Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Lyapunov’s Stability Theory

  • Hassan K. KhalilEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_77-1

Abstract

Lyapunov’s theory for characterizing and studying the stability of equilibrium points is presented for time-invariant and time-varying systems modeled by ordinary differential equations.

Keywords Asymptotic stability, Equilibrium point, Exponential stability, Global asymptotic stability, Hurwitz matrix, Invariance principle, Linearization, Lipschitz condition, Lyapunov function, Lyapunov surface, Negative (semi-) definite function, Perturbed system, Positive (semi-) definite function, Region of attraction, Stability, Time-invariant system, Time-varying system

Introduction

Stability theory plays a central role in systems theory and engineering. For systems represented by state models, stability is characterized by studying the asymptotic behavior of the state variables near steady-state solutions, like equilibrium points or periodic orbits. In this article, Lyapunov’s method for determining the stability of equilibrium points is introduced. The attractive features of the method include a solid theoretical foundation, the ability to conclude stability without knowledge of the solution (no extensive simulation effort), and an analytical framework that makes it possible to study the effect of model perturbations and design feedback control. Its main drawback is the need to search for an auxiliary function that satisfies certain conditions.

Stability of Equilibrium Points

We consider a nonlinear system represented by the state model
$${\dot x} = f(x)$$
(1)
where the n-dimensional locally Lipschitz function f(x) is defined for all x in a domain DR n . A function f(x) is locally Lipschitz at a point x 0 if it satisfies the Lipschitz conditionf(x) − f(y) ∥ ≤ Lxy ∥ for all x, y in some neighborhood of x 0, where L is a positive constant and \(\|x\| = \sqrt{x_{1 }^{2 } + x_{2 }^{2 } + \cdots + x_{n }^{2}}\). The Lipschitz condition guarantees that Eq. (1) has a unique solution for given initial state x(0). Suppose \(\bar{x} \in D\) is an equilibrium point of Eq. (1); that is, \(f(\bar{x}) = 0\). Whenever the state of the system starts at \(\bar{x}\), it will remain at \(\bar{x}\) for all future time. Our goal is to characterize and study the stability of \(\bar{x}\). For convenience, we take \(\bar{x} = 0\). There is no loss of generality in doing so because any equilibrium point \(\bar{x}\) can be shifted to the origin via the change of variables \(y = x -\bar{ x}\). Therefore, we shall always assume that f(0) = 0 and study stability of the origin x = 0.
The equilibrium point x = 0 of Eq. (1) is stable if for each ε > 0, there is δ = δ(ε) > 0 such that ∥ x(0) ∥ < δ implies that ∥ x(t) ∥ < ε, for all t ≥ 0. It is asymptotically stable if it is stable and δ can be chosen such that ∥ x(0) ∥ < δ implies that x(t) converges to the origin as t tends to infinity. When the origin is asymptotically stable, the region of attraction (also called region of asymptotic stability, domain of attraction, or basin) is defined as the set of all points x such that the solution of Eq. (1) that starts from x at time t = 0 approaches the origin as t tends to . When the region of attraction is the whole space, we say that the origin is globally asymptotically stable. A stronger form of asymptotic stability arises when there exist positive constants c, k, and λ such that the solutions of Eq. (1) satisfy the inequality
$$\|x(t)\| \leq k\|x(0)\|{e}^{-\lambda t},\ \ \ \forall \ t \geq 0$$
(2)
for all ∥ x(0) ∥ < c. In this case, the equilibrium point x = 0 is said to be exponentially stable. It is said to be globally exponentially stable if the inequality is satisfied for any initial state x(0).

Linear Systems

For the linear time-invariant system
$${\dot x} = Ax$$
(3)
the stability properties of the origin can be determined by the location of the eigenvalues of A. The origin is stable if and only if all the eigenvalues of A satisfy Re[λ i ] ≤ 0 and for every eigenvalue with Re[λ i ] = 0 and algebraic multiplicity q i ≥ 2, rank(Aλ i I) = nq i , where n is the dimension of x and q i is the multiplicity of λ i as a zero of det(λIA). The origin is globally exponentially stable if and only if all eigenvalues of A have negative real parts; that is, A is a Hurwitz matrix. For linear systems, the notions of asymptotic and exponential stability are equivalent because the solution is formed of exponential modes. Moreover, due to linearity, if the origin is exponentially stable, then the inequality of Eq. (2) will hold for all initial states.

