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Chance constraints ensure the satisfaction of constraints under uncertainty with a desired probability. This scheme is unfortunately sensitive to assumptions of the probability distribution of the uncertainty, which are difficult to verify. The uncertainty in the probability distribution can be considered via ambiguity sets that include all distributions close to a nominal distribution with respect to different ambiguity metrics. One simple methodology to deal with distributional ambiguity is to solve the original chance constrained problem with a specifically lowered risk level, called perturbed risk level In this paper, we compare different ambiguity metrics for which a perturbed risk level exists. The comparison highlights that one particular ambiguity metric is especially well suited to handle low risk levels, which are typically needed in control applications. To approximate the solution of the ambiguous chance constrained problem, we present a general method to achieve distributionally robust bounds with methods like the scenario approach and derive closed-form expressions for the confidence level and one-level probability bounds., Comment: Submitted to "IEEE Transactions on Automatic Control", Code: https://github.com/MoritzHein/DistriRobRiskLev

The main benefit of model predictive control (MPC) is its ability to steer the system to a given reference without violating the constraints while minimizing some objective. Furthermore, a suitably designed MPC controller guarantees asymptotic stability of the closed-loop system to the given reference as long as its optimization problem is feasible at the initial state of the system. Therefore, one of the limitations of classical MPC is that changing the reference may lead to an unfeasible MPC problem. Furthermore, due to a lack of deep knowledge of the system, it is possible for the user to provide a desired reference that is unfeasible or non-attainable for the MPC controller, leading to the same problem. This chapter summarizes MPC formulations recently proposed that have been designed to address these issues. In particular, thanks to the addition of an artificial reference as decision variable, the formulations achieve asymptotic stability and recursive feasibility guarantees regardless of the reference provided by the user, even if it is changed online or if it violates the system constraints. We show a recent formulation which extends this idea, achieving better performance and larger domains of attraction when working with small prediction horizons. Additional benefits of these formulations, when compared to classical MPC, are also discussed and highlighted with illustrative examples., Comment: (21 pages, 8 figures)

This paper provides a comprehensive tutorial on a family of Model Predictive Control (MPC) formulations, known as MPC for tracking, which are characterized by including an artificial reference as part of the decision variables in the optimization problem. These formulations have several benefits with respect to the classical MPC formulations, including guaranteed recursive feasibility under online reference changes, as well as asymptotic stability and an increased domain of attraction. This tutorial paper introduces the concept of using an artificial reference in MPC, presenting the benefits and theoretical guarantees obtained by its use. We then provide a survey of the main advances and extensions of the original linear MPC for tracking, including its non-linear extension. Additionally, we discuss its application to learning-based MPC, and discuss optimization aspects related to its implementation., Comment: (15 pages, 1 figure)

Model Predictive Control (MPC) is a popular control approach due to its ability to consider constraints, including input and state restrictions, while minimizing a cost function. However, in practice, these constraints can result in feasibility issues, either because the system model is not accurate or due to the existence of external disturbances. To mitigate this problem, a solution adopted by the MPC community is the use of soft constraints. In this article, we consider a not-so-typical methodology to encode soft constraints in a particular MPC formulation known as MPC for Tracking (MPCT), which has several advantages when compared to standard MPC formulations. The motivation behind the proposed encoding is to maintain the semi-banded structure of the ingredients of a recently proposed solver for the considered MPCT formulation, thus providing an efficient and fast solver when compared to alternative approaches from the literature. We show numerical results highlighting the benefits of the formulation and the computational efficiency of the solver.

The main objective of tracking control is to steer the tracking error, that is the difference between the reference and the output, to zero while the plant's operation limits are satisfied. This requires that some assumptions on the evolution of the future values of the reference must be taken into account. Typically a simple evolution of the reference is considered, such as step, ramp, or parabolic reference signals. It is important to notice that the tracking problem considers possible variations in the reference to be tracked, such as steps or slope variations of the ramps. Then the tracking control problem is inherently uncertain, since the reference may differ from what is expected. If the value of the reference is changed, then there is no guarantee that the feasibility and stability properties of the resulting control law hold. This report presents the MPC for tracking (MPCT) approach, which ensures recursive feasibility and asymptotic stability of the setpoint when the value of the reference is changed.

