Model order reduction aims to determine a low-order approximation of high-order models with least possible approximation errors. For application to physical systems, it is crucial that the reduced order model (ROM) is robust to any disturbance that acts on the full order model (FOM) – in the sense that the output of the ROM remains a good approximation of that of the FOM, even in the presence of such disturbances. In this work, we present a framework for online model order reduction for a class of continuous-time linear systems that ensures this property for any ℒ_2 disturbance. Apart from robustness to disturbances in this sense, the proposed framework also displays other desirable properties for model order reduction: (1) a provable bound on the error defined as the L_2 norm of the difference between the output of the ROM and FOM, (2) preservation of stability, (3) compositionality properties and a provable error bound for arbitrary interconnected systems, (4) a provable bound on the output of the FOM when the controller designed for the ROM is used with the FOM, and finally, (5) compatibility with existing approaches such as balanced truncation and moment matching. Property (4) does not require computation of any gap metric and property (5) is beneficial as existing approaches can also be equipped with some of the preceding properties. The theoretical results are corroborated on numerical case studies, including on a building model.

The paper deals with the setting where two viruses (say virus 1 and virus 2) coexist in a population, and they are not necessarily mutually exclusive, in the sense that infection due to one virus does not preclude the possibility of simultaneous infection due to the other. We develop a coupled bi-virus susceptible-infected-susceptible (SIS) model from a 4n-state Markov chain model, where n is the number of agents (i.e., individuals or subpopulation) in the population. We identify a sufficient condition for both viruses to eventually die out, and a sufficient condition for the existence, uniqueness and asymptotic stability of the endemic equilibrium of each virus. We establish a sufficient condition and multiple necessary conditions for local exponential convergence to the boundary equilibrium (i.e., one virus persists, the other one dies out) of each virus. Under mild assumptions on the healing rate, we show that there cannot exist a coexisting equilibrium where for each node there is a nonzero fraction infected only by virus 1; a nonzero fraction infected only by virus 2; but no fraction that is infected by both viruses 1 and 2. Likewise, assuming that healing rates are strictly positive, a coexisting equilibrium where for each node there is a nonzero fraction infected by both viruses 1 and 2, but no fraction is infected only by virus 1 (resp. virus 2) does not exist. Further, we provide a necessary condition for the existence of certain other kinds of coexisting equilibria. We show that, unlike the competitive bivirus model, the coupled bivirus model is not monotone. Finally, we illustrate our theoretical findings using an extensive set of in-depth simulations.

In this paper we consider a Deterministic Annealing (DA) framework for optimizing reservoir placement and associated water transport costs in Water Distribution Networks (WDNs). We introduce a capacity-constrained formulation to ensure that each junction in the network serves a limited number of demand points, preventing overload and improving system stability. Moreover, we propose update rules for secondary Lagrange multipliers corresponding to capacity inequality constraints. Finally, we extend the framework to utilize real-world pipe networks and validate our algorithm through a case study on the Modena, Italy network. Experimental results demonstrate that our DA-based method achieves lower operational costs compared to previous approach.

Default risk spreading processes in inter-banking networks are commonly viewed as contagion processes, with inter-bank loans as a direct spreading channel and overlapping investment portfolios as an indirect channel. In this paper, we propose a multi-layer network default risk contagion model to incorporate additional panic contagions in the networks of depositors as a novel augmentation of previous models, allowing for the direct characterization of the “bank run” phenomenon, where many depositors simultaneously issue withdrawal requests. Our model is calibrated with post-COVID pandemic data, accounting for macroeconomic factors such as fluctuating interest rates and asset bubbles. Using system identification methods, we analyze relationships between federal interest rates and market prices, and formulate an optimal control problem to mitigate default risk via liquidity ratio requirements in stress tests. Long-term simulation results are presented to reveal threshold structures under varying contagion parameters.

We study finite-time performance of a recently proposed distributed dual subgradient (DDSG) method for convex constrained multi-agent optimization problems. The algorithm enjoys performance guarantees on the last primal iterate, as opposed to those derived for ergodic means for vanilla DDSG algorithms. Our work improves the recently published convergence rate of $\Ocal(\log T/\sqrt{T})$ with decaying step-sizes to $\Ocal(1/\sqrt{T})$ with constant step-size on a metric that combines suboptimality and constraint violation. We then numerically evaluate the algorithm on three grid optimization problems. Namely, these are tie-line scheduling in multi-area power systems, coordination of distributed energy resources in radial distribution networks, and joint dispatch of transmission and distribution assets. The DDSG algorithm applies to each problem with various relaxations and linearizations of the power flow equations. The numerical experiments illustrate various properties of the DDSG algorithm--comparison with vanilla DDSG, impact of the number of agents, and why Nesterov-style acceleration fails in DDSG settings.

