Search results
Results from the WOW.Com Content Network
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [ 1 ] [ 2 ] In particular, the discrete-time Lyapunov equation (also known as Stein equation ) for X {\displaystyle X} is
The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
If the derivative at a is exactly 1 or −1, then more information is needed in order to decide stability. There is an analogous criterion for a continuously differentiable map f : R n → R n with a fixed point a , expressed in terms of its Jacobian matrix at a , J a ( f ) .
The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state x ≠ 0 {\displaystyle x\neq 0} in some domain D , then the state will remain in D for all time.
To introduce Lyapunov exponent consider a fundamental matrix () (e.g., for linearization along a stationary solution in a continuous system), the fundamental matrix is (() |) consisting of the linearly-independent solutions of the first-order approximation of the system.
Minimizing the drift of a quadratic Lyapunov function leads to the backpressure routing algorithm for network stability, also called the max-weight algorithm. [1] [2] Adding a weighted penalty term to the Lyapunov drift and minimizing the sum leads to the drift-plus-penalty algorithm for joint network stability and penalty minimization.
where and are components of the system state, is a matrix that represents the linear dynamics of , and : and : represent higher-order nonlinear terms. If all eigenvalues of the matrix A {\displaystyle A} have negative real parts, and X ( x , y ), Y ( x , y ) vanish when x = 0, then the solution x = 0, y = 0 of this system is stable with respect ...