Spectral Stability Criterion of Nonlinear Control Systems
A spectral stability criterion is formulated for nonlinear control systems, continuous and discrete, whose matrices have a simple structure. The spectrum providing global and exponential stability of a continuous nonlinear system is called acceptable. The spectral stability criterion is formulated as a sufficient condition for the admissibility of the matrix spectrum. For continuous nonlinear systems, the elements of the matrix spectrum are represented by the sum of two components, the first (main) is an arbitrarily selected negative scalar common to the entire spectrum, the second (the so–called increment) is constructed as functions that differ for all elements of the spectrum. The conditions for the admissibility of the matrix spectrum are reduced to a restriction on the maximum absolute value of the increment. A formula has been developed that determines the exact upper bound of this value, which ensures the acceptability of the spectrum. The spectral stability criterion of discrete nonlinear systems is based on the developed criterion for continuous systems. A discrete system with a Lyapunov function in the form of a quadratic form with a constant matrix is compared by a given formula to a continuous system with a Lyapunov function of the same structure. In this case, the formulation of the spectral stability criterion for a nonlinear discrete system is reduced to the spectral stability criterion for a constructed correlated continuous system. The solution of the stabilization problem for a wide class of nonlinear control systems based on the formulated spectral stability criteria is obtained. The disadvantages of the proposed solution are noted.
