Publications
399 results found
Jiang J, Astolfi A, Parisini T, 2021, Robust traffic wave damping via shared control, Transportation Research Part C: Emerging Technologies, Vol: 128, Pages: 1-23, ISSN: 0968-090X
The traffic wave damping problem in a circular single lane track is addressed and solved via a shared control technique which takes a model of the human drivers’ driving habits into consideration. A formal analysis shows that the effectiveness of the proposed shared controller does not depend on the parameters of the human driver’s model, which is an important property in the implementation of the shared controller, since these parameters are difficult to measure, and vary from one human driver to another and from one driving situation to another. In addition, the proposed control law is robust: the stop-and-go wave can be dampened and there is no collisions among vehicles even if there is noise on the information each vehicle receives from the higher level traffic control center. A comparison between performances of the vehicles with and without the proposed control scheme demonstrates the robustness and the effectiveness of the shared control solution.
Parisini T, 2021, CSS Conferences: A "New Normal"?, IEEE CONTROL SYSTEMS MAGAZINE, Vol: 41, Pages: 9-10, ISSN: 1066-033X
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- Citations: 3
Rezaee H, Parisini T, Polycarpou M, 2021, Resiliency in dynamic leader-follower multiagent systems, Automatica, Vol: 125, Pages: 1-10, ISSN: 0005-1098
Resilient control of multiagent systems (MASs) in the presence of dynamic leaders is studied in this paper. We consider a network of agents consisting of a leader, a set of healthy agents, and a set of attacked malicious agents. The objective is developing a control strategy for the healthy agents to follow the trajectory of the leader, while they are in interaction with the unknown malicious agents. The main contribution of this paper is resilient leader-follower control of MASs when a dynamic leader determines a continuous time-varying trajectory for the MAS. By defining the concept of r-robust leader-follower graphs, we propose and analyze sufficient conditions on interaction among the agents such that the mentioned objective is achieved. Numerical examples verify the accuracy of the proposed control scheme.
Parisini T, 2021, The IEEE Control Systems Society: An Inclusive Ecosystem for Volunteers, IEEE CONTROL SYSTEMS MAGAZINE, Vol: 41, Pages: 8-10, ISSN: 1066-033X
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- Citations: 2
Bin M, Cheung PYK, Crisostomi E, et al., 2021, Post-lockdown abatement of COVID-19 by fast periodic switching, PLoS Computational Biology, Vol: 17, Pages: 1-34, ISSN: 1553-734X
COVID-19 abatement strategies have risks and uncertainties which could lead to repeating waves of infection. We show—as proof of concept grounded on rigorous mathematical evidence—that periodic, high-frequency alternation of into, and out-of, lockdown effectively mitigates second-wave effects, while allowing continued, albeit reduced, economic activity. Periodicity confers (i) predictability, which is essential for economic sustainability, and (ii) robustness, since lockdown periods are not activated by uncertain measurements over short time scales. In turn—while not eliminating the virus—this fast switching policy is sustainable over time, and it mitigates the infection until a vaccine or treatment becomes available, while alleviating the social costs associated with long lockdowns. Typically, the policy might be in the form of 1-day of work followed by 6-days of lockdown every week (or perhaps 2 days working, 5 days off) and it can be modified at a slow-rate based on measurements filtered over longer time scales. Our results highlight the potential efficacy of high frequency switching interventions in post lockdown mitigation. All code is available on Github at https://github.com/V4p1d/FPSP_Covid19. A software tool has also been developed so that interested parties can explore the proof-of-concept system.
Parisini T, 2021, Welcome from the President ofthe IEEE Control Systems Society, ISSN: 0743-1546
Ghosh A, Scandella M, Bin M, et al., 2021, Traffic-Light Control at Urban Intersections Using Expected Waiting-Time Information, 60th IEEE Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 1953-1959, ISSN: 0743-1546
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- Citations: 1
Gallo AJ, Boem F, Parisini T, 2021, Distributed cyber-attack isolation for large-scale interconnected systems, European Control Conference (ECC), Publisher: IEEE, Pages: 48-53
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- Citations: 1
Salvato E, Ghosh A, Fenu G, et al., 2021, Control of a Mixed Autonomy Signalised Urban Intersection: An Action-Delayed Reinforcement Learning Approach, IEEE Intelligent Transportation Systems Conference (ITSC), Publisher: IEEE, Pages: 2042-2047, ISSN: 2153-0009
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- Citations: 1
Bin M, Parisini T, 2020, A distributed methodology for approximate uniform global minimum sharing
The paper deals with the distributed minimum sharing problem, in which anetwork of decision-makers - exchanging information through a communicationnetwork - computes the minimum of some local quantities of interest in adistributed and decentralized way. The problem is equivalently cast into acost-coupled distributed optimization problem, and an adjustable approximate(or sub-optimal) solution is presented which enjoys several properties ofcrucial importance in applications. In particular, the proposed solution isscalable in that the dimension of the state space does not grow with the sizeor topology of the communication network. Moreover, a global and uniform (bothin the initial time and in the initial condition) asymptotic stability resultis provided, as well as an attractiveness property towards a steady state whichcan be made arbitrarily close to the sought minimum. Exact asymptoticconvergence is also recovered at the price, however, of loosing uniformity ofthe convergence with respect to the initial time.
