Introduction

The most of the dynamical systems in technology and nature are nonlinear dynamical systems. The 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.

There are several methods in the literature to identify candidates for Lyapunov functions [Krasovsky59]: decomposition of the sum of squares [Papach02]; using the Gröbner basis to select parameters [Forsman91]; use of homotopy operators for decomposition of the vector field of states of the system [Chukanov12,Edelen85]; the assumption that the derivative of the Lyapunov function is negative definite, and then obtain by integration and check the positive definiteness (gradient method).

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; finding 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.

The theory of control of dynamic objects can be divided into two subgroups [Khalil02]:

(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

The objects in the theory of Gröbner bases are polynomial ideals and algebraic varieties [Nesic02]. Let p 1 , ..., p s be multidimensional polynomials in variables x 1 , ..., x n , whose coefficients lie in the field k (we will consider the field of real numbers R). The variables x 1 , ..., x n are considered "place markers" in the polynomials: p 1 , . . . , p s ∈ R[x 1 , . . . , x n ]. Algebraic variety defined by the polynomials p 1 , . . . , p s is the collection of all solutions in R n of the system of equations:

p 1 (x 1 , ..., x n ) = 0, . . . p s (x 1 , ..., x n ) = 0.

(1) Formally:

V (p 1 , ..., p s ) := {(a 1 , ..., a n ) ∈ R n : p i (x 1 , ..., x n ) = 0, i = 1, ..., s}(2)

The polynomial ideal I, which is generated by p 1 , ..., p s , is a set of polynomials obtained by combining these polynomials by multiplying and adding with other polynomials:

I = ⟨p 1 , ..., p s ⟩ := { f = s ∑ i=1 g i p i : g i ∈ R[x 1 , ..., x n ] } (3)

The polynomials p i , i = 1, . . . , s form the basis of the ideal I. A useful interpretation of the polynomial ideal I is in terms of the equations (3). Multiplying g i by arbitrary polynomials g i ∈ R[x 1 , ..., x n ] and adding them, we get the consequence from (1):

f = g 1 p 1 + . . . + g s p s = 0,

and f ∈ I. Therefore, I = ⟨p 1 , . . . , p s ⟩ the ideal contains all "polynomial consequences" of the equations (3).

The 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 coefficient, familiar for one-dimensional polynomials, to multidimensional polynomials. Let's consider lexicographic or lex order [Nesic02]. Let α, β be two n -tuples of integers α

= (α 1 , . . . , α n ) ∈ N n , β = (β 1 , . . . , β n ) ∈ N n . n -tuple α follows β (in lex order), which is denoted as α ≻ β, if in the difference of vectors α−β = (α 1 − β 1 , . . . , α n − β n ) the leftmost nonzero element is positive. For the polynomial f = x 3 1 x 2 x 3 3 + 2x 3 1 x 4 3 using lex order x 1 ≻ x 2 ≻ x 3 results in x 3 1 x 2 x 3 3 follows x 3 1 x 4 3

, since the multidegrees of monomials satisfy: (3, 1, 3) ≻ (3, 0, 4). In this order, the leading coefficient and the leading term are respectively LC(f ) = 1 and LT (f ) = x 3 1 x 2 x 3 3 . When using lex of order

x 3 ≻ x 2 ≻ x 1 senior term: LT (f ) = 2x 3 1 x 4 3 , since (4, 0, 3) ≻ (3, 1, 3

). The ideal I has no unique basis, but for any two different bases ⟨p 1 , ..., p s ⟩ and ⟨g 1 , ..., g m ⟩ of the ideal I, the varieties V (p 1 , . . . , p s ) and V (g 1 , . . . , g m ) are equal; the variety depends only on the ideal generated by its defining 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 I and a given monomial order, we denote the set of leading terms of elements I as LT (I). The ideal generated by elements from LT (I) is denoted by ⟨LT (I)⟩. The Gröbner basis is formally defined as a set of polynomials g 1 , . . . , g m , for which ⟨LT (I)⟩ = ⟨LT (g 1 ), . . . , LT (g m )⟩. When calculating Gröbner bases, a monomial order is specified. We note two properties of Gröbner bases for a given monomial order:

1. Each ideal I ⊂ R[x 1 , ..., x n ], different from the trivial ⟨0⟩, has a Gröbner basis. 2. For the ideal I ⊂ R[x 1 , ..., x n ], different from the trivial ⟨0⟩, the Gröbner basis of the ideal I can be calculated using a finite number of algebraic operations.

