Keywords:Machine Learning and Control Abstract: There is a close analogy between deep learning and optimal control. This analogy can be exploited to develop deep learning-based algorithms for optimal control, and optimal control-based algorithms for deep learning. I will discuss the progress made along these directions.

Keywords:Multidimensional Systems Abstract: The aim of this contribution is to characterize the set of feasible initial conditions on a diagonal line in order to compute the solutions of a 2D discrete state-space system (defined over Z^2) on a half-plane of the 2D grid. This characterization is given in terms of the system matrices for the state updating.

Keywords:Multidimensional Systems, Linear Systems, Algebraic Systems Theory Abstract: In this paper, we consider two classes of spatially interconnected systems (including ladder circuits) modelled as mixed discrete-continuous linear 2D systems. Within the algebraic analysis approach to linear systems theory, we prove that these systems can always be transformed into equivalent implicit Roesser models. Moreover we show that ladder circuits can also be transformed into equivalent implicit Fornasini-Marchesini models. An advantage of our results compared to previous ones obtained through the zero coprime system equivalence approach is that the dimension of the state vector of the equivalent Roesser (or Fornasini-Marchesini) models is significantly smaller.

Keywords:Multidimensional Systems, Algebraic Systems Theory, Computations in Systems Theory Abstract: For nD systems, initial data is obtained by specifying trajectories on special subsets of the domain, known as characteristic sets. In this paper, we consider a special class of systems that admits a union of a coordinate sublattice and finitely many parallel translates of it as a characteristic set; we call such systems as strongly relevant systems. Using the discrete Noether's normalization lemma, every discrete autonomous nD system can be transformed to a strongly relevant system. For a strongly relevant system, the set of allowable initial conditions, obtained by restricting trajectories on the characteristic set, is characterized. We then provide an implementable algorithm, based on Groebner basis, for obtaining a representation of the set of allowable initial conditions. Once such a representation is obtained, important deductions, such as, arbitrary assignability of initial data can be easily made.

Keywords:Algebraic Systems Theory, Mathematical Theory of Networks and Circuits, Optimal Control Abstract: A new spin is given on the classical optimal control problem with piecewise differentiable dynamics and performance index with respect to the state variables. While in each domain of differentiability, the necessary conditions for optimality are easily established, their interpretation at the boundaries between domains is not well-understood. In this paper we show that in order to make sense of the Euler-Lagrange equation at this interface one needs to transcend the classical theory of Schwartz distributions and make suitable extensions to allow for the questionable behavior of impulses multiplied by discontinuities, and the notion of partial derivatives at a discontinuity. Such a theory has been developed, in the Colombeau, Oberguggenberger and Rosinger theory of Generalized Functions in 1990, going back to ideas from Nonstandard Analysis (NSA). We develop an alternative NSA based approach applicable to impulsive dynamics and optimal control.

Keywords:Dissipativity, Stability, Large Scale Systems Abstract: In this paper we consider stability of large scale interconnected nonlinear systems that satisfy a strict dissipativity property in terms of local storage and supply functions. Existing compositional stability criteria certify global stability by constructing a global Lyapunov function as the (weighted) sum of local storage functions. We generalize these results by unifying spatial composition, i.e., (weighted) sum of local supply functions is neutral, with temporal composition, i.e., (weighted) sum of supply functions over a time cycle is neutral. Two benchmark examples illustrate the benefits of the developed compositional stability criteria in terms of reducing conservatism and constrained distributed stabilization.

Department of Mathematics, University of Stuttgart

Keywords:Dissipativity, Stability, Computations in Systems Theory Abstract: We develop a novel convex parametrization of integral quadratic constraints with a terminal cost for subdifferentials of convex functions, involving general O'Shea-Zames-Falb multipliers. We show the benefit of our results for the reduction of conservatism of existing techniques, and sketch applications to the analysis of optimization algorithms or the stability analysis of neural network controllers. The development is prepared by providing a novel link between the convex integrability of a multivariable mapping and dissipativity theory.

Keywords:Nonlinear Systems and Control, Large Scale Systems, Infinite Dimensional Systems Theory Abstract: We prove that a network of input-to-state stable infinite-dimensional systems is input-to-state stable, provided that the gain operator of the network satisfies the monotone limit property. This property is equivalent to the strong small-gain condition in the case of finite networks. We prove our small-gain theorem for a very general class of networks, including networks of nonlinear partial and delay differential equations. It also recovers the classical nonlinear small-gain theorems for finitely many finite-dimensional systems as a special case.

Keywords:Dissipativity, Nonlinear Systems and Control, Hybrid Systems Abstract: We consider feedback interconnections of two dynamical systems in feedback configuration. The dynamics of individual systems are modeled by a differential inclusion, and the corresponding set-valued mapping is (anti-) maximal monotone with respect to the state of the system for each fixed value of the external signal that defines the interconnection. We provide conditions on these mappings under which the dynamics of the resulting interconnected system are (anti-) maximal monotone. An interpretation of our main results is provided: firstly, by considering dynamical systems defined by the subgradient of a saddle function, and secondly, by considering an interconnection of incrementally passive systems. In the same spirit, when we associate more structure to the individual systems by considering linear complementarity systems, we allow for more flexibility in describing the interconnections and derive more specific sufficient conditions in terms of system matrices that result in the overall system being described by a maximal monotone operator.

Keywords:Machine Learning and Control, Artificial Intelligence Abstract: Dynamical systems across many disciplines, including Physics and Biology are modeled as interacting particles or agents, with interaction rules that depend on the states of pairs of agents, but in fact may truly depend only on a very small number of variables (e.g. pairwise distances, pairwise differences of phases, etc...). These relatively simple interaction rules still can determine complex emergent behaviors (clustering, flocking, swarming, etc.) in so-called self organization dynamics. We propose a learning technique that, given observations of states and corresponding derivatives along trajectories of the agents, it estimates both the variables upon which the interaction kernel depends, and the interaction kernel itself. This yields an effective dimension reduction which avoids the curse of dimensionality from the high-dimensional observation data (states and corresponding derivatives of all the agents). We demonstrate the learning capability of our method to a variety of first-order interacting systems.

