Krasovskii-LaSalle principle

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The Krasovskii-Lasalle principle is a criterion for the asymptotic stability of a (possibly nonlinear) dynamical system.

Contents

[edit] The global Krasovskii-LaSalle principle

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 C1 function V(\mathbf x) can be found such that

V( \mathbf x) > 0 , for all \mathbf x \neq \underline 0 (positive definite)
\dot{V}(\mathbf x) \le 0 for all \mathbf x (negative semidefinite)

and

V( \mathbf 0) = \dot{V} (\mathbf 0) = 0

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

[edit] Local version of the Krasovskii-LaSalle principle

If

V( \mathbf x) > 0 , when \mathbf x \neq \underline 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 x(t)=0, t \geq 0, then the local version of the Krasovskii-Lasalle principle states that the origin is locally asymptotically stable.

[edit] 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 Krasovskii-Lasalle principle gives a criterion for asymptotic stability in the case when \dot{V} ( \mathbf x) is only negative semidefinite.

[edit] Example: the pendulum with friction

Gravity acting on the pendulum
Gravity acting on the pendulum

This section will apply the Krasovskii-LaSalle principle to establish the local asymptotic stability of a simple system, the pendulum with friction. This system can be modeled with the differential equation [1]

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

where θ 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 Krasovskii-LaSalle principle, it can be shown that all trajectories which begin in a ball of certain size around the origin x1 = x2 = 0 asymptotically converge to the origin. We define V(x1,x2) as

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

This V(x1,x2) is simply the scaled energy of the system [2] Clearly, V(x1,x2) is positive definite in an open ball of radius π 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. 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 = {(x1,x2) | x2 = 0}

does not contain any trajectory of the system, except the trivial trajectory x = 0. Indeed, if at some time t, x2(t) = 0, then because x1 must be less π 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 Krasovskii-LaSalle 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 [3].

[edit] History

While LaSalle was the first author in the West to publish this theorem in 1960, its first publication was in 1952 by Barbashin and Krasovskii in a special case, and in 1959 by Krasovskii in the general case [4].

[edit] See also

[edit] Original papers

  • Barbashin, E.A, Krasovskii, N. N. , On the stability of motion as a whole, (Russian), Dokl. Akad. Nauk, 86, pp.453-456, 1952.
  • Krasovskii, N. N. Problems of the Theory of Stability of Motion, (Russian), 1959. English translation: Stanford University Press, Stanford, CA, 1963.
  • Lasalle, J.P. Some extensions of Liapunov's second method, IRE Transactions on Circuit Theory, CT-7, pp. 520-527, 1960.

[edit] References

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