For a given set of polynomials P , there is an algorithm that computes the Gröbner basis for the (ideal generated by) P in a finite number of steps [Buchberger70]. 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. This algorithm was implemented on computers in symbolic computation programs using Gröbner bases for solving systems of polynomial equations [?, Wolfram03,Demenkov15].

Finding equilibrium states of a nonlinear dynamical system

The use of the Gröbner basis in finding 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: p 1 = x 1 − x 2 2 = 0, p 2 = x 2 + x 2 3 = 0, p 3 = x 3 − 2x 2 1 = 0, to a triangular form using the Gröbner basis method for lex order: x 1 ≻ x 2 ≻ x 3 . In the WOLFRAM MATHEMATICA package, the function call GroebnerBasis[{p1, p2, p3}, {x1, x2, x3}, {}] leads to a triangular Gaussian form of polynomial equations:

x 1 − x 4 3 = 0, x 2 + x 2 3 = 0, −x 3 + 2x 8 3 = 0,

which allows us to get a solution to this system. Consider a nonlinear system without inputs ẋ(t) = f (x(t)); x, f ∈ R n , t ∈ R, where f (x) = 0 is a vector of polynomials in x. The equilibrium states for this polynomial system are obtained as solutions of a nonlinear system of polynomial equations: f (x) = 0.

Example 1

Equilibrium states of the [Nesic02] polynomial system:

ẋ1 = x 1 + x 2 − x 2 3 , ẋ2 = x 2 1 + x 2 − x 3 , ẋ3 = −x 1 + x 2 2 + x 3 ,

can be obtained as solutions of system polynomial equations:

p 1 := x 1 + x 2 − x 2 3 = 0, p 2 := x 2 1 + x 2 − x 3 = 0, p 3 := −x 1 + x 2

2 + x 3 = 0. The Gröbner basis for the ideal (p 1 , p 2 , p 3 ) using lex order: x 1 ≻ x 2 ≻ x 3 , has the form:

g 1 := 4x 1 − 2x 2 1 − 4x 3 1 + x 4 1 + x 6 1 , g 2 := −x 2 1 + x 4 1 − 2x 2 + 2x 2 1 x 2 , g 3 := −x 1 + x 2 1 + x 2 + x 2 2 , g 4 := −x 2 1 − x 2 + x 3 .

Algebraic equations g i = 0, i = 1, 2, 3, 4 has the same solutions as p j = 0, j = 1, 2, 3. The polynomial g 4 depends only on x 3 ; from the algebraic equation g 4 (x 3 ) = 0, you can determine x 3 . If the numerical value of x 3 substitute in g 3 (x 2 , x 3 ) = 0, then you can define x 2 ; from g 2 (x 1 , x 2 , x 3 ) = 0 you can define x 1 .

In the WOLFRAM MATHEMATICA package: Form an ideal of polynomials:

p1 = x1 + x2 − x3 2 ; p2 = x1 2 + x2 − x3; p3 = −x1 + x2 2 + x3.

Let us define the Gröbner basis:

grbas = GroebnerBasis[{p1, p2, p3}, {x3, x2, x1}, {}], grbas = {4x1 − 2x1 2 − 4x1 3 + x1 4 + x1 6 , −x1 2 + x1 4 − 2x2 + 2x1 2 x2, −x1 + x1 2 + x2 + x2 2 , −x1 2 − x2 + x3}.

To find the roots of x 1 , we define the reduced Gröbner basis:

grbas = GroebnerBasis [{p1, p2, p3} , {x3, x2, x1} , {x3, x2}] , grbas = { 4x1 − 2x1 2 − 4x1 3 + x1 4 + x1 6 } . Let's execute: Roots[4x1 − 2x1 2 − 4x1 3 + x1 4 + x1 6 == 0, x1].