Keywords:System Identification, Systems on Graphs, Discrete Event Systems Abstract: Analytical approximations of the macroscopic behavior of agent-based models (e.g. via mean-field theory) often introduce a significant error, especially in the transient phase. For an example model called continuous-time noisy voter model, we use two data-driven approaches to learn the evolution of collective variables instead. The first approach utilizes the SINDy method to approximate the macroscopic dynamics without prior knowledge, but has proven itself to be not particularly robust. The second approach employs an informed learning strategy which includes knowledge about the agent-based model. Both approaches exhibit a considerably smaller error than the conventional analytical approximation.

Keywords:Stochastic Modeling and Stochastic Systems Theory, System Identification, Neural Networks Abstract: Macroscopic, coarse descriptions of microscopic, agent-based dynamical systems are useful for tasks such as optimization, bifurcation analysis, and control. Once suitable coarse variables are defined, their dynamics can be either derived analytically or approximated in a data-driven fashion. For many agent-based systems, this coarse-graining procedure requires appropriate closure terms or stochastic elements on the macroscopic scale to summarize degrees of freedom of the agents. In this contribution, we identify effective stochastic differential equations (SDE) for coarse observables of agent-based simulations. These SDE then act as surrogate models on the macroscopic scale. We approximate the drift and diffusivity functions for these SDE through neural networks. Based on earlier work, the loss function is inspired by the structure of established stochastic numerical integrators, in particular Euler-Maruyama and Milstein schemes. We consider cases where the coarse collective observables are known in advance, and where they must be found with data-driven methods. We demonstrate the feasibility on data from an egress simulation of pedestrians in two-dimensional continuous space (with the crowd simulation software Vadere).

Keywords:Nonlinear Systems and Control Abstract: We introduce a dynamic disequilibrium agent-based model(ABM) that was used to forecast the economics of the Covid-19 pandemic. This model was designed to understand the upstream and downstream propagation of the industry-specific demand and supply shocks caused by Covid-19, which were exceptional in their severity, suddenness and heterogeneity across industries. We used this model to forecast sectoral and aggregate economic activity for the United Kingdom during the early phase of the pandemic. This work demonstrates that an out of equilibrium model calibrated against national accounting data can serve as a useful real time policy evaluation and forecasting tool. We further extend this modeling framework to a large-scale,data-driven ABM of the New York metropolitan area that simulates both, epidemic and economic outcomes across industries, occupations, and income levels. This coupled epidemic-economic model is designed to address the potential tradeoff between economy and health which has been a key issue faced by policymakers. Our results show that lockdown policies affect different social groups very heterogeneously in terms of income and infections.

Keywords:Machine Learning and Control, Optimal Control, Neural Networks Abstract: Recent work has demonstrated the potential of applying supervised learning to train neural networks which approximate optimal feedback laws. In this talk, we show that some neural networks with good test accuracy can fail to even locally stabilize the dynamics. To address this challenge, we propose some novel neural network architectures which guarantee local asymptotic stability while still closely approximating optimal feedback laws on large domains.

Keywords:Machine Learning and Control, Neural Networks, Computations in Systems Theory Abstract: Numerical methods for computing control Lyapunov functions often suffer from the so-called curse of dimensionality, i.e., an exponential growth of the numerical effort in the state dimension. It is known that deep neural networks can approximate compositional functions without suffering from this curse of dimensionality. In this talk, we extend the results for computing Lyapunov functions presented in [Grüne, L. (2021). Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 8(2), 131–152] to the case of control Lyapunov functions. To this end, we discuss the use of methods from nonlinear control theory that yield the existence of a compositional control Lyapunov function. Moreover, we develop a suitable network architecture and a training algorithm for an efficient approximation of such a control Lyapunov function.

Keywords:Optimal Control, Machine Learning and Control, Neural Networks Abstract: A deep learning approach for the approximation of the Hamilton-Jacobi-Bellman partial differential equation (HJB PDE) associated to the Nonlinear Quadratic Regulator (NLQR) problem. A state-dependent Riccati equation control law is first used to generate a gradient-augmented synthetic dataset for supervised learning. The resulting model becomes a warm start for the minimization of a loss function based on the residual of the HJB PDE. The combination of supervised learning and residual minimization avoids spurious solutions and mitigate the data inefficiency of a supervised learning-only approach. Numerical tests validate the different advantages of the proposed methodology.

Keywords:Feedback Control Systems Abstract: A learning approach for optimal feedback gains for nonlinear continuous time control systems is proposed and analysed. Numerical results demonstrate the feasibilty of the approach, which allows to obtain suboptimal feedback gains, without focusing on directly solving the underlying Hamilton Jacobi Belman equation.

Keywords:Adaptive Control, Feedback Control Systems, Linear Systems Abstract: We consider tracking control of linear minimum phase systems with known arbitrary relative degree which are subject to possible output measurement losses. We provide a control law which guarantees the evolution of the tracking error within a (shifted) prescribed performance funnel whenever the output signal is available. The result requires a maximal duration of measurement losses and a minimal time of measurement availability, which both strongly depend on the internal dynamics of the system, and are derived explicitly.

Keywords:Adaptive Control, Machine Learning and Control Abstract: This paper concerns the problem of bounded l2-gain adaptive control with noisy measurements for linear time-invariant systems with uncertain parameters belonging to a finite set. We show that it is necessary and sufficient to consider observer-based control with a multiple-observer structure consisting of one H-infinity-observer paired with a model fitness metric per candidate model.

