Krasovskii-LaSalle principle

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

1 The global Krasovskii-LaSalle principle
2 Local version of the Krasovskii-LaSalle principle
3 Relation to Lyapunov theory
4 Example: the pendulum with friction
5 History
6 See also
7 Original papers
8 References

[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 $C 1$ 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

$V( \mathbf x) > 0$ , 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

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 $x 1 = x 2 = 0$ 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 ^[2] Clearly, $V (x 1, x 2)$ 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 = {(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 $π$ 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

Lyapunov stability

[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

^ 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.