Function Reference

lyap

Solve continuous-time Lyapunov equations

Syntax

• X = lyap(A,Q)
  X = lyap(A,B,C)
  

Description

lyap solves the special and general forms of the Lyapunov matrix 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

where and are square matrices of identical sizes. The solution X is a symmetric matrix if is.

X = lyap(A,B,C) solves the generalized Lyapunov equation (also called Sylvester equation).

The matrices must have compatible dimensions but need not be square.

Algorithm

lyap transforms the and matrices to complex Schur form, computes the solution of the resulting triangular system, and transforms this solution back [1].

Limitations

The continuous Lyapunov equation has a (unique) solution if the eigenvalues of and of satisfy

If this condition is violated, lyap produces the error message

• Solution does not exist or is not unique.
  

See Also
covar       Covariance of system response to white noise

dlyap       Solve discrete Lyapunov 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] Bryson, A.E. and Y.C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975. pp. 328-338.

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