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Storage functionals and Lyapunov functions for passive dynamical systems

Borukhov V., Shnip A., Zelenyak D.

Systems Science

2001

Vol. 27, nr 2

27-35

For nonlinear time-invariant input-output dynamical systems the passivity conditions are obtained under some restrictions. The conditions imply storage functions satisfying dissipation inequality. A class of storage functions allowing unique reconstruction of a passive dynamical system is defined. These results are illustrated by an example of a linear system with fading memory. An important, for practical application, class of the linear relaxation systems without direct input-output interaction is considered. A new necessary condition for dynamical systems to be of the relaxation type is obtained for this class. The condition is connected with the existence of a unique quadratic Lyapunov function satisfying the complete monotonicity condition. This unique Lyapunov function corresponds to a "standard" thermodynamic potential in a compact family of potentials in the non-equilibrium thermodynamics. The results obtained can be useful in automatic control, mechanics of viscoelastic materials, and various applications in physics and the system theory.

Subordinated systems and dissipative inequality for linear finite-dimensional passive systems.

Borukhov V., Shnip Al.

Systems Science

1999

Vol. 25, nr 4

15-24

A quadric storage functional for a passive input-output dynamical system is defined through the outputs of another dynamical system. The latter is called the subordinated system to the former one. The necessary and sufficient conditions for the system subordination are obtained. A necessary condition for the system subordination relation the related systems dimensions is proven. A practically important case of the systems without direct input-output action is considered and more detailed results on subordinated systems are given for this case.