Keywords:Adaptive Control, Nonlinear Systems and Control Abstract: We consider tracking control for uncertain nonlinear multi-input, multi-output systems modelled by functional differential equations, in the presence of input constraints. The objective is to guarantee the evolution of the tracking error within a performance funnel with prescribed asymptotic shape. We design a novel funnel controller which, in order to satisfy the input constraints, contains a dynamic component which widens the funnel boundary whenever the input saturation is active. This design is model-free, of low-complexity and extends earlier funnel control approaches.

Keywords:Algebraic Systems Theory, Operator Theoretic Methods in Systems Theory, Mathematical Theory of Networks and Circuits Abstract: The goal of the paper is two-fold. The first of which is to derive an explicit formula to compute the generating series of a closed-loop system when a plant, given in a Chen-Fliess series description is in multiplicative output feedback connection with another system given in Chen-Fliess series description. In addition, the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant. The second of the two-part goal of this paper is same as the first part albeit when the Chen-Fliess series in the feedback is replaced by a memoryless map, so called multiplicative static feedback connection.

Keywords:Infinite Dimensional Systems Theory, Stability, Operator Theoretic Methods in Systems Theory Abstract: We introduce the notions of semi-uniform input-to-state stability and polynomial input-to-state stability for infinite-dimensional systems. A characterization of semi-uniform input-to-state stability is developed based on attractivity properties as in the uniform case. We also provide sufficient conditions for linear systems to be polynomially input-to-state stable.

Keywords:Nonlinear Systems and Control, Stability, Infinite Dimensional Systems Theory Abstract: We study input-to-state stability (ISS) of systems with a linear control and a bilinear feedback term, depending on the state trajectory itself and the output of the system. Both, the control and the bilinear feedback enter the system through possibly unbounded operators. Further, the observation operator, associated to the output, is also considered to be unbounded. We derive sufficient conditions for a global in time well-posedness result for small initial data as well as sufficient conditions for an ISS-estimate. This extends recent investigations on bilinear systems, where a second control was considered instead of an output. The developed results are applied to a Burgers equation.

Keywords:Operator Theoretic Methods in Systems Theory, Infinite Dimensional Systems Theory Abstract: Input-to-state stability is characterised by admissibility for linear systems, governed by strongly continuous semigroups. Yet, in some applications semigroups may fail to be strongly continuous with respect to the norm of the underlying Banach space. Typical examples are given by shift-semigroups and the Gauß-Weierstraß semigroup on spaces of bounded continuous functions as well as dual semigroups. This requires a suitable theory for this general setting within the framework of so-called bi-continuous semigroups, including proper admissibility concepts. Our contribution mainly focuses on non-trivial variants of results from the classical case. For instance, the recently shown fact that the generator of a strongly continuous semigroup is only admissible if it is a bounded operator, fails for bi-continuous semigroups.

Keywords:Optimal Control Abstract: This paper concerns an optimal control problem on the space of probability measuresover a compact Riemannian manifold. The motivation behind it is to model certain situations where the central planner of a deterministic controlled system has only a probabilistic knowledge of the initial condition. The lack of information here is very specific. In particular, we show that the value function verifies a dynamic programming principle and we prove that it is the unique viscosity solution to a suitable Hamilton Jacobi Bellman equation. The notion of viscosity is defined using test functions that are directionally differentiable in the the space of probability measures.

Keywords:Optimal Control, Infinite Dimensional Systems Theory, Nonlinear Systems and Control Abstract: We introduce an abstract framework for the study of general mean field game and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set-valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set-valued map expressing the admissible trajectories for the microscopical agents. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system.

Keywords:Optimal Control, Multidimensional Systems Abstract: Several optimal control problems in R^d, like systems with uncertainty, control of flock dynamics, or control of multiagent systems, can be naturally formulated in the space of probability measures in R^d. The compatibility of such control systems with a state constraint can be studied by an Hamilton-Jacobi-Bellman equation stated in the Wasserstein space of probability measure. We show that the dynamic is compatible with the constraint when the distance function satisfies the Hamilton Jacobi inequality in a suitable viscosity sense.

Keywords:Nonlinear Systems and Control, Numerical and Symbolic Computations, Optimal Control Abstract: We introduce a new numerical method to approximate the solutions of a class of static Hamilton-Jacobi-Bellman equations arising from minimum time optimal control problems. We rely on several grid approximations, and look for the optimal trajectories by using the coarse grid approximations to reduce the search space for the optimal trajectories in fine grids. This may be thought of as an infinite dimensional version, for PDE, of the “highway hierarchy” method which has been developed to solve discrete shortest path problems. We obtain, for each level, an approximate value function on a sub-domain of the state space. We show that the sequence obtained in this way does converge to the viscosity solution of the HJB equation. Moreover, the number of arithmetic operations that we need to obtain an error of O(e) is bounded by O(1/e^(2d/(1+b))), to be compared with O(1/e^(2d)) for ordinary grid-based methods. Here b depends on the "stiffness" of the value function around the optimal trajectories. Under a regularity condition on the dynamics, we obtain a bound of O(1/e^((1-b)d)) operations, for b<1, and this bound becomes O(|loge|) for b=1. This allowed us to solve HJB PDE of Eikonal type up to dimension 7.

Keywords:Linear Systems Abstract: Nevanlinna-Pick interpolation developed from a topic in classical complex analysis to a useful tool for solving various problems in control theory and electrical engineering. Over the years many extensions of the original problem were considered, including extensions to different function spaces, nonstationary problems, several variable settings and interpolation with matrix and operator points. In this talk we discuss a variation on Nevanlinna-Pick interpolation for positive real odd functions evaluated in real matrix points. This problem was studied by Cohen and Lewkowicz using convex invertible cones and the Lyapunov order, making interesting connections with stability theory. The solution requires an analysis of linear matrix maps using representations that go back to work of R.D. Hill from the 1970s and focusses, in particular, on the question when positive linear matrix maps are completely positive. If time permits, some possible extensions to multidimensional systems will briefly be discussed.

