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3.1 Lyapunov Equations
Control System Professional (Section 12.2) contains the functions LyapunovSolve and DiscreteLyapunovSolve. The function LyapunovSolve solves the continuous-time Lyapunov and Sylvester equations:
The function DiscreteLyapunovSolve solves the discrete-time Lyapunov equation:
The scope of the function DiscreteLyapunovSolve is now extended to solve the discrete-time Sylvester equation:
Extension to the function DiscreteLyapunovSolve.
The Lyapunov and Sylvester equations occur in many control applications. These include feedback stabilization (Section 5.3); model reduction (Chapter 7); controllability and observability tests (Section 7.1 of Control System Professional); stability and robust stability analysis; computation of  and  norms; and numerical solutions of the algebraic Riccati equations using the Newton method (Section 3.2.4).
New numerical methods are implemented in Advanced Numerical Methods to solve Eqs. (3.1) to (3.4). These methods are: the Schur method by Bartels and Stewart (1972) to solve Eqs. (3.1) and (3.3), based on the decomposition of  to the real Schur form; and the Hessenberg-Schur method by Golub, Nash, and Van Loan (1979) to solve Eqs. (3.2) and (3.4), based on the decomposition of the smaller of two matrices  and  to the real Schur form and the other matrix to the Hessenberg form (see Datta (2003) for details). These methods are invoked on numerical matrices for the default setting SolveMethod -> Automatic.
New option values for LyapunovSolve and DiscreteLyapunovSolve.
Make sure the application is loaded.
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Load the collection of test examples.
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Here is the state-space model of an L-1011 aircraft.
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This solves the Lyapunov equation  .
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Computing the relative residual norm verifies the solution accuracy.
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This is the state-space system of a simple process control of a paper machine.
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This solves the discrete Lyapunov equation  .
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Computing the relative residual norm verifies the solution accuracy.
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