Keywords:Opinion Dynamics Abstract: The talk will focus on the main mechanisms influencing opinion dynamics, like homophily, mutual appraisal, and bounded confidence. Some classic opinion dynamics models, as well as some recent ones, will be presented. Interesting open problems as well as promising research directions will be proposed.

Keywords:Infinite Dimensional Systems Theory, Stability, Operator Theoretic Methods in Systems Theory Abstract: We review Linear-Matrix-Inequality criteria for various notions of stability and performance for discrete-time autonomous (state/output) linear systems. Here we discuss analogues of all these ideas for a certain class of time-varying discrete-time, state/output linear systems, where output-stability requires that the Z-transform of the output sequence be in a weighted Bergman space over the unit disk rather than in the usual Hardy space over the unit disk.

Keywords:Operator Theoretic Methods in Systems Theory Abstract: This is an extended abstract of a paper ``Positivity is undecidable in tensor products of free algebras'' which is currently in preparation. In quantum information, we are interested in tensor products of free algebras and related algebras, since these tensor products model spatially separated subsystems with entanglement. The recent MIP*=RE result shows that it is undecidable to determine whether an element in the tensor product of two free group algebras is positive in all finite-dimensional representations. This shows that this tensor product is not RFD, resolving the Connes embedding problem. In this work, we show that these tensor products are also not archimedean-closed, by showing that it is undecidable to determine if an element of the tensor product is positive. The result also holds for tensor products of related algebras, like algebra of *-polynomials or the group algebra of a free product of abelian groups.

Keywords:Signal Processing Abstract: The best low rank tensor approximation problem occurs in a wide variety of applications; however, this problem is strictly speaking not well posed. Indeed, best low rank tensor approximations can fail to exist. In the case that a best low rank approximation fails to exist, computing a near optimal low rank approximation is highly numerically ill-conditioned.

In this talk we will consider the best low rank approximation problem for the special class of tensors which are positive definite. We will show that the set of low rank tensors that are positive definite is relatively closed as a subset of the set of tensors that are positive definite. Using this fact, we will provide a deterministic bound for the existence of a best low rank approximation of a positive definite tensor. We will illustrate through numerical experiments that our bound is highly predictive of numerical errors when attempting to compute a best low rank approximation of a measured tensor.

Keywords:Operator Theoretic Methods in Systems Theory, Algebraic Systems Theory Abstract: The talk considers evaluations of linear matrix pencils on matrix tuples. It is shown that ranks of linear matrix pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. Variants of this property for other classical symmetries are discussed, and the relation between rank inequalities and orbit closure membership is addressed.

Keywords:Applications of Algebraic and Differential Geometry in Systems Theory, Quantum Control, Operator Theoretic Methods in Systems Theory Abstract: The foundations of classical Algebraic Geometry and Real Algebraic Geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades the basic analogous theorems for matrix and operator theory (noncommutative variables) have emerged. This paper concerns commuting operator strategies for nonlocal games, recalls NC Nullstellensatz which are helpful, extends these, and applies them to a very broad collection of games.

The main results of this procedure are two characterizations, based on Nullstellensatz, which apply to games with perfect commuting operator strategies. The first applies to all games and reduces the question of whether or not a game has a perfect commuting operator strategy to a question involving left ideals and sums of squares. Previously, Paulsen and others translated the study of perfect synchronous games to problems entirely involving a ∗-algebra. The characterization we present is analogous, but works for all games. The second characterization is based on a new Nullstellensatz we derive in this paper. It applies to a class of games we call torically determined games, special cases of which are XOR and linear system games. For these games we show the question of whether or not a game has a perfect commuting operator strategy reduces to instances of the subgroup membership problem.

Keywords:Hybrid Systems, Optimal Control, Model Predictive Control Abstract: We consider finite horizon optimal control problems for hybrid plants that are modeled as hybrid equations. To determine key properties of the problem, such as existence and regularity of the optimal cost, we formulate a Mayer form that is tailored to hybrid systems. Within the setting of nominally outer well-posed hybrid plants, and under mild (and standard) regularity conditions, establishing existence of optimal solutions and nice (upper semicontinuous and continuous) dependence of the optimal cost is enabled by the proposed Mayer form. The advantage of the proposed approach is that it does not require additional properties that are typically required in the literature, such as assumptions on the continuous dynamics or that the terminal cost is a control Lyapunov function on the terminal constraint set. The proposed new form is illustrated in examples.

Keywords:Nonlinear Systems and Control, Hybrid Systems Abstract: We investigate the scenario where a perturbed nonlinear system communicates its output measurements to a remote observer via a network. The sensors are grouped into N nodes and each of these nodes decides when its measured data is transmitted over the network independently. Given a (continuous-time) observer, we present an approach to design local (dynamic) transmission policies to obtain accurate state estimates, while only sporadically using the communication network. We prove a practical convergence property to the origin for the estimation error and we show there exists a uniform strictly positive minimum inter-event time for each local triggering rule under mild conditions on the plant. The analysis relies on hybrid Lyapunov tools. The efficiency of the proposed techniques is illustrated on a numerical case study of a flexible robotic arm.

Keywords:Stability, Hybrid Systems Abstract: We present and develop tools to analyze stability properties of discrete-time switched linear systems driven by shuffled switching signals. A switching signal is said to be shuffled if all modes of the system are activated infinitely often. We establish a notion of joint spectral radius related to these systems: the shuffled joint spectral radius (SJSR) which intuitively measures the impact of shuffling on the decay rate of the system’s state. We show how this quantity relates to stability properties of such systems. Specifically, from the SJSR, we can build a lower bound on the minimal shuffling rate in order to stabilize an unstable system. Then, we present several methods to approximate the SJSR, mainly by computing lower and upper bounds using Lyapunov methods and some automata theoretic techniques.

Keywords:Hybrid Systems, Stability, Nonlinear Systems and Control Abstract: We introduce a new class of hybrid Lur'e dynamical systems where a sector nonlinearity may affect both the continuous-time evolution and the reset map acting on suitable closed-loop states, under a time-regularization mechanism ensuring dwell time of solutions. For this class of systems we characterize Lyapunov-based exponential stability conditions exploiting homogeneity of the closed loop. In particular, we show that, with quadratic Lyapunov certificates these conditions can be cast as linear matrix inequalities. We then focus on the control design problem, where both the feedback gains acting on the continuous evolution and the reset action must be designed, in addition to the sets where such resets are triggered, expressed by sign-indefinite quadratic forms. For this control design problem we also show that the synthesis can be performed by solving a set of linear matrix inequalities.

Keywords:Hybrid Systems, Nonlinear Systems and Control Abstract: Abstract: We study tree-like networks of leaderless Kuramoto-like non-identical oscillators having time-varying natural frequencies taking values in a compact set. We interconnect the oscillators via a novel class of hybrid coupling rules inducing uniform global practical asymptotic stability of the synchronization set, thereby ensuring global uniform convergence. Moreover, we show that the synchronization set is uniformly globally finite-time stable whenever the coupling function is discontinuous at the origin. Numerical simulation results illustrate the advantage of the proposed model with respect to non-uniform behavior typically found with classical Kuramoto models.

