# Lyapunov Equations (G Dataflow)

Version:
Last Modified: January 9, 2017

Solves the Lyapunov matrix equation.  ## matrix type

The matrix type of matrix A.

Specifying the matrix type allows this node to execute more quickly by avoiding unnecessary computations, which could introduce numerical inaccuracy.

Name Value Description
General 0 A matrix that you cannot describe with one of the other categories.
Upper Triangular 3 An upper triangular.

Default: Upper Triangular ## matrix A

Matrix A in the Lyapunov equation.

This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.

This input must be a square matrix or upper quasi-triangular matrix in canonical Schur form. ## matrix B

Matrix B in the Lyapunov equation.

This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers. ## equation type

Type of Lyapunov equation.

Name Value Description
Continuous 0 Solves the continuous Lyapunov equation.
Discrete 1 Solves the discrete Lyapunov equation.

Default: Continuous ## error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error ## matrix X

Solution to the Lyapunov equation. ## scale

Scaling factor of the Lyapunov equation. ## error out

Error information. The node produces this output according to standard error behavior.

## Solving Lyapunov Matrix Equations

The following equation defines the continuous Lyapunov equation:

AX + XAH = αB

where AH is the conjugate transpose of A and α is a scaling factor used to avoid overflow in X.

The continuous Lyapunov equation has a unique solution if and only if λi + λ*j ≠ 0 for all eigenvalues of A, where λ* is the complex conjugate of λ.

The following equation defines the discrete Lyapunov equation:

AXAH - X = αB

where AH is the conjugate transpose of A and alpha is a scaling factor used to avoid overflow in X.

The discrete Lyapunov equation has a unique solution if and only if λiλ*j ≠ 1 for all eigenvalues of A, where λ* is the complex conjugate of λ.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported