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Time-varying optimization problems are central to many engineering applications, where performance metrics and system constraints evolve dynamically with time. A number of algorithms have been proposed in recent years to solve such problems; a common feature of all these methods is that they implicitly require precise knowledge of the temporal variability of the solutions in order to exactly track the optimizers. In this paper, we seek to lift these stringent assumptions. Our main result is a fundamental characterization, showing that an algorithm can track an optimal trajectory if and only if it contains a model of the temporal variability of the problem. We refer to this concept to as the internal model principle of time-varying optimization. By recasting the optimization objective as a nonlinear regulation problem and using tools from center manifold theory, we provide necessary and sufficient conditions both for an optimization algorithm to achieve exact asymptotic tracking and for such an algorithm to exist. We illustrate the applicability of the approach numerically on both synthetic problems as well as practical problems in transportation.

Feedback optimization is an increasingly popular control paradigm to optimize dynamical systems, accounting for control objectives that concern the system's operation at the steady-state. Existing feedback optimization techniques heavily rely on centralized system and controller architectures, and thus suffer from scalability and privacy issues when systems become large-scale. In this paper, we propose a distributed architecture for feedback optimization inspired by distributed gradient descent, whereby each agent updates its local control variable by combining the average of its neighbors with a local negative gradient step. Under convexity and smoothness assumptions for the cost, we establish convergence of the control method to a critical optimization point. By reinforcing the assumptions to restricted strong convexity, we show that our algorithm converges linearly to a neighborhood of the optimal point, where the size of the neighborhood depends on the choice of the stepsize. Simulations corroborate the theoretical results., Comment: 8 pages, 4 figures

Governments, regulatory bodies, and manufacturers are proposing plans to accelerate the adoption of electric vehicles (EVs), with the goal of reducing the impact of greenhouse gases and pollutants from internal combustion engines on human health and climate change. In this context, the paper considers a scenario where ride-sharing enterprises utilize a 100%-electrified fleet of vehicles, and seeks responses to the following key question: How can renewable-based EV charging be maximized without disrupting the quality of the ride-sharing services? We propose a new mechanism to promote EV charging during hours of high renewable generation, and we introduce the concept of charge request, which is issued by a power utility company. Our mechanism is inspired by a game-theoretic approach where the power utility company proposes incentives and the ride-sharing platform assigns vehicles to both ride and charge requests; the bargaining mechanism leads to prices and EV assignments that are aligned with the notion of Nash equilibria. Numerical results show that it is possible to shift the EV charging during periods of high renewable generation and adapt to intermittent generation while minimizing the impact on the quality of service. The paper also investigates how the users' willingness to ride-share affects the charging strategy and the quality of service., Comment: Article published in Scientific Reports

The exact pole placement problem concerns computing a feedback gain that will assign the poles of a system, controlled via static state feedback, at a set of pre-specified locations. This is a classic problem in feedback control and numerous methodologies have been proposed in the literature for cases where a model of the system to control is available. In this paper, we study the problem of computing feedback gains for pole placement (and, more generally, eigenstructure assignment) directly from experimental data. Interestingly, we show that the closed-loop poles can be placed exactly at arbitrary locations without relying on any model description but by using only finite-length trajectories generated by the open-loop system. In turn, these findings imply that classical control objectives, such as feedback stabilization or meeting transient performance specifications, can be achieved without first identifying a system model. Numerical experiments demonstrate the benefits of the data-driven pole-placement approach as compared to its model-based counterpart.