Keywords:Hybrid Systems, Stability, Nonlinear Systems and Control Abstract: We consider mobile robots described through unicycle dynamics equipped with range sensors and cameras, one in the front and one in the back providing measurements of the distance and misalignment to a target. We derive locally asymptotically stabilizing control laws driving the robot to the target position and orientation. The local control laws are combined into a hybrid global stabilizer, switching between control laws relying on the measurements from the front and rear sensors. Using Lyapunov arguments in the local setting as well as in the hybrid systems formulation, we prove global asymptotic stability of the target set for the hybrid closed-loop system. The results are illustrated on numerical examples.

Keywords:Coding Theory, Information Theory Abstract: The problem of decoding a random-like linear block code is recognized as one of the most important mathematical problems that apparently will remain hard even with the availability of solvers based on quantum computers. This motivates an increasing interest in code-based cryptography as a solution for the design of post-quantum cryptographic primitives. However, while several robust and efficient code-based systems exist for asymmetric encryption and key exchange, mostly stemming from the McEliece and Niederreiter original cryptosystems, devising robust and efficient code-based signature schemes is a far more challenging task. This work provides an overview of past and current approaches to the problem of designing secure and practical code-based signature schemes following two main directions: adapting the McEliece and Niederreiter schemes to the digital signature setting following the classical hash-and-sign approach or deriving digital signatures from code-based identification schemes through suitable transformations.

Keywords:Coding Theory, Information Theory Abstract: We exhibit a way to reduce the public-key size of a rank metric based public-key cryptosystem. This approach does not use a structural property of the code but exhibit some particular generator matrices that have a quasi-cyclic like structure.

Keywords:Coding Theory, Information Theory Abstract: Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of the Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. This can be used to solve the interpolation step of interpolation-based decoding of (interleaved) Gabidulin, linearized Reed—Solomon and skew Reed—Solomon codes efficiently, which have various applications in coding theory and code-based quantum-resistant cryptography.

Keywords:Coding Theory Abstract: The prospect and demand of using the Lee metric to construct public-key encryption schemes and digital signature schemes have immensely grown in recent times. This leads the researchers to ask about the hardness of decoding a general Lee metric code. In this work, we answer this question by showing that the syndrome decoding problem over the Lee metric is NP-complete. Moreover, we will quantify the computational hardness of the syndrome decoding problem with respect to the best-known ISD algorithms.

Keywords:Coding Theory Abstract: Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. It can be efficiently stored in memory because it is fully specified by its first row. The ring of n ×n circulant matrices can be identified with F[x]/(x n − 1). In consequence, the strong algebraic structure of the ring F[x]/(x n − 1) can be used to study properties of the collection of all n × n circulant matrices. The ring F[x]/(x n − 1) is a special case of a group algebra and elements of any finite dimensional group algebra can be represented with square matrices which are specified by a single column. In this paper we study this representation and prove that it is an injective Hamming weight preserving homomorphism of F-algebras and classify it in the case where the underlying group is abelian. Our work is motivated by the desire to generalize the BIKE cryptosystem (a contender in the NIST competition to get a new post-quantum standard for asymmetric cryptography). Group algebras can be used to design similar cryptosystems or, more generally, to construct low density or moderate density parity-check matrices for codes.

Keywords:Information Theory, Algebraic Systems Theory, Computations in Systems Theory Abstract: This paper proposes an application of a new observer theory for non-linear systems developed previously to solve the Cryptanalysis problem of a special class of pseudorandom generators which are commonly used in Cryptography. The Cryptanalysis problem addressed here is that of the recovery of internal state of the non-linear dynamic stream generator from the output stream. The proposed methodology is termed as emph{observability attack}. It is also shown that for a special class of generators, the computations are of complexity O(D^4) in pre-computation and of O(D) for online computation, where D = sum_{i=0}^{d} {n choose i} for this class of stream generators with n states and d the degree of the output function. The attack is technically applicable over general finite fields as well as most dynamic systems arising from models of stream ciphers and appropriate bounds on computation are estimated. From these complexity bounds, it follows that this attack is feasible in realistic cases and gives important estimates of time and memory resources required for Cryptanalysis of a class of stream ciphers.

Keywords:Machine Learning and Control, Nonlinear Systems and Control, Optimal Control Abstract: In this study, we explore the relationship between the complexity of neural networks and the internal compositional structure of the function to be approximated. The results shed light on the reason why using neural network approximation helps to avoid the curse of dimensionality (COD). In Section 2, we discuss the challenge of COD in feedback control. In Section 3, we introduce four compositional features that determine the complexity and error upper bound of neural network approximation for dynamical and control systems. In Section 4, several examples are given to illustrate the widely observed phenomenon in science and engineering that complicated functions are formed by the composition of simple ones.

Keywords:Machine Learning and Control, Optimal Control Abstract: We investigate model-free reinforcement learning of linear quadratic control from an optimization viewpoint. In the first part, we consider distributed reinforcement learning of decentralized linear quadratic control, where we propose a zeroth-order distributed policy optimization algorithm, analyze its performance, and point out some limitations of the theoretical guarantees. The second part is on the optimization landscape analysis of the LQG problem, where we study 1) the connectivity of the set of stabilizing controllers, and 2) the structural properties of stationary points of the LQG cost function.

Keywords:Machine Learning and Control, Neural Networks, Nonlinear Systems and Control Abstract: We propose new mathematical connections between Hamilton-Jacobi (HJ) partial differential equations (PDEs) with initial data and neural network architectures. Specifically, we prove that some classes of neural networks correspond to representation formulas of HJ PDE solutions whose Hamiltonians and initial data are obtained from the parameters or the activation functions of the neural networks. These results do not require any learning stage. In addition these results do not rely on universal approximation properties of neural networks; rather, our results show that some classes of neural network architectures naturally encode the physics contained in some HJ PDEs. Our results naturally yield efficient neural network-based methods for evaluating solutions of some HJ PDEs in high dimension without using grids or numerical approximations.

Keywords:Neural Networks, Machine Learning and Control, Linear Systems Abstract: This extended abstract summarizes a series of recent works on the approximations theory of deep learning methods for time series modelling and analysis. The primary aim is to develop the mathematical foundations for modelling sequential relationships with neural networks, which guides the practical implementation and design of such architectures. In particular, we place on concrete mathematical footing on when and how certain architectures (recurrent neural networks, encoder-decoder structures, dilated convolutions, etc) can adapt to corresponding data structures in the temporal relationships to be learned (memory, rank, sparsity, etc). These form the first step towards principled neural network architecture design and selection for practical machine learning of temporal relationships.

