This chapter covers the analysis of static systems under probabilistic input uncertainty. The first part of the chapter is devoted to analyzing linear and nonlinear static systems when the first and second moments of the input vector are known, and it provides techniques for characterizing the first and second moments of the state vector. For the linear case, the techniques provide the exact moment characterization, whereas for the nonlinear case, the characterization, which is based on a linearization of the system model, is approximate. The second part of the chapter provides techniques for the analysis of both linear and nonlinear static systems when the pdf of the input vector is known. The techniques included provide exact characterizations of the state pdf for both linear and nonlinear systems. In both cases, the inversion of the input-to-state mapping is required, which in the linear case involves the computation of the inverse of a matrix; however, for the nonlinear, it involves obtaining an analytical expression for the input-to-state mapping. The chapter concludes by utilizing the techniques developed to study the power flow problem under active power injection uncertainty.

This chapter covers the analysis of static systems under set-theoretic input uncertainty. In the first part of the chapter, we assume that the input belongs to an ellipsoid and analyze both linear and nonlinear systems. For the linear case, we provide techniques to exactly characterize the set containing all possible values that the state can take. For the nonlinear case, we again resort to linearization to approximately characterize the set containing all possible values that the state can take. The second part of the chapter considers linear and nonlinear systems when the input is known to belong to a zonotope. For the linear case, we are able to compute the exact set containing all possible values the state can take, whereas for the nonlinear case, we settle for an approximation thereof obtained via linearization. The techniques developed are utilized to analyze the power flow problem under uncertain active power injections.

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 studies static systems under structural uncertainty. The first part of the chapter is devoted to the development of a model describing the system stochastic behavior. To this end, we assume that the system can only adopt a finite number of input-to-state mappings, and that transitions among these different mappings are random and governed by a Markov chain. We consider both discrete- and continuous-time settings and provide expressions governing the evolution of the probability distribution associated with the resulting Markov chains. The second part of the chapter tailors the techniques developed earlier to analyze multi-component systems subject to component failures and repairs. Techniques for constructing the system input-to-state model are extensively covered, as this is in general the most difficult part of the analysis when analyzing systems with a large number of components.