Motivated by perception-based control problems in autonomous systems, this paper addresses the problem of developing feedback controllers to regulate the inputs and the states of a dynamical system to optimal solutions of an optimization problem when one has no access to exact measurements of the system states. In particular, we consider the case where the states need to be estimated from high-dimensional sensory data received only at discrete time intervals. We develop a sampled-data feedback controller that is based on adaptations of a projected gradient descent method, and that includes neural networks as integral components to estimate the state of the system from perceptual information. We derive sufficient conditions to guarantee (local) input-to-state stability of the control loop. Moreover, we show that the interconnected system tracks the solution trajectory of the underlying optimization problem up to an error that depends on the approximation errors of the neural network and on the time-variability of the optimization problem; the latter originates from time-varying safety and performance objectives, input constraints, and unknown disturbances. As a representative application, we illustrate our results with numerical simulations for vision-based autonomous driving., Comment: This is an extended version of the paper accepted to IFAC World Congress 2023 for publication, containing proofs, and recently updated to address a typo in Assumption 3

Given a network of agents, we study the problem of designing a distributed algorithm that computes k independent weighted means of the network's initial conditions (namely, the agents agree on a k-dimensional space). Akin to average consensus, this problem finds applications in distributed computing and sensing, where agents seek to simultaneously evaluate k independent functions at a common point by running a single coordination algorithm. We show that linear algorithms can agree on quantities that are oblique projections of the vector of initial conditions, and we provide techniques to design protocols that are compatible with a pre-specified communication graph. More broadly, our results show that a single agreement algorithm can solve $k$ consensus problems simultaneously at a fraction of the complexity of classical approaches but, in general, it requires higher network connectivity.

This paper considers the problem of controlling a dynamical system when the state cannot be directly measured and the control performance metrics are unknown or partially known. In particular, we focus on the design of data-driven controllers to regulate a dynamical system to the solution of a constrained convex optimization problem where: i) the state must be estimated from nonlinear and possibly high-dimensional data; and, ii) the cost of the optimization problem -- which models control objectives associated with inputs and states of the system -- is not available and must be learned from data. We propose a data-driven feedback controller that is based on adaptations of a projected gradient-flow method; the controller includes neural networks as integral components for the estimation of the unknown functions. Leveraging stability theory for perturbed systems, we derive sufficient conditions to guarantee exponential input-to-state stability (ISS) of the control loop. In particular, we show that the interconnected system is ISS with respect to the approximation errors of the neural network and unknown disturbances affecting the system. The transient bounds combine the universal approximation property of deep neural networks with the ISS characterization. Illustrative numerical results are presented in the context of control of robotics and epidemics., Comment: This is an extended version of the paper published in the IEEE Open Journal of Control Systems - Special Section on Machine Learning with Control, containing proofs with a correction in Section II.A

We revisit here a landmark five parameter SIR-type model of [DvdD93, Sec. 4], which is maybe the simplest example where a complete picture of all cases, including non-trivial bistability behavior, may be obtained using simple tools. We also generalize it by adding essential vaccination and vaccination-induced death parameters, with the aim of revealing the role of vaccination and its possible failure. The main result is Theorem 5, which describes the stability behavior of our model in all possible cases.

In this work we study the stability properties of the equilibrium points of deterministic epidemic models with nonconstant population size. Models with nonconstant population have been studied in the past only in particular cases, two of which we review and combine. Our main result shows that for simple "matrix epidemic models" introduced in [1], an explicit general formula for the reproduction number and the corresponding "weak stability alternative" still holds, under small modifications, for models with nonconstant population size, and even when the model allows for vaccination and loss of immunity. The importance of this result is clear once we note that the models of [1] include a large number of viral and bacterial models of epidemic propagation, including for example the totality of homogeneous COVID-19 models. To better understand the nature of the result, we emphasize that the models proposed in [1] and considered here are extensions of the SIR-PH model, which is essentially characterized by a phase-type distribution that models transitions between the "disease/infectious compartments". In these cases, the reproduction number and a certain Lyapunov function for the disease free equilibrium are explicitly expressible. Not surprisingly, accounting for varying demography, loss of immunity, and vaccinations lead to several challenges. One of the most important is that a varying population size leads to multiple endemic equilibrium points: this is in contrast with "classic models" which in general admit unique disease-free and endemic equilibria. As a special case of our analysis, we consider a "first approximation" (FA) of our model, which coincides with the constant-demography model often studied in the literature, and for which more explicit results are available. Furthermore, we propose a second heuristic approximation named "intermediate approximation" (IA).

