This chapter analyzes linear and nonlinear discrete-time systems described by a discrete-time state-space model whose inputs are uncertain but known to belong to an ellipsoid. For the linear case, even if the input set is an ellipsoid, the set containing all possible values that the state can take is not an ellipsoid in general, but it can be upper bounded by an ellipsoid. We develop techniques for recursively computing a family of such upper-bounding ellipsoids. Within this family, we then show how to choose ellipsoids that are optimal in some sense, e.g., they have minimum volume. For the nonlinear case, we will again resort to linearization techniques to approximately characterize the set containing all possible values that the state can take. The application of the techniques presented is illustrated using the same inertia-less AC microgrid model used in Chapter 5.

This chapter starts by reviewing important concepts from probability theory and stochastic processes. Subsequent chapters on probabilistic input and structural uncertainty make heavy use of random vectors and vector-valued stochastic process, so the reader should be familiar with the material included on these concepts. Next, the chapter provides a review of set-theoretic notions. The material on sets in Euclidean space included in this part is key to understanding the set-theoretic approach to input uncertainty modeling. The chapter concludes with a review of several fundamental concepts from the theory of discrete- and continuous-time linear dynamical systems.

We prove the converse of Stone’s theorem.

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.

We answer some natural mathematical questions concerning representations. We develop the theory of induced representations for finite groups, which sheds considerable light on the structure of the induced representation of the Poincaré group studied in Chapter 8.

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 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.

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 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.

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.