To find the roots of x 2 with known x 1 , we define the reduced Gröbner basis:

grbas = GroebnerBasis [{p1, p2, p3} , {x3, x2, x1} , {x3}] , grbas = { −x1 + x1 2 + x2 + x2 2 } . Let's execute: Roots[−x1 + x1 2 + x2 + x2 2 == 0, x2].

To find the roots of x 3 with known x 1 , x 2 , execute:

Roots[−x1 2 − x2 + x3 == 0, x3].

The results are shown in Table 1.

Table 1.

x 1

x 2 x 3 Solution 1 :

x 1 = −1, x 2 = 0.5 − 1.32i, x 3 = 0.5 − 1.32i, Solution 2 :

x 1 = 0, x 2 = 0, x 3 = 0, Solution 3 :

x 1 = 1, x 2 = 0, x 3 = 1, Solution 4 :

x 1 = 1.18, x 2 = −0.69, x 3 = 0.70, Solution 5 :

x 1 = −0.59 − 1.74i, x 2 = −2.35 + 1.03i, x 3 = −5.04 + 3.09i, Solution 6 : x 1 = −0.59 + 1.74i, x 2 = 1.35 − 1.03i, x 3 = −1.35 − 3.09i, Solution 7 : x 1 = −1, x 3 = −0.5 + 1.32i,

x 3 = 0.5 + 1.32i, Solution 8 :

x 1 = 0, x 2 = −1, x 3 = −1, Solution 9 : x 1 = 1, x 2 = −1,

x 3 = 0, Solution 10 :

x 1 = 1.18, x 2 = −0.31, x 3 = 1.08, Solution 11 :

x 1 = −0.59 − 1.74i, x 2 = 1.35 − 1.03i, x 3 = −1.35 + 1.03i, Solution 12 : x 1 = −0.59 + 1.74i, x 2 = −2.35 + 1.03i, x 3 = −5.04 − 1.03i.

3 Application of the Gröbner basis method in the theory of the method of Lyapunov functions

Estimation of the area of attraction

The set of all initial conditions of a dynamical system, which converge to the same equilibrium state, is called the area of attraction of this equilibrium state [Forsman91,Sidorov19]. One way to get an estimate of the domain of attraction is to use the Lyapunov functions.

The standard result of Lyapunov's theory is that if x = 0 is an equilibrium point for a system with continuous time: ẋ = f (x), x ∈ D ⊂ R n , is a domain containing x = 0 and V : D → R is a continuously differentiable Lyapunov function such that V (0) = 0 and V (x) > 0, V = V x f (x) < 0, ∀x ∈ D − {0} ; then the point x = 0 is asymptotically stable. For such a Lyapunov function, consider the sets Ω = {x ∈ R n : V x f (x) < 0} and

B d = {x ∈ R n : V (x) ≤ d} .

If there is a value d > 0 such that B d ⊂ Ω, then the set B d is an estimate of the domain of attraction.

For polynomial systems with a polynomial Lyapunov function V , the Gröbner basis can be used to determine B d . You can determine the largest B d by finding a d such that B d ⊂ Ω. For polynomial systems with polynomial Lyapunov functions, V (x) − d and V x f (x) are polynomials and the boundaries of the sets B d and Ω are varieties Z(V − d) and Z(V x f (x)), respectively. At the points of contact Z(V − d) and Z(V x f (x)), the gradients V and V x f (x) are parallel [Luenberger16]. Using this information, we obtain a system of n + 2 polynomial equations in n + 2 variables (x 1 , . . . , x n , d, λ), where λ is the Lagrange multiplier (see Appendix):

V − d = 0, V x f = 0, ∇(V x f ) − λ∇V = 0. (4)

In the case of the vector of Lagrange multipliers λ = (λ 1 , . . . , λ m ) T ∈ R m we obtain a system of n + m + 1 equations from n + m + 1 variables x 1 , ..., x n , d, λ 1 , . . . , λ m .

Calculating the Gröbner basis for this system, where the variable d has the lowest rank in the lex order, we obtain a polynomial equation for d. The smallest positive solution of this equation (the value d min > 0), is the best estimate of the area of attraction.

Conclusion

The paper considers methods for estimating stability using Lyapunov functions, applied to nonlinear systems. The 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 finding the critical points of a given nonlinear system is proposed. The coordination of input-output signals of the system based on the construction of Gröbner bases is considered.