Model Predictive Control (MPC) for tracking formulation presents numerous advantages compared to standard MPC, such as a larger domain of attraction and recursive feasibility even when abrupt changes in the reference are produced. As a drawback, it includes some extra decision variables in its related optimization problem, leading to a semi-banded structure that differs from the banded structure encountered in standard MPC. This semi-banded structure prevents the direct use of the efficient algorithms available for banded problems. To address this issue, we present an algorithm based on the alternating direction method of multipliers that explicitly takes advantage of the underlying semi-banded structure of the MPC for tracking.

Harmonic model predictive control (HMPC) is a recent model predictive control (MPC) formulation for tracking piece-wise constant references that includes a parameterized artificial harmonic reference as a decision variable, resulting in an increased performance and domain of attraction with respect to other MPC formulations. This article presents an extension of the HMPC formulation to track periodic harmonic references and discusses its use to track arbitrary references. The proposed formulation inherits the benefits of its predecessor, namely its good performance and large domain of attraction when using small prediction horizons, and that the complexity of its optimization problem does not depend on the period of the periodic reference. We show closed-loop results discussing its performance and comparing it to other MPC formulations., Comment: (11 pages, 14 figures)

Model Predictive Control (MPC) is typically characterized for being computationally demanding, as it requires solving optimization problems online; a particularly relevant point when considering its implementation in embedded systems. To reduce the computational burden of the optimization algorithm, most solvers perform as many offline operations as possible, typically performing the computation and factorization of its expensive matrices offline and then storing them in the embedded system. This improves the efficiency of the solver, with the disadvantage that online changes on some of the ingredients of the MPC formulation require performing these expensive computations online. This article presents an efficient algorithm for the factorization of the key matrix used in several first-order optimization methods applied to linear MPC formulations, allowing its prediction model and cost function matrices to be updated online at the expense of a small computational cost. We show results comparing the proposed approach with other solvers from the literature applied to a linear time-varying system., Comment: (6 pages, 2 figures)

Supervised classification recognizes patterns in the data to separate classes of behaviours. Canonical solutions contain misclassification errors that are intrinsic to the numerical approximating nature of machine learning. The data analyst may minimize the classification error on a class at the expense of increasing the error of the other classes. The error control of such a design phase is often done in a heuristic manner. In this context, it is key to develop theoretical foundations capable of providing probabilistic certifications to the obtained classifiers. In this perspective, we introduce the concept of probabilistic safety region to describe a subset of the input space in which the number of misclassified instances is probabilistically controlled. The notion of scalable classifiers is then exploited to link the tuning of machine learning with error control. Several tests corroborate the approach. They are provided through synthetic data in order to highlight all the steps involved, as well as through a smart mobility application., Comment: 13 pages, 4 figures, 1 table, submitted to IEEE TNNLS

Harmonic model predictive control (HMPC) is a model predictive control (MPC) formulation which displays several benefits over other MPC formulations, especially when using a small prediction horizon. These benefits, however, come at the expense of an optimization problem that is no longer the typical quadratic programming problem derived from most linear MPC formulations due to the inclusion of a particular class of second order cone constraints. This article presents a method for efficiently dealing with these constraints in operator splitting methods, leading to a computation time for solving HMPC in line with state of the art solvers for linear MPC. We show how to apply this result to the alternating direction method of multipliers algorithm, presenting a solver which we compare against other solvers from the literature, including solvers for other linear MPC formulations. The results show that the proposed solver, and by extension the HMPC formulation, is suitable for its implementation in embedded systems., Comment: Accepted version of the article published in IEEE Transactions on Automatic Control (8 pages, 5 figures)

In this paper, we address the probabilistic error quantification of a general class of prediction methods. We consider a given prediction model and show how to obtain, through a sample-based approach, a probabilistic upper bound on the absolute value of the prediction error. The proposed scheme is based on a probabilistic scaling methodology in which the number of required randomized samples is independent of the complexity of the prediction model. The methodology is extended to address the case in which the probabilistic uncertain quantification is required to be valid for every member of a finite family of predictors. We illustrate the results of the paper by means of a numerical example., Comment: 8 pages, 2 figure

Model Predictive Control (MPC) typically includes a terminal constraint to guarantee stability of the closed-loop system under nominal conditions. In linear MPC this constraint is generally taken on a polyhedral set, leading to a quadratic optimization problem. However, the use of an ellipsoidal terminal constraint may be desirable, leading to an optimization problem with a quadratic constraint. In this case, the optimization problem can be solved using Second Order Cone (SOC) programming solvers, since the quadratic constraint can be posed as a SOC constraint, at the expense of adding additional slack variables and possibly compromising the simple structure of the solver ingredients. In this paper we present a sparse solver for linear MPC subject to a terminal ellipsoidal constraint based on the alternating direction method of multipliers algorithm in which we directly deal with the quadratic constraints without having to resort to the use of a SOC constraint nor the inclusion of additional decision variables. The solver is suitable for its use in embedded systems, since it is sparse, has a small memory footprint and requires no external libraries. We compare its performance against other approaches from the literature., Comment: Accepted version of the article published in IEEE Transactions on Control Systems Technology (8 pages, 5 figures)

