Fabian R. Wirth

Andre R. Fioravanti, Jakub Marecek, Robert N. Shorten et al.

Across smart-grid and smart-city application domains, there are many problems where an ensemble of agents is to be controlled such that both the aggregate behaviour and individual-level perception of the system's performance are acceptable. In many applications, traditional PI control is used to regulate aggregate ensemble performance. Our principal contribution in this note is to demonstrate that PI control may not be always suitable for this purpose, and in some situations may lead to a loss of ergodicity for closed-loop systems. Building on this observation, a theoretical framework is proposed to both analyse and design control systems for the regulation of large scale ensembles of agents with a probabilistic intent. Examples are given to illustrate our results.

Roman Geiselhart, Fabian R. Wirth

In this paper ISS small-gain theorems for discrete-time systems are stated, which do not require input-to-state stability (ISS) of each subsystem. This approach weakens conservatism in ISS small-gain theory, and for the class of exponentially ISS systems we are able to prove that the proposed relaxed small-gain theorems are non-conservative in a sense to be made precise. The proofs of the small-gain theorems rely on the construction of a dissipative finite-step ISS Lyapunov function which is introduced in this work. Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of ISS Lyapunov functions, are shown to be sufficient and necessary to conclude ISS of the overall system.

Roman Geiselhart, Fabian R. Wirth

In the stability analysis of large-scale interconnected systems it is frequently desirable to be able to determine a decay point of the gain operator, i.e., a point whose image under the monotone operator is strictly smaller than the point itself. The set of such decay points plays a crucial role in checking, in a semi-global fashion, the local input-to-state stability of an interconnected system and in the numerical construction of a LISS Lyapunov function. We provide a homotopy algorithm that computes a decay point of a monotone op- erator. For this purpose we use a fixed point algorithm and provide a function whose fixed points correspond to decay points of the monotone operator. The advantage to an earlier algorithm is demonstrated. Furthermore an example is given which shows how to analyze a given perturbed interconnected system.

Björn S. Rüffer, Fabian R. Wirth

A monotone self-mapping of the nonnegative orthant induces a monotone discrete-time dynamical system which evolves on the same orthant. If with respect to this system the origin is attractive then there must exists points whose image under the monotone map is strictly smaller than the original point, in the component-wise partial ordering. Here it is shown how such points can be found numerically, leading to a recipe to compute order intervals that are contained in the region of attraction and where the monotone map acts essentially as a contraction. An important application is the numerical verification of so-called generalized small-gain conditions that appear in the stability theory of large-scale systems.

Jakub Marecek, Michal Roubalik, Ramen Ghosh et al.

In power systems, one wishes to regulate the aggregate demand of an ensemble of distributed energy resources (DERs), such as controllable loads and battery energy storage systems. We suggest a notion of predictability and fairness, which suggests that the long-term averages of prices or incentives offered should be independent of the initial states of the operators of the DER, the aggregator, and the power grid. We show that this notion cannot be guaranteed with many traditional controllers used by the load aggregator, including the usual proportional-integral (PI) controller. We show that even considering the non-linearity of the alternating-current model, this notion of predictability and fairness can be guaranteed for incrementally input-to-state stable (iISS) controllers, under mild assumptions.