Continuous Lyapunov equation solution

2023-05-30 00:07| 来源: 网络整理| 查看: 265

lyap

Syntax

lyap X = lyap(A,Q) X = lyap(A,B,C) X = lyap(A,Q,[],E)

Description

lyap solves the special and general forms of the Lyapunov equation. Lyapunov equations arise in several areas of control, including stability theory and the study of the RMS behavior of systems.

X = lyap(A,Q) solves the Lyapunov equation

AX+XAT+Q=0

where A and Q represent square matrices of identical sizes. If Q is a symmetric matrix, the solution X is also a symmetric matrix.

X = lyap(A,B,C) solves the Sylvester equation

AX+XB+C=0

The matrices A, B, and C must have compatible dimensions but need not be square.

X = lyap(A,Q,[],E) solves the generalized Lyapunov equation

AXET+EXAT+Q=0

where Q is a symmetric matrix. You must use empty square brackets [] for this function. If you place any values inside the brackets, the function errors out.

Limitations

The continuous Lyapunov equation has a unique solution if the eigenvalues α1,α2,...,αn of A and β1,β2,...,βn of B satisfy

αi+βj≠0 for all pairs (i,j)

If this condition is violated, lyap produces the error message:

Solution does not exist or is not unique. ExamplesExample 1

Solve Lyapunov Equation

Solve the Lyapunov equation

AX+XAT+Q=0

where

A=[12−3−4] Q=[3111]

The A matrix is stable, and the Q matrix is positive definite.

A = [1 2; -3 -4]; Q = [3 1; 1 1]; X = lyap(A,Q)These commands return the following X matrix:X = 6.1667 -3.8333 -3.8333 3.0000You can compute the eigenvalues to see that X is positive definite.eig(X)

The command returns the following result:

ans = 0.4359 8.7308Example 2

Solve Sylvester Equation

Solve the Sylvester equation

AX+XB+C=0

where

A=5 B=[4343] C=[21]

A = 5; B = [4 3; 4 3]; C = [2 1]; X = lyap(A,B,C)

These commands return the following X matrix:

X = -0.2000 -0.0500Algorithms

lyap uses SLICOT routines SB03MD and SG03AD for Lyapunov equations and SB04MD (SLICOT) and ZTRSYL (LAPACK) for Sylvester equations.

References

[1] Bartels, R.H. and G.W. Stewart, "Solution of the Matrix Equation AX + XB = C," Comm. of the ACM, Vol. 15, No. 9, 1972.

[2] Barraud, A.Y., “A numerical algorithm to solve A XA - X = Q,” IEEE® Trans. Auto. Contr., AC-22, pp. 883–885, 1977.

[3] Hammarling, S.J., “Numerical solution of the stable, non-negative definite Lyapunov equation,” IMA J. Num. Anal., Vol. 2, pp. 303–325, 1982.

[4] Penzl, T., ”Numerical solution of generalized Lyapunov equations,” Advances in Comp. Math., Vol. 8, pp. 33–48, 1998.

[5] Golub, G.H., Nash, S. and Van Loan, C.F., “A Hessenberg-Schur method for the problem AX + XB = C,” IEEE Trans. Auto. Contr., AC-24, pp. 909–913, 1979.

Version History

Introduced before R2006a