Many popular approaches in the field of robust model predictive control (MPC) are based on nominal predictions. This paper presents a novel formulation of this class of controller with proven input-to-state stability and robust constraint satisfaction. Its advantages are: (i) the design of its main ingredients are tractable for medium to large-sized systems, (ii) the terminal set does not need to be robust with respect to all the possible system uncertainties, but only for a reduced set that can be made arbitrarily small, thus facilitating its design and implementation, (iii) under certain conditions the terminal set can be taken as a positive invariant set of the nominal system, allowing us to use a terminal equality constraint, which facilitates its application to large-scale systems, and (iv) the complexity of its optimization problem is comparable to the non-robust MPC variant. We show numerical closed-loop results of its application to a multivariable chemical plant and compare it against other robust MPC formulations., Comment: Accepted version of article in Journal of Process Control (13 pages, 25 figures)

This article presents the real-time implementation of the model predictive control for tracking formulation to control a two-wheeled inverted pendulum robot. This formulation offers several advantages over standard MPC formulations at the expense of the addition of a small number of decision variables, which complicates the inner structure of the matrices of the optimization problem. We implement a sparse solver, based on an extension of the alternating direction method of multipliers, in the system's embedded hardware. The results indicate that the solver is suitable for controlling a real system with sample times in the range of milliseconds using current, readily-available hardware., Comment: 6 pages, 9 figures

This survey analyses the role of data-driven methodologies for pandemic modelling and control. We provide a roadmap from the access to epidemiological data sources to the control of epidemic phenomena. We review the available methodologies and discuss the challenges in the development of data-driven strategies to combat the spreading of infectious diseases. Our aim is to bring together several different disciplines required to provide a holistic approach to epidemic analysis, such as data science, epidemiology, and systems-and-control theory. A 3M-analysis is presented, whose three pillars are: Monitoring, Modelling and Managing. The focus is on the potential of data-driven schemes to address three different challenges raised by a pandemic: (i) monitoring the epidemic evolution and assessing the effectiveness of the adopted countermeasures; (ii) modelling and forecasting the spread of the epidemic; (iii) making timely decisions to manage, mitigate and suppress the contagion. For each step of this roadmap, we review consolidated theoretical approaches (including data-driven methodologies that have been shown to be successful in other contexts) and discuss their application to past or present epidemics, such as Covid-19, as well as their potential application to future epidemics., Comment: 24 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2006.01731

Accelerated first order methods, also called fast gradient methods, are popular optimization methods in the field of convex optimization. However, they are prone to suffer from oscillatory behaviour that slows their convergence when medium to high accuracy is desired. In order to address this, restart schemes have been proposed in the literature, which seek to improve the practical convergence by suppressing the oscillatory behaviour. This paper presents a restart scheme for accelerated first order methods for which we show linear convergence under the satisfaction of a quadratic functional growth condition, thus encompassing a broad class of non-necessarily strongly convex optimization problems. Moreover, the worst-case convergence rate is comparable to the one obtained using a (generally non-implementable) optimal fixed-rate restart strategy. We compare the proposed algorithm with other restart schemes by applying them to a model predictive control case study., Comment: Accepted version of the article published in IEEE Transactions on Automatic Control (8 pages, 1 figure)

In this paper, a sample-based procedure for obtaining simple and computable approximations of chance-constrained sets is proposed. The procedure allows to control the complexity of the approximating set, by defining families of simple-approximating sets of given complexity. A probabilistic scaling procedure then allows to rescale these sets to obtain the desired probabilistic guarantees. The proposed approach is shown to be applicable in several problem in systems and control, such as the design of Stochastic Model Predictive Control schemes or the solution of probabilistic set membership estimation problems., Comment: 16 pages, 11 figures, submitted to Automatica

This work presents a new methodology to obtain probabilistic interval predictions of a dynamical system. The proposed strategy uses stored past system measurements to estimate the future evolution of the system. The method relies on the use of dissimilarity functions to estimate the conditional probability density function of the outputs. A family of empirical probability density functions, parameterized by means of two scalars, is introduced. It is shown that the proposed family encompasses the multivariable normal probability density function as a particular case. We show that the presented approach constitutes a generalization of classical estimation methods. A validation scheme is used to tune the two parameters on which the methodology relies. In order to prove the effectiveness of the presented methodology, some numerical examples and comparisons are provided., Comment: 10 pages, 3 figures

