LyapunovSolve
LyapunovSolve[a,c]
finds a solution of the matrix Lyapunov equation .
LyapunovSolve[a,b,c]
solves .
LyapunovSolve[{a,d},c]
solves .
LyapunovSolve[{a,d},{b,e},c]
solves .
Details
- LyapunovSolve solves the continuous-time Lyapunov and Sylvester equations.
- LyapunovSolve works on both numerical and symbolic matrices.
Examples
open allclose allScope (7)
Applications (7)
Test the stability of by checking if the solution of is positive definite for a negative definite :
As expected, the eigenvalues are in the left half-plane:
Compute the controllability Gramian of a stable continuous-time system:
Compute the observability Gramian of a stable continuous-time system:
Compute the norm of an asymptotically stable continuous-time system:
Compute the feedback gains that place poles at desired locations:
For MIMO systems, the feedback gains are not unique:
Construct an observer for a StateSpaceModel:
First, choose an appropriate and such that the Lyapunov equation yields a nonsingular solution:
Then construct the observer as , , where is the observer state vector, is the output, is the input, and is the estimated state vector:
Compute the estimated state trajectories for a UnitStep input:
Compute the actual state trajectories for a UnitStep input:
Properties & Relations (5)
The equation , with a negative definite , yields a unique positive definite solution if and only if the eigenvalues of are in the closed left half-plane:
The definite integral is the solution to if is asymptotically stable:
Compute the infinite-horizon quadratic cost for the asymptotically stable system :
Compute using direct integration:
LinearSolve gives the same solution:
Solve the Lyapunov equation using LinearSolve:
LyapunovSolve gives the same solution:
Text
Wolfram Research (2010), LyapunovSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/LyapunovSolve.html.
CMS
Wolfram Language. 2010. "LyapunovSolve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LyapunovSolve.html.
APA
Wolfram Language. (2010). LyapunovSolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LyapunovSolve.html