Search results
Results from the WOW.Com Content Network
In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is A X A H − X + Q = 0 {\displaystyle AXA^{H}-X+Q=0} where Q {\displaystyle Q} is a Hermitian matrix and A H {\displaystyle A^{H}} is the conjugate transpose of A {\displaystyle A} , while the continuous-time Lyapunov equation is
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).
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:
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 ...
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.
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 ...
Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi-dimensional vector. A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state.
Lyapunov proved that if the system of the first approximation is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable. Later, it was stated by O. Perron that the requirement of regularity of the ...