finds the numeric solution of the discrete matrix equation .
- DiscreteLyapunovSolve solves the discrete-time Lyapunov and Stein equations.
- DiscreteLyapunovSolve works on both numerical and symbolic matrices.
Examplesopen allclose all
Test the stability of by checking if the solution of is positive definite for a negative definite :
As expected, the eigenvalues are inside the unit circle:
Compute the controllability Gramian of a stable discrete-time system:
Compute the observability Gramian of a stable discrete-time system:
Properties & Relations (5)
The equation , with a negative definite , yields a unique positive definite solution if and only if the eigenvalues of are within the unit circle:
The indefinite sum is the solution to if is asymptotically stable:
Compute the infinite-horizon quadratic cost for the asymptotically stable system :
Compute the same using direct summation:
LinearSolve gives the same solution:
Solve the equation using LinearSolve:
DiscreteLyapunovSolve gives the same solution:
Wolfram Research (2010), DiscreteLyapunovSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html.
Wolfram Language. 2010. "DiscreteLyapunovSolve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html.
Wolfram Language. (2010). DiscreteLyapunovSolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html