Keywords:Model Predictive Control, Nonlinear Systems and Control, Optimal Control Abstract: In model predictive control, it is a natural idea that not only one but multiple objectives have to be optimized. This leads to the formulation of a multiobjective optimal control problem (MO OCP). In this talk we introduce a multiobjective MPC algorithm, which yields on the one hand performance estimates for all considered objective functions and on the other hand stability results of the closed-loop solution. To this end, we build on the results in Zavala and Flores-Tlacuahuac (2012); Grüne and Stieler (2019) and introduce a simplified version of the algorithm presented in Grüne and Stieler (2019). Compared to Grüne and Stieler (2019), we allow for more general MO OCPs than in Grüne and Stieler (2019) and get rid of restrictive assumption on the existence of stabilizing stage and termi- nal costs in all cost components. Compared to Zavala and Flores-Tlacuahuac (2012), we obtain rigorous performance estimate for the MPC closed loop.

Keywords:Model Predictive Control, Stability, Adaptive Control Abstract: Recently, a unifying approach to the stability analysis of nonlinear model predictive controllers (MPC) with arbitrary positive definite cost functions has been presented based on dissipativity theory. We have established that regardless of the choice of the positive definite cost function, the resulting value function always satisfies a dissipation inequality. This led to less conservative stability conditions for nonlinear MPC that do not require monotonic decrease of the optimal cost function along closed-loop trajectories. In this extended abstract we recall these results and we analyze recursive feasibility, which has not yet been addressed. To this end we use a control contractive terminal set and an adaptive prediction horizon, without adding a terminal cost.

Keywords:Model Predictive Control Abstract: In this work, we study economic model predictive control (MPC) in situations where the optimal operating behavior is periodic. In such a setting, the performance of a plain economic MPC scheme without terminal conditions can generally be far from optimal even with arbitrarily long prediction horizons. Whereas there are modified economic MPC schemes that guarantee optimal performance, all of them are based on prior knowledge of the optimal period length or of the optimal periodic orbit itself. In contrast to these approaches, we propose to achieve optimality by multiplying the stage cost by a linear discount factor. This modification is not only easy to implement but also independent of any system- or cost-specific properties, making the scheme robust against online changes therein. Under standard dissipativity and controllability assumptions, we can prove that the resulting linearly discounted economic MPC without terminal conditions achieves optimal asymptotic average performance up to an error that vanishes with growing prediction horizons. Moreover, we can guarantee practical asymptotic stability of the optimal periodic orbit under slightly stronger assumptions.

Keywords:Model Predictive Control, Stability, Computations in Systems Theory Abstract: This paper presents a Matlab toolbox that implements methods for computing stabilizing terminal costs and sets for nonlinear model predictive control (NMPC). Given a discrete-time nonlinear model provided by the user, the toolbox computes quadratic/ellipsoidal terminal costs/sets and local control laws for the following options: (i) cyclically time-varying or standard terminal ingredients; (ii) first or quasi-second order Taylor approximation of the dynamics; (iii) linear or nonlinear local control laws. The YALMIP toolbox and the MOSEK solver are used for solving linear matrix inequalities and the IPOPT solver (with global search) is used for nonlinear programming. Simulation of the resulting stabilizing NMPC algorithms is provided using the CasADi toolbox.

Keywords:Model Predictive Control, Adaptive Control Abstract: For constrained linear systems with bounded disturbances and parametric uncertainty, we propose a robust adaptive Model Predictive Control scheme with online parameter estimation. Constraints enforcing persistent excitation in closed loop operation are introduced to ensure asymptotic parameter convergence. The algorithm requires the online solution of a convex optimisation problem, satisfies constraints robustly, and ensures recursive feasibility and input-to-state stability. Almost sure convergence to the actual system parameters is obtained under mild conditions on stabilisability and the tightness of disturbance bounds.

Keywords:Nonlinear Systems and Control Abstract: This paper tackles the problem of nonlinear systems, with sublinear growth but unbounded control, under perturbation of some time-varying state constraints. It is shown that, given a trajectory to be approximated, one can find a neighboring one that lies in the interior of the constraints, and which can be made arbitrarily close to the reference trajectory both in L^infty-distance and L^2-control cost. This result is an important tool to prove the convergence of approximation schemes of state constraints based on interior solutions and is applicable to control-affine systems.

Keywords:Optimal Control, Infinite Dimensional Systems Theory, Nonlinear Systems and Control Abstract: In some applied models (as for instance of flocking or of the crowd control) it is more natural to deal with elements of a metric space (as for instance a family of subsets of a vector space endowed with the Hausdorff metric) rather than with vectors of a normed vector space. We consider a generalized control system on a metric space and investigate necessary and sufficient conditions for viability and invariance of proper subsets, describing state constraints. As examples of application we study controlled continuity equations on the metric space of probability measures, endowed with the Wasserstein distance, and controlled morphological systems on the space of nonempty compact subsets of the Euclidean space endowed with the Hausdorff metric. We also provide sufficient conditions for the existence and uniqueness of contingent solutions to the Hamilton-Jacobi-Bellman equation on a proper metric space.

Keywords:Nonlinear Systems and Control, Stability, Systems Biology Abstract: A basic dynamical model for (clinical) depression is proposed that describes the time evolution of two coupled states: a depression symptom and the memory of past symptoms. The model consists of a system of two coupled first order differential equations with unit coefficients that qualitatively captures different courses of illness.

Keywords:Optimization : Theory and Algorithms Abstract: We study a continuous-time system that solves optimization problems over the set of orthonormal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but asymptotically lands on the manifold. We introduce a generalized Stiefel manifold to which we extend the canonical metric of the Stiefel manifold. We show that the vector field of the proposed flow can be interpreted as the sum of a Riemannian gradient on a generalized Stiefel manifold and a normal vector. Moreover, we prove that the proposed flow globally converges to the set of critical points, and any local minimum and isolated critical point is asymptotically stable.

Keywords:Optimization : Theory and Algorithms Abstract: We are interested in determining the worst performance exhibited by a given first-order optimization method on the class of quadratic functions. Thanks to the Performance Estimation Problem (PEP) methodology, it is possible to compute the exact worst-case performance of a first-order optimization method on a given class of functions. Since it has been introduced, PEP can be solved for a large number of classes. We extend the PEP framework to the class of quadratic functions. We apply it to analyze the difference of performance of the gradient method between convex quadratic and general smooth convex functions.

Keywords:Control of Distributed Parameter Systems, Nonlinear Systems and Control, Stability Abstract: We study global finite-dimensional observer-based stabilization of a 1D heat equation with a known globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We then design a finite-dimensional nonlinear Luenberger observer, which takes into account the known semilinearity. The proposed controller is based on this observer. Our Lypunov H^1-stability analysis leads to LMIs, which are feasible for a large enough observer dimension and small enough Lipschitz constant.

