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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
A Lyapunov function for an autonomous dynamical system {: Ë™ = ()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 Ë™ = is non positive (these conditions are required on some region containing the origin).
This example shows a system where a Lyapunov function can be used to prove Lyapunov stability but cannot show asymptotic stability. Consider the following equation, based on the Van der Pol oscillator equation with the friction term changed:
In this case, A = -1 and X(0, y) = Y(0, y) = 0 for all y, so this system satisfy the hypothesis of Lyapunov-Malkin theorem. The figure below shows a plot of this vector field along with some trajectories that pass near (0,0).
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 ...
In that paper the relation to solvability of the Lur’e equations was also established. Both papers considered scalar-input systems. The constraint on the control dimensionality was removed in 1964 by Gantmakher and Yakubovich [3] and independently by Vasile Mihai Popov. [4] Extensive reviews of the topic can be found in [5] and in Chapter 3 ...
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