This article presents a sparse, low-memory footprint optimization algorithm for the implementation of the model predictive control (MPC) for tracking formulation in embedded systems. This MPC formulation has several advantages over standard MPC formulations, such as an increased domain of attraction and guaranteed recursive feasibility even in the event of a sudden reference change. However, this comes at the expense of the addition of a small amount of decision variables to the MPC's optimization problem that complicates the structure of its matrices. We propose a sparse optimization algorithm, based on an extension of the alternating direction method of multipliers, that exploits the structure of this particular MPC formulation. We describe the controller formulation and detail how its structure is exploited by means of the aforementioned optimization algorithm. We show closed-loop simulations comparing the proposed solver against other solvers and approaches from the literature., Comment: Accepted version of the article published in IEEE Transactions on Control Systems Technology (8 pages, 5 figures)

This document analyzes the role of data-driven methodologies in Covid-19 pandemic. We provide a SWOT analysis and a roadmap that goes from the access to data sources to the final decision-making step. We aim to review the available methodologies while anticipating the difficulties and challenges in the development of data-driven strategies to combat the Covid-19 pandemic. A 3M-analysis is presented: Monitoring, Modelling and Making decisions. The focus is on the potential of well-known datadriven schemes to address different challenges raised by the pandemic: i) monitoring and forecasting the spread of the epidemic; (ii) assessing the effectiveness of government decisions; (iii) making timely decisions. Each step of the roadmap is detailed through a review of consolidated theoretical results and their potential application in the Covid-19 context. When possible, we provide examples of their applications on past or present epidemics. We do not provide an exhaustive enumeration of methodologies, algorithms and applications. We do try to serve as a bridge between different disciplines required to provide a holistic approach to the epidemic: data science, epidemiology, controltheory, etc. That is, we highlight effective data-driven methodologies that have been shown to be successful in other contexts and that have potential application in the different steps of the proposed roadmap. To make this document more functional and adapted to the specifics of each discipline, we encourage researchers and practitioners to provide feedback. We will update this document regularly.

In recent years, the increasing interest in Stochastic model predictive control (SMPC) schemes has highlighted the limitation arising from their inherent computational demand, which has restricted their applicability to slow-dynamics and high-performing systems. To reduce the computational burden, in this paper we extend the probabilistic scaling approach to obtain low-complexity inner approximation of chance-constrained sets. This approach provides probabilistic guarantees at a lower computational cost than other schemes for which the sample complexity depends on the design space dimension. To design candidate simple approximating sets, which approximate the shape of the probabilistic set, we introduce two possibilities: i) fixed-complexity polytopes, and ii) $\ell_p$-norm based sets. Once the candidate approximating set is obtained, it is scaled around its center so to enforce the expected probabilistic guarantees. The resulting scaled set is then exploited to enforce constraints in the classical SMPC framework. The computational gain obtained with the proposed approach with respect to the scenario one is demonstrated via simulations, where the objective is the control of a fixed-wing UAV performing a monitoring mission over a sloped vineyard.

This paper presents a novel model predictive control (MPC) formulation for set-point tracking. Stabilizing predictive controllers based on terminal ingredients may exhibit stability and feasibility issues in the event of a reference change for small to moderate prediction horizons. In the MPC for tracking formulation, these issues are solved by the addition of an artificial equilibrium point as a new decision variable, providing a significantly enlarged domain of attraction and guaranteeing recursive feasibility for any reference change. However, it may suffer from performance issues if the prediction horizon is not large enough. This paper presents an extension of this formulation where a harmonic artificial reference is used in place of the equilibrium point. The proposed formulation achieves even greater domains of attraction and can significantly outperform other MPC formulations when the prediction horizon is small. We prove the asymptotic stability and recursive feasibility of the proposed controller, as well as provide guidelines for the design of its main ingredients. Finally, we highlight its advantages with a case study of a ball and plate system., Comment: Accepted version of the article published in IEEE Transactions on Automatic Control (14 pages, 11 figures)

In this paper a constructive method to determine and compute probabilistic reachable and invariant sets for linear discrete-time systems, excited by a stochastic disturbance, is presented. The samples of the disturbance signal are not assumed to be uncorrelated, only a bound on the correlation matrices is supposed to be known. The concept of correlation bound is introduced and employed to determine probabilistic reachable sets and probabilistic invariant sets. Constructive methods for their computation, based on convex optimization, are given.