Keywords:Adaptive Control Abstract: The control objective in funnel control is output feedback control such that the norm of the error e(t) of the closed-loop system remains inside a prespecified funnel with boundary ψ^{-1}(t), i.e. ‖e(t)‖ < ψ^{-1}(t) for all t > 0. In other words, prescribed transient behaviour as well as asymptotic accuracy is achieved. Typical features of funnel control are:

- Simplicity of the feedback law. The feedback does not invoke any identification scheme, but is – for example in the relative degree one case – a time-varying error feedback of the form u(t) = − 1 / (1 − ψ(t) ‖e(t)‖) e(t), e(t) := y(t) − y_{ref}(t), where y_{ref}(·) denotes a sufficiently smooth bounded signal with bounded derivative. Note that the gain k(t) = − 1 / (1 − ψ(t) ‖e(t)‖) is large if, and only if, the error is close to the funnel boundary.

- Funnel control is feasible for a whole class of input-output systems, which is characterized by structural assumptions, e.g., well-defined relative degree and stable zero dynamics.

After two decades of high-gain adaptive control, funnel control was introduced in 2002. First results were on linear, single-input, single-output, time-invariant systems with relative degree one and being minimum phase. From then on feasibility of funnel control was shown for other system classes such as multi-input, multi-output, nonlinear, infinite dimensional, perturbed systems, unknown control directions – provided they have stable zero dynamics and satisfy certain assumptions on the high-frequency gain. A particular challenge was to show feasibility for systems with higher relative degree, and to design a funnel controllers for systems described by partial differential equations. Funnel control was applied to various applications such as control in chemical reactor models, industrial servo-systems, wind turbine systems, electrical circuits, to name but a few. Recently, funnel control has been investigated in combination with model predictive control and applied to magnetic levitation systems.

Keywords:Optimization : Theory and Algorithms Abstract: Approaches to decision making and learning mainly rely on optimization techniques to achieve “best” values for parameters and decision variables. In most practical settings, however, the optimization takes place in the presence of uncertainty about model correctness, data relevance, and numerous other factors that influence the resulting solutions. For complex processes modeled by nonlinear ordinary and partial differential equations, the incorporation of these uncertainties typically results in high or even infinite dimensional problems in terms of the uncertain parameters as well as the optimization variables, which in many cases are not solvable with current state of the art methods. One promising potential remedy to this issue lies in the approximation of the forward problems using novel techniques arising in uncertainty quantification and machine learning. We propose in this talk a general framework for machine learning based optimization under uncertainty and inverse problems. Our approach replaces the complex forward model by a surrogate, e.g. a neural network, which is learned simultaneously in a one-shot sense when estimating the unknown parameters from data or solving the optimal control problem. By establishing a link to the Bayesian approach, an algorithmic framework is developed which ensures the feasibility of the parameter estimate / control w.r. to the forward model. This is joint work with Philipp Guth (U Mannheim) and Simon Weissmann (U Heidelberg).

Keywords:Algebraic Systems Theory, Hybrid Systems, Stability Abstract: In this paper, we prove a Lie algebraic criterion for stability of switched differential algebraic equations (DAEs). We show that if the Lie algebra generated by the differential flows associated with the DAE subsystems of an impulse free switched DAE with descriptor matrices that share a common kernel can be decomposed into the solvable ideal and a compact Lie algebra, then the switched DAE is globally uniformly exponentially stable. Furthermore, we show that the proposed Lie algebraic result completely generalizes an existing result in the literature. We also present a conjecture regarding the stability of switched DAEs.

Laboratoire Des Signaux Et Systèmes, Université Paris Saclay, Ce

Keywords:Nonlinear Systems and Control, Feedback Control Systems, Applications of Algebraic and Differential Geometry in Systems Theory Abstract: In this paper, we consider networks of two synaptically coupled excitatory-inhibitory neural modules. It has been shown that the connection strengths may slowly vary with respect to time and that they can actually be considered as inputs of the network. The problem that we are studying is which connection strengths should be modified (in other words, which connection strengths should be considered as inputs), in order to achieve flatness for the resulting control system. Flatness of the control network depends on the number of inputs and, for all possible values of the number of connection strengths acting as controls, we identify and geometrically describe all flat configurations of the system. In particular, for each case we study whether there are interactions between the two subnetworks or between the excitatory and inhibitory populations that are not allowed (translating into zero connection strengths) or, on the contrary, that necessarily have to take place (translating into nonzero connection strengths), and provide a geometric characterization of each case.

Keywords:Algebraic Systems Theory, Applications of Algebraic and Differential Geometry in Systems Theory Abstract: This extended abstract presents several recent results and generalizations that have been obtained in the theory of collision-freeness studied in Zerz and Herty (2019). A nonlinear ODE system x'(t)=f(x(t)) is called collision-free if the solution to the initial value problem with x(0)=x^0 has distinct components for all times t whenever the initial state x^0 has distinct components. This is an important structural property of particle systems. Here, we address the case where the state of the i-th particle has d components x_{ik}, and a collision occurs if there exist t and distinct i,j such that x_{ik}(t)=x_{jk}(t) for all k in {1,...,d}.

Keywords:Delay Systems, Stability, Feedback Control Systems Abstract: It is well known that rational approximation theory involves degenerate hypergeometric functions and, in particular, the Padé approximation of the exponential function is closely related to Kummer hypergeometric functions. Recently, in the context of the study of the exponential stability of the trivial solution of delay-differential equations, a new link between the degenerate hypergeometric function and the zeros distribution of the characteristic function associated with linear delay-differential equations was emphasized. Such a link allowed the characterization of a property of time-delay systems known as multiplicity-induced-dominancy (MID), which opened a new direction in designing low-complexity controllers for time-delay systems by using a partial pole placement idea. Thanks to their relations to hypergeometric functions, we explore in this paper links between the spectrum of delay-differential equations and Padé approximations of the exponential function. This note exploits and further comments recent results from [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Comptes Rendus. Mathématique] and [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Bulletin des Sciences Mathématiques].

