How do you calculate Lyapunov equation?
Definition of the Lyapunov Function. X′=f(X)ordxidt=fi(x1,x2,…,xn),i=1,2,…,n, with the zero equilibrium X≡0. V(x1,x2)=ax21+bx22,V(x1,x2)=ax21+bx42,a,b>0.
What is Lyapunov Theorem?
Lyapunov vector-measure theorem, theorem in measure theory that the range of any real-valued, non-atomic vector measure is compact and convex. Lyapunov–Malkin theorem, a mathematical theorem detailing nonlinear stability of systems.
What is Q in Lyapunov equation?
• The discrete-time Lyapunov equation has a unique solution P, for any Q = QT , if and only if λi(A)λj(A) = 1, for i, j = 1,…,n. • If A is stable, Lyapunov equation has a unique solution P, for any Q = QT .
How do you determine if a system is asymptotically stable?
A time-invariant system is asymptotically stable if all the eigenvalues of the system matrix A have negative real parts. If a system is asymptotically stable, it is also BIBO stable.
How is Lyapunov function derived?
We calculate the derivative of the function V(X) by virtue of the system: dVdt = ∂V∂xdxdt+∂V∂ydydt=2x⋅y+2y⋅(−x)≡0. Thus, the derivative is identically zero. Hence, the function V(X) is a Lyapunov function and the zero solution of the system is stable in the sense of Lyapunov.
How do you solve the Sylvester equation?
A Sylvester equation has a unique solution for X exactly when there are no common eigenvalues of A and −B. More generally, the equation AX + XB = C has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space….External links.
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What is meant by asymptotically stable?
Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium. Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate .
How do you determine if a system is Lyapunov stable?
It is clear that to find a stability using the Lyapunov method, we need to find a positive definite Lyapunov function defined in some region of the state space containing the equilibrium point whose derivative V ˙ = d v d x f ( x ) is negative semidefinite along the system trajectories.
How do you calculate Lyapunov exponents?
In practice, Lyapunov exponents can be computed by exploiting the natural tendency of an n-dimensional volume to align along the n most expanding subspace. From the expansion rate of an n-dimensional volume, one obtains the sum of the n largest Lyapunov exponents.