Rezaee H, Parisini T, Polycarpou M, 2020, Almost Sure Resilient Consensus Under Stochastic Interaction: Links Failure and Noisy Channels, IEEE Transactions on Automatic Control, Vol: 66, ISSN: 0018-9286
The resilient consensus problem over a classof discrete-time linear multiagent systems is addressed.Because of external cyber-attacks, some agents are assumed to be malicious and not following a desired cooperative behavior. Thus, the objective consists in designing acontrol strategy for the healthy agents to reach consensusupon their state vectors, while due to interaction among theagents, the malicious agents try to prevent them to achieveconsensus. Although this problem has been investigatedby some researchers, under the existing approaches in theliterature, achieving consensus is only guaranteed whenthe information exchange among the agents is deterministic. Based on this motivation, the main contribution ofthe paper is on almost sure resilient consensus control ofa network of healthy agents in the presence of stochastic links failure and communication noises. We design adiscrete-time protocol for the set of the healthy agents, andwe show that under some probabilistic conditions on interaction among the agents, achieving almost sure consensusamong the healthy agents can be guaranteed. The resultsalso are verified by numerical examples
Ding H, Gao RX, Isaksson AJ, et al., 2020, State of AI-based monitoring in smart manufacturing and introduction to focused section, IEEE/ASME Transactions on Mechatronics, Vol: 25, Pages: 2143-2154, ISSN: 1083-4435
Over the past few decades, intelligentization, supported by artificial intelligence (AI) technologies, has become an important trend for industrial manufacturing, accelerating the development of smart manufacturing. In modern industries, standard AI has been endowed with additional attributes, yielding the so-called industrial artificial intelligence (IAI) that has become the technical core of smart manufacturing. AI-powered manufacturing brings remarkable improvements in many aspects of closed-loop production chains from manufacturing processes to end product logistics. In particular, IAI incorporating domain knowledge has benefited the area of production monitoring considerably. Advanced AI methods such as deep neural networks, adversarial training, and transfer learning have been widely used to support both diagnostics and predictive maintenance of the entire production process. It is generally believed that IAI is the critical technologies needed to drive the future evolution of industrial manufacturing. This article offers a comprehensive overview of AI-powered manufacturing and its applications in monitoring. More specifically, it summarizes the key technologies of IAI and discusses their typical application scenarios with respect to three major aspects of production monitoring: fault diagnosis, remaining useful life prediction, and quality inspection. In addition, the existing problems and future research directions of IAI are also discussed. This article further introduces the papers in this focused section on AI-based monitoring in smart manufacturing by weaving them into the overview, highlighting how they contribute to and extend the body of literature in this area.
Higgins M, Teng F, Parisini T, 2020, Stealthy MTD against unsupervised learning-based blind FDI Attacks in power systems, IEEE Transactions on Information Forensics and Security, Vol: 16, Pages: 1275-1287, ISSN: 1556-6013
This paper examines how moving target defenses (MTD) implemented in power systems can be countered by unsupervised learning-based false data injection (FDI) attack and how MTD can be combined with physical watermarking to enhance the system resilience. A novel intelligent attack, which incorporates dimensionality reduction and density-based spatial clustering, is developed and shown to be effective in maintaining stealth in the presence of traditional MTD strategies. In resisting this new type of attack, a novel implementation of MTD combining with physical watermarking is proposed by adding Gaussian watermark into physical plant parameters to drive detection of traditional and intelligent FDI attacks, while remaining hidden to the attackers and limiting the impact on system operation and stability.
