This chapter covers the analysis of linear and nonlinear continuous-time dynamical systems described by a continuous-time state-space model whose input belongs to an ellipsoid. Similar to the linear discrete-time case, the set containing all possible values that the state can take is not an ellipsoid in general, but it can be upper bounded by a family of ellipsoids whose evolution is governed by a differential equation that can be derived from the system state-space model. As in the discrete-time case, it is possible to choose ellipsoids within this family that are optimal in some sense. The nonlinear case is again handled using linearization. The techniques developed in the chapter are used to analyze the performance of a buck DC-DC power converter. In addition, we show how the techniques can be used to assess the effect of variability associated with renewable-based electricity generation on bulk power system dynamics, with a focus on time-scales involving electromechanical phenomena.

This chapter studies continuous-time dynamical systems described by a continuous-time state-space model whose input is subject to probabilistic uncertainty. The first part of the chapter is devoted to the analysis of linear systems and provides techniques for computing the first and second moments of the state vector when the evolution of the input vector is governed by a "white noise" process with known mean and covariance functions. Then, by additionally imposing this white noise process to be Gaussian, we provide a partial differential equation whose solution yields the pdf of the state vector. The second part of the chapter extends these techniques to the analysis of nonlinear systems, with a special focus on the case when the white noise governing the evolution of the input vector is Gaussian. The third part of the chapter illustrates the application of the techniques developed to the analysis of inertia-less AC microgrids when the measurements utilized by the frequency control system are corrupted by additive disturbances.