Keywords:Algebraic Systems Theory, Linear Systems, Applications of Algebraic and Differential Geometry in Systems Theory Abstract: In this paper, we initiate a new algebraic analysis approach to linear differential systems based on rings of integro-differential operators. Within this algebraic analysis approach, we first interpret the method of variations of constants as an operator identity. Using this result, we show that the module associated with a state-space representation of a linear system is the same as the one associated with its standard convolution representation. This finitely presented module over the ring of integro-differential operators is proved to be stably free. Finally, we show how the reachability property can be expressed within this algebraic analysis approach.

Keywords:Optimal Control, Networked Control Systems, Mechanical Systems Abstract: We reconsider a classic geometric problem and find some interesting generalizations and applications. The problem in question is Fermat's problem, where for a given triangle, ABC, on finds the point, T, such that the sum of the distances from T to the vertices is minimal. This point T is called the Torricelli point. We generalize the problem to finding this Torricelli point for an arbitrary collection of points in mathbb{R}^2 and make connections with the study of linkages. %2D and 3D. We determine the symmetries that leave T invariant. % and solve an associated minimal sensitivity problem. We then propose a balancing problem: Migrate a given set of points minimally so that the Torricelli-point coincides with the origin, and analyze some variants under different objectives and constraints. We illustrate the relevance in foraging and applications in swarm configurations where minimum distance or minimal average communication delay may be important.

Keywords:System Identification, Dissipativity, Linear Systems Abstract: The concept of dissipativity is a cornerstone of systems and control theory. Typically, dissipativity properties are verified by resorting to a mathematical model of the system under consideration. In this extended abstract, we aim at assessing dissipativity by computing storage functions for linear systems directly from measured data. As our main contributions, we provide conditions under which dissipativity can be ascertained from a finite collection of noisy data samples. Different noise models will be considered that can capture a variety of situations, including the cases that the data samples are noise-free, the energy of the noise is bounded, or the individual noise samples are bounded. All of our conditions are phrased in terms of data-based linear matrix inequalities, which can be readily solved using existing software packages.

Keywords:Port-Hamiltonian Systems, Dissipativity, Linear Systems Abstract: For linear time-invariant descriptor systems it is well known that port-Hamiltonian systems are passive and that passive systems are positive real. In our contribution we study under which assumptions also the converse implications hold. We also study the relationship between passivity, KYP inequalities and a finite required supply.

Keywords:Dissipativity, Feedback Control Systems, Mathematical Theory of Networks and Circuits Abstract: Lossless trajectories of a passive system are the trajectories that satisfy the dissipation inequality with equality. In other words, for a suitable input-state-output representation, these are trajectories for which the rate of change of stored energy is equal to the power supplied to the system. In this paper, we present a method to design feedback control strategies that restrict the trajectories of a passive system to its lossless trajectories. In particular, we deal with passive systems that do not admit an Algebraic Riccati Equation (ARE) arising from the dissipation inequality; we call such systems singularly passive systems. We show that suitably designed PD feedback controllers help us restrict the trajectories of the system to its lossless trajectories. The design method of the controller is linked to the LMI arising from the KYP Lemma corresponding to a passive system.

Keywords:Nonlinear Systems and Control, Dissipativity, Operator Theoretic Methods in Systems Theory Abstract: This abstract summarizes our recent results on reachability analysis using dissipation inequalities. We first outline a method to outer-approximate forward reachable sets (FRS) on finite horizons for uncertain polynomial systems. This method makes use of time-dependent polynomial storage functions that satisfy appropriate dissipation inequalities that account for L2 disturbances, uncertain parameters, and perturbations characterized by time-domain, integral quadratic constraints (IQC). By introducing IQCs to reachability analysis, we now allow for various types of uncertainty, including unmodeled dynamics.We next discuss backward reachable sets (BRS), and decompose control synthesis process into two steps: first we construct storage functions whose sublevel sets are used for BRS estimation, and then we compute control laws using these storage functions through quadratic programs (QP). In a separate result we simultaneously compute an under-approximation to the BRS, as well as an explicit control law in order to incorporate input saturation limits. These methods make use of the generalized S-procedure and Sum-of-Squares techniques to derive algorithms with the goal of finding the tightest approximation to the reachable sets.

Keywords:Port-Hamiltonian Systems, Systems on Graphs Abstract: In this paper, we study output consensus of coupled linear port-Hamiltonian systems on graphs in the presence of constant disturbances, where couplings are allowed to be both static and dynamic. Utilizing port-Hamiltonian structures, we present dynamic controllers achieving output consensus where the consensus values are determined by the disturbances. Finally, the utility of the proposed controller is illustrated by applying it to current sharing of DC microgrids.

Keywords:Networked Control Systems, Port-Hamiltonian Systems, Nonlinear Systems and Control Abstract: This extended abstract proposes a passivity-based approach using bearing and velocity information for an angle-based formation control with a class of underlying triangulated Laman graphs. The controller is designed using virtual couplings on the relative measurements related to the edges. The different embedding of the graph is mapped by the measurement Jacobian, which is calculated by the time-evolution of the measurement. Furthermore, to avoid unavailable distance measurements in the control law, an estimator is designed based on the port-Hamiltonian theory using bearing and velocity measurements. The stability analysis of the closed-loop system is provided and simulations are performed to illustrate the effectiveness of the approach.

Keywords:Coding Theory Abstract: We investigate the connections between rank-metric codes and evasive {mathbb F}_q-subspaces of {mathbb F}_{q^m}^k. We show how the parameters of a rank-metric code are related to special geometric properties of the associated evasive subspace and construct new MRD-codes.