We provide an insight into the open data resources pertinent to the study of the spread of Covid-19 pandemic and its control. We identify the variables required to analyze fundamental aspects like seasonal behaviour, regional mortality rates, and effectiveness of government measures. Open data resources, along with data-driven methodologies, provide many opportunities to improve the response of the different administrations to the virus. We describe the present limitations and difficulties encountered in most of the open-data resources. To facilitate the access to the main open-data portals and resources, we identify the most relevant institutions, at a world scale, providing Covid-19 information and/or auxiliary variables (demographics, mobility, etc.). We also describe several open resources to access Covid-19 data-sets at a country-wide level (i.e. China, Italy, Spain, France, Germany, U.S., etc.). In an attempt to facilitate the rapid response to the study of the seasonal behaviour of Covid-19, we enumerate the main open resources in terms of weather and climate variables. CONCO-Team: The authors of this paper belong to the CONtrol COvid-19 Team, which is composed of different researches from universities of Spain, Italy, France, Germany, United Kingdom and Argentina. The main goal of CONCO-Team is to develop data-driven methods for the better understanding and control of the pandemic., Comment: 30 pages Minor improvements

In this paper, we consider a stochastic Model Predictive Control able to account for effects of additive stochastic disturbance with unbounded support, and requiring no restrictive assumption on either independence nor Gaussianity. We revisit the rather classical approach based on penalty functions, with the aim of designing a control scheme that meets some given probabilistic specifications. The main difference with previous approaches is that we do not recur to the notion of probabilistic recursive feasibility, and hence we do not consider separately the unfeasible case. In particular, two probabilistic design problems are envisioned. The first randomization problem aims to design \textit{offline} the constraint set tightening, following an approach inherited from tube-based MPC. For the second probabilistic scheme, a specific probabilistic validation approach is exploited for tuning the penalty parameter, to be selected \textit{offline} among a finite-family of possible values. The simple algorithm here proposed allows designing a \textit{single} controller, always guaranteeing feasibility of the online optimization problem. The proposed method is shown to be more computationally tractable than previous schemes. This is due to the fact that the sample complexity for both probabilistic design problems depends on the prediction horizon in a logarithmic way, unlike scenario-based approaches which exhibit linear dependence. The efficacy of the proposed approach is demonstrated with a numerical example., Comment: Submitted as Contributed Paper to IFAC2020 Congress

Solving nonlinear model predictive control problems in real time is still an important challenge despite of recent advances in computing hardware, optimization algorithms and tailored implementations. This challenge is even greater when uncertainty is present due to disturbances, unknown parameters or measurement and estimation errors. To enable the application of advanced control schemes to fast systems and on low-cost embedded hardware, we propose to approximate a robust nonlinear model controller using deep learning and to verify its quality using probabilistic validation techniques. We propose a probabilistic validation technique based on finite families, combined with the idea of generalized maximum and constraint backoff to enable statistically valid conclusions related to general performance indicators. The potential of the proposed approach is demonstrated with simulation results of an uncertain nonlinear system., Comment: 23 pages, 4 figures

Fast Iterative Shrinking-Threshold Algorithm (FISTA) is a popular fast gradient descent method (FGM) in the field of large scale convex optimization problems. However, it can exhibit undesirable periodic oscillatory behaviour in some applications that slows its convergence. Restart schemes seek to improve the convergence of FGM algorithms by suppressing the oscillatory behaviour. Recently, a restart scheme for FGM has been proposed that provides linear convergence for non strongly convex optimization problems that satisfy a quadratic functional growth condition. However, the proposed algorithm requires prior knowledge of the optimal value of the objective function or of the quadratic functional growth parameter. In this paper we present a restart scheme for FISTA algorithm, with global linear convergence, for non strongly convex optimization problems that satisfy the quadratic growth condition without requiring the aforementioned values. We present some numerical simulations that suggest that the proposed approach outperforms other restart FISTA schemes., Comment: This paper constitutes an extended and revised version of "Restart FISTA with Global Linear Convergence" by Teodoro Alamo et. al. presented at the European Control Conference (ECC), 2019. v2: typos corrected and includes the proof of Prop. 1(ii) of "Gradient Based Restart FISTA" by Teodoro Alamo et. al. presented at the Conference on Decision and Control (CDC), 2019. (12 pages, 4 figures)