A state-space view on locally-stable, globally-unstable nonlinear models driven by Gaussian burst inputs

In this paper, the behaviour of nonlinear dynamic systems driven by stationary random excitations is studied from a model-based perspective - i.e. starting from a perfect knowledge of the system under study and its driving random input - over a finite time interval (a burst excitation is assumed). For a given discrete-time nonlinear state-space model operating in the neighbourhood of a stable equilibrium, a “blow-up” is seen as the event of escaping out of a region of attraction. Based on Laplace integration, a method is outlined to approximate a future state´s probability density function (pdf) at low excitation amplitudes. Inspection of this pdf can reveal additional insights into the complex behaviour of an abstract state-space model, compared with the simulation approach. The probability of staying inside the region of attraction (viz. obtaining a bounded operation subject to an input active in a finite time interval) can be obtained by integration of this pdf. The state pdf estimation is illustrated with numerical Monte-Carlo simulation experiments.