# DiscreteLyapunovSolve

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 .

# Examples

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## Basic 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:

Wolfram Research (2010), DiscreteLyapunovSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html.

#### 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_2024_discretelyapunovsolve, author="Wolfram Research", title="{DiscreteLyapunovSolve}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html}", note=[Accessed: 18-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_discretelyapunovsolve, organization={Wolfram Research}, title={DiscreteLyapunovSolve}, year={2010}, url={https://reference.wolfram.com/language/ref/DiscreteLyapunovSolve.html}, note=[Accessed: 18-September-2024 ]}