Pin G, Fenu G, Casagrande V, et al., 2020, Robust stabilization of a class of nonlinear systems controlled over communication networks, IEEE Transactions on Automatic Control, Vol: 66, Pages: 3036-3051, ISSN: 0018-9286
The paper deals with the stabilization of nonlin-ear systems in which the loop is closed over a lossy non-acknowledged communication network. Given a Regional Input-to-State (ISS) stabilizing state-feedback control law, designedwithout accounting for the network-induced delays, we proposea non-acknowledged communication policy that allows to deploythe above controller over the network without any modification,while preserving the Regional ISS property. The time-varyingdelays and packet dropouts occurring on both the up-link andthe down-link are compensated through a model-based predictionscheme and a packet-management policy based on time-stamping.The consistency of the prediction, which is a major issue inthe context of nonlinear systems with an embedded networkedcontroller, is guaranteed through the exploitation of a novel move-blocking strategy for computing the command sequence to beforwarded to the actuators.
Gallo A, Turan M, Boem F, et al., 2020, A distributed cyber-attack detection scheme with application to DC microgrids, IEEE Transactions on Automatic Control, Vol: 65, Pages: 3800-3815, ISSN: 0018-9286
DC microgrids often present a hierarchical control architecture, requiring integration of communication layers. This leads to the possibility of malicious attackers disrupting the overall system. Motivated by this application, in this paper we present a distributed monitoring scheme to provide attack-detection capabilities for linear Large-Scale Systems. The proposed architecture relies on a Luenberger observer together with a bank of Unknown-Intput Observers (UIOs) at each subsystem, providing attack detection capabilities. We describe the architecture and analyze conditions under which attacks are guaranteed to be detected, and, conversely, when they are stealthy . Our analysis shows that some classes of attacks cannot be detected using either module independently; rather, by exploiting both modules simultaneously, we are able to improve the detection properties of the diagnostic tool as a whole. Theoretical results are backed up by simulations, where our method is applied to a realistic model of a low-voltage DC microgrid under attack.
Barboni A, Rezaee H, Boem F, et al., 2020, Detection of covert cyber-attacks in interconnected systems: a distributed model-based approach, IEEE Transactions on Automatic Control, Vol: 65, Pages: 3728-3741, ISSN: 0018-9286
Distributed detection of covert attacks for linear large-scale interconnected systems is addressed in this article. Existing results consider the problem in centralized settings. This article focuses on large-scale systems subject to bounded process and measurement disturbances, where a single subsystem is under a covert attack. A detection methodology is proposed, where each subsystem can detect the presence of covert attacks in neighboring subsystems in a distributed manner. The detection strategy is based on the design of two model-based observers for each subsystem using only local information. An extensive detectability analysis is provided and simulation results on a power network benchmark are given, showing the effectiveness of the proposed methodology for the detection of covert cyber-attacks.
Yilmaz S, Dudkina E, Bin M, et al., 2020, Kemeny-based testing for COVID-19
Testing, tracking and tracing abilities have been identified as pivotal inhelping countries to safely reopen activities after the first wave of theCOVID-19 virus. Contact tracing apps give the unprecedented possibility toreconstruct graphs of daily contacts, so the question is who should be tested?As human contact networks are known to exhibit community structure, in thispaper we show that the Kemeny constant of a graph can be used to identify andanalyze bridges between communities in a graph. Our "Kemeny indicator" is thechange in Kemeny constant when a node or edge is removed from the graph. Weshow that testing individuals who are associated with large values of theKemeny indicator can help in efficiently intercepting new virus outbreaks, whenthey are still in their early stage. Extensive simulations provide promisingresults in early identification and in blocking possible "super-spreaders"links that transmit disease between different communities.
Cervellera C, Maccio D, Parisini T, 2020, Learning robustly stabilizing explicit model predictive controllers: A non-regular sampling approach, IEEE Control Systems Letters, Vol: 4, Pages: 737-742, ISSN: 2475-1456
Off-line supervised learning from data of robustly-stabilizing nonlinear explicit model predictive controllers (EMPC) is dealt with in this letter. The learning procedure relies on the construction of a suitably large set of specifically chosen sampling points of the state space in which the values of the optimal EMPC control function have to be computed. When bounding the magnitude of approximation errors is important for stability or performance specifications, regular gridding techniques are not feasible due to the curse of dimensionality arising from the structural exponential growth of the number of points with the state dimension. In this note, we consider non-regular sampling techniques – namely, i.i.d. sampling with uniform distribution, low-discrepancy sequences and lattice point sets – that offer a good covering of the state space without suffering from an unfeasible growth of the number of points, while preserving at the same time the method guarantees in terms of robustness and stability. Some theoretical properties of the proposed sampling schemes are briefly discussed, and their successful application is showcased in a practically-relevant optimal heating problem involving a 21-dimensional state space that rules out the use of regular gridding techniques.
