DiscreteLyapunovSolve [a,c]
finds the numeric solution of the discrete matrix equation .
DiscreteLyapunovSolve [a,b,c]
solves .
DiscreteLyapunovSolve [{a,d},c]
solves .
DiscreteLyapunovSolve [{a,d},{b,e},c]
solves .
DiscreteLyapunovSolve
DiscreteLyapunovSolve [a,c]
finds the numeric solution of the discrete matrix equation .
DiscreteLyapunovSolve [a,b,c]
solves .
DiscreteLyapunovSolve [{a,d},c]
solves .
DiscreteLyapunovSolve [{a,d},{b,e},c]
solves .
Details
- DiscreteLyapunovSolve solves the discrete-time Lyapunov and Stein equations.
- DiscreteLyapunovSolve works on both numerical and symbolic matrices.
Examples
open all close allBasic Examples (1)
Solve the discrete Lyapunov equation :
Scope (7)
Solve a discrete Lyapunov equation:
Verify the solution:
Solve an equation with symbolic matrices:
Solve for coefficient matrices having different dimensions:
Solve :
Solve :
Solve the discrete Lyapunov equation with symbolic coefficients:
Obtain the symbolic solution of :
Applications (4)
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:
An unstable system:
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:
An unstable system:
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:
Solve the matrix equation :
LinearSolve gives the same solution:
Solve the equation using LinearSolve :
DiscreteLyapunovSolve gives the same solution:
See Also
Related Guides
History
Text
Wolfram Research (2010), DiscreteLyapunovSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html.
CMS
Wolfram Language. 2010. "DiscreteLyapunovSolve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html.
APA
Wolfram Language. (2010). DiscreteLyapunovSolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html
BibTeX
@misc{reference.wolfram_2025_discretelyapunovsolve, author="Wolfram Research", title="{DiscreteLyapunovSolve}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html}", note=[Accessed: 24-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_discretelyapunovsolve, organization={Wolfram Research}, title={DiscreteLyapunovSolve}, year={2010}, url={https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html}, note=[Accessed: 24-November-2025]}