Keywords:Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Stability Abstract: This paper is devoted to the study of the robustness properties of the 1-D wave equation for an elastic vibrating string under four different damping mechanisms that are usually neglected in the study of the wave equation: (i) friction with the surrounding medium of the string (or viscous damping), (ii) thermoelastic phenomena (or thermal damping), (iii) internal friction of the string (or Kelvin-Voigt damping), and (iv) friction at the free end of the string (the so-called passive damper). The passive damper is also the simplest boundary feedback law that guarantees exponential stability for the string. We study robustness with respect to distributed inputs and boundary disturbances in the context of Input-to-State Stability (ISS). By constructing appropriate ISS Lyapunov functionals, we prove the ISS property expressed in various spatial norms.

Keywords:Linear Systems, Infinite Dimensional Systems Theory, Control of Distributed Parameter Systems Abstract: We consider reachability properties of families of parameter-dependent linear systems, where the inputs are restricted to be independent of the parameter. If for every family of parameter-dependent target states and every neighborhood of it there is an input such that the zero state can be steered simultaneously into the given neighborhood the parameter-dependent system is called ensemble reachable. Recently, a lot of effort has been spent on the derivation of necessary and sufficient conditions for ensemble reachability. Here we tackle the subsequent question how to determine a suitable input if the target family and the neighborhood is given. We present two methods for discrete-time linear systems which are based on complex approximation theory. We will also point out that one of the polynomial techniques can also be applied to certain continuous-time systems.

Keywords:Linear Systems, Stability, Infinite Dimensional Systems Theory Abstract: We prove small-gain type criteria of exponential stability for positive linear discrete-time systems in ordered Banach spaces that have a normal and generating positive cone. Such criteria play an important role in the finite-dimensional systems theory but are rather unexplored in the infinite-dimensional setting, yet. Furthermore, we show that our stability criteria can be considerably strengthened if the cone has non-empty interior or if the operator inducing the discrete-time system is quasi-compact.

Keywords:Linear Systems, Large Scale Systems Abstract: Variation diminishment -- the reduction in the number of sign changes and local extrema in a signal -- is an intrinsic system property that lies at the heart of positive systems theory and over- and undershooting analysis in controlled systems. While, for general system operators, this property is difficult to verify, we show that it can be readily verified for the controllability and observability operators of finite-dimensional linear time-invariant systems under an internal k-positivity assumption. This complements earlier results on verifying this property for Hankel and Toeplitz operators, and establishes a bridge to internally positive systems theory. Our results provide a new framework for upper bounding the number of over- and undershoots in step responses, as well as a new realization theory of externally positive systems.

Keywords:Networked Control Systems, Linear Systems, Large Scale Systems Abstract: The quantification of controllability has gained renewed interest in the context of large, complex network dynamical systems. In some application areas such as computational neuroscience, there is a large interest in modal controllability, which describes the ability of an input to control the modes of a system. In case of a linear system, the modes of the system are given by the left eigenvectors associated with the system matrix. In this work, we identify mode specific and gross metrics for modal controllability for discrete linear time invariant systems. Our metrics are based on energy requirements to move along a given mode and find applications in problems involving selection of driver nodes for minimizing control effort along particular modes of the network. We conclude by studying the properties of the metrics.

Keywords:Large Scale Systems, Networked Control Systems, Mathematical Theory of Networks and Circuits Abstract: The adverse effect of increasing penetration of distributed energy resources has resulted in increased vulnerabilities to resilience, defined as the ability of the grid to preserve the original properties under disruptive scenarios which were unforeseen in the traditional power grid. Hence it necessitates the development of accurate and reliable resilience metrics to have deeper insight under any disturbance resulting in modification of structural properties. Especially, considering the critical role of local structure and its inherent underlying geometry makes the impact analysis more challenging. In view of this, the proposed Persistent homologybased resilience enhancement (PHRE) technique utilizes the concept of Topological Data Analysis, particularly Betti numbers (identifying the most vulnerable buses) and persistent homology, extracting the longer-lasting topological features of the graphs through the network filtration at various spatial resolutions characterizing the structural functionality of the network. The proposed PHRE technique is validated using a benchmark system.

Keywords:Networked Control Systems, System Identification, Large Scale Systems Abstract: We propose a Kron-based model reduction method for open chemical reaction networks with constant inflow and proportional outflow, which guarantees the preservation of network structures and interlacing property of the reduced-order model. We further show that the proposed Kron-based model reduction method achieves zero-moment matching for nonlinear systems. Lastly, for single-species single-substrate open chemical reaction networks, we present a Gramian-based approach to optimally select the removed nodes and provide the corresponding a priori error bound.

Keywords:System Identification, Systems Biology, Nonlinear Systems and Control Abstract: Differential algebra-based theory and software have been widely used to study the {em a priori} structural identifiability of nonlinear systems. This technique however fails to provide definitive answers for complex reaction networks which involve several reactions and species. In this work, for reaction systems following mass action kinetics, using the theory of reaction extents, we show that identifiability can be ascertained by determining the rank of a matrix. Further, we show that for systems involving most bi-molecular reactions, the parameters are guaranteed to be identifiable, if R (where R= number of independent reactions) species that satisfy a rank condition are measured.

Keywords:System Identification, Networked Control Systems, Linear Systems Abstract: The knowledge of the underlying topology is essential for understanding and manipulating power grids, water distribution networks, and biological networks. At times, the topology may be reported (or recorded) erroneously, mostly owing to human mistakes in reporting. The networks can be represented as a graph in which entities are represented as nodes and interactions among nodes as edges. This work focuses on the study of a specific type of error in topology that occurs when the incidence of an edge in the network is incorrectly reported. We propose a methodology to detect, isolate, and rectify this type of error using a single noisy measurement of flows along all the edges of a conserved network. We first show that this type of error generates specific error signatures, which enables error diagnosis when the data is noise-free. An approach based on a series of statistical tests is developed to handle noisy data for online error detection and rectification. Simulation studies are performed to test the robustness of the proposed methodology.

Keywords:Operator Theoretic Methods in Systems Theory, Networked Control Systems Abstract: The solution to the Hamilton Jacobi equation plays a central role in various problems in system theory, including dissipative theory. In particular, the storage function verifying the dissipative property of the input-output control system can be obtained as a solution to the Hamilton Jacobi equation. The dissipative theory provides a systematic and scalable approach for the stability verification of network control systems. We establish a connection between the spectral theory of the Koopman operator and the Hamilton Jacobi equation. In particular, the solution to the Hamilton Jacobi equation can be constructed using the eigenfunctions of the Hamiltonian system associated with the Hamilton Jacobi equation. This connection between the HJ solution, Koopman spectrum, on the one hand, and the dissipative theory, network system, on the other hand, allows us to discover spectral Koopman theory for the stability analysis of large scale network system.