Li P, Boem F, Pin G, et al., 2020, Kernel-based simultaneous parameter-state estimation for continuous-time systems, IEEE Transactions on Automatic Control, Vol: 65, Pages: 3053-3059, ISSN: 0018-9286
In this note, the problem of jointly estimating thestate and the parameters of continuous-time systems is addressed.Making use of suitably designed Volterra integral operators,the proposed estimator does not need the availability of time-derivatives of the measurable signals and the dependence ontheunknown initial conditions is removed. As a result, the estimatesconverge to the true values in arbitrarily short time in noise-freescenario. In the presence of bounded measurement and processdisturbances, the estimation error is shown to be bounded. Thenumerical implementation aspects are dealt with and extensivesimulation results are provides showing the effectivenessof theestimator.
Wang Y, Pin G, Serrani A, et al., 2020, Removing SPR-Like Conditions in Adaptive Feedforward Control of Uncertain Systems, IEEE TRANSACTIONS ON AUTOMATIC CONTROL, Vol: 65, Pages: 2309-2324, ISSN: 0018-9286
The paper considers the problem of designing Adaptive Feedforward Control (AFC) systems for uncertain SISO linear systems perturbed by multi-sinusoidal disturbances of known frequencies. The proposed approach removes the longstanding assumption that either the sign of the real part or the imaginary part of the transfer function of a stable plant at the frequency of excitation be known for AFC to be applicable, which is referred to in this paper as an SPR-like condition. Notable features of the solution are that persistence of excitation is not required and stability analysis tools based on averaging are avoided; hence, the requirement of an exponentially stable equilibrium for the closed-loop system is circumvented.
Barrere M, Hankin C, Nicolau N, et al., 2020, Measuring cyber-physical security in industrial control systems via minimum-effort attack strategies, Journal of Information Security and Applications, Vol: 52, Pages: 1-17, ISSN: 2214-2126
In recent years, Industrial Control Systems (ICS) have become increasingly exposed to a wide range of cyber-physical attacks, having massive destructive consequences. Security metrics are therefore essential to assess and improve their security posture. In this paper, we present a novel ICS security metric based on AND/OR graphs and hypergraphs which is able to efficiently identify the set of critical ICS components and security measures that should be compromised, with minimum cost (effort) for an attacker, in order to disrupt the operation of vital ICS assets. Our tool, META4ICS (pronounced as metaphorics), leverages state-of-the-art methods from the field of logical satisfiability optimisation and MAX-SAT techniques in order to achieve efficient computation times. In addition, we present a case study where we have used our system to analyse the security posture of a realistic Water Transport Network (WTN).
Boem F, Gallo A, Raimondo DM, et al., 2020, Distributed fault-tolerant control of large-scale systems: An active fault diagnosis approach, IEEE Transactions on Control of Network Systems, Vol: 7, Pages: 288-301, ISSN: 2325-5870
The paper proposes a methodology to effectively address the increasingly important problem of distributed faulttolerant control for large-scale interconnected systems. The approach dealt with combines, in a holistic way, a distributed fault detection and isolation algorithm with a specific tube-based model predictive control scheme. A distributed fault-tolerant control strategy is illustrated to guarantee overall stability and constraint satisfaction even after the occurrence of a fault. In particular, each subsystem is controlled and monitored by a local unit. The fault diagnosis component consists of a passive set-based fault detection algorithm and an active fault isolation one, yielding fault-isolability subject to local input and state constraints. The distributed active fault isolation module - thanks to a modification of the local inputs - allows to isolate the fault that has occurred avoiding the usual drawback of controllers that possibly hide the effect of the faults. The Active Fault Isolation method is used as a decision support tool for the fault tolerant control strategy after fault detection. The distributed design of the tube-based model predictive control allows the possible disconnection of faulty subsystems or the reconfiguration of local controllers after fault isolation. Simulation results on a well-known power network benchmark show the effectiveness of the proposed methodology.
