This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# LyapunovSolve

 LyapunovSolve finds a solution x of the matrix Lyapunov equation . LyapunovSolvesolves . LyapunovSolvesolves . LyapunovSolvesolves .
• LyapunovSolve solves the continuous-time Lyapunov and Sylvester equations.
Solve the Lyapunov equation :
Solve the Lyapunov equation :
 Out[1]=
 Scope   (7)
Solve a Lyapunov equation:
Verify the solution:
Solve :
Solve for coefficient matrices with different dimensions:
Solve :
Solve :
Solve the Lyapunov equation with symbolic coefficients:
Obtain the symbolic solution of :
 Applications   (7)
Test the stability of by checking if the solution of is positive definite for a negative definite c:
As expected, the eigenvalues are in the left half plane:
An unstable system:
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:
Verify the solution:
For MIMO systems, the feedback gains are not unique:
Construct an observer for a state-space model:
First, choose an appropriate and such that the Lyapunov equation yields a non-singular solution:
Then construct the observer as , , where is the observer state vector, is the output, is the input, and is the estimated state vector:
Plot the actual and estimated states for a unit step input:
The equation , with a negative definite c, yields a unique positive definite solution if and only if the eigenvalues of a are in the closed left-half plane:
A stable system:
The definite integral is the solution to if a is asymptotically stable:
Compute the infinite-horizon quadratic cost for the asymptotically stable system :
Compute using direct integration:
Solve the matrix equation :
LinearSolve gives the same solution:
Solve the Lyapunov equation using LinearSolve:
LyapunovSolve gives the same solution:
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