Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 859578, 5 pages http://dx.doi.org/10.1155/2013/859578

Research Article

Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System

Zhenyi Ji,1,2 Wenyuan Wu,2 Yong Feng,2 and Guofeng Zhang3

1 Laboratory of Computer Reasoning and Trustworthy Computation, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

2 Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Science, Chongqing 401120, China

3 L.A.S Department ofChengDu College, University of Electronic Science and Technology of China, Chengdu 611731, China

Correspondence should be addressed to Zhenyi Ji; zyji001@163.com Received 24 July 2013; Accepted 21 November 2013 Academic Editor: Bo-Qing Dong

Copyright © 2013 Zhenyi Ji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose an approach for constructing Lyapunov function in quadratic form of a differential system. First, positive polynomial system is obtained via the local property of the Lyapunov function as well as its derivative. Then, the positive polynomial system is converted into an equation system by adding some variables. Finally, numerical technique is applied to solve the equation system. Some experiments show the efficiency of our new algorithm.

1. Introduction

Analysis of the stability of dynamical systems plays a very important role in control system analysis and design. For linear systems, it is easy to verify the stability of equilibria. For nonlinear dynamical systems, proving stability of equilibria of nonlinear systems is more complicated than linear systems. One can use the Lyapunov function at the equilibria to determine the stability.

For an autonomous polynomial system of differential equations, how to compute the Lyapunov function at equilibria is a basic problem. In [1, 2], the author transformed the problem of computing the Lyapunov function into a quantifier elimination problem. The disadvantage of the method is that the computation complexity of quantifier elimination is doubly exponential in the number of total variables. In order to avoid this problem, She et al. [3] propose a symbolic method; they first construct a special semialgebraic system using the local properties of a Lyapunov function as well as its derivative and solving these inequations using cylindrical algebraic decomposition (CAD) introduced by Collins in [4]. The algorithm in [5] uses semidefinite programming to search for Lyapunov function. There are also other algorithms, see [6, 7] for more details.

In this paper, we suppose Lyapunov function has quadratic form and some coefficients of Lyapunov function are unknown numbers. Some positive polynomials are obtained using the technique mentioned in [3] first, then a positive dimensional polynomial system is constructed by adding some new variables. The parameter in Lyapunov function is computed through solving the real root of the positive dimensional system using the numerical method.

The rest of this paper is organized as follows: Definitions and preliminaries about the Lyapunov function and the asymptotic stability analysis of differential system are given in Section 2. Section 3 reviews some methods for solving the real root of positive dimensional polynomial system. The new algorithm to compute the Lyapunov function and some experiments are shown in Section 4. In Section 5, some examples are given to illustrates the efficiency of our algorithm. Finally, Section 6 draws a conclusion of this paper.

2. Stability Analysis of Differential Equations

In this section, some preliminaries on the stability analysis of differential equations are presented.

In this paper, we consider the following differential equations:

= fi (x) = fl (X)

*„ = f„ (x) ,

where x = (x1,x2,..., xn), ft e R[x], and xt = xt(t), xt = dxJdt.Apointx = (x1,x2,...,xn) in the n-dimensional real Euclidean space R" is called an equilibrium of differential system (1) if /¡(x) = 0 for all i e {1,2,..., n}. Without loss of generality, we suppose the origin is an equilibrium of the given system in this paper.

In general, there exists two techniques to analyze the stability of an equilibrium: the Lyapunov's first method with the technique of linearization which considers the eigenvalues of the Jacobian matrix at equilibrium.

Theorem 1. Let JF(x) denote the Jacobian matrix of system {f1,...,fn} at point x. If all the eigenvalues of JF(x) have negative real parts, then x is asymptotically stable. If the matrix Jp(x) has at least one eigenvalue with positive real part, then x is unstable.

For a small system, it is easy to obtain the eigenvalues of the matrix JF(x); then one can analyze the stability of the equilibrium using Theorem 1. For a high-dimensional system, solving the characteristic polynomial to get the exact zeros is a difficult problem. Indeed, to answer the question on stability of an equilibrium, we only need to know whether all the eigenvalues have negative real parts or not. Therefore, the theorem of Routh-Hurwitz [8] serves to determine whether all the roots of a polynomial have negative real parts.

