necessary to use nonlinear control methods. To improve the quality of the dynamic object control system, it is necessary to take into account the nonlinear features of the system.

Gröbner bases
Formally: e polynomial ideal , which is generated by 1 , ..., , is a set of polynomials obtained by combining these polynomials by multiplying and adding with other polynomials: e polynomials , = 1, … , form the basis of the ideal . A useful interpretation of the polynomial ideal is in terms of the equations (3). Multiplying by arbitrary polynomials ∈ ℝ[ 1 , ..., ] and adding them, we get the consequence from (1): and ∈ . erefore, = ⟨ 1 , … , ⟩ the ideal contains all "polynomial consequences" of the equations (3). e Gröbner basis method is based on the concept of monomial ordering (a monomial is a polynomial consisting of one term), since it introduces a corresponding extension of the concept of a leading term and a leading coe cient, familiar for one-dimensional polynomials, to multidimensional polynomials. Let's consider lexicographic or lex order [8]. . When using lex of order 3 ≻ 2 ≻ 1 senior term: . e ideal has no unique basis, but for any two di erent bases ⟨ 1 , ..., ⟩ and ⟨ 1 , ..., ⟩ of the ideal , the varieties ( 1 , … , ) and ( 1 , … , ) are equal; the variety depends only on the ideal generated by its de ning equations. If all polynomials in a given basis of an ideal have a degree lower than the degree of any other polynomial in an ideal, then this basis is the simplest. For an ideal and a given monomial order, we denote the set of leading terms of elements as ( ). e ideal generated by elements from ( ) is denoted by ⟨ ( )⟩. e Gröbner basis is formally de ned as a set of polynomials 1 , … , , for which ⟨ ( )⟩ = ⟨ ( 1 ), … , ( )⟩. When calculating Gröbner bases, a monomial order is speci ed. We note two properties of Gröbner bases for a given monomial order: 1. Each ideal ⊂ ℝ[ 1 , ..., ], di erent from the trivial ⟨0⟩, has a Gröbner basis. 2. For the ideal ⊂ ℝ[ 1 , ..., ], di erent from the trivial ⟨0⟩, the Gröbner basis of the ideal can be calculated using a nite number of algebraic operations.
For a given set of polynomials , there is an algorithm that computes the Gröbner basis for the (ideal generated by) in a nite number of steps [9]. 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].
2. Finding equilibrium states of a nonlinear dynamical system e use of the Gröbner basis in nding solutions to a nonlinear system of polynomial equations is similar to the application of the Gauss method for solving a quadratic system of linear equations. Consider an example of reducing a nonlinear system of polynomial equations: 1 = 1 − 2 2 = 0, 2 = 2 + 2 3 = 0, 3 = 3 − 2 2 1 = 0, to a triangular form using the Gröbner basis method for lex order: 1 ≻ 2 ≻ 3 . In the WOLFRAM MATHEMATICA package, the function call [{ 1, 2, 3}, { 1, 2, 3}, {}] leads to a triangular Gaussian form of polynomial equations: which allows us to get a solution to this system. Consider a nonlinear system without inputṡ ( ) = ( ( )); , ∈ ℝ , ∈ ℝ, where ( ) = 0 is a vector of polynomials in . e equilibrium states for this polynomial system are obtained as solutions of a nonlinear system of polynomial equations: ( ) = 0.
Calculating the Gröbner basis for this system, where the variable has the lowest rank in the lex order, we obtain a polynomial equation for . e smallest positive solution of this equation (the value min > 0), is the best estimate of the area of a raction.

Example 2
Consider a second-order system: We will assume that ( ) = ℍ ( ) is a scalar potential.
It is necessary to choose such a functioṅ , so that the obtained from (7) is positive de nite; then the equilibrium state = 0 of the system (5) is stable. Consider the variable gradient method. Using this method, you can nd the Lyapunov function, assuming that = = , where = ( ) must be determined; can be a function of , and ( ) = ( ). e choice of = ( ) must ensure the ful llment of the condition (6) and ( ) must be positive de nite. If ( ) = ( ) , theṅ takes the forṁ = ( ) , where ( ) = ( ). For the matrix ( ), the following condition is selected: (ℎ < 0) ∨ (ℎ ≤ 0) , ℎ + ℎ = 0; = 1, … , , to ensure thaṫ is negative de nite. e function ( ) can be de ned by the integral (7) and check whether it is positive de nite. Performing the integration, we obtain:

Conversions of input-output signals of a nonlinear system
Consider a di erential ring -a ring on which the di erentiation operation is de ned. It is assumed that di erentiation is carried out with respect to the implicit variable . A di erential ideal is an ideal that is closed under di erentiation.
A polynomial system in the state space is a system of di erential equations: where =̇ − ( , ), = 1, … , . e problem of transformation from the state space to the input-output form: let be a di erential ideal; nd a generator for the di erential ideal ∩ ℝ [ , ].
Suppose that it is necessary to nd a di erential relationship between and from the description in the state space of the system:̇ 1 = −2 1 + 2 2 ;̇ 2 = − 1 2 + ; = 2 .

Example 6
Let's consider a method for nding the observability of the components of the state vector of a system based on the construction of a reduced Gröbner basis. Let us choose the system from Example 5. Suppose that it is necessary to nd the in uence of the variation of the component of the state vector 1 on the output signal . Di erentiating the equations of the system with respect to and replacinġ by , we obtain the polynomials 1 , 2 , 3 similar to the polynomials of Example 5 in the absence of ,̇ . Replace → 0 ,̇ → 1 ,̈ → 2 in , calculate the Gröbner basis for: ( 0 − 1 , 1 + 1 2 , 2 ( 2 2 − 2 1 ) 2 − 2 1 2 ) relative to lex order: 0 ≺ 1 ≺ 2 ≺ 1 ≺ 2 .
In the WOLFRAM MATHEMATICA package:  e canonical relations of a nonlinear system are approximated by polynomials of the components of the state and control vectors. To assess the stability, Gröbner bases are used. A method for nding the critical points of a given nonlinear system is proposed. e coordination of input-output signals of the system based on the construction of Gröbner bases is considered.
Let * be a regular point of constraints ℎ( ) = 0 and a point of local extremum of the function, taking into account these constraints. en for ∈ ℝ , satisfying ∇ℎ( * ) = 0, must hold: ∇ ( * ) = 0. is means that ∇ ( * ) is a linear combination of gradients ∇ℎ in * ; relations lead to the need to introduce the vector of Lagrange multipliers .