Krasovskii–LaSalle principle

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

The global Krasovskii–LaSalle principle

Given a representation of the system

: $dot\{mathbf\{x\; =\; f\; left(mathbf\; x\; ight)$

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

: $f\; left(\; mathbf\; 0\; ight)\; =\; mathbf\; 0$

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

: $V(\; mathbf\; x)\; >\; 0$, for all $mathbf\; x\; eq\; underline\; 0$ (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.

Local version of the Krasovskii–LaSalle principle

If : $V(\; mathbf\; x)\; >\; 0$, when $mathbf\; x\; eq\; underline\; 0$: $dot\{V\}(mathbf\; x)\; le\; 0$

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

:

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.

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.

Example: the pendulum with friction

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 ref|nd1

:$m\; l\; ddot\{\; heta\}\; =\; -\; m\; g\; sin\; heta\; -\; k\; l\; dot\{\; heta\}$

where $heta$ 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 ref|nd2 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 $V(0)\; =\; dot\{V\}\; =\; 0$. If it were true that , 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 $pi$ away from the origin, $sin\; x\_1\; eq\; 0$ and $dot\{x\_2\}(t)\; eq\; 0$. 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 $t\; ightarrow\; infty$ ref|tw1.

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 ref|vid.

See also

* Lyapunov stability

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.

References

# [http://www.nd.edu/~lemmon/courses/ee580/ Lecture notes on nonlinear control] , University of Notre Dame, Instructor: Michael Lemmon, lecture 4.
# ibid.
# [http://cc.ee.ntu.edu.tw/~fengli/Teaching/NonlinearSystems/ 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.