On the Region of Asymptotic Stability of Nonlinear Quadratic Systems F. Amato†, , C. Cosentino† , A. Merola†
Abstract— This paper considers the following problem: given a nonlinear quadratic system and a certain box containing the origin of the state space, determine whether this box belongs to the Region of Asymptotic Stability of the zero equilibrium point of the system under consideration. Quadratic systems play an important role in the modelling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is of mandatory importance not only to determine whether the origin of the state space is locally asymptotically stable but also to ensure that the operative range is included into the convergence region of the equilibrium. The proposed algorithm requires the solution of a suitable feasibility problem involving Linear Matrix Inequalities constraints. Some examples illustrate the effectiveness of the proposed procedure.
I. I NTRODUCTION Nonlinear quadratic systems provide an appropriate tool for modeling phenomena in a wide range of applications, either in engineering (electric power systems, chemical reactors, and robots), or in other areas such as biology, ecology and economics. In biology, for example, quadratic dynamical models can be used to describe interactions among genes/proteins involved in intra-cellular signaling and regulatory networks. Over the years, several papers have focused on the estimate of the region of asymptotic stability (RAS) of the zero equilibrium point of nonlinear (quadratic, cubic and, more in general, polynomial) autonomous systems. Many of them present methods suitable for low order systems, like in [1] where a Lyapunov-based procedure is proposed, that is able to compute an ellipsoidal estimate of the RAS of a second order nonlinear system containing either linear and quadratic or linear and cubic terms. For general n-th order systems, gridding techniques have been employed in [2],[3],[4], whereas [5],[6] provide an estimate of the RAS based on topological considerations. An alternative approach, exploited to deal with nonlinear polynomial systems, is based on the solution of a convex optimization problem [7]. More recently, the problem of determining the RAS of polynomial systems has been solved by recasting it as a Linear Matrix Inequality (LMI) feasibility problem [8]. In these papers, the problem is tackled by means of a twosteps procedure: i) choice of a quadratic Lyapunov function, which proves local asymptotic stability of the equilibrium; ii) computation of the estimate of the RAS associated to that particular Lyapunov function. However, the choice of
the optimal Lyapunov function is not a trivial task and may severely affect the conservatism of the estimate. Moreover, higher order systems, such as biological systems, require a heavy computational burden for estimating the whole RAS. However, in many practical situations, it is sufficient to verify the stability properties only in the admissible operative range of the system, which is typically assigned giving the variation interval of each state variable. In this paper we focus on nonlinear quadratic systems; assigned a certain box containing the origin of the state space, the proposed approach deals with the problem of determining whether this box belongs to the RAS of the system under consideration. A fundamental point is that our approach optimizes over the set of quadratic Lyapunov functions, thus reducing the conservatism of the previous approaches. The solution requires to solve a feasibility problem whose constraints are LMIs. The paper is organized as follows: In Section II the problem we deal with is precisely stated and some preliminary notation is provided. Section III contains the description of the proposed methodology, whose effectiveness is then illustrated in Section IV through some examples. II. P ROBLEM S TATEMENT In this paper we consider a quadratic system, that is a nonlinear system in the form x˙ = Ax + B(x) , where x ∈ Rn is the system state and T x B1 x xT B2 x B(x) = . ..
(2)
xT Bn x
with Bi ∈ Rn×n , i = 1, . . . , n. First note that the study of the stability properties of a nonzero equilibrium point of system (1) can be always reduced to the study of the corresponding properties of the zero state of a suitable fictitious quadratic system. Indeed assume that xe = 0 is an equilibrium point for system (1), then
†
F. Amato, C. Cosentino and A. Merola are with the School of Computer and Biomedical Engineering, Universit`a degli Studi Magna Græcia di Catanzaro, Via T. Campanella 115, 88100 Catanzaro, Italy Corresponding author, e-mail:
[email protected], tel:+39-0961-369-4082, fax:+39-0961-369-4090
(1)
Axe + B(xe ) = 0 .
(3)
z = x − xe ;
(4)
Now let
it is readily seen that, from (3), T xe B1 xTe B2 z˙ = A + 2 . z + B(z) + Axe + B(xe ) .. xT Bn eT xe B1 xTe B2 (5) = A + 2 . z + B(z) , .. xTe Bn
which is a quadratic system in the form (1). On the basis of this observation, we shall focus on the stability properties of the zero equilibrium point of system (1). Also, with slight abuse of terminology, we shall refer to the stability properties of system (1), in place of the stability properties of the zero equilibrium point of system (1). To check local asymptotic stability of system (1) is rather simple, since it amounts to evaluate the eigenvalues location of the linearized system x˙ = Ax. In practical engineering applications, however, establishing the simple local asymptotic stability is often not sufficient. Therefore in this paper we will try to solve the following problem. Problem 1: Given a box R ⊂ Rn , 0 ∈ R, defined as R := [x1 , x1 ] × [x2 , x2 ] × · · · [xn , xn ] ,
(6)
establish whether R belongs to the RAS of system (1). ♦ The reason for considering a box in Problem 1 follows from the practical consideration that the operative range of a nonlinear system is typically assigned in terms of the variation range of the state variables. It is simple to recognize that the proposed technique can be easily extended to general polytopic regions. Note that alternative equivalent descriptions of the box (6) are the following: R = conv x(1) , x(2) , . . . , x(2n ) (7a) n T = x ∈ R : ak x ≤ 1 , k = 1, 2, . . . , 2n , (7b) where x(i) denotes the i-th vertex of R and conv{·} denotes the operation of taking the convex hull of the argument. For example the box in R2 R := [−1, 2] × [−1, 3] , can be also described in the form (7a) with
T
T x(1) = 2 −1 , x(2) = 2 3
T
T x(3) = −1 3 , x(4) = −1 −1 or in the form (7b) with
aT1 = 12 0 , aT2 = −1 0
aT3 = 0 13 , aT4 = 0 −1 .