Khalili M, Zhang X, Cao Y, et al., 2020, Distributed fault-tolerant control of multi-agent systems: An adaptive learning approach, IEEE Transactions on Neural Networks and Learning Systems, Vol: 31, Pages: 420-432, ISSN: 2162-2388
This paper focuses on developing a distributed leader-following fault tolerant tracking control scheme for a class of high-order nonlinear uncertain multi-agent systems. Neural network based adaptive learning algorithms are developed to learn unknown fault functions, guaranteeing the system stability and cooperative tracking even in the presence of multiple simultaneous process and actuator faults in the distributed agents. The time-varying leader’s command is only communicated to a small portion of follower agents through directed links, and each follower agent exchanges local measurement information only with its neighbors through a bidirectional but asymmetric topology. Adaptive fault-tolerant algorithms are developed for two cases, i.e., with full-state measurement and with only limited output measurement, respectively. Under certain assumptions, the closed-loop stability and asymptotic leader-follower tracking properties are rigorously established.
Zoppoli R, Sanguineti M, Gnecco G, et al., 2020, From functional optimization to nonlinear programming by the extended ritz method, Communications and Control Engineering, Pages: 39-88
This chapter describes the approximate solution of infinite-dimensional optimization problems by the “Extended Ritz Method” (ERIM). The ERIM consists in substituting the admissible functions with fixed-structure parametrized (FSP) functions containing vectors of “free” parameters. The larger the dimensions, the more accurate the approximations of the optimal solutions of the original functional optimization problems. This requires solving easier nonlinear programming problems. In the area of function approximation, we review the definition of approximating sequences of sets, which enjoy the property of density in the sets of functions one wants to approximate. Then, we provide the definition of polynomially complex approximating sequences of sets, which are able to approximate functions provided with suitable regularity properties by using, for a desired arbitrary accuracy, a number of “free” parameters increasing at most polynomially when the number of function arguments grows. In the less studied area of approximate solution of infinite-dimensional optimization problems, the optimizing sequences and the polynomially complex optimizing sequences of FSP functions are defined. Results are presented that allow to conclude that, if appropriate hypotheses occur, polynomially complex approximating sequences of sets give rise to polynomially complex optimizing sequences of FSP functions, possibly mitigating the curse of dimensionality.
Zoppoli R, Sanguineti M, Gnecco G, et al., 2020, The basic infinite-dimensional or functional optimization problem, Communications and Control Engineering, Pages: 1-38
In infinite-dimensional or functional optimization problems, one has to minimize (or maximize) a functional with respect to admissible solutions belonging to infinite-dimensional spaces of functions, often dependent on a large number of variables. As we consider optimization problems characterized by very general conditions, optimal solutions might not be found analytically and classical numerical methods might fail. In these cases, one must use suitable approximation methods. This chapter describes an approximation method that uses families of nonlinear approximators – including commonly used shallow and deep neural networks as special cases – to reduce infinite-dimensional problems to finite-dimensional nonlinear programming ones. It is named “Extended Ritz Method” (ERIM). This term originates from the classical Ritz method, which uses linear combinations of functions as an approximation tool. It does not seem that the Ritz method has met with important successes as regards problems whose admissible solutions depend on large number of variables. This might be ascribed to the fact that this method can be subject to one of the forms of the so-called “curse of dimensionality.” However, theoretical and numerical results show that the ERIM might mitigate this drawback. Examples of infinite-dimensional optimization problems are presented that can be approximated by nonlinear programming ones.
Zoppoli R, Sanguineti M, Gnecco G, et al., 2020, Optimal control problems over an infinite Horizon, Communications and Control Engineering, Pages: 471-511
Optimal control problems over an infinite number of decision stages are considered with emphasis on the deterministic scenario. Both the open-loop and the closed-loop formulations are given and conditions for the existence of a stationary optimal control law are provided. Unless strong assumptions are made on the dynamic system and on the random variables (if present), the design of the optimal infinite-horizon controllers is an almost impossible task. Then, the well-known “receding-horizon” (RH) approximation is considered and the optimal control problem is restated accordingly. In the second part of the chapter, we consider the fundamental issue of closed-loop stability that arises owing to the infinite number of decision stages. More specifically, we address the stability properties of the closed-loop deterministic system under the action of approximate RH control laws obtained by the “extended Ritz method” and implemented through fixed-structure parametrized functions containing vectors of “free” parameters. Conditions are established on the maximum allowable approximation errors so as to ensure the boundedness of the state trajectories.
