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.

In this chapter, we first provide some motivation for the type of modeling problems we address in this book. Then we provide an overview of the type of mathematical models used to describe the behavior of the classes of systems of interest. We also describe the types of uncertainty models adopted and how they fit into the mathematical models describing system behavior. In addition, we provide a preview of the applications discussed throughout the book, mostly centered around electric power systems. We conclude the chapter by providing a brief summary of the content of subsequent chapters.

This chapter provides techniques for analyzing discrete-time dynamical systems under probabilistic input uncertainty. Here, the relation between the input and the state is described by a discrete-time state-space model. The input vector is modeled as a vector-valued stochastic process with known first and second moments (or known pdf). The first part of the chapter is devoted to the analysis of linear systems and provides techniques for characterizing the first and second moments and the pdf of the state vector. The second part deals with the analysis of nonlinear systems, where we use the techniques developed in Chapter 4 to exactly characterize the distribution of the state vector when the pdf of the input vector is given. In addition, we rely on linearization techniques to obtain expressions that approximately characterize the first and second moments and the pdf of the state vector. The third part of the chapter illustrates the application of the techniques developed to the analysis of inertia-less AC microgrids under random active power injections.