Set Theory and Algorithms for Dynamical Systems

Abstract

In this project, we propose to use set theory to develop methods and algorithms for modeling and analyzing complex physical systems, particularly to address uncertainty.

Coordinators

Mirko Fiacchini (GIPSA-lab)

Thao Dang (VERIMAG)

Results

With regard to the stabilization of switched systems, we have found new conditions for discrete-time linear switched systems. We used SMT (Satisfiability Modulo Theories) and the Z3 tool to address the problem of testing for inclusion in the union of ellipsoids. With regard to discrete-time nonlinear systems, we studied a systematic method for computing approximations of the attractor set and convex Lyapunov functions for certain classes of discrete-time nonlinear systems. This method was applied to dynamic models of cancer under chemotherapy.

Another outcome of the project is the development of efficient algorithms for set operations that appear in verification and synthesis procedures. A new algorithm was designed for the accessibility operators of discrete-time polynomial systems required by a safe parameter synthesis technique proposed in our previous work. This algorithm has been successfully applied to population models in macrobiology. Based on the temporal logic of signals, we designed an algorithm to synthesize the parameters of a polynomial system so that it satisfies a given STL property. We developed a new accessibility analysis method to verify the robustness of given controllers, designed from nominal models, against the nonlinearities and uncertainties of real systems. We also proposed a set-membership-based approach to design reliable static stabilization controllers for nonlinear dynamic systems. The project led to the ANR COMPACS project on the implementation of computation-aware control systems.

Notable publications

Nacim Meslem, Nacim Ramdani. "A Reliable Stabilizing Controller Based on Set-Value Parameter Synthesis." *IMA Journal of Mathematical Control and Information*, Oxford University Press (OUP), 2015, pp. 1–20.

Nassim Loukkas, Nacim Meslem, John Jairo Martinez Molina. Set-Membership Tests to Evaluate the Performance of Nominal Feedback Control Laws. IEEE Conference on Control Applications (CCA 2016).

Rachid Riah, Mirko Fiacchini, Mazen Alamir. Estimation of the domain of attraction for a cancer chemotherapy model affected by state-proportional uncertainty. European Control Conference (ECC 2016).

Mirko Fiacchini, Antoine Girard, and Marc Jungers. “On the Stabilizability of Discrete-Time Switched Linear Systems: Novel Conditions and Comparisons.” IEEE Transactions on Automatic Control 61.5 (2016): 1181–1193.

Tommaso Dreossi, Thao Dang, Carla Piazza. Reachability Computation for Polynomial Dynamical Systems. Formal Methods in System Design, 2017.