Linearization

Suppose the function f(x) of Eq. (1) is continuously differentiable in a domain D containing the origin. The Jacobian matrix [∂f∂x] is an n ×n matrix whose (i, j) element is ∂f i ∂x j . Let A be the Jacobian matrix evaluated at the origin x = 0. It can be shown that
$$f(x) = [A + G(x)]x,\ \ \ \ \mbox{ where}\ \lim _{x\rightarrow 0}G(x) = 0$$
This suggests that in a small neighborhood of the origin we can approximate the nonlinear system \({\dot x} = f(x)\) by its linearization about the origin \({\dot x} = Ax\). Indeed, we can draw conclusions about the stability of the origin as an equilibrium point for the nonlinear system by examining the eigenvalues of A. The origin of Eq. (1) is exponentially stable if and only if A is Hurwitz. It is unstable if Re[λ i ] > 0 for one or more of the eigenvalues of A. If Re[λ i ] ≤ 0 for all i, with Re[λ i ] = 0 for some i, we cannot draw a conclusion about the stability of the origin of Eq. (1).

Lyapunov’s Method

Let V (x) be a continuously differentiable scalar function defined in a domain DR n that contains the origin. The function V (x) is said to be positive definite if V (0) = 0 and V (x) > 0 for x≠0. It is said to be positive semidefinite if V (x) ≥ 0 for all x. A function V (x) is said to be negative definite or negative semidefinite if − V (x) is positive definite or positive semidefinite, respectively. The derivative of V along the trajectories of Eq. (1) is given by
$$ {\dot V}(x) =\displaystyle\sum _{ i=1}^{n} \frac{\partial V } {\partial x_{i}}{\dot x}_{i} = \frac{\partial V } {\partial x} f(x)$$
where [∂V∂x] is a row vector whose ith component is ∂V∂x i .

Lyapunov’s stability theorem states that the origin is stable if there is a continuously differentiable positive definite function V (x) so that \( {\dot V}(x)\) is negative semidefinite, and it is asymptotically stable if \( {\dot V}(x)\) is negative definite. A function V (x) satisfying the conditions for stability is called a Lyapunov function. The surface V (x) = c, for some c > 0, is called a Lyapunov surface or a level surface.

When \( {\dot V}(x)\) is only negative semidefinite, we may still conclude asymptotic stability of the origin if we can show that no solution can stay identically in the set \(\{ {\dot V}(x) = 0\}\), other than the zero solution x(t) ≡ 0. Under this condition, V (x(t)) must decrease toward 0, and consequently x(t) converges to zero as t tends to infinity. This extension of the basic theorem is known as the invariance principle.

Lyapunov functions can be used to estimate the region of attraction of an asymptotically stable origin, that is, to find sets contained in the region of attraction. Let V (x) be a Lyapunov function that satisfies the conditions of asymptotic stability over a domain D. For a positive constant c, let \(\Omega _{c}\) be the component of {V (x) ≤ c} that contains the origin in its interior. The properties of V guarantee that, by choosing c small enough, \(\Omega _{c}\) will be bounded and contained in D. Then, every trajectory starting in \(\Omega _{c}\) remains in \(\Omega _{c}\) and approaches the origin as t. Thus, \(\Omega _{c}\) is an estimate of the region of attraction. If D = R n and V (x) is radially unbounded, that is, ∥ x ∥ → implies that V (x) → , then any point xR n can be included in a bounded set \(\Omega _{c}\) by choosing c large enough. Therefore, the origin is globally asymptotically stable if there is a continuously differentiable, radially unbounded function V(x) such that for all x ∈ R n, V (x) is positive definite and \( {\dot V}(x)\) is either negative definite or negative semidefinite but no solution can stay identically in the set \(\{ {\dot V}(x) = 0\}\) other than the zero solution x(t) ≡ 0.

Time-Varying Systems

Equation (1) is time-invariant because f does not depend on t. The more general time-varying system is represented by
$${\dot x} = f(t,x)$$
(4)
In this case, we may allow the Lyapunov function candidate V to depend on t. Let V (t, x) be a continuously differentiable function defined for all t ≥ 0 and xD. The derivative of V along the trajectories of Eq. (4) is given by
$$ {\dot V}(t,x) = \frac{\partial V } {\partial t} + \frac{\partial V } {\partial x} f(t,x)$$
If there are positive definite functions W 1(x), W 2(x), and W 3(x) such that
$$\displaystyle\begin{array}{rcl} W_{1}(x) \leq V (t,x) \leq W_{2}(x)& &\end{array}$$
(5)
$$\displaystyle\begin{array}{rcl} {\dot V}(t,x) \leq -W_{3}(x)& &\end{array}$$
(6)
for all t ≥ 0 and all xD, then the origin is uniformly asymptotically stable, where “uniformly” indicates that the ε–δ definition of stability and the convergence of x(t) to zero are independent of the initial time t 0. Such uniformity annotation is not needed with time-invariant systems since the solution of a time-invariant state equation starting at time t 0 depends only on the difference tt 0, which is not the case for time-varying systems. If the inequalities of Eqs. (5) and (6) hold globally and W 1(x) is radially unbounded, then the origin is globally uniformly asymptotically stable. If W 1(x) = k 1x a , W 2(x) = k 2x a , and W 3(x) = k 3x a for some positive constants k 1, k 2, k 3, and a, then the origin is exponentially stable.