We present an analysis of epidemiological compartment models that explicitly capture the dynamics of asymptomatic but infectious individuals. Our models can be viewed as an extension to classic SIR models, to which a distinct Asymptomatic compartment is added. We discuss both a group compartment model capturing a Susceptible-Asymptomatic-Infected-Recovered-Susceptible (SAIRS) epidemic process, and also introduce and evaluate SAIRS dynamics evolving over networks. We investigate equilibria and stability properties that include both disease-free and endemic equilibria states for these models, providing sufficient conditions for convergence to these equilibria. Model parameter estimation results based on local test-site and Peoria county clinic data are given, and a number of simulations illustrating the effects of asymptomatic-infected individuals and network structure on the spread and/or persistence of the disease are presented.

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We consider a discrete-time dynamical system over a discrete state-space, which evolves according to a structured Markov model called Bernoulli autoregressive (BAR) model. Our goal is to obtain sample complexity bounds for the problem of estimating the parameters of this model using an indirect maximum likelihood estimator. Our sample complexity bounds exploit the structure of the BAR model and are established using concentration inequalities for random matrices and Lipschitz functions.

Diagnostic tests have proven to be a critical tool in controlling the progression of a virus. In this paper, we formulate the testing of a homogeneous population as an optimal control problem. The population state, given by the distribution of agents’ viral states in a compartmental model, is assumed to be unknown. Information regarding the population state is provided via noisy tests, which are allocated from a stockpile whose size is updated via a stochastic process. The objective of the control problem is to allocate tests so as to minimize uncertainty of the underlying population state over a finite horizon. As such, the control problem is cast as a POMDP with a negative entropy reward function. We study various heuristic policies and investigate conditions under which each heuristic performs best.

In this paper, we incorporate wake steering models into a multiobjective wind farm model for improving power extraction, specifically under the framework of handling maintenance of individual turbines. We expand upon a previous heuristic method to find axial induction factors for the far-field wake problem and using the results of the algorithm as parameters for a model predictive control framework. Simulation results are discussed, demonstrating improved power output over algorithms that do not account for maintenance.

When facing the global health threat posed by an infectious disease, predictive mathematical models are crucial not only to understand and forecast the epidemic evolution, but also to plan effective control strategies that contrast the disease and its spread in the population. This tutorial aims to give a broad overview of the fundamental developments enabled by systems-and-control methodologies in modelling and controlling epidemiological dynamics across scales, from infection dynamics within hosts to contagion dynamics between hosts. The first part is focused on modelling and control of infectious diseases in the host, capturing the dynamic interplay between pathogens and the immune system, and discussing control strategies to design tailored therapies and treatments to optimally clear the infection. The second part deals with the spread of contagion between hosts: epidemic dynamics are modelled resorting to networked systems where the nodes represent individuals and the links represent interactions that can lead to contagion, and a comparison to compartmental models is carried out. The third part surveys multi-scale models and multi-pronged approaches to contrast the spread of infectious diseases: a holistic perspective is adopted, including behavioural and socio-economic aspects along with public health issues, to discuss optimal epidemic control across scales.

This special section of the SIAM Journal on Control and Optimization (SICON) addresses the fundamental challenges inherent in the mathematical modeling, analysis, and control of epidemics. The ongoing COVID-19 pandemic has brought into the spotlight the critical importance of understanding complex epidemic processes. Yet the modeling, analysis, estimation, and control of these processes presents several formidable challenges. Epidemics inherently exhibit nonlinear dynamics as they spread through populations, requiring analytical techniques that can accurately capture both the transient and the steady-state aspects of the disease. Furthermore, careful attention must be paid to the resolution at which to model the epidemics, ranging from coarse mass-action models to metapopulation and individual-level models. Each of these approaches provides different insights and challenges for analysis and control. Models that operate at finer resolutions will also come with an increase in complexity, necessitating tractable abstractions ranging from mean-field models to deterministic and stochastic ODEs and PDEs. In the context of estimation of epidemics, the key challenge is to determine the important parameters of the epidemic (often in real time) based on data gathered from the affected population, in conjunction with dynamical models of the spreading process. Finally, formulating techniques to optimally control and address an ongoing epidemic (through either a centralized intervention, decentralized incentive mechanisms, or resource allocation policies) remains a critical challenge. The special section gathers contributions from the intersection of the fields of systems and control theory and the mathematical study of epidemic spread processes. A total of 16 papers represents a wide range of topics in modeling, identification, dynamic analysis, control, and optimization. Regarding modeling problems, the section covers contributions on Polya contagion networks, networked bivirus epidemic models, and age-differentiated compartmental models. For dynamic analysis, the section highlights contributions on explicit solutions and control design for simplified models, convergence and equilibria analysis, as well as the role of delays and saturations in closed loop settings. For identification and estimation problems, contributions are offered on the identifiability of model parameters and on parameter estimation using limited measurements. Most of the contributed articles focus on control design, optimization, and game theoretic problems. Centralized and distributed strategies are proposed. Actuation mechanisms include edge deletion, test allocation, optimal incentives, optimal switching between lockdown and opening the economy, screening, and curing policies. We believe this special section presents an excellent cross section of current research and hope that it will be of great interest to the broad readership of SICON. We offer our deepest thanks to all authors who submitted their work and to all reviewers who contributed their time and energy to the peer-review process. We would also like to thank the proficient and timely editorial support provided by Brian Fauth and the insightful and gracious advice that we received from the SICON Editor-in-Chief George Yin. Carolyn Beck University of Illinois at Urbana-Champaign Francesco Bullo University of California, Santa Barbara Giacomo Como Politecnico di Torino Kimon Drakopoulos University of Southern California Dang H. Nguyen University of Alabama Cameron Nowzari George Mason University Victor M. Preciado University of Pennsylvania Shreyas Sundaram Purdue University, Guest editors

