Abstract-We propose and implement an algorithm based on reachability analysis to estimate the region of attraction (ROA) of an equilibrium point for nonlinear systems. The stability region is obtained via the computation of forward reachable sets. We compare our results with well-established techniques in this area. In particular, we consider the optimization of the Lyapunov function (LF) sub-level set using sum-of-squares (SOS) decomposition, and the computation of backward reachable sets of a target set using the viscosity solution of a time-dependant Hamilton-Jacobi-Isaacs (HJI) formulation. Our method can overcome many limitations imposed on the applicability of Lyapunov-based approaches, such as conservatism in estimating the stability region, and difficulties associated with choosing a suitable LF. This is due to the fact that our reachability algorithm does not require a LF in order to provide an estimate of the ROA. Various numerical examples show that our proposed approach can estimate the exact ROA quite accurately, and more importantly, scales moderately with the system dimension compared to alternative techniques.

In this paper, the computation of closed-form convex combinations is considered. In many control tasks, convex combinations play a crucial role, thus requiring an efficient computation. This is the case for online control of fast dynamical systems, in which the control algorithms rely on convex combinations, for example robust control of linear parametervarying systems. On the other hand, for formal verification, it is necessary that the closed-loop behavior can be expressed in closed-form, which is not possible if the convex combination is expressed as constraints. In this paper, we provide closedform expressions for any kind of polytope with finitely many extreme points. For special types of polytopes, such as simplices and parallelotopes, we provide especially efficient, analytical closed-form expressions. Numerical experiments show that the closed-form expressions are significantly faster in all randomlygenerated cases and thus enable shorter sampling intervals of control schemes involving convex combinations.

Abstract-While a number of efficient methods have been proposed for approximating backward reachable sets, no synthesis method via backward reachable sets has been developed for estimating and enlarging the region of attraction (RA). This paper shows how to use backward reachable sets to enlarge the estimate of the RA of linear discrete-time systems, by using an optimal static feedback controller. Two controller design methods are provided: the first method enlarges the estimate of the RA via invariant sets, whose existence is ensured by zonotope containment; the second method provides the optimal control input by using Lyapunov stability and quadratic stabilization. The backward reachable set is represented by zonotopes which give a good compromise between accuracy and efficiency. The effectiveness of both methods is illustrated by a numerical example.

Abstract-The main challenge associated with the analysis of power systems via the computation of reachable sets is improving the algorithmic efficiency to scale towards industrially relevant problem sizes. In this paper, we present a compositional algorithm that can drastically reduce the computational effort required to assess the dynamical response of power systems during transients using reachability analysis. The main reason for the algorithmic efficiency is that we reformulate the transmission network into a set of subsystems, each consisting of a synchronous generator connected to a generator bus, whose algebraic constraints are unknown-but-bounded within some confidence intervals. This makes it possible to parallelize the computation of reachable sets for transient stability analysis and, more importantly, preserve the interaction and correlation between different machines connected to the grid. The applicability of the proposed compositional algorithm is illustrated on several benchmark examples and compared to other algorithms that compute the reachable set without employing any compositional techniques.

Abstract-The load-following capabilities of power plants became increasingly important in recent years as a means of ensuring a reliable operation of future power systems. In this work, we propose a generic approach, based on reachability analysis, to rigorously verify the safety of critical components that often pose limitations on the flexibility of conventional power plants to perform fast load changes. The proposed reachability algorithm makes it possible to compute the bounds of all possible trajectories for a range of operating conditions while simultaneously meeting the practical requirements of a real power plant. As an example, we consider the verification of the water level inside a drum unit. In contrast to previous work, our results are based on measurement data of a realistic configuration of a boiler system located within a 450 MW combined cycle plant in Germany. We use an abstract model which considers the modelling errors to ensure that all dynamic behaviors of the process are replicated by the abstraction. Through the implementation of our abstract model, we formally guarantee that the water level inside the drum always remains within safe limits for load changes equivalent to 40 MW which, as a result, exploits the power plant's adaptability and load-following capabilities.