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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