This paper considers the problem of regulating a linear dynamical system to the solution of a convex optimization problem with an unknown or partially-known cost. We design a data-driven feedback controller - based on gradient flow dynamics - that (i) is augmented with learning methods to estimate the cost function based on infrequent (and possibly noisy) functional evaluations; and, concurrently, (ii) is designed to drive the inputs and outputs of the dynamical system to the optimizer of the problem. We derive sufficient conditions on the learning error and the controller gain to ensure that the error between the optimizer of the problem and the state of the closed-loop system is ultimately bounded; the error bound accounts for the functional estimation errors and the temporal variability of the unknown disturbance affecting the linear dynamical system. Our results directly lead to exponential input-to-state stability of the closed-loop system. The proposed method and the theoretical bounds are validated numerically., Comment: This is an extended version of the paper accepted to the 4th Annual Learning for Dynamics & Control Conference (L4DC 2022), containing proofs

Since the early months of 2020, non-pharmaceutical interventions (NPIs) -- implemented at varying levels of severity and based on widely-divergent perspectives of risk tolerance -- have been the primary means to control SARS-CoV-2 transmission. We seek to identify how risk tolerance and vaccination rates impact the rate at which a population can return to pre-pandemic contact behavior. To this end, we develop a novel feedback control method for data-driven decision-making to identify optimal levels of NPIs across geographical regions in order to guarantee that hospitalizations will not exceed a given risk tolerance. Results are shown for the state of Colorado, and they suggest that: coordination in decision-making across regions is essential to maintain the daily number of hospitalizations below the desired limits; increasing risk tolerance can decrease the number of days required to discontinue NPIs, at the cost of an increased number of deaths; and if vaccination uptake is less than 70\%, at most levels of risk tolerance, return to pre-pandemic contact behaviors before the early months of 2022 may newly jeopardize the healthcare system., Comment: A preliminary version of this work appeared on medrXiv on April 1, 2021. Available at: https://www.medrxiv.org/content/10.1101/2021.04.20.21255350v1

This paper proposes a data-driven control framework to regulate an unknown, stochastic linear dynamical system to the solution of a (stochastic) convex optimization problem. Despite the centrality of this problem, most of the available methods critically rely on a precise knowledge of the system dynamics (thus requiring off-line system identification and model refinement). To this aim, in this paper we first show that the steady-state transfer function of a linear system can be computed directly from control experiments, bypassing explicit model identification. Then, we leverage the estimated transfer function to design a controller -- which is inspired by stochastic gradient descent methods -- that regulates the system to the solution of the prescribed optimization problem. A distinguishing feature of our methods is that they do not require any knowledge of the system dynamics, disturbance terms, or their distributions. Our technical analysis combines concepts and tools from behavioral system theory, stochastic optimization with decision-dependent distributions, and stability analysis. We illustrate the applicability of the framework on a case study for mobility-on-demand ride service scheduling in Manhattan, NY.

This paper investigates the problem of tracking solutions of stochastic optimization problems with time-varying costs that depend on random variables with decision-dependent distributions. In this context, we propose the use of an online stochastic gradient descent method to solve the optimization, and we provide explicit bounds in expectation and in high probability for the distance between the optimizers and the points generated by the algorithm. In particular, we show that when the gradient error due to sampling is modeled as a sub-Weibull random variable, then the tracking error is ultimately bounded in expectation and in high probability. The theoretical findings are validated via numerical simulations in the context of charging optimization of a fleet of electric vehicles., Comment: To appear in IEEE L-CSS