Another method to determine asymptotic stability is to check if there exists a Lyapunov function at the point x, which is defined in the following.

Definition 2. Given a differential system and a neighborhood U of the equilibrium, a Lyapunov function with respect to the differential system is a continuously differential function F : U ^ R such that

(1) : F(0) = 0 and F(x) > 0 whenever x = 0;

(2) : (d/dt)F(0) = 0 and (d/dt)F(x) < 0 whenever x = 0.

3. Solving the Real Roots of

Positive Dimensional Polynomial System

Solving polynomial system has been one of the central topics in computer algebra. It is required and used in many scientific and engineering applications. Indeed, we only care about the real roots of a polynomial system arising from many practical problems. For zero dimensional system, homotopy continuation method [9, 10] is a global convergence algorithm. For positive dimensional system, computing real roots of this system is a difficult and extremely important problem.

Due to the importance of this problem, many approaches have been proposed. The most popular algorithm which solves this problem is CAD; another is the so-called critical point methods, such as Seidenberg's approach of computing critical points of the distance function [11]. The algorithm in [12] uses the idea of Seidenberg to compute the real root of a positive dimensional defined by a signal polynomial; and extends it to a random polynomial system in [13]. Actually, these algorithms depend on symbolic computations, so they are restricted to small size systems because of the high complexity of the symbolic computation. In order to avoid this problem, homotopy method has been used to compute real root of polynomial system in [14,15].

Recently, Wu and Reid [16] propose a new approach, which is different from the critical point technique. In order to facilitate the description of this algorithm, we suppose polynomial system g = {g1, g2,..., gk}; the system has k polynomials, n variables, and k < n. First, n-k hyperplanes h = {h1,... ,hn_k} in R[x] are chosen randomly. Note that {g1,...,gk ,h1,..., hn_k} is a square system; then witness points are computed by homotopy method and verified by the following theorem.

Theorem 3 (see [17]). Let f(x) : R" ^ R" be a polynomial system, and x e R". Let DR be the set of real intervals, and DR" and DR"X" be the set of real interval vectors and real interval matrices, respectively. Given X e DR" with 0 e X and M e DR"X" satisfies V/;(x + X) c M^fori = 1,2,..., n. Denote by In the identity matrix and assume

-F-1 (x)F(x) + (ln -Fx (x) M) X c int (X), (2)

where Fx(x) is the Jacobian matrix of F(x) at x. Then there is a unique x e X such that f(x) = 0. Moreover, every matrix M e M is nonsingular, and the Jacobian matrix Fx(x) is nonsingular.

There may exist some components which have no intersection with these random hyperplanes. Some points on these components must be the solutions of the Lagrange optimization problem:

f = 0, = n. (3)

Here n is a random vector in R". The system has n + k equations and n+k variables; thus we can find real points through solving system (3).

4. Algorithm for Computing the Lyapunov Function

In this section, we will present an algorithm for constructing the Lyapunov function. Our idea is to compute positive polynomial system which satisfies the definition of Lyapunov function first. Then we solve the polynomial system deduced from the positive polynomial system using homotopy algorithm; at this step, we use the famous package hom4ps2 [18].

Given a quadratic polynomial F(x), the following theorem gives a sufficient condition for the polynomial to be a Lyapunov function.

Theorem 4 (see [3]). Let F(x) be a quadratic polynomial, for a given differential system; if F(x) satisfies the fact that Hess(F)\x=0 is positive definite and Hess((d/dt)F)\x=0 is negative definite, then F(x) is a Lyapunov function.

By the theory of linear algebra, one knows that the symmetric matrix Hess(F)\x=0 is positive definite if and only if all its eigenvalues are positive, and Hess((d/dt)F)\x=0 is negative definite if and only if all its eigenvalues are negative.

