This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 BUILT-IN MATHEMATICA SYMBOL

# DiscreteLyapunovSolve

 DiscreteLyapunovSolve finds the numeric solution of the discrete matrix equation . DiscreteLyapunovSolvesolves . DiscreteLyapunovSolvesolves . DiscreteLyapunovSolvesolves .
Solve the discrete Lyapunov equation :
Solve the discrete Lyapunov equation :
 Out[1]=
 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:
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:
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