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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.
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,
System is called globally asymptotically stable at zero (0-GAS) if the corresponding system with zero input ...
The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand ...
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 is
Then Aizerman's conjecture is that the system is stable in large (i.e. unique stationary point is global attractor) if all linear systems with f(e)=ke, k ∈(k1,k2) are asymptotically stable. There are counterexamples to Aizerman's conjecture such that nonlinearity belongs to the sector of linear stability and unique stable equilibrium coexists ...
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 Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix has all the eigenvalues inside the unit circle. For control purposes the output of the observer system is fed back to the input of both the observer and the plant through the gains matrix K {\displaystyle K} .