Perturbed Systems

Consider the system
$${\dot x} = f(t,x) + g(t,x)$$
(7)
where f and g are continuous in t and locally Lipschitz in x, for all t ≥ 0 and xD, in which DR n is a domain that contains the origin x = 0. Suppose f(t, 0) = 0 and g(t, 0) = 0 so that the origin is an equilibrium point of Eq. (7). We think of the system (7) as a perturbation of the nominal system (4). The perturbation term g(t, x) could result from modeling errors, uncertainties, or disturbances. In a typical situation, we do not know g(t, x), but we know some information about it, like knowing an upper bound on ∥ g(t, x) ∥. Suppose the nominal system has an exponentially stable equilibrium point at the origin, what can we say about the stability of the origin as an equilibrium point of the perturbed system? A natural approach to address this question is to use a Lyapunov function for the nominal system as a Lyapunov function candidate for the perturbed system.
Let V (t, x) be a Lyapunov function that satisfies
$$\displaystyle\begin{array}{rcl} c_{1}\|{x\|}^{2} \leq V (t,x) \leq c_{ 2}\|{x\|}^{2}& &\end{array}$$
(8)
$$\displaystyle\begin{array}{rcl} \frac{\partial V } {\partial t} + \frac{\partial V } {\partial x} f(t,x) \leq -c_{3}\|{x\|}^{2}& &\end{array}$$
(9)
$$\displaystyle\begin{array}{rcl} \left \|\frac{\partial V } {\partial x} \right \| \leq c_{4}\|x\|& &\end{array}$$
(10)
for all xD for some positive constants c 1, c 2, c 3, and c 4. Suppose the perturbation term g(t, x) satisfies the linear growth bound
$$\|g(t,x)\| \leq \gamma \| x\|,\ \ \forall \ t \geq 0,\ \forall \ x \in D$$
(11)
where γ is a nonnegative constant. We use V as a Lyapunov function candidate to investigate the stability of the origin as an equilibrium point for the perturbed system. The derivative of V along the trajectories of Eq. (7) is given by
$$ {\dot V}(t,x) = \frac{\partial V } {\partial t} + \frac{\partial V } {\partial x} f(t,x) + \frac{\partial V } {\partial x} g(t,x)$$
The first two terms on the right-hand side are the derivative of V (t, x) along the trajectories of the nominal system, which is negative definite and satisfies the inequality of Eq. (9). The third term, [∂V∂x]g, is the effect of the perturbation. Using Eqs. (9) through (11), we obtain
$$\displaystyle\begin{array}{rcl} {\dot V}(t,x)& \leq & -c_{3}\|{x\|}^{2} + \left \|\frac{\partial V } {\partial x} \right \|\ \|g(t,x)\| \\ & \leq & -c_{3}\|{x\|}^{2} + c_{ 4}\gamma \|{x\|}^{2} \\ \end{array}$$
If γ < c 3c 4, then
$$\leq -(c_{3} -\gamma c_{4})\|{x\|}^{2},\ \ (c_{ 3} -\gamma c_{4})> 0$$
which shows that the origin is an exponentially stable equilibrium point of the perturbed system (7).

Summary

Lyapunov’s method is a powerful tool for studying the stability of equilibrium points. However, there are two drawbacks of the method. First, there is no systematic procedure for finding Lyapunov functions. Second, the conditions of the theory are only sufficient; they are not necessary. Failure of a Lyapunov function candidate to satisfy the conditions for stability or asymptotic stability does not mean that the origin is not stable or asymptotically stable. These drawbacks have been mitigated by a long history of using the method in the analysis and design of engineering systems, where various techniques for finding Lyapunov functions for specific systems have been determined.

Cross-References

For forced systems of the form \({\dot x} = f(x,u)\), where u is an input, the asymptotic behavior of the state variables is characterized by the notion of Input-to-State Stability, which is presented in a separate article. This notion reduces to asymptotic stability of the origin when u = 0 and is characterized by Lyapunov functions. The use of Lyapunov’s theory in the design of feedback control for nonlinear systems is described in the articles on Feedback Stabilization of Nonlinear Systems and Regulation and Tracking of Nonlinear Systems.

Recommended Reading

For an introduction to Lyapunov’s stability theory at the level of first-year graduate students, the textbooks Khalil (2002), Sastry (1999), Slotine and Li (1991), and (Vidyasagar 2002) are recommended. The books by Bacciotti and Rosier (2005) and Haddad and Chellaboina (2008) cover a wider set of topics at the same introductory level. A deeper look into the theory is provided in the monographs Hahn (1967), Krasovskii (1963), Rouche et al. (1977), Yoshizawa (1966), and (Zubov 1964). Lyapunov’s theory for discrete-time systems is presented in Haddad and Chellaboina (2008) and Qu (1998). The monograph Michel and Wang (1995) presents Lyapunov’s stability theory for general dynamical systems, including functional and partial differential equations.

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© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringMichigan State UniversityEast LansingUSA