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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.
Lyapunov stability means that if the system starts in a state in some domain D, then the state will remain in D for all time. For asymptotic stability, the state is also required to converge to =. A control-Lyapunov function is used to test whether a system is asymptotically stabilizable, that is whether for any state x there exists a control ...
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.
Lyapunov equation, used in many branches of control theory, such as stability analysis and optimal control Lyapunov fractal , bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population periodically switches between two values
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory.
ISS unified the Lyapunov and input-output stability theories and revolutionized our view on stabilization of nonlinear systems, design of robust nonlinear observers, stability of nonlinear interconnected control systems, nonlinear detectability theory, and supervisory adaptive control. This made ISS the dominating stability paradigm in ...
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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