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Chlorine is a widely used disinfectant and proxy for water quality (WQ) monitoring in water distribution networks (WDN). Chlorine-based WQ regulation and control aims to maintain pathogen-free water. Chlorine residual evolution within WDN is commonly modeled using the typical single-species decay and reaction dynamics that account for network-wide, spatiotemporal chlorine concentrations only. Prior studies have proposed more advanced and accurate descriptions via multi-species dynamics. This paper presents a host of novel state-space, control-theoretic representations of multi-species water quality dynamics. These representations describe decay, reaction, and transport of chlorine and a fictitious reactive substance to reflect realistic complex scenarios in WDN. Such dynamics are simulated over space- and time-discretized grids of the transport partial differential equation and the nonlinear reaction ordinary differential equation. To that end, this paper (i) provides a full description on how to formulate a high fidelity model-driven state-space representation of the multi-species water quality dynamics and (ii) investigates the applicability and performance of different Eulerian-based schemes (Lax-Wendroff, backward Euler, and Crank- Nicolson) and Lagrangian-based schemes (method of characteristics) in contrast with EPANET and its EPANET-MSX extension. Numerical case studies reveal that the Lax-Wendroff scheme and method of characteristics outperform other schemes with reliable results under reasonable assumptions and limitations.

In this survey, we describe controlled interacting particle systems (CIPS) to approximate the solution of the optimal filtering and the optimal control problems. Part I of the survey is focussed on the feedback particle filter (FPF) algorithm, its derivation based on optimal transportation theory, and its relationship to the ensemble Kalman filter (EnKF) and the conventional sequential importance sampling-resampling (SIR) particle filters. The central numerical problem of FPF -- to approximate the solution of the Poisson equation -- is described together with the main solution approaches. An analytical and numerical comparison with the SIR particle filter is given to illustrate the advantages of the CIPS approach. Part II of the survey is focussed on adapting these algorithms for the problem of reinforcement learning. The survey includes several remarks that describe extensions as well as open problems in this subject.

The fundamental lemma by Jan C. Willems and co-workers, which is deeply rooted in behavioral systems theory, has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style paper combines recent insights into stochastic and descriptor-system formulations of the lemma to further extend and broaden the formal basis for behavioral theory of stochastic linear systems. We show that series expansions -- in particular Polynomial Chaos Expansions (PCE) of $L^2$-random variables, which date back to Norbert Wiener's seminal work -- enable equivalent behavioral characterizations of linear stochastic systems. Specifically, we prove that under mild assumptions the behavior of the dynamics of the $L^2$-random variables is equivalent to the behavior of the dynamics of the series expansion coefficients and that it entails the behavior composed of sampled realization trajectories. We also illustrate the short-comings of the behavior associated to the time-evolution of the statistical moments. The paper culminates in the formulation of the stochastic fundamental lemma for linear (descriptor) systems, which in turn enables numerically tractable formulations of data-driven stochastic optimal control combining Hankel matrices in realization data (i.e. in measurements) with PCE concepts., 31 pages, 7 figures

Convergent sequences of real numbers play a fundamental role in many different problems in system theory, e.g., in Lyapunov stability analysis, as well as in optimization theory and computational game theory. In this survey, we provide an overview of the literature on convergence theorems and their connection with Fejer monotonicity in the deterministic and stochastic settings, and we show how to exploit these results.

The purpose of this tutorial is to give a brief introduction to linear quantum control systems. The mathematical model of linear quantum control systems is presented first, then some fundamental control-theoretic notions such as stability, controllability and observability are given, which are closely related to several important concepts in quantum information science such as decoherence-free subsystems, quantum non-demolition variables, and back-action evasion measurements. After that, quantum Gaussian states are introduced, in particular, an information-theoretic uncertainty relation is presented which often gives a better bound for mixed Gaussian states than the well-known Heisenberg uncertainty relation. The quantum Kalman filter is presented for quantum linear systems, which is the quantum analogy of the Kalman filter for classical (namely, non-quantum-mechanical) linear systems. The quantum Kalman canonical decomposition for quantum linear systems is recorded, and its application is illustrated by means of a recent experiment. As single- and multi-photon states are useful resources in quantum information technology, the response of quantum linear systems to these types of input is presented. Finally, coherent feedback control of quantum linear systems is briefly introduced, and a recent experiment is used to demonstrate the effectiveness of quantum linear systems and networks theory.dback control of quantum linear systems is briefly introduced, and a recent experiment is used to demonstrate the effectiveness of quantum linear systems and networks theory., Comment: 55 pages, 4 figures, to appear in Annual Reviews in Control