1 n , n-1 ,

n = S + tn-1s + ••• + t0

be a characteristic polynomial of a matrix; the following theorem deduced from the Descartes' rule of signs [19]canbe used to determine whether h has only positive roots or not.

Theorem 5 (see [3]). Suppose all the roots of a real polynomial h are real; then its roots are all positive if and only if for all 1<i< n,(-1)'tn_ ; > 0.

Combine Theorems 4 and 5, finding that the Lyapunov function in quadratic form can be converted into solving the real root of some positive polynomial system, denoting it by

Inequ = {gx >0, ß2 >0,..., gn > 0}.

Suppose we have obtained the positive polynomial system as in (5), and denote the variable in the system by a. In order to obtain one value of a using numerical technique, we first convert the positive equation into equation. A simple ideal is to add new variable set x = (x1,x2,..xn), and construct the equation system as follows:

ps = {g±-x\,g2-x\,...,

If we find one real point (a, x) of system (6) such that there has nonzero element in x, then it is easy to see that the point a satisfies

{gx (a) > 0,g2 (a) > 0,...,gn (a) > 0},

which means the differential system exists a Lyapunov function at the equilibrium.

Note that the number of variable is more than the number of equation in system (6); then the system ps must be a positive dimensional polynomial system.

Recall the algorithm mentioned in Section 3; all of the algorithms obtain at least one real point in each connect component, and they use Theorem 3 to verify the existence of real root which deduces the low efficiency. However, in this paper, we only need one real point of system (6) to ensure the establishment of these inequalities in (7), so we verify the establishment of these inequalities using the residue of inequalities at the real part of every approximate real root of the system (6).

In the following we propose an algorithm to determine if there exists a Lyapunov function at the equilibrium.

Algorithm 6. Input: a differential system as defined in (1) and a tolerance e.

Output: a Lyapunov function or UNKNOW.

(1) Construct the positive polynomial.

(2) Convert the positive polynomial system into positive dimensional system defined in system (6).

(3) We choose n random point (x^ ,X2,..., xn) and n random vector v1, v2,..., vn; then construct n hyperplane in Rn through x; with normal v; for i = \,2,...,n. Denote the set of this hyperplane by ps2.

(4) Let ps = {ps1,ps2}, and solve the square system using homotopy continuation algorithm, denoting solution of ps by roots.

(5) for s = 1 : length(roots)

(a) if the norm of imaginary part of roots[s} is smaller than e, then substitute the real part of roots[s} into {g1,..gn}, and denote the value by {v1, v2,..., vn}. If v > 0 for all i e {1,2,..., n}, then return the real part of roots{s} and break the program.

(6) End for.

(7) Construct polynomial system ps3 = Yn=1 XtVf] = v, where Xt is new variable and v are chosen from {v1,..., vn} randomly.

(8) Solve {ps1, ps3} using homotopy continuation algorithm, denote its solution by roots, and go to Step 4.

(9) return UNKNOW.

In the following, we present a simple example to illustrate our algorithm.

Example 7. This is an example from [20]

T 3 ~ 4

x = -x + 2y -2y y = -x - y + xy.

Let Lyapunov function F(x, y) = x2 + axy + by2.

Step 1. We obtain the positive polynomial using Theorems 4 and 5 as follows:

[2b + 2> 0, -a2 + 4b>0, 2a + 4b + 4>0, 4a2 + 4b2 - 16b > 0].

Step 2. Convert system (9) into the following system:

2b+ 2-x\ = 0 -a2 +4b- x22 = 0 2a +4b + 4- x23 = 0 4a2 + 4b2 - 16b -x24 = 0.

Step 3. Construct two hyperplanes [h1,h2] in R6 randomly, where

h1 = 0.09713178123584754a + 0.04617139063115394b + 0.27692298496089x1 + 0.8234578283272926x2 + 0.694828622975817x3 + 0.3170994800608605x4 + 0.9502220488383549, h2 = 0.3815584570930084« + 0.4387443596563982b

+ 0.03444608050290876x1 + 0.7655167881490024x2 + 0.7951999011370632x3 + 0.1868726045543786x4

+ 0.4897643957882311.