Keywords:Coding Theory, Information Theory Abstract: In this work, the multi-cover metric is introduced. It is defined as a Cartesian product of classical cover metrics. A Singleton bound is given and maximum multi-cover distance (MMCD) codes are defined. Puncturing and shortening of linear MMCD codes are studied. It is shown that the dual of a linear MMCD code is not necessarily MMCD, and those satisfying this duality condition are defined as dually MMCD codes. Finally, constructions of dually MMCD codes are given, which also include some new linear codes attaining the Singleton bound for the classical cover metric and classical crisscross error correction.

Keywords:Coding Theory, Information Theory, Mathematical Theory of Networks and Circuits Abstract: We initiate the study of the one-shot capacity of communication (coded) networks with an adversary having access only to a proper subset of the network edges. We introduce the Diamond Network as a minimal example to show that known cut-set bounds are not sharp in general, and that their non-sharpness comes precisely from restricting the action of the adversary to a region of the network. We give a capacity-achieving scheme for the Diamond Network that implements an adversary detection strategy. We also show that linear network coding does not suffice in general to achieve capacity, proving a strong separation result between the one-shot capacity and its linear version. We then give a sufficient condition for tightness of the Singleton Cut-Set Bound in a family of two-level networks. Finally, we discuss how the presence of nodes that do not allow local encoding and decoding does or does not affect the one-shot capacity.

Keywords:Coding Theory, Linear Systems Abstract: A constant dimension subspace code can be viewed geometrically as a subset of the Grassmann variety defined over a finite field.

There exist few algebraic constructions for constant dimension subspace codes. A major technique is the 'lifting technique' of a rank metric code with a good distance. For rank metric codes exist several good algebraic constructions. First and for most one should mention the technique of constructing Gabidulin codes which can be seen as the image of a linear space of linearized functions under an evaluation map. The technique of constructing Gabidulin codes naturally generalizes the construction of AG-codes such as Reed-Solomon codes and more general geometric Goppa codes.

To our knowledge no similar constructions of subspace codes using evaluation maps is known.

In this talk we will show how convolutional codes give raise in a natural way to constant dimension subspace codes. Using the evaluation over some extension field it is possible to achieve subspace codes with maximal possible distance.

Keywords:Coding Theory Abstract: In the context of Network Coding, flag codes can be seen as an extension of constant dimension codes. In this case, the codewords are sequences of nested subspaces (flags) of a finite dimensional vector space over a finite field. As with constant dimension codes, a particularly interesting way of constructing flag codes is through group actions, which produces codes with an orbital structure. The aim of this note is to present two specific constructions of flag codes having maximum distance through the action of Singer groups. To this end, we will make use of the transitive action of these groups on lines and hyperplanes.

Università Degli Studi Della Campania "Luigi Vanvitelli"

Keywords:Coding Theory, Information Theory Abstract: Cyclic subspace codes gained a lot of attention especially because they may be used in random network coding for correction of errors and erasures. Roth, Raviv and Tamo in 2018 established a connection between cyclic subspace codes (with certain parameters) and Sidon spaces, introduced by Bachoc, Serra and Zémor in 2017 in relation with the linear analogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo to construct large classes of cyclic subspace codes with one or more orbits. In this abstract we will consider an extension of the notion of Sidon space, which turns out to be related to cyclic subspace codes with more than one orbit. Moreover, we will also use the geometry of linear sets to provide some bounds on the parameters of a cyclic subspace code.

Keywords:Networked Control Systems, Nonlinear Systems and Control, Adaptive Control Abstract: In this work, we consider multi-agent systems (MAS) operating under a collective constraint, i.e., a constraint that involves the collective states of MAS, using control barrier function (CBF) technique. CBF-based control design usually consists of designing a task-achieving controller, and modifying it minimally in a quadratic program (QP) to satisfy the state constraint. Despite its success for single-agent systems, most existing CBF-based control designs for MAS are either centralized or sub-optimal. Our proposed distributed CBF-based control scheme guarantees that the optimal to the QP control signals are obtained in finite time, and the collective constraint is satisfied for all time. The result is valid for a large class of MAS (linear or nonlinear, homogeneous or heterogeneous), underlying tasks (consensus, formation, coverage, etc), and collective constraints. We also analyze another scheme with some comparative remarks. Several numerical examples are shown.

Keywords:Networked Control Systems, Nonlinear Systems and Control Abstract: We design a state-feedback controller to impose prescribed performance attributes on the output stabilization error for uncertain nonlinear systems, in the presence of unknown time-varying delays appearing both to the state and control input signals, provided that an upper bound on those delays is known. The proposed controller achieves pre-specified minimum convergence rate and maximum steady-state error, and keeps bounded all signals in the closed-loop. We proved that the error is confined strictly within a delayed version of the constructed performance envelope. Nevertheless, the maximum value of the output error at steady state remains unaltered, exactly as pre-specified by the constructed performance functions. Furthermore, the controller does not incorporate knowledge regarding the nonlinearities of the controlled system, and is of low-complexity in the sense that no hard calculations (analytic or numerical) are required to produce the control signal. Simulation results validate the theoretical findings.

Keywords:Adaptive Control, Linear Systems, Stability Abstract: We consider adaptive ouput feedback tracking control of linear time-invariant systems which are not necessarily minimum phase. The zero dynamics is split into a stable and an unstable part, we show that a flat output of the unstable part can contribute to the design of a funnel controller of the system. More precisely, we consider an auxiliary output based of the "true output" of the system and the flat output of the unstable part of the zero dynamics. The funnel controller is designed for this auxiliary output, and the consequences for the true output are discussed.

