Title: About observers of hybrid systems
Abstract: I recently reviewed the very nice paper [1] by Pauline Bernard and Ricardo Sanfelice about an observer design for hybrid systems, whose ideas I would like to share with you. I will briefly recall the hybrid systems framework proposed by Goebel, Sanfelice and Teel [2]. Afterwards I will highlight the difficulties in designing an observer for these kind of systems, in particular, when the jump times cannot be observed directly. Then I explain the actual observer design proposed in [1].
[1] Pauline Bernard and Ricardo Sanfelice, “Semi-global high-gain hybrid observer for a class of hybrid dynamical systems with unknown jump times,”
Submitted to Transaction on Automatic Control, full version available at https://hal. archives-ouvertes.fr/hal-03736135, 2022.
[2] R. Goebel, R. Sanfelice, and A. Teel, “Hybrid Dynamical Systems: Modeling, Stability and Robustness,” Princeton University Press, 2012
Title: System analysis with scaled relative graphs
Abstract: Scaled relative graphs (SRGs) were recently introduced by Ryu, Hannah and Yin to analyze the convergence of optimisation algorithms using two dimensional Euclidean geometry. In this seminar, I’ll show how the SRG can be used to study incremental input/output properties of feedback systems. The SRG of an LTI transfer function is closely related to its Nyquist diagram. The SRG may be plotted or approximated for arbitrary nonlinear operators, and allows classical Nyquist techniques to be applied to nonlinear systems. I will give a generalisation of the Nyquist criterion for stable systems, where the Nyquist diagram and the point −1 are replaced by SRGs of two feedback-interconnected nonlinear operators. The distance between the two SRGs is a nonlinear stability margin, and is the reciprocal of the incremental gain of the feedback system. This theorem generalises a range of existing results, and has advantages over classical input/output techniques.
Title: In LMIs We Trust
Abstract: Problems encountered in Systems and Control are often considered “solved” once they have been reformulated as linear matrix inequalities (LMIs). Indeed, the conventional wisdom is that LMIs can be solved using interior point methods, for which existing software packages are readily available. Some hands-on experience, however, shows that LMI solvers can be numerically unstable. Computation time also does not scale well with the number of unknowns. At the same time, the usual suspects of LMIs (arising e.g. in Lyapunov and dissipativity theory) have a particular structure that could be exploited by specialised solvers. In this talk, we take a closer look at an LMI that arises in data-driven stabilization. As a preliminary result, we will discuss a simple iterative algorithm to solve this LMI.
Title: Scalability and robustness in interconnected systems/Multi-Agent Systems
Abstract: A lot of work on interconnected or large-scale systems focus on ensuring stability, a non-trivial task when system scale and complexity grows. However, robustness towards external disturbance has often been left open, though this is crucial to prevent cascaded failures. For fixed size systems this may be guaranteed.
In our work we aim to find conditions which ensure robustness in networks of systems, regardless of system scale, connectivity or structure. In this talk I will present our work regarding robust and scalable analysis and controller design for linear systems, with single integrator, SISO and MIMO dynamics, and our outlook on expanding this to non-linear systems.
Title: Background study on the biology and mathematics behind the olfactory sense
Abstract: The presentation will provide the basics of the physiology of the olfactory sense and would locate the main areas of interest for the research. This would come together with the existing mathematical model and the discrepancies that exist between this and the biological features. A brief mention of electronic noses will also be given.
Title: On data informativity: distributed control with performance guarantees and noisy data with cross-covariance bounds
Abstract: Informativity is an important concept in data-driven control and identification of interconnected systems. The concept of informativity allows the use of data that is not necessarily persistently exciting of a sufficient order for identification, but is informative enough for, e.g., stabilization or performance-oriented controller synthesis. Conditions for informativity for linear systems with exact data and disturbed data with quadratic noise bounds have been studied in the literature. In this presentation we study informativity of data from interconnected systems for the synthesis of stabilizing distributed controllers with a guaranteed H-infinity performance. The presented approach enables scalable analysis and control of large-scale interconnected systems from noisy input-state data. Alternative to data with quadratic noise bounds, we further investigate informativity with a different prior knowledge on the noise in the form of ellipsoidal and polyhedral cross-covariance bounds. We provide informativity conditions for input-output and input-state data with cross-covariance noise bounds for stabilization and H-2/H-infinity control. Simulation experiments show that the considered noise characterization can reduce conservatism in the informativity analysis.