Step 4. Compute the roots of the augmented system [ps1 = 0, h1 = 0,h2 = 0} using homotopy method, and we find the system has only 16 roots.

Step 5. We obtain the first approximate real root of the system

x = [-2.407604610156789,4.633115716668555,

3.356520733339377,3.568739680591174, (12)

-4.209186815331512, -5.909266734956268].

Substituting a = -2.407604610156789, b = 4.633115716668555 into the left of the positive polynomial in (9), we obtain the following result:

[11.26623143,12.73590291,17.71725365, 34.91943333].

This ensure the establishment of inequality in (9). Thus,

F (x, y) = x2 + 4.633115716668555y2 - 2.407604610156789xy

is a Lyapunov function.

If the random hyperplanes {h1, h2} are as follows:

h1 = -3a -b + x1 + 2x2 - 2x3 - 2x4 - 3, h2 = 3a - 3b - x1 - 2x2 + x3 + 2x4 - 2,

we find that polynomial system {h1 = 0,h2 = 0, ps = 0} has no real root; then we go to Step 7 in Algorithm 6 and obtain the following system:

Ps3 = ■

-2\2a + 2\3 + 8\4a -1 = 0

2X1 + 4X2 + 4X 3 + X 4 (8b -16)-3 = 0

-2X 1x1 + 1=0

-2X 2x2 + 2 = 0

-2X3x3 -2 = 0

-2X4x4 -3 = 0.

Solving the system {ps1 = 0,ps3 = 0}, we find the first approximate real root and substitute the value of a = 1.3053335232048229, b = 0.4314538107033688 into the left of the positive polynomial in (9) and we obtain the following result:

[2.862907621406738, 0.021919636011159,

8.336482289223121,0.656931019037197].

This ensures the establishment of inequality in (9). Thus,

F (x, y) = x2 + 0.4314538107033688/ + 1.3053335232048229xy is a Lyapunov function.

5. Experiments

In this section, some examples are given to illustrate the efficiency of our algorithm.

Example 8. This is an example from [7]

x = y, y = z,

z = -4x - 3y - 2z + x2y + x2z.

We assume that F(x,y,z) = x2 + y2 + z2 + axy + bxz+cyz. Algorithm 6 returns a Lyapunov function

F (x, y, z) = x2 + y2 + z2 + 1.370502803658027xy

+ 0.655753434727512xz (20)

+ 0.632220465746607yz,

at Step 4 using only 1.085175 s. If the algorithm does not terminate at Step 4, it returns

F (x, y, z)=x2 + y2 + z2 + 0.566986159377122xy

+ 1.934844270891010xz (21)

+ 0.065341301862036yz,

using about 21.285095 s.

Example 9. This is an example from a classic ODE's textbook:

x = -x - 3y + 2y + yz,

y = 3x-y-z + xz, (22)

z = -2x + y - z + xy.

Assume that F(x, y, z) = x2 + axy + xz + cy2 + dyz + ez2. With about 2.4 s, we got a real root for the parameters that form the coefficients of F.Indeed, thispoint wasobtained from Step 4. If there is no real point at Step 4, this program returns one real root using about 267 s, which is also more efficient than 1800 s in [3].

Example 10. This is another example from an ODE's textbook:

X = -x + y + xz - X ,

y = x-y + z2 - y3, (23)

z = -yz - z .

Assume that F = x2 + bxz + cy2 + dyz + ez2. For this program, our algorithm stops at Step 3, using about 1.24475 s. In [3], they use about 840 s.

6. Conclusion

For a differential system, based on the technique of computing real root of positive dimensional polynomial system, we present a numerical method to compute the Lyapunov function at equilibria. According to the relationship between the positive dimensional system and the Lyapunov function, we know we just need only one real root of this system, so we convert the algorithm into two steps. At each step, rather than using interval Newton's method to verify the existence of real root, we use the residue of the positive polynomial system at approximate real root to verify the correctness of the positive polynomial system.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (11171053) and the National Natural Science Foundation of China Youth Fund Project (11001040) and cstc2012ggB40004.

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