Search results
Results from the WOW.Com Content Network
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).
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 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.
A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state. Typically, the function is defined to grow large when the system moves towards undesirable states. System stability is achieved by taking control actions that make the Lyapunov function drift in the negative direction towards zero.
An important result due to D. Angeli, E. Sontag and Y. Wang is that system is integral ISS if and only if there exists an iISS-Lyapunov function for it. Note that in the formula above is assumed to be only positive definite. It can be easily proved, [13] that if is an iISS-Lyapunov function with , then is actually an ISS-Lyapunov function for a ...
The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts. Let (X, d) be a metric space and f : X → X a continuous function. A point x in X is said to be Lyapunov stable, if,
If, in addition, all eigenvalues of have negative real parts (is stable), and the unique solution of the Lyapunov equation + = is positive definite, the system is controllable. The solution is called the Controllability Gramian and can be expressed as W c = ∫ 0 ∞ e A τ B B T e A T τ d τ {\displaystyle {\boldsymbol {W_{c}}}=\int _{0 ...
Comparison functions are primarily used to obtain quantitative restatements of stability properties as Lyapunov stability, uniform asymptotic stability, etc. These restatements are often more useful than the qualitative definitions of stability properties given in ε - δ {\displaystyle \varepsilon {\text{-}}\delta } language.