Keywords:Stability, Nonlinear Systems and Control, Adaptive Control Abstract: Funnel control (FC) in combination with internal model (IM) achieves asymptotic tracking but feasibility of IM-FC in presence of input saturation (if e.g. a feasibility condition is satisfied) is unclear. Here, both aspects are brought together to obtain a closed-loop system comprising of funnel controller, serial interconnection of internal model with anti-windup and input-saturated high-gain stabilizable system which achieves prescribed transient and asymptotic accuracy if a feasibility conditions is satisfied (Hackl, 2017, Chapter 10). It will be illustrated that in presence of actuator saturation, funnel control with internal model but without anti-windup might exhibit integrator windup deteriorating control performance and resulting in instability of the closed-loop system. The theory of funnel control of input-saturated systems will be extended to allow for the application of funnel control with internal model to input-saturated systems by introducing an anti-windup strategy called conditional integration. The proposed approach is implemented and illustrated for a simple relative-degree-two system.

Keywords:Adaptive Control, Linear Systems Abstract: This paper presents a novel high-gain adaptive control approach for stabilization of unknown minimum-phase linear systems with output constraints. The proposed approach unites the high-gain adaptive control technique and the tool of barrier function to guarantee the output constraint, without resorting to a priori knowledge of the system matrices. Simulation results performed on a numerical example is illustrated to testify the effectiveness of the developd controller.

Keywords:Adaptive Control, Feedback Control Systems, Nonlinear Systems and Control Abstract: This note explores a property of the internal dynamics that is essential for the functioning of classical funnel control. We relax the property of internal dynamics bounded-input bounded-output (BIBO) stability by concentrating on a narrower class of output reference trajectories. Our ideas have a connection with the recently introduced concept of induced contraction, which is motivated by the phenomenon of noise-induced synchronization in neuronal oscillators. We remark that, contrary to previous works, we only have to modify the class of admissible output reference trajectories to handle non-minimum phase systems. We also propose a way (specific to nonlinear systems) to choose an output reference trajectory (possibly under output constraints) for classical funnel control to work for a certain class of unknown systems.

Keywords:Optimization : Theory and Algorithms, Large Scale Systems, Systems on Graphs Abstract: This note discusses an essentially decentralized interior point method, which is well suited for optimization problems arising in energy networks. Advantages of the proposed method are guaranteed and fast local convergence for problems with non-convex constraints. Moreover, our method exhibits a small communication footprint and it achieves a comparably high solution accuracy with a limited number of iterations. Furthermore, the local subproblems are of low computational complexity. We illustrate the performance of the proposed method on an optimal power flow problem with 708 buses.

Keywords:Optimal Control, Optimization : Theory and Algorithms, Process Control Abstract: In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas.

For the operation of gas transport networks, it is important to take into account the randomness of the consumers' demand, where often information on the probability distribution is available. Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H2-norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause embrittlement of the pipeline metal.

Keywords:Optimal Control, Stochastic Control and Estimation Abstract: We present a stochastic optimal control problem for a serial network. The dynamics of the network are governed by transport equations with a special emphasis on the nonlinear damping function. The demand profile at the network sink is modeled by a stochastic differential equation. An explicit optimal inflow into the network is determined and numerical simulations are presented to show the effects for different choices of the nonlinear damping.

Keywords:Nonlinear Systems and Control, Stability, Multidimensional Systems Abstract: Most network dynamical systems are out of equilibrium and externally driven by fluctuations. Linear response theory generically characterizes systems responses to such fluctuations for small driving amplitudes yet cannot capture response properties that are either due to strong driving or intrinsically nonlinear. For oscillation-driven systems, we here report average response offsets that scale quadratically with asymptotically small amplitudes. At some critical driving amplitude, responses cease to stay close to a given operating point and may diverge. Standard response theory fails to predict these amplitudes even at arbitrarily high orders. We propose a new method for predicting critical amplitudes based on an integral self-consistency condition that captures the full nonlinear system dynamics. We illustrate our approach for a minimal one-dimensional model and capture the nonlinear shift of voltages in the phase, frequency and voltage dynamics of AC power grid networks. Our approach may help to quantitatively predict intrinsically nonlinear response dynamics as well as bifurcations emerging at large driving amplitudes in non-autonomous dynamical systems.

Keywords:Networked Control Systems, Stochastic Control and Estimation, Transportation Systems Abstract: In the past decade, bike-sharing system has attracted significant attention as eco-friendly and healthy system of transportation. However, rebalancing the number of bicycles parked at port stations in networks is a serious challenge to provide comfortable service for users. One method to rebalance bicycles is by incentivizing users to rent or return their bicycles to appropriate port stations. In this paper, we first introduce a mathematical model to describe the dynamics of the number of parked bicycles, in which the behavior of incentivized users is stochastic. Then, we propose a distributed stochastic feedback-control law of the incentive, based on the idea of the quantized gossip algorithm to rebalance bicycles at all port stations in a network. We prove the convergence around the references in a probabilistic manner and discuss the results.

Keywords:Mathematical Theory of Networks and Circuits Abstract: This paper is concerned with the feasibility of the power flow in DC power power grids with constant power loads. Necessary and sufficient matrix inequalities are derived that guarantee a minimal p-norm distance between a configuration of power demands and the infeasibility boundary in the space of power demands. The (non)convexity of these matrix inequalities is studied subsequently.

Keywords:Optimal Control, Control of Distributed Parameter Systems, Linear Systems Abstract: Recent theoretical and numerical results on a constrained optimal control problem of Bolza type for a class of parabolic equations obtained by the authors are presented. We study the case when the cost functional is quadratic and comprises the norm of a control and the distance of the system trajectory from the desired evolution profile, the constraint is imposed on the final state that should be steered within a prescribed distance to a given target and the control enters the system through the initial condition.The theoretical results provide a formula for the optimal control and the numerical algorithm is based on efficient rational Krylov approximation techniques.

Keywords:Optimization : Theory and Algorithms, Operator Theoretic Methods in Systems Theory, Computations in Systems Theory Abstract: The splitting algorithms of monotone operator theory find zeros of sums of relations. This corresponds to solving series or parallel one-port electrical circuits, or the negative feedback interconnection of two subsystems. One-port circuits with series and parallel interconnections, or block diagrams with multiple forward and return paths, give rise to current-voltage relations consisting of nested sums and inverses. In this extended abstract, we present new splitting algorithms specially suited to these structures, for interconnections of monotone and antimonotone relations.