Keywords:Optimal Control Abstract: Many algorithms in everyday use implicitly employ the Euclidean inner product of the underlying space. While this is convenient and user-friendly on the one hand, it also turns out that the Euclidean metric may not be the one yielding the best performance of the respective algorithm. In this talk we revisit the role of the metric in a number of well- known algorithms in numerical linear algebra and optimization, and demonstrate the potential of user-defined metrics in each case.

Keywords:Stability Abstract: It has long been known that asymptotic controllability of a nonlinear system to a desired equilibrium or target set require discontinuous controllers for feedback stabilization, which, in turn, is equivalent to the existence of a nonsmooth control Lyapunov function. More recently, results combining stabilization and safety, captured by so-called barrier functions, have been proposed. This also gives rise to the need for discontinuous feedback controllers, though for slightly different reasons. In this talk, we summarise these results and present a hybrid feedback solution to the combined stabilization and safety problem for a non-trivial class of systems.

Keywords:Control for distributed parameter systems Abstract: This talk is concerned with the control of distributed parameter systems defined on a 1D spatial domain using the port Hamiltonian framework. We consider two different cases: when actuators and sensors are located within the spatial domain and when the actuator is situated at the boundary of the spatial domain, leading to a boundary control system (BCS). In the first case we show how dynamic extensions and structural invariants can be used to change the internal properties of the system when the system is fully actuated, and how it can be done in an approximate way when the system is actuated using piecewise continuous actuators stemming from the use of patches. Asymptotic stability is achieved using damping injection. In the boundary-controlled case we show how the closed loop energy function can be partially shaped, modifying the minimum and a part of the shape of this function and how damping injection can be used to guarantee asymptotic convergence. We end with some some extensions of the proposed results to irreversible thermodynamic systems.

Keywords:Linear Systems, Operator Theoretic Methods in Systems Theory Abstract: We extend Carleson interpolation Theorem to sequences of matrices (of eventually unbounded dimensions) with spectra in the unit disc. As for the multi-variable case, we see how an analogous of the Pick property enjoyed by the NC (non commutative) Drury-Arveson space allows one to characterize interpolating sequences on the NC unit ball in terms of some Riesz system-type conditions on NC kernels. We discuss examples, and some possible directions for some future research in the topic.

Keywords:Operator Theoretic Methods in Systems Theory Abstract: A fundamental result of Löwner gives that a real-valued function is matrix monotone if and only if it extends to an analytic function with a rather rigid structure. Löwner's student Kraus essentially showed that functions that are matrix convex are analytic functions with a similarly rigid form.

We consider recent work in the realization theory of matrix monotone and matrix convex functions that generalize the classical results to the setting of noncommutative function theory. We also discuss continuation results for such realizations. Further topics include partially matrix convex functions and plurisubharmonic functions.

Keywords:Operator Theoretic Methods in Systems Theory, Multidimensional Systems, Optimization : Theory and Algorithms Abstract: It is well known that noncommutative (nc) rational functions regular at the origin admit a good realization (or linearization) theory. This is very useful both conceptually and for a variety of applications since it often essentially reduces the study of these rational functions to a study of linear pencils.By translation the method can be applied to nc rational functions that are regular at some scalar point, but not beyond.

In this talk we discuss the realization problem for nc rational functions regular at an arbitrary given matrix point using the nc difference--differential calculus and the general Taylor--Taylor series of nc function theory and provide a solution which is the analogue of the classical Hankel realization.

Keywords:Operator Theoretic Methods in Systems Theory Abstract: The study of Optimal Polynomial Approximants (OPAs) in weighted Dirichlet-type spaces has seen a great deal of success in the past decade. In more than one variable, not as much is known, and the failure of the famous Shanks Conjectur shows that there may be issues with the typical Drury-Arveson approach. A remediation is to recast the multivariable into the noncommutative setting where the deficiencies of the Drury-Arveson space are not found. This paper outlines the introductory ideas behind noncommutative Optimal Polynomial Approximants, as well as a couple of tangible conjectures and potential approaches inspired by classical techniques in the freely noncommutative setting.

Keywords:Operator Theoretic Methods in Systems Theory, Multidimensional Systems, Optimization : Theory and Algorithms Abstract: Ando's classical characterization of the unit ball in the numerical radius norm was generalized by Farenick, Kavruk, and Paulsen using the free joint numerical radius of a tuple of Hilbert space operators (X1, ..., Xm). In particular, the characterization leads to a positive definite completion problem. In this paper, we study various aspects of Ando's result in this generalized setting. Among other things, this leads to the study of finding a positive definite solution L to a certain matrix equation, which may be viewed as a fixxed point equation. Once such a fixed point is identified, the desired positive definite completion is easily obtained. Along the way we derive other related results including basic properties of the free joint numerical radius and an easy way to determine the free joint numerical radius of a tuple of generalized permutations. Finally, we present some open problems.

Keywords:Optimal Control, Dissipativity Abstract: Jan Willems introduced the system-theoretic notion of dissipativity in his foundational two-part paper which appeared in the Archive of Rational Mechanics and Analysis in 1972. Even earlier, in a likewise pivotal 1971 IEEE Transactions on Automatic Control paper, he investigated infinite-horizon least-squares optimal control and the algebraic Riccati equation from a dissipativity point of view. This note revisits infinite-horizon optimal control leveraging strict dissipativity. We discuss the interplay between dissipativity and stability properties in continuous-time infinite-horizon problems without assuming linear dynamics or quadratic cost functions. Finally, we compare our recent results from Faulwasser and Kellett (2021) to the original findings of Willems (1971).

Keywords:Port-Hamiltonian Systems, Optimal Control, Dissipativity Abstract: We consider the problem of minimizing the entropy, energy, or exergy production for state transitions of irreversible port-Hamiltonian systems subject to control constraints. Via a dissipativity-based analysis we show that optimal solutions exhibit the manifold turnpike phenomenon with respect to the manifold of thermodynamic equilibria. We illustrate our analytical findings via numerical results for a heat exchanger.

Keywords:Model Predictive Control, Dissipativity, Optimal Control Abstract: During the last couple of years the theory of why and when Model Predictive Control (MPC) generates stable, feasible and near optimal closed-loop solutions has significantly matured. In this talk we give a survey about the contribution of the dissipativity concept in this line of research.

Keywords:Dissipativity, Robust and H-Infinity Control, Large Scale Systems Abstract: Electrical networks constructed out of resistors (R), inductors (L), capacitors (C), transformers (T), and gyrators (G) are used throughout engineering and the applied sciences to model physical processes. Synthesising RLCTG networks for control purposes is also important, since in a number application domains the corresponding controllers can be implemented without an energy source. We show that if a process can be modelled by an LCTG network, a controller that maximises robustness with respect to normalised coprime factor perturbations can be synthesised by a decentralised resistive network. The results are illustrated on an example centred on the iterative solution to constrained least squares problems.