This paper proposes a data-driven framework to solve time-varying optimization problems associated with unknown linear dynamical systems. Making online control decisions to regulate a dynamical system to the solution of an optimization problem is a central goal in many modern engineering applications. Yet, the available methods critically rely on a precise knowledge of the system dynamics, thus mandating a preliminary system identification phase before a controller can be designed. In this work, we leverage results from behavioral theory to show that the steady-state transfer function of a linear system can be computed from data samples without any knowledge or estimation of the system model. We then use this data-driven representation to design a controller, inspired by a gradient-descent optimization method, that regulates the system to the solution of a convex optimization problem, without requiring any knowledge of the time-varying disturbances affecting the model equation. Results are tailored to cost functions satisfy the Polyak-\L ojasiewicz inequality.

We study the stability properties of a control system composed of a dynamical plant and a feedback controller, the latter generating control signals that can be compromised by a malicious attacker. We consider two classes of feedback controllers: a static output-feedback controller, and a dynamical gradient-flow controller that seeks to steer the output of the plant towards the solution of a convex optimization problem. In both cases, we analyze the stability properties of the closed-loop system under a class of switching attacks that persistently modify the control inputs generated by the controllers. Our stability analysis leverages the framework of hybrid dynamical systems, Lyapunov-based arguments for switching systems with unstable modes, and singular perturbation theory. Our results reveal that, under a suitable time-scale separation between plant and controllers, the stability of the interconnected system can be preserved when the attack occurs with "sufficiently low frequency" in any bounded time interval. We present simulation results in a power-grid example that corroborate the technical findings., Comment: Appeared in Nonlinear Analysis: Hybrid Systems, pp. 101152, vol. 44, may. 2022

This paper investigates the problem of regulating in real time a linear dynamical system to the solution trajectory of a time-varying constrained convex optimization problem. The proposed feedback controller is based on an adaptation of the saddle-flow dynamics, modified to take into account projections on constraint sets and output-feedback from the plant. We derive sufficient conditions on the tunable parameters of the controller (inherently related to the time-scale separation between plant and controller dynamics) to guarantee exponential and input-to-state stability of the closed-loop system. The analysis is tailored to the case of time-varying strongly convex cost functions and polytopic output constraints. The theoretical results are further validated in a ramp metering control problem in a network of traffic highways.

This paper studies the design of feedback controllers to steer a switching linear time-invariant dynamical system towards the solution trajectory of a time-varying convex optimization problem. We propose two types of controllers: (i) a continuous controller inspired by the online gradient descent method, and (ii) a hybrid controller that can be interpreted as an online version of Nesterov's accelerated gradient method with restarts of the state variables. By design, the controllers continuously steer the system towards the time-varying optimizer without requiring knowledge of exogenous disturbances affecting the system. For cost functions that are smooth and satisfy the Polyak-\L ojasiewicz inequality, we demonstrate that the online gradient-flow controller ensures uniform global exponential stability when the time scales of the system and controller are sufficiently separated and the switching signal of the system varies slowly on average. For cost functions that are strongly convex, we show that the hybrid accelerated controller outperforms the continuous gradient descent method. When the cost function is not strongly convex, we show that the the hybrid accelerated method guarantees global practical asymptotic stability.

Advanced traffic navigation systems, which provide routing recommendations to drivers based on real-time congestion information, are nowadays widely adopted by roadway transportation users. Yet, the emerging effects on the traffic dynamics originating from the widespread adoption of these tools have remained largely unexplored until now. In this paper, we propose a dynamic model where drivers imitate the path preferences of previous drivers, and we study the properties of its equilibrium points. Our model is a dynamic generalization of the classical traffic assignment framework, and extends it by accounting for dynamics both in the path decision process and in the network's traffic flows. We show that when travelers learn shortest paths by imitating other travelers, the overall traffic system benefits from this mechanism and transfers the maximum admissible amount of traffic demand. On the other hand, we demonstrate that when the travel delay functions are not sufficiently steep or the rates at which drivers imitate previous travelers are not adequately chosen, the trajectories of the traffic system may fail to converge to an equilibrium point, thus failing asymptotic stability. Illustrative numerical simulations combined with empirical data from highway sensors illustrate our findings.

