Abstract
Lyapunov functions for linear multivariable time-invariant systems are developed by use of construction techniques which are based on Lyapunov's second method. Two methods of construction are studied. In the first method (direct method) the Lyapunov function is obtained by directly solving the Lyapunov matrix equation while in the second method (indirect method) it is generated by reducing the system matrix by two successive similarity transformations to the Schwarz canonical form. The conditions of stability on the resulting matrix from Lyapunov's second method are the same as the Routh-Hurwitz conditions. Both construction techniques are applied to a freely-spinning dual-spin system, with viscous rate damping assumed about both principal transverse axes. It is seen that when the rotor is spinning relative to the main part, damping about only one of the principal transverse axes is required for stability. The Lyapunov functions constructed are quadratic forms whose different coefficients, collectively, involve all of the system parameters including the damping coefficients.
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