Keywords:Dissipativity, Optimal Control, Linear Systems Abstract: We extend the classical discrete-time bounded real lemma to the general case of systems that need not be controllable or observable, and its relationship to the related discrete-time optimal control problem. In the talk accompanying this extended abstract, we will further discuss the analogies in discrete time to the recent continuous time development of an assumption free theory of linear passive and non-expansive systems that draws on the behavioral framework of Jan Willems and collaborators.

Keywords:Port-Hamiltonian Systems, Linear Systems, Numerical and Symbolic Computations Abstract: Port-Hamiltonian DAE systems are discussed that are the composition of a Dirac structure and a Lagrangian subspace, where the latter is generalizing the Hamiltonian function expressing energy storage. The algebraic constraints in such systems are correspondingly divided into Dirac and Lagrange algebraic constraints. The relations between different representations of the same port-Hamiltonian DAE system are discussed.

Keywords:Coding Theory Abstract: We present a new variant of the McEliece cryptosystem that possesses several interesting properties, including a reduction of the public key for a given security level. In contrast to the classical McEliece cryptosystems, where block codes are used, we propose the use of a convolutional encoder to be part of the public key. The secret key is constituted by a Generalized Reed-Solomon code and two Laurent polynomial matrices that contain large parts that are generated completely at random. In this setting the message is a sequence of messages instead of a single block message and the errors are added randomly throughout the sequence. We analyse its security against ISD attacks in the first instants and when the whole message is transmitted, as well as against structural attacks.

Keywords:Coding Theory, Information Theory Abstract: In network coding, a flag code is a collection of flags, that is, sequences of nested subspaces of a vector space over a finite field. If the sequence of subspace dimensions is complete, we speak about full flag codes. In this work we present some combinatorial tools coming from the classical theory of partitions that can be naturally associated with full flag codes in order to extract relevant information about them. In particular, we state a combinatorial characterization of those full flag codes that attain the maximum possible distance.

Keywords:Coding Theory, Algebraic Systems Theory, Information Theory Abstract: Quasi-cyclic codes over finite fields are an important class of linear block codes. A fundamental problem in the theory of these codes is to describe their algebraic structure. In this paper it is shown that every quasi-cyclic code is the subfield code and the trace code of a quasi-cyclic code over an extension field. The latter is defined by a parity check matrix obtained from a spectral analysis of a reduced Gröbner basis of the former. Moreover, it is shown that the quasi-cyclic code over the extension field and the one under consideration have the same length, dimension and minimum Hamming distance. Furthermore, we show that under certain conditions it is possible to construct a generator matrix of the quasi-cyclic code over the extension field using similar techniques to construct its parity check matrix. We illustrate that this construction is attainable for some good quasi-cyclic low density parity check codes like the [155,64,20] binary Tanner code.

Keywords:Coding Theory, Information Theory Abstract: Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the code distance. This result enables the use of permutations as representation scheme, in contrast to the usual discrete representation, which makes the search of the optimum polynomial time dependent from the base field. In particular, we have implemented genetic and CHC algorithms using this representation as a proof of concept. Experimental results have been carried out employing codes over fields with two and eight elements, which suggests that evolutionary algorithms with our proposed permutation encoding are competitive with regard to existing methods in the literature. As a by-product, we have found and amended some inaccuracies in the Magma Computational Algebra System concerning the stored distances of some linear codes.

Keywords:Coding Theory, Mathematical Theory of Networks and Circuits, Information Theory Abstract: In the recent history of the theory of network coding the multi-shot network coding has been prove as a good alternative for the classical one-shot network theory which is managed by using block codes. To perform communications in this multi-shot context we have, among others, rank-metric convolutional codes and concatenated codes (using a convolutional code as an outer code and a rank-metric code as inner code). In this work we analyse their performance over the rank deficiency channel (described by Gilbert-Elliot channel model) in terms of the correction capabilities and the complexity of the two decoding schemes.

Keywords:Coding Theory, Information Theory Abstract: LDPC codes constructed from permutation matrices have recently attracted the interest of many researchers. A crucial point when dealing with such codes is trying to avoid cycles of short length in the associated Tanner graph, i.e. obtaining a possibly large girth. In this paper, we provide a framework to obtain constructions of such codes. We relate criteria for the existence of cycles of a certain length with some number-theoretic concepts, in particular with the so-called Sidon sets. In this way we obtain examples of LDPC codes with a certain girth. Finally, we extend our constructions to also obtain irregular LDPC codes.

Keywords:Aerospace and Avionic Systems, System Identification, Adaptive Control Abstract: Learn-to-Fly is a framework for incorporating learning methods into the development cycle of new aircraft. This paper provides an overview of the concept and summarizes some results from previous test campaigns. It will also discuss some perceived benefits of Learn-to-Fly and improvements to the procedure for more practical and widespread use. Ongoing efforts are needed to continue to develop the necessary underlying technologies for integration into this framework.

Keywords:Systems Biology, Machine Learning and Control Abstract: Controlling biological systems presents challenges not typically dealt with in traditional control theoretic approaches but also gives way to leniences not traditionally tolerated. Here, we present a holistic view to this new research area and current developments integrating various data-driven approaches for modeling and control.

Keywords:Machine Learning and Control, Artificial Intelligence, Control of Distributed Parameter Systems Abstract: We study the stationary points and local geometry of gradient play for stochastic games (SGs), where each agent tries to maximize its own total discounted reward by making decisions independently based on current state information which is shared between agents. Policies are directly parameterized by the probability of choosing a certain action at a given state. We show that Nash equilibria (NEs) and first-order stationary policies are equivalent in this setting by establishing a gradient domination condition for SGs. We characterize the structure of strict NEs and show that gradient play locally converges to strict NEs within finite steps. Further, for a subclass of SGs called Markov potential games, we prove that strict NEs are local maxima of the total potential function, thus locally stable under gradient play, and fully-mixed NEs are strict saddle points, thus unstable under gradient play.

Keywords:Feedback Control Systems, Optimal Control Abstract: We study the optimal landing problem for aerial vehicles under (1) a fixed landing time horizon or (2) the minimum time horizon. Both problems can be framed into solving the corresponding two-point boundary value problems. However, solving the boundary value problem in numerics is challenging, primarily due to the lack of good initial conditions. We present a space-marching scheme combined with machine learning techniques to provide good initial conditions for the boundary value problem solver. The algorithm greatly improves the solver's performance by increasing the success rate and reducing the computation time.

Keywords:Model Predictive Control, Machine Learning and Control, Robotics Abstract: Safe collision-free operation of autonomous systems, such as mobile robots in crowded, uncertain, only partially known environments, is challenging. We propose learning a collision-free corridor from demonstration via heteroscedastic Gaussian processes. We incorporate available deterministic obstacle information in the learning procedure to derive safety guarantees for the corridor. The learned passage is utilized in a model predictive path planning controller that steers the system safely through the environment. The achievable results are underlined in simulations considering a mobile robot.

