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 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.