III. M AIN R ESULT First we recall the following theorem. Theorem 1 ([9]): A given closed set E ⊂ Rn , 0 ∈ E, is an estimate of the RAS of system (1) if i) E is an invariant set; ii) there exists a Lyapunov function v(x) such that a) v(x) is positive definite on E; b) v(x) ˙ is negative definite on E. ♦ Remark 1: Recall that the RAS is defined as the largest connected set Ω containing the origin and such that every solution starting in Ω converges to zero. Any closed region E, 0 ∈ E, contained in Ω is an estimate of the RAS. Note that, in order to ensure that a given set E is an estimate of the RAS, it is not sufficient to find a positive definite Lyapunov function with negative definite derivative on E. Indeed, as stated in Theorem 1, we need to ensure that the set E is an invariant set, i. e. that every solution of (1) starting in E remains in E for all t. This assumption is necessary in order to ensure that a given solution starting in E cannot escape to infinity. ♦ In this paper, as usual, we consider quadratic Lyapunov functions in the form v(x) = xT P x; the following lemma allows to establish if a given Lyapunov function satisfies condition ii) in Theorem 1 when E is a box. Lemma 1: Given the quadratic Lyapunov function v(x) = xT P x and the set R defined in (6), v(x) satisfies the hypothesis ii) of Theorem 1 on R if a) P > 0 (P is a symmetric positive definite matrix); b) the symmetric matrix function T x B1 xT B2 AT P + P A + P . .. xT Bn T
+ B1 x B2T x · · · BnT x P (8) is negative definite on the vertices of the box R. Proof: Condition a) is obviously needed in order to guarantee the positive definiteness of v(x) on R. To prove that the negative definiteness of v(x) ˙ is implied by condition b) consider the derivative of v(x) along the trajectories of system (1), v(x) ˙ = x˙ T P x + xT P x˙
= xT AT + B1T x B2T x . . . BnT x P T x B1 xT B2 (9) +P A + . x < 0 . .. xT Bn Note that the bracketed expression is an affine function of the state variables x1 , . . . , xn , therefore it is negative definite on R if the property holds on the vertices of R [10]. Given the set R, the existence of a quadratic Lyapunov function which satisfies condition ii) of Theorem 1 on R
is not sufficient to guarantee that R belongs to the RAS. This because boxes are not invariant sets. Our idea is that of enclosing R into an invariant set which does belong to the RAS. More precisely, let us consider a box R ⊃ R and a quadratic function v(x) = xT P x such that v(x) satisfies the hypotheses of Lemma 1 on R . Then, given c > 0, any ellipsoidal region (10) Ec := x ∈ Rn : xT P x ≤ c ⊂ R belongs to the RAS. Clearly, if it is possible to prove that Ec ⊃ R for some positive scalar c, a solution to Problem 1 has been found. In the following we shall show that the double constraint R ⊃ Ec ⊃ R
(11)
can be translated into LMI conditions, therefore Problem 1 can be translated (with some conservatism) into a feasibility problem involving LMI constraints [11] and solved via available software [12]. Let us refine this idea; define ρR, ρ > 1, as the set obtained by multiplying by ρ the dimensions of R. Roughly speaking the set ρR has the same shape of the set R, but its vertices are amplified by a factor ρ. In the following we are interested in finding a quadratic Lyapunov function v(x) = xT P x, a corresponding ellipsoidal region Ec containing the set R and a factor ρ > 1 such that: a) v(x) satisfies the hypothesis of Lemma 1 on the amplified version of the set R, namely ρR; b) ρR contains Ec . Everything should be translated into LMIs condition. First we focus on the condition Ec ⊃ R. Since, in the final formulation of the problem, we shall also optimize over the choice of the quadratic Lyapunov function, without loss of generality we can look for ellipsoidal regions with c = 1; indeed c = 1 is equivalent to rescale P . Looking to (10) and (7a) we have that the condition E := E1 ⊃ R i = 1, 2, . . . , 2n ,
(13)
which are 2n LMIs in the matrix variable P . Now note that the set ρR can be equivalently described as ρR = x ∈ Rn : aTk x ≤ ρ , k = 1, 2, . . . , 2n aTk n x ≤ 1 , k = 1, 2, . . . , 2n . (14) = x∈R : ρ Therefore, according to [11], p. 70, we have that the condition ρR ⊃ E (15) can be equivalently rewritten aTk −1 ak P ≤ 1, ρ ρ
k = 1, 2, . . . , 2n ,
On the basis of the above discussion, Problem 1 is solvable if the following Generalized Eigenvalue Problem (GEVP) ([11], p. 10) admits a feasible solution. Problem 2: Find a scalar γ and a symmetric matrix P such that
1 γak
0