Software Defined Networking (SDN) is a recent paradigm in telecommunication networks that disentangles data and control planes and brings more flexibility to the network. The Controller Placement (CP) problem in SDN, which typically has a specific optimality criteria, is one of the primary problems in the SDN systems. Dynamic Controller Placement (DCP) enables a placement that is adaptable to inherent variability in network components. DCP has gained much attention in recent years, yet most solutions proposed in the literature cannot be implemented in real-time, which is a critical concern especially in UAV/drone based SDN networks where mobility is high and real-time updates are necessary. As conventional methods fail to be relevant to such scenarios, we propose a real-time control placement (RCP) algorithm. Namely, we propose a temporal clustering algorithm that provides real-time solutions for DCP, based on a control theoretic framework that is exponentially stable and converges to optimal placement of controllers. RCP has linear ${\mathcal{O}}(N)$ iteration complexity with respect to the underlying network size (N), and also leverages the maximum entropy principle from information theory. This approach results in high quality solutions that are practically immune from getting stuck in poor local optima, which is a serious drawback conventional methods. We compare our work with a frame-by-frame approach and show its superiority, both in terms of speed and incurred cost, via simulations. According to our simulations RCP can be up to 25 times faster than the conventional frame-by-frame methods.

Wind turbine arrays can be viewed as large coupled networks, wherein wake effects limit the available power extraction of turbines downstream. In this paper, we incorporate wake steering and time dependent wind estimation models into a multiobjective wind farm control problem for improving power extraction. We further aim to mitigate the effects of turbulence and power spikes caused by wind passing through upstream turbines. We expand upon a previous heuristic method for the far-field wake problem and apply the algorithm on a model predictive control framework. Simulation results are given, demonstrating improved power output as compared to algorithms that do not incorporate wake steering or wind estimation models.

Motivated by broad applications in various fields of engineering, we study a network resource allocation problem where the goal is to optimally allocate a fixed quantity of the resources over a network of nodes. We consider large scale networks with complex interconnection structures, thus any solution must be implemented in parallel and based only on local data resulting in a need for distributed algorithms. In this article, we study a distributed Lagrangian method for such problems. By utilizing the so-called distributed subgradient methods to solve the dual problem, our approach eliminates the need for central coordination in updating the dual variables, which is often required in classic Lagrangian methods. Our focus is to understand the performance of this distributed algorithm when the number of resources is unknown and may be time-varying. In particular, we obtain an upper bound on the convergence rate of the algorithm to the optimal value, in expectation, as a function of network topology. The effectiveness of the proposed method is demonstrated by its application to the economic dispatch problem in power systems, with simulations completed on the benchmark IEEE-14 and IEEE-118 bus test systems.

We provide corrections to two typos (in Assumption 3 and Corollary 2 ) and three misstatements (in Corollary 1 and Theorems 3 and 4 ) of [1] .

Epidemic processes are used commonly for modeling and analysis of biological networks, computer networks, and human contact networks. The idea of competing viruses has been explored recently, motivated by the spread of different ideas along different social networks. Previous studies of competitive viruses have focused only on two viruses and on static networks. In this paper, we consider multiple competing viruses over static and dynamic graph structures, and investigate the eradication and propagation of diseases in these systems. Stability analysis for the class of models we consider is performed and an antidote control technique is proposed.

We provide corrections to two typos (in Assumption 3 and Corollary 2) and three misstatements (in Corollary 1 and Theorems 3 and 4) of [1].

We derive two upper bounds for the probability of deviation of a vector-valued Lipschitz function of a collection of random variables from its expected value. The resulting upper bounds can be tighter than bounds obtained by a direct application of a classical theorem due to Bobkov and Götze.

We present an epidemiological compartment model, SAIR(S), that explicitly captures the dynamics of asymptomatic infected individuals in an epidemic spread process. We first present a group model and then discuss networked versions. We provide an investigation of equilibria and stability properties for these models, and present simulation results illustrating the effects of asymptomatic-infected individuals on the spread of the disease. We also discuss local isolation effects on the epidemic dynamics in terms of the networked models. Finally, we provide initial parameter estimation results based on simple least-squares approaches and local test-site data.