Zoppoli R, Sanguineti M, Gnecco G, et al., 2020, Stochastic optimal control with imperfect state information over a finite Horizon, Communications and Control Engineering, Pages: 383-426
Discrete-time stochastic optimal control problems are considered. These problems are stated over a finite number of decision stages. The state vector is assumed to be observed through a noisy measurement channel. Because of the very general assumptions under which the problems are stated, obtaining analytically optimal solutions is practically impossible. Note that the controller has to retain the vector of all the measures and of all the controls in memory, up to the most recent decision stage. Such measures and controls constitute the “information vector” that the control function depends on. The increasing dimension of the information vector makes it practically impossible to use dynamic programming. Then, we resort to the “extended Ritz method” (ERIM). The ERIM consists in substituting the admissible functions with fixed-structure parametrized functions containing vectors of “free” parameters. Of course, if the number of decision stages is large, the application of the ERIM is also impossible. Therefore, an approximate approach is followed by truncating the information vector and retaining in the memory only a suitable “limited-memory information vector.”
Zoppoli R, Sanguineti M, Gnecco G, et al., 2020, Deterministic optimal control over a finite Horizon, Communications and Control Engineering, Pages: 255-298
This chapter addresses discrete-time deterministic problems, where optimal controls have to be generated at a finite number of decision stages. No random variables influence either the dynamic system or the cost function. Then, there is no necessity of estimating the state vector. Such optimization problems are stated for their intrinsic practical importance and to describe the basic concepts of dynamic programming. As the problems are formulated under very general assumptions, their optimal solutions cannot be found in an analytical form. Therefore, we resort to an approximation consisting of the discretization of the state space into suitable grids at each decision stage. The discretization by regular grids is the simplest approach (and the one most widely used until some time ago). However, unless a small dimension of the state space is considered, this approach leads to an exponential growth of the number of samples, and thus to the curse of dimensionality. Therefore, the discretization by deterministic sequences of samples is addressed, which spread the samples in the most uniform way. Specifically, low-discrepancy sequences are considered, like quasi-Monte Carlo sequences. We also point out that the optimization problem can even be viewed as a nonlinear programming problem solvable by gradient-based descent techniques.
Zoppoli R, Sanguineti M, Gnecco G, et al., 2020, Stochastic optimal control with perfect state information over a finite Horizon, Communications and Control Engineering, Pages: 299-382
Discrete-time stochastic optimal control problems are stated over a finite number of decision stages. The state vector is assumed to be perfectly measurable. Such problems are infinite-dimensional as one has to find control functions of the state. Because of the general assumptions under which the problems are formulated, two approximation techniques are addressed. The first technique consists of an approximation of dynamic programming. The approximation derives from the fact that the state space is discretized. Instead of using regular grids, which lead to an exponential growth of the number of samples (and thus to the curse of dimensionality), low-discrepancy sequences (as quasi-Monte Carlo ones) are considered. The second approximation technique is given by the application of the “Extended Ritz Method” (ERIM). The ERIM consists in substituting the admissible functions with fixed-structure parametrized functions containing vectors of “free” parameters. This requires solving easier nonlinear programming problems. If suitable regularity assumptions are verified, such problems can be solved by stochastic gradient algorithms. The computation of the gradient can be performed by resorting to the classical adjoint equations, which solve deterministic optimal control problems with the addition of one term, dependent on the chosen family of fixed-structure parametrized functions.
Zoppoli R, Sanguineti M, Gnecco G, et al., 2020, Team optimal control problems, Communications and Control Engineering, Pages: 427-469
We consider discrete-time stochastic optimal control problems over a finite number of decision stages in which several controllers share different information and aim at minimizing a common cost functional. This organization can be described within the framework of “team theory.” Unlike the classical optimal control problems, linear-quadratic-Gaussian hypotheses are sufficient neither to obtain an optimal solution in closed-loop form nor to understand whether an optimal solution exists. In order to obtain an optimal solution in closed-loop form, additional suitable assumptions must be introduced on the “information structure” of the team. The “information structure” describes the way in which each controller’s information vector is influenced by the stochastic environment and by the decisions of the other controllers. Dynamic programming cannot be applied unless the information structure takes particular forms. On the contrary, the “extended Ritz method” (ERIM) can be always applied. The ERIM consists in substituting the admissible functions with fixed-structure parametrized functions containing vectors of “free” parameters. The ERIM is tested in two case studies. The former is the well-known Witsenhausen counterexample. The latter is an optimal routing problem in a store-and-forward packet-switching network.
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