Keywords:Model Predictive Control, Neural Networks, Machine Learning and Control Abstract: In this extended abstract we present a general procedure to quantify the performance of rectified linear unit (ReLU) neural network (NN) controllers that preserve the desirable properties of a designed model predictive control (MPC) scheme. First, by quantifying the approximation error between NN and MPC state-to-input mappings, we establish suitable conditions involving the worst-case error and the Lipschitz constant that guarantee the stability of the closed-loop system. Then, we develop an offline, mixed-integer (MI) optimization-based method to compute those quantities exactly, thus providing an analytical tool to certify the stability and performance of a ReLU-based approximation of an MPC control law.

Keywords:Model Predictive Control, Machine Learning and Control, Optimal Control Abstract: We show that the explicit solution of a data-driven predictive control scheme for deterministic LTI systems may not be as intractable as previously assumed. By comparing the structure of resulting parametric quadratic programs for the data-driven and classical model-based formulation, we analyze similarities and redundancies that ultimately lead to related structures of the respective explicit solutions. More precisely, some observations indicate a one-to-one relationship of these solutions that will be explored in future work. We illustrate this result by a thorough analysis of a simple example.

Keywords:Model Predictive Control, Robust and H-Infinity Control Abstract: We present a data-driven predictive control scheme for the stabilization of unknown LTI systems subject to process disturbances. The scheme uses Willems’ lemma for the prediction of future system trajectories and can be set up using only a priori measured input-output data of the disturbed system and an upper bound on its order. The main contribution is the introduction of a novel constraint tightening, which purely based on data guarantees closed-loop constraint satisfaction and recursive feasibility, even in the presence of process disturbances. Furthermore, a pre-stabilizing controller can be integrated into the scheme which ensures applicability for unstable systems.

Swiss Federal Institute of Technology (ETH) Zurich

Keywords:System Identification, Linear Systems, Model Predictive Control Abstract: The fundamental lemma by Willems and coauthors facilitates a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays an important role in data-driven simulation and control. Under the hood, the fundamental lemma works by applying a persistently exciting input to the system. This ensures that the Hankel matrix of resulting input/output data has the “right” rank, meaning that its columns span the entire subspace of trajectories. However, such binary rank conditions are known to be fragile in the sense that a small additive noise could already cause the Hankel matrix to have full rank. Therefore, in this extended abstract we present a robust version of the fundamental lemma. The idea behind the approach is to guarantee certain lower bounds on the singular values of the data Hankel matrix, rather than mere rank conditions. This is achieved by designing the inputs of the experiment such that the minimum singular value of a deeper input Hankel matrix is sufficiently large. This inspires a new quantitative and robust notion of persistency of excitation. The relevance of the result for data-driven control will also be highlighted through comparing the predictive control performance for varying degrees of persistently exciting data.

S.P. Timoshenko Institute of Mechanics of NAS of Ukraine

Keywords:Stability, Hybrid Systems, Linear Systems Abstract: This article proposes an approach to constructing the Lyapunov function for a linear coupled impulsive system consisting of two time-invariant subsystems. In this case, in contrast to various variants of small-gain stability conditions for coupled systems, the presence of the asymptotic stability property of independent subsystems is not assumed. To analyze the asymptotic stability of a coupled system, the direct Lyapunov method is used in combination with the discretization method. The periodic case and the case when the Floquet theory is not applicable at all are considered separately. The main results are illustrated with examples.

Keywords:Hybrid Systems, Computations in Systems Theory Abstract: Mathematical models of hybrid systems may exhibit Zeno behaviour, in which infinitely many discrete events occur in a finite time interval. Although this may be considered physically unrealistic, since Zeno behaviour is useful in modelling real-world systems, it is important for computational tools to be able to handle it. In particular, verification tools should use a well-defined semantics under which the Zeno behaviour is effectively and rigorously computable. In this paper, we give a semantics for systems with events similar to mechanical impacts and show that computation of the behaviour is possible, even beyond the Zeno time. If we allow perturbations outside this class of impacting systems, then it may not be possible to handle Zeno behaviour in a realistic way.

S.P. Timoshenko Institute of Mechanics of NAS of Ukraine

Keywords:Stability, Hybrid Systems, Infinite Dimensional Systems Theory Abstract: We recall two recently published approaches to study stability properties of nonlinear infinite dimensional impulsive systems and apply them to finite and infinite dimensional systems. Both approaches cover the case, when discrete and continuous dynamics are not stable simultaneously. We illustrate these approaches by means of several examples. In particular we demonstrate that our approaches can be used in situations where the existing results cannot be applied. In particular, we will derive sufficient conditions for the ISS property of a linear and spatially non-homogeneous parabolic system with impulsive actions.

Keywords:Linear Systems, Computations in Systems Theory Abstract: We propose a model reduction approach for singular linear switched systems in discrete time with a fixed mode sequence based on a balanced truncation reduction method for linear time-varying discrete-time systems. The key idea is to use the one-step map to find an equivalent time-varying system with an identical input-output behavior, and then adapt available balance truncation methods for (discrete) time-varying systems. The proposed method is illustrated with a low-dimensional academic example.

Keywords:Optimal Control, Linear Systems Abstract: In this abstract the finite horizon linear quadratic optimal control problem with constraints on the terminal state for switched differential algebraic equations is considered. Furthermore, we seek for an optimal solution that is impulse-free. In order to solve the problem, a non standard finite horizon problem for non-switched DAEs is considered. Necessary and sufficient conditions on the initial value for solvability of this non standard problem are stated. Based on these results a sequence of subspaces can be defined which lead to necessary and sufficient conditions for solvability of the finite horizon optimal control problem for switched DAEs.

Keywords:Delay Systems, Stability, Linear Systems Abstract: Recently, a number of Lyapunov matrix-based necessary and sufficient stability tests which require a finite set of operations to be verified were presented for linear time-invariant time delay systems, see Egorov et al. (2017), Gomez et al. (2019) and Bajodek et al. (2021). Motivated by those works, in this contribution we revisit the early paper Medvedeva & Zhabko (2015) and develop the idea presented there to construct a necessary and sufficient finite stability test for a single-delay system as well. The approach relies on a simple piecewise linear approximation of the arguments of Lyapunov--Krasovskii functionals based on the Lyapunov matrix, and shows its competitiveness at least in the case of small delays.

Keywords:Delay Systems, Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory Abstract: Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE representation has limitations. In network models with delay, the delayed channels are typically low-dimensional and accounting for this heterogeneity is challenging in the DDE framework. In addition, DDEs cannot be used to model difference equations. In this paper, we examine alternative representations for networked systems with delay and provide formulae for conversion between representations. First, we examine the Differential-Difference (DDF) formulation which allows us to represent the low-dimensional nature of delayed information. Next, we consider the coupled ODE-PDE framework and extend this to the recently developed Partial-Integral Equation (PIE) representation. The PIE framework has the advantage that it allows the H infinity-optimal estimation and control problems to be solved efficiently using the recently developed software package PIETOOLS. In each case, we consider a very general class of networks, specifically accounting for four sources of delay - state delay, input delay, output delay, and process delay. Finally, we use a scalable network model of temperature control to show that the use of the DDF/PIE formulation allows for optimal control of a network with 40 users, 80 states, 40 delays, 40 inputs, and 40 disturbances.