Title: Towards Impossibility and Possibility Results in String Stability of Vehicles Platoon
Title: Optimal control of switched differential algebraic equations and more…
Abstract: The first part of this presentation will be concerned with linear quadratic regulator (LQR) problem for switched differential algebraic equations (sDAEs). Due to the linearity of the dynamics and the quadratic cost functional it can be shown that the optimal cost is a quadratic function of the initial value and the optimal controller is linear in the state. Taking dynamic programming approach it will be shown that the LQR problem for sDAEs can be reduced to a repeated finite horizon LQR problem for non-switched DAEs. However, as this LQR problem is considered in the context of sDAEs additional constraints on the terminal state and the terminal cost matrix have to be imposed. As a consequence of these constraints, there generally does not exist an optimal solution for all initial values, but only for initial values contained in a certain subspace. As a main result, it is shown how to characterize this subspace and how to compute an optimal control if it exists. This part of the presentation will be concluded with some simulations to illustrate the results.
In the second part of the presentation, the platooning problem will briefly be addressed. It will be shown that if a leader-follower structure is adopted, the problem of finding a decentralized controller can be posed as a disturbance decoupling problem combined with output stabilization. Within this geometrical context, string stability follows from the spacing policy. Necessary and sufficient conditions on the existence of a decentralized controller are given and the results are illustrated through simulation results.
Title: A behavioral approach to data-driven control with noisy input-output data
Abstract: Data-driven analysis and control is a research topic that has received a lot of attention in the past few years. The idea that lies at the core of this research area is to use data obtained from an unknown dynamical system to verify certain system properties and to design feedback control laws for that system. The main challenge is to do the analysis and design without the usual first step of establishing a mathematical model of the system (for example by using first principles modeling or system identification), but work directly with the data instead. This has been the subject of many recent publications in the area, for the most part in the context of input-output systems in state space form.
In this talk, we will abandon the paradigm of systems in state space form, and will, instead, use as model class the set of all input-output systems described by higher order difference equations, also called auto-regressive (AR) systems. The unknown dynamical system that we want to analyse or control is assumed to be a member of this model class of AR systems. We will assume that noisy input-output data on a given finite time-interval have been obtained from this unknown AR system. These data are employed to check stability or to verify whether a dynamic feedback controller exists that stabilizes the unknown system and, if so, to compute a stabilizing controller.
We will discuss data-based tests to tackle these analysis and design problems. To do this, we will heavily rely on methods from the behavioral approach to systems and control. In particular, we will adopt the notion of quadratic difference form (QDF) as a framework for Lyapunov functions for autonomous systems described by higher order difference equations. Our main results will be necessary and sufficient condition for data informativity in terms of feasibility of certain linear matrix inequalities (LMIs) obtained from the data. Important tools in deriving these conditions are recent results on quadratic matrix inequalties (QMIs) and a matrix version of Yakubovich’s S-procedure.
Title: Control problems and stability of hierarchical moment models for fluid dynamics
Abstract: Applications of atmospheric re-entry and geophysical flows are characterized by a large variety of separate models for specific tasks. This model variety poses significant difficulties both for the analysis and for the numerical solution. We thus need to rethink mathematical modelling for future numerical simulations. In this talk, I will introduce hierarchical moment models as a flexible way to derive hierarchies of models in fluid dynamics and other applications. The general derivation procedure results in structural similarities of the models, which facilitate physical insight, model adaptivity, and the development of suitable numerical methods. Based on rarefied gases and shallow flows, I will exemplify the hierarchical moment approach and highlight runtime and accuracy improvements. From a control theory perspective, I will discuss problems for the future extension towards adaptive self-learning models that make use of the hierarchical structure of the models. Lastly, I will discuss the derivation and stability analysis of a hierarchical model for a linear delay differential equation with uncertain parameters.