This paper investigates the use of Infrastructure-To-Vehicle (I2V) communication to generate routing suggestions for drivers in transportation systems, with the goal of optimizing a measure of overall network congestion. We define link-wise levels of trust to tolerate the non-cooperative behavior of part of the driver population, and we propose a real-time optimization mechanism that adapts to the instantaneous network conditions and to sudden changes in the levels of trust. Our framework allows us to quantify the improvement in travel time in relation to the degree at which drivers follow the routing suggestions. We then study the resilience of the system, measured as the smallest change in routing choices that results in roads reaching their maximum capacity. Interestingly, our findings suggest that fluctuations in the extent to which drivers follow the provided routing suggestions can cause failures of certain links. These results imply that the benefits of using Infrastructure-To-Vehicle communication come at the cost of new fragilities, that should be appropriately addressed in order to guarantee the reliable operation of the infrastructure., Comment: Accepted for presentation at the IEEE 2019 American Control Conference

This paper studies, for the first time, the trajectory planning problem in adversarial environments, where the objective is to design the trajectory of a robot to reach a desired final state despite the unknown and arbitrary action of an attacker. In particular, we consider a robot moving in a two-dimensional space and equipped with two sensors, namely, a Global Navigation Satellite System (GNSS) sensor and a Radio Signal Strength Indicator (RSSI) sensor. The attacker can arbitrarily spoof the readings of the GNSS sensor and the robot control input so as to maximally deviate his trajectory from the nominal precomputed path. We derive explicit and constructive conditions for the existence of undetectable attacks, through which the attacker deviates the robot trajectory in a stealthy way. Conversely, we characterize the existence of secure trajectories, which guarantee that the robot either moves along the nominal trajectory or that the attack remains detectable. We show that secure trajectories can only exist between a subset of states, and provide a numerical mechanism to compute them. We illustrate our findings through several numerical studies, and discuss that our methods are applicable to different models of robot dynamics, including unicycles. More generally, our results show how control design affects security in systems with nonlinear dynamics., Comment: Accepted for publication in Automatica

This paper proposes a simplified version of classical models for urban transportation networks, and studies the problem of controlling intersections with the goal of optimizing network-wide congestion. Differently from traditional approaches to control traffic signaling, a simplified framework allows for a more tractable analysis of the network overall dynamics, and enables the design of critical parameters while considering network-wide measures of efficiency. Motivated by the increasing availability of real-time high-resolution traffic data, we cast an optimization problem that formalizes the goal of minimizing the overall network congestion by optimally controlling the durations of green lights at intersections. Our formulation allows us to relate congestion objectives with the problem of optimizing a metric of controllability of an associated dynamical network. We then provide a technique to efficiently solve the optimization by parallelizing the computation among a group of distributed agents. Lastly, we assess the benefits of the proposed modeling and optimization framework through microscopic simulations on typical traffic commute scenarios for the area of Manhattan. The optimization framework proposed in this study is made available online on a Sumo microscopic simulator based interface [1].

This paper studies the observability radius of network systems, which measures the robustness of a network to perturbations of the edges. We consider linear networks, where the dynamics are described by a weighted adjacency matrix, and dedicated sensors are positioned at a subset of nodes. We allow for perturbations of certain edge weights, with the objective of preventing observability of some modes of the network dynamics. To comply with the network setting, our work considers perturbations with a desired sparsity structure, thus extending the classic literature on the observability radius of linear systems. The paper proposes two sets of results. First, we propose an optimization framework to determine a perturbation with smallest Frobenius norm that renders a desired mode unobservable from the existing sensor nodes. Second, we study the expected observability radius of networks with given structure and random edge weights. We provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks, showing that line networks are inherently more robust than star networks., Comment: 8 pages, 3 figures