Keywords:Nonlinear Filtering and Estimation, Infinite Dimensional Systems Theory Abstract: A semilinear infinite-dimensional system with a disturbance input is considered. The observation is modelled by an affine linear map with a different disturbance. An observer, based on the extended Kalman filter (EKF), is constructed and its well-posedness is proven under mild conditions. Moreover, local exponential stability of the error dynamics is shown. Thus, if the error in the initial condition is small enough, the estimation error converges to zero. This is a first generalization of the EKF to infinite-dimensional systems. Since only detectability, not observability, is assumed, this result is new even for finite-dimensional systems. An implementation is provided for a magnetic drug-delivery system and numerical results support the effectiveness of the observer.

Keywords:Optimal Control, Nonlinear Systems and Control, Machine Learning and Control Abstract: In this paper, how to successfully and efficiently condition a target population of agents towards consensus is discussed. To overcome the curse of dimensionality, the mean field formulation of the consensus control problem is considered. Although such formulation is designed to be independent of the number of agents, it is feasible to solve only for moderate intrinsic dimensions of the agents space. For this reason, the solution is approached by means of a Boltzmann procedure, i.e. quasi-invariant limit of controlled binary interactions as approximation of the mean field PDE. The need for an efficient solver for the binary interaction control problem motivates the use of a supervised learning approach to encode a binary feedback map to be sampled at a very high rate. A gradient augmented feedforward neural network for the Value function of the binary control problem is considered and compared with direct approximation of the feedback law.

Keywords:Control of Distributed Parameter Systems, Stochastic Modeling and Stochastic Systems Theory, Stability Abstract: Recently, qualitative methods for finite-dimensional boundary state-feedback control were introduced for stochastic 1D parabolic PDEs. In this paper, we present constructive and efficient design conditions for state-feedback control of stochastic 1D heat equations driven by a nonlinear multiplicative noise. We consider the Neumann actuation and apply modal decomposition with either trigonometric or polynomial dynamic extension. The controller design employs a finite number of comparatively unstable modes. We provide mean-square L^2 stability analysis of the full-order closed-loop system, where we employ It^{o}'s formula, leading to linear matrix inequality (LMI) conditions for finding the controller gain and as large as possible noise intensity for the mean-square stabilizability. We prove that the LMIs are always feasible for small enough noise intensity. We further show that in the case of linear multiplicative noise, the system is stabilizable for noise intensities that guarantee the stabilizability of the stochastic finite-dimensional part of the closed-loop system. Numerical simulations illustrate the efficiency of our method.

Keywords:Networked Control Systems, Systems on Graphs, Large Scale Systems Abstract: We consider the prototypical networked control problem of distributed consensus in networks of agents with integrator dynamics of order two or higher (n>=2). We assume all feedback to be localized in the sense that each agent has a bounded number of neighbors and consider a scaling of the network through the addition of agents. We show that standard consensus algorithms that rely on relative state feedback and fixed gains can be subject to scale fragilities, meaning that stability is lost as the network grows. For high-order agents (n>=3), we prove that no consensus algorithm is what we term scalably stable. That is, while a given algorithm may allow a small network to converge, it causes instability if the network grows beyond a certain finite size. This holds in families of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size (equivalently, non-expanding graphs). For second-order consensus (n = 2), we prove that the same scale fragility applies to classes of directed graphs that have a complex Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). We derive algebraic conditions for the affected graphs, and discuss how the consensus algorithm can be modified to retrieve scalable stability.

Keywords:Networked Control Systems, Stochastic Modeling and Stochastic Systems Theory, Discrete Event Systems Abstract: We derive fundamental limitations on the performance that can be achieved by intrinsic averaging algorithms in open multi-agent systems, which are systems subject to random arrivals and departures of agents. Each agent holds an intrinsic value, and their objective is to collaboratively estimate the average of the values of the agents presently in the system. We provide a lower bound on the expected Mean Square Error for such algorithms where we assume that the size of the system remains constant. Our derivation is based on the error obtained with an algorithm that achieves optimal performance under a set of restrictions on the way agents obtain information about one another. This error represents a lower bound on the error obtained with any other algorithm that can be implemented under the same restrictions. This approach is then applied to derive lower bounds on the performance of the well-known Gossip algorithm by considering restrictions that allow implementing it.

Keywords:Optimization : Theory and Algorithms, Systems on Graphs, Discrete Event Systems Abstract: The resource allocation problem consists in the optimal distribution of a budget between a group of agents. We consider a version of this optimization problem in open systems where agents can be replaced, resulting in variations of the budget and the total cost function to be minimized. We analyze the performance of the Random Coordinate Descent algorithm (RCD) in that setting using natural performance indexes which are related to those used in online optimization. We show that, in a simple setting, both the accumulated error obtained from using the RCD as compared to the optimal solution and the accumulated gain obtained from using the RCD instead of not collaborating grow linearly with the number of iterations considered for the computation, so that in expectation an error cannot be avoided, but remains bounded.

Keywords:Networked Control Systems, Nonlinear Systems and Control, Stability Abstract: Dynamic consensus is a property of networked systems that pertains to the case in which all the interconnected systems synchronise their motions and a collective behaviour arises. If the coupling strength is large such behaviour may be modelled by a single system, but if it is weak, the behaviour is best modelled by a reduced-order network. For networks of homogeneous Stuart-Landau oscillators under weak coupling, we characterise the dimension and dynamics of such reduced-order network in function of the coupling strength.

Keywords:Nonlinear Systems and Control, Stability, Feedback Control Systems Abstract: This paper proposes staircase-like feedback laws for control of human infectious diseases based on the SIQR model in the situation where vaccine administration alone cannot be sufficient. The control inputs consisting of isolation, contact regulation, and vaccination are designed to achieve input-to-state stability (ISS) with respect to uncertainty that includes perturbation of immigrants and newborns. The designed controller is semi-global so that the ISS guarantee can be obtained on an arbitrary large compact set in the state space. To accomplish the practical control implementation, this paper qualifies a simple separable function as a control Lyapunov function.

Keywords:Nonlinear Systems and Control Abstract: We study the dynamical behaviour of the SIR network model at individual nodes. In two particular cases of a network consisting of only two nodes, we show how this behaviour differs from the epidemic outbreak and monotonic decreasing trend that occur in the homogeneous case. The first case deals with a network in which contact is only direct from one node to another, while the second treats a network with all contacts equal to one. The result shown for this scenario remains true by continuity for all those networks sufficiently close in parameter space.