On Stability Radii of Slowly Time-Varying Systems (original) (raw)
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Exponential Stability of Slowly Time-Varying Nonlinear Systems
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Let _ x x ¼ f ðx; t; t=aÞ be a time-varying vector field depending on t containing a regular and a slow time scale (a large). Assume there exist a kðtÞ b 1 and a gðtÞ such that kx t ðt; t 0 ; x 0 Þk a kðtÞe ÀgðtÞðtÀt0Þ kx 0 k, with x t ðt; t 0 ; x 0 Þ the solution of the parametrized system _
Stability of Time Varying Systems
Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems
The stability behavior of time varying systems can be studied using the concept of Lyapunov exponents and their corresponding Lyapunov subspaces. For linear time varying systems the entire Lyapunov spectrum can be approximated by the Floquet exponents of periodic systems. This leads to a variety of stability results, including the characterization of stability radii. Furthermore, a structural stability type theorem shows that stability features of time varying hyperbolic systems persist under small perturbations. For nonlinear time varying systems a stable manifold theorem allows us to interpret the linear results for the nonlinear system locally around an equilibrium point.
A Note on Exponential Stability of Partially Slowly Time-Varying Nonlinear Systems
Consider a system _ x = f(x; t; t ) with a timevarying vector eld which contains a regular and a slow time scale ( large). Assume there exists ( ) such that kx (t; t 0 ; x 0 )k K( )kx 0 ke ( )(t?t0) where x (t; t 0 ; x 0 ) is the solution of the system _ x = f(x; t; ) with initial state x 0 at t 0 . We show that for su ciently large, _ x = f(x; t; t ) is exponentially stable when the average of ( ) is negative. This result can be used to extend the circle criterion i.e. to obtain a su cient condition for exponential stability of a feedback interconnection of a slowly time-varying linear system and a sector nonlinearity. An example is included which shows that the technique can be used to obtain an exponential stability result for a pendulum with a nonlinear partially slowly time-varying friction attaining positive and negative values.