Abdalla Swikir

Abdalla Swikir, Antoine Girard, Majid Zamani

In this work, we introduce a compositional framework for the construction of finite abstractions (a.k.a. symbolic models) of interconnected discrete-time control systems. The compositional scheme is based on the joint dissipativity-type properties of discrete-time control subsystems and their finite abstractions. In the first part of the paper, we use a notion of so-called storage function as a relation between each subsystem and its finite abstraction to construct compositionally a notion of so-called simulation function as a relation between interconnected finite abstractions and that of control systems. The derived simulation function is used to quantify the error between the output behavior of the overall interconnected concrete system and that of its finite abstraction. In the second part of the paper, we propose a technique to construct finite abstractions together with their corresponding storage functions for a class of discrete-time control systems under some incremental passivity property. We show that if a discrete-time control system is so-called incrementally passivable, then one can construct its finite abstraction by a suitable quantization of the input and state sets together with the corresponding storage function. Finally, the proposed results are illustrated by constructing a finite abstraction of a network of linear discrete-time control systems and its corresponding simulation function in a compositional way. The compositional conditions in this example do not impose any restriction on the gains or the number of the subsystems which, in particular, elucidates the effectiveness of dissipativity-type compositional reasoning for networks of systems.

Abdalla Swikir, Majid Zamani

In this paper, we introduce a compositional scheme for the construction of finite abstractions (a.k.a. symbolic models) of interconnected discrete-time control systems. The compositional scheme is based on small-gain type reasoning. In particular, we use a notion of so-called alternating simulation functions as a relation between each subsystem and its symbolic model. Assuming some small-gain type conditions, we construct compositionally an overall alternating simulation function as a relation between an interconnection of symbolic models and that of original control subsystems. In such compositionality reasoning, the gains associated with the alternating simulation functions of the subsystems satisfy a certain "small-gain" condition. In addition, we introduce a technique to construct symbolic models together with their corresponding alternating simulation functions for discrete-time control subsystems under some stability property. Finally, we apply our results to the temperature regulation in a circular building by constructing compositionally a finite abstraction of a network containing $N$ rooms for any $N\geq3$. We use the constructed symbolic models as substitutes to synthesize controllers compositionally maintaining room temperatures in a comfort zone. We choose $N=1000$ for the sake of illustrating the results. We also apply our proposed techniques to a nonlinear example of fully connected network in which the compositionality condition still holds for any number of components. In these case studies, we show the effectiveness of the proposed results in comparison with the existing compositionality technique in the literature using a dissipativity-type reasoning.

Abdalla Swikir, Majid Zamani

In this paper, we provide a compositional methodology for constructing symbolic models for networks of discrete-time switched systems. We first define a notion of so-called augmented-storage functions to relate switched subsystems and their symbolic models. Then we show that if some dissipativity type conditions are satisfied, one can establish a notion of so-called alternating simulation function as a relation between a network of symbolic models and that of switched subsystems. The alternating simulation function provides an upper bound for the mismatch between the output behavior of the interconnection of switched subsystems and that of their symbolic models. Moreover, we provide an approach to construct symbolic models for discrete-time switched subsystems under some assumptions ensuring incremental passivity of each mode of switched subsystems. Finally, we illustrate the effectiveness of our results through two examples.

David Boetius, Abdelrahman Abdelnaby, Ashok Kumar et al.

Learning stable dynamical systems from data is crucial for safe and reliable robot motion planning and control. However, extending stability guarantees to trajectories defined on Riemannian manifolds poses significant challenges due to the manifold's geometric constraints. To address this, we propose a general framework for learning stable dynamical systems on Riemannian manifolds using neural ordinary differential equations. Our method guarantees stability by projecting the neural vector field evolving on the manifold so that it strictly satisfies the Lyapunov stability criterion, ensuring stability at every system state. By leveraging a flexible neural parameterisation for both the base vector field and the Lyapunov function, our framework can accurately represent complex trajectories while respecting manifold constraints by evolving solutions directly on the manifold. We provide an efficient training strategy for applying our framework and demonstrate its utility by solving Riemannian LASA datasets on the unit quaternion (S^3) and symmetric positive-definite matrix manifolds, as well as robotic motions evolving on \mathbb{R}^3 \times S^3. We demonstrate the performance, scalability, and practical applicability of our approach through extensive simulations and by learning robot motions in a real-world experiment.