The Stefan PDE system is a representative model for thermal phase change phenomena, such as melting and solidification, arising in numerous science and engineering processes. The mathematical description is given by a Partial Differential Equation (PDE) of the temperature distribution defined on a spatial interval with a moving boundary, where the boundary represents the liquid-solid interface and its dynamics are governed by an Ordinary Differential Equation (ODE). The PDE-ODE coupling at the boundary is nonlinear and creates a significant challenge for state estimation with provable convergence and robustness. This tutorial article presents a state estimation method based on PDE backstepping for the Stefan system, using measurements only at the moving boundary. PDE backstepping observer design generates an observer gain by employing a Volterra transformation of the observer error state into a desirable target system, solving a Goursat-form PDE for the transformation's kernel, and performing a Lyapunov analysis of the target observer error system. The observer is applied to models of problems motivated by climate change and the need for renewable energy storage: a model of polar ice dynamics and a model of charging and discharging in lithium-ion batteries. The numerical results for polar ice demonstrate a robust performance of the designed estimator with respect to the unmodeled salinity effect in sea ice. The results for an electrochemical PDE model of a lithium-ion battery with a phase transition material show the elimination of more than 15 \% error in State-of-Charge estimate within 5 minutes even in the presence of sensor noise., 45 pages, 18 figures, submitted to Annual Reviews in Control

Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is therefore to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks., Annual Reviews in Control, Accepted, Oct. 1st

Testing is a crucial control mechanism in the beginning phase of an epidemic when the vaccines are not yet available. It enables the public health authority to detect and isolate the infected cases from the population, thereby limiting the disease transmission to susceptible people. However, despite the significance of testing in epidemic control, the recent literature on the subject lacks a control-theoretic perspective. In this paper, an epidemic model is proposed that incorporates the testing rate as a control input and differentiates the undetected infected from the detected infected cases, who are assumed to be removed from the disease spreading process in the population. After estimating the model on the data corresponding to the beginning phase of COVID-19 in France, two testing policies are proposed: the so-called best-effort strategy for testing (BEST) and constant optimal strategy for testing (COST). The BEST policy is a suppression strategy that provides a minimum testing rate that stops the growth of the epidemic when implemented. The COST policy, on the other hand, is a mitigation strategy that provides an optimal value of testing rate minimizing the peak value of the infected population when the total stockpile of tests is limited. Both testing policies are evaluated by their impact on the number of active intensive care unit (ICU) cases and the cumulative number of deaths for the COVID-19 case of France., 50 pages, 26 figures

For his work in the economics of climate change, Professor William Nordhaus was a co-recipient of the 2018 Nobel Memorial Prize for Economic Sciences. A core component of the work undertaken by Nordhaus is the Dynamic Integrated model of Climate and Economy, known as the DICE model. The DICE model is a discrete-time model with two control inputs and is primarily used in conjunction with a particular optimal control problem in order to estimate optimal pathways for reducing greenhouse gas emissions. In this paper, we provide a tutorial introduction to the DICE model and we indicate challenges and open problems of potential interest for the systems and control community., Comment: 18 pages, 4 figures

We investigate adaptive strategies to robustly and optimally control the COVID-19 pandemic via social distancing measures based on the example of Germany. Our goal is to minimize the number of fatalities over the course of two years without inducing excessive social costs. We consider a tailored model of the German COVID-19 outbreak with different parameter sets to design and validate our approach. Our analysis reveals that an open-loop optimal control policy can significantly decrease the number of fatalities when compared to simpler policies under the assumption of exact model knowledge. In a more realistic scenario with uncertain data and model mismatch, a feedback strategy that updates the policy weekly using model predictive control (MPC) leads to a reliable performance, even when applied to a validation model with deviant parameters. On top of that, we propose a robust MPC-based feedback policy using interval arithmetic that adapts the social distancing measures cautiously and safely, thus leading to a minimum number of fatalities even if measurements are inaccurate and the infection rates cannot be precisely specified by social distancing. Our theoretical findings support various recent studies by showing that 1) adaptive feedback strategies are required to reliably contain the COVID-19 outbreak, 2) well-designed policies can significantly reduce the number of fatalities compared to simpler ones while keeping the amount of social distancing measures on the same level, and 3) imposing stronger social distancing measures early on is more effective and cheaper in the long run than opening up too soon and restoring stricter measures at a later time., Comment: This is the accepted version of the paper in Annual Reviews in Control, 2020

Recent years have witnessed a significant trend towards filling the gap between Social Network Analysis (SNA) and control theory. This trend was enabled by the introduction of new mathematical models describing dynamics of social groups, the development of algorithms and software for data analysis and the tremendous progress in understanding complex networks and multi-agent systems (MAS) dynamics. The aim of this tutorial is to highlight a novel chapter of control theory, dealing with dynamic models of social networks and processes over them, to the attention of the broad research community. In its first part (Proskurnikov & Tempo, 2017), we have considered the most classical models of social dynamics, which have anticipated and to a great extent inspired the recent extensive studies on MAS and complex networks. This paper is the second part of the tutorial, and it is focused on more recent models of social processes that have been developed concurrently with MAS theory. Future perspectives of control in social and techno-social systems are also discussed.