LaSalle's invariance principle

LaSalle's invariance principle (also known as the invariance principle,^[1] Barbashin-Krasovskii-LaSalle principle,^[2] or Krasovskii-LaSalle principle ) is a criterion for the asymptotic stability of an autonomous (possibly nonlinear) dynamical system.

Global version

Given a representation of the system

\dot{\mathbf{x}} = f \left(\mathbf x \right)

where $\mathbf {x}$ is the vector of variables, with

f\left(\mathbf {0} \right)=\mathbf {0} .

If a $C^{1}$ function $V(\mathbf x)$ can be found such that

\dot{V}(\mathbf x) \le 0

for all

\mathbf x

(negative semidefinite),

then the set of accumulation points of any trajectory is contained in ${\mathcal I}$ where ${\mathcal I}$ is the union of complete trajectories contained entirely in the set $\{\mathbf x : \dot{V}( \mathbf x) = 0 \}$ .

If we additionally have that the function $V$ is positive definite, i.e.

V( \mathbf x) > 0

, for all

\mathbf x \neq \mathbf 0

V( \mathbf 0) = 0

and if ${\mathcal I}$ contains no trajectory of the system except the trivial trajectory $\mathbf x(t) = \mathbf 0$ for $t\geq 0$ , then the origin is asymptotically stable.

Furthermore, if $V$ is radially unbounded, i.e.

V(\mathbf x) \to \infty

, as

\Vert \mathbf x \Vert \to \infty

then the origin is globally asymptotically stable.

Local version

V( \mathbf x) > 0

, when

\mathbf x \neq \mathbf 0

\dot{V}(\mathbf x) \le 0

hold only for $\mathbf x$ in some neighborhood $D$ of the origin, and the set

\{ \dot{V}( \mathbf x) = 0 \} \bigcap D

does not contain any trajectories of the system besides the trajectory $\mathbf x(t)=\mathbf 0, t \geq 0$ , then the local version of the invariance principle states that the origin is locally asymptotically stable.

Relation to Lyapunov theory

If $\dot{V} ( \mathbf x)$ is negative definite, the global asymptotic stability of the origin is a consequence of Lyapunov's second theorem. The invariance principle gives a criterion for asymptotic stability in the case when $\dot{V} ( \mathbf x)$ is only negative semidefinite.

Example: the pendulum with friction

This section will apply the invariance principle to establish the local asymptotic stability of a simple system, the pendulum with friction. This system can be modeled with the differential equation

m l \ddot{\theta} = - m g \sin \theta - k l \dot{\theta}

where $\theta$ is the angle the pendulum makes with the vertical normal, $m$ is the mass of the pendulum, $l$ is the length of the pendulum, $k$ is the friction coefficient, and g is acceleration due to gravity.

This, in turn, can be written as the system of equations

\dot{x}_1 = x_2

\dot{x}_2 = -\frac{g}{l} \sin x_1 - \frac{k}{m} x_2

Using the invariance principle, it can be shown that all trajectories which begin in a ball of certain size around the origin $x_1 = x_2 = 0$ asymptotically converge to the origin. We define $V(x_1,x_2)$ as

V(x_1,x_2) = \frac{g}{l} (1 - \cos x_1) + \frac{1}{2} x_2^2

This $V(x_1,x_2)$ is simply the scaled energy of the system Clearly, $V(x_1,x_2)$ is positive definite in an open ball of radius $\pi$ around the origin. Computing the derivative,

\dot{V}(x_1,x_2) = \frac{g}{l} \sin x_1 \dot{x}_1 + x_2 \dot{x}_2 = - \frac{k}{m} x_2^2

Observe that $V(0) = \dot{V}(0) = 0$ . If it were true that $\dot{V} < 0$ , we could conclude that every trajectory approaches the origin by Lyapunov's second theorem. Unfortunately, $\dot{V} \leq 0$ and ${\dot {V}}$ is only negative semidefinite. However, the set

S = \{ (x_1,x_2) | \dot{V}(x_1,x_2) = 0 \}

which is simply the set

S = \{ (x_1,x_2) | x_2 = 0 \}

does not contain any trajectory of the system, except the trivial trajectory x = 0. Indeed, if at some time $t$ , $x_2(t)=0$ , then because $x_{1}$ must be less than $\pi$ away from the origin, $\sin x_1 \neq 0$ and $\dot{x}_2(t) \neq 0$ . As a result, the trajectory will not stay in the set $S$ .

All the conditions of the local version of the invariance principle are satisfied, and we can conclude that every trajectory that begins in some neighborhood of the origin will converge to the origin as $t \rightarrow \infty$ .

History

The general result was independently discovered by J.P. LaSalle (then at RIAS) and N.N. Krasovskii, who published in 1960 and 1959 respectively. While LaSalle was the first author in the West to publish the general theorem in 1960, a special case of the theorem was in communicated in 1952 by Barbashin and Krasovskii, followed by a publication of the general result in 1959 by Krasovskii .

Original papers

LaSalle, J.P. Some extensions of Liapunov's second method, IRE Transactions on Circuit Theory, CT-7, pp. 520–527, 1960. (PDF)
Barbashin, E. A.; Nikolai N. Krasovskii (1952). Об устойчивости движения в целом [On the stability of motion as a whole]. Doklady Akademii Nauk SSSR (in Russian). 86: 453–456.
Krasovskii, N. N. Problems of the Theory of Stability of Motion, (Russian), 1959. English translation: Stanford University Press, Stanford, CA, 1963.

Text books

LaSalle, J.P.; Lefschetz, S. (1961). Stability by Liapunov's direct method. Academic Press.
Haddad, W.M.; Chellaboina, VS (2008). Nonlinear Dynamical Systems and Control, a Lyapunov-based approach. Princeton University Press. ISBN 9780691133294.
Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
Wiggins, S. (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos (2 ed.). New York City: Springer Verlag. ISBN 0-387-00177-8.

Lectures

Texas A&M University notes on the invariance principle (PDF)
NC State University notes on LaSalle's invariance principle (PDF).
Caltech notes on LaSalle's invariance principle (PDF).
MIT OpenCourseware notes on Lyapunov stability analysis and the invariance principle (PDF).
Purdue University notes on stability theory and LaSalle's invariance principle (PDF).

References

↑ Khalil, Hasan (2002). Nonlinear Systems (3rd ed.). Upper Saddle River NJ: Prentice Hall.
↑ Wassim, Haddad; Chellaboina, VijaySekhar (2008). Nonlinear Dynamical Systems and Control, a Lyapunov-based approach. Princeton University Press.

^ Lecture notes on nonlinear control, University of Notre Dame, Instructor: Michael Lemmon, lecture 4.
^ ibid.
^ Lecture notes on nonlinear analysis, National Taiwan University, Instructor: Feng-Li Lian, lecture 4-2.
^ Vidyasagar, M. Nonlinear Systems Analysis, SIAM Classics in Applied Mathematics, SIAM Press, 2002.

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