Keywords:Delay Systems, Infinite Dimensional Systems Theory, Robust and H-Infinity Control Abstract: The H2 norm of an exponentially stable system described by Delay Differential Algebraic Equations (DDAEs) might be infinite, due to the existence of hidden feedthrough terms and it might become infinite as a result of infinitesimal changes to the delay parameters. We first introduce the notion of strong H2 norm of semi-explicit DDAEs, a robustified measure that takes into account delay perturbations, and we analyze its properties. Next, we discuss necessary and sufficient finiteness criteria for the strong H2 norm in terms of a frequency sweeping test over a hypercube, and in terms of a finite number of equalities involving multi-dimensional powers of a finite set of matrices. Finally, we show that if the H2 norm of the DDAE is finite, it is possible to construct an exponentially stable neutral delay-differential equation which has the same transfer matrix as the DDAE, without any need for differentiation of inputs or outputs. This connected with a neutral system enables the framework of Lyapunov matrices for computing the H2 norm.

Keywords:Infinite Dimensional Systems Theory, Delay Systems, Linear Systems Abstract: Coprime factorizations of transfer functions play various important roles, e.g., minimality of realizations, stabilizability of systems, etc. This paper studies the Bezout condition over the ring E'(R-) of distributions of compact support and the ring M(R-) of measures with compact support. These spaces are known to play crucial roles in minimality of state space representations and controllability of behaviors. We give a detailed review of the results obtained thus far, as well as discussions on a new attempt of deriving general results from that for measures. It is clarified that there is a technical gap in generalizing the result for M(R-) to that for E'(R-). A detailed study of a concrete example is given.

Keywords:Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Nonlinear Systems and Control Abstract: We extend a well-known non-controllability result of Ball, Marsden, and Slemrod on infinite-dimensional bilinear systems (with bounded control term) to control-affine semi-linear systems whose linear part generates an analytic semigroup and whose control term is possibly unbounded as well. Here control inputs are assumed to lie in some suitable Lp-space, p > 1. The result allows an application to PDEs whose control terms include not only the state but also lower oder derivatives compared to the uncontrolled leading linear part. The proof relies on an abstract compactness principle for parameter dependent fixed point maps.

Keywords:Infinite Dimensional Systems Theory, Nonlinear Systems and Control, Stability Abstract: In this paper we study ISS-like properties of infinite dimensional discrete time systems and compare them with their continuous time counterparts available in the literature. New characterizations of such properties are derived in this work. We discuss differences between discrete and continuous time systems in view of their robust stability and demonstrate the corresponding properties by means of examples.

Keywords:Optimal Control, Nonlinear Systems and Control Abstract: We consider deterministic mean field games in which the agents control their acceleration and are constrained to remain in a region of R^n. We study relaxed equilibria in the Lagrangian setting; they are described by a probability measure on trajectories. The main results of the paper concern the existence of relaxed equilibria under suitable assumptions. The fact that the optimal trajectories of the related optimal control problem solved by the agents do not form a compact set brings a difficulty in the proof of existence. The proof also requires closed graph properties of the map which associates to initial conditions the set of optimal trajectories.

Keywords:Optimal Control Abstract: A family of continuous-time regulator problems, subject to linear time-varying dynamics and convex state constraints, is parameterized with respect to a space of semiconcave terminal costs. By embedding the state constraints into an extended real-valued running cost, a family of relaxations is employed to develop a min-plus convolution-based fundamental solution concept for the family of regulator problems. A representation of the fundamental solution concept is provided in terms of solutions of associated differential Riccati and related equations, and possible avenues for efficient numerical computation are explored.

Keywords:Optimal Control, Mathematical Theory of Networks and Circuits Abstract: We study deterministic Mean Field Games with finite horizon in which the state space of the players is a network. In these games, the generic agent can control its dynamics: inside each edge, it can choose its velocity (which coincides with its control) and it can also choose, when it arrives at a vertex, the edge in which it enters. It will pay a cost which is formed by a running cost and a terminal cost; both these costs depend on the trajectory that it has chosen and on the evolution of the distribution of all agents. On the other hand, its position cannot affect the distribution of the whole population. As in the Lagrangian approach, we introduce a relaxed notion of Mean Field Games equilibria which relies on probability measures on trajectories on the network instead of probability measures on the network. Our main result is to establish the existence of such Mean Field equilibria.

With such an equilibrium at hand, we can introduce the value function and we prove that this function is a generalized solution to the associated first order Hamilton-Jacobi problem on the network.

Keywords:Mathematical Theory of Networks and Circuits, Numerical and Symbolic Computations, Systems on Graphs Abstract: The purpose of this presentation is to study the well posedness of a time–dependent Hamilton–Jacobi equation, coupled with suitable additional conditions, posed on a network. We consider a connected network Γ embedded in R N with a finite number of arcs γ, which are regular simple curves parametrized in [0, 1], linking points of RN called vertices, which make up a set we denote by V. We define a Hamiltonian on Γ as a collection of Hamiltonians Hγ : [0, 1] × R → R, indexed by arcs, with the crucial feature that Hamiltonians associated to arcs possessing different support, are totally unrelated.

Keywords:Large Scale Systems, Systems on Graphs, Stochastic Control and Estimation Abstract: An analysis of infinite horizon linear quadratic Gaussian (LQG) Mean Field Games is given within the general framework of Graphon Mean Field Games (GMFG) on dense infinite graphs (or networks) introduced in Caines and Huang (2018). For a class of LQG-GMFGs, analytical expressions are derived for the infinite horizon Nash values at the nodes of the infinite graph. Furthermore, under specific conditions on the network and the initial population means, it is shown that the nodes with strict local maximal infinite network degree are also nodes with strict local minimal cost at equilibrium.