Tube-based coalitional MPC with plug-and-play features

IFAC PapersOnLine 56-2 (2023) 7704–7709 ScienceDirect Available online at www.sciencedirect.com 2405-8963 Copyright © 2023 The Authors. This is an open access article under the CC BY-NC-ND license . Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2023.10.1173 10.1016/j.ifacol.2023.10.1173 2405-8963 Copyright © 2023 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚ Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚ ,P. R. Baldivieso-Monasterios ˚˚ , J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Ag. NAg. 2Ag. 1 Subs. 1 Subs. 2 Physical Coupling Subs. N Multi-agent system: Control Network Comm. link ... ... cC Coalitions of agents Contr. 1 c1 plug-in plug-out Plug-and-play subsystems Subs. i Contr. i Contr. N Contr. 2 ··· Fig. 1. Proposed scheme with plug-and-play features for coupled subsystems, where there is a set of coalitions (c1,c 2,...,c C) of local controllers. automatically reconfigure when a new component is added to or removed from the system (Stoustrup, 2009). Multiple interpretations arise around the term plug-and-play. In fault-tolerant control, it is used for automatic recovery of the control objective after a failure (Bodenburg et al., 2014). In micro-grids applications, PnP operations are allowed to adapt control actions in real-time according to varying network conditions (D¨orfler et al., 2014), and are more related to the concept of neutral interactions among the distributed generation units Riverso et al. (2014b), rather than on robustness against subsystem couplings. In this latter context, which is more relevant to the topic of this work, Riverso et al. (2014a) and Lucia et al. (2015) propose control strategies capable of handling PnP subsystems offline, provided that feasibility and stability is not compromised. Otherwise, PnP operations are rejected to ensure overall system’s recursive feasibility and stability. This work presents a tube-based coalitional model predictive control scheme with PnP features for linear multiagent systems (see Fig. 1). We consider that subsystems can join or leave the system in real time, and switching dynamics may be introduced. The formation of coalitions allows for automatic reconfiguration of the control system and avoids the rejection of PnP operations, unlike the studies mentioned above. The set of coalitions is selected in real time to accommodate disturbances and find a trade-off between performance and cooperation burden. Similarly to (Trodden and Maestre, 2017), agents can scale down their constraint sets and share their scaling factors among neighbors to reconfigure the disturbance sets. Our approach goes a step further in exploring an inherent robustness gap to cover disturbances not considered by tubes (e.g., PnP events) and avoid regrouping agents. The robustness gap is created by having each agent employ two different scaling factors: a public one that is implemented and broadcast to the neighborhood, and a private one that contains confidential information. Finally, a case study demonstrates the benefits of this coalitional approach. The paper is structured as follow: §2 introduces the problem setting, §3 describes the tube-based coalitional MPC approach, §4 presents the control algorithm, §5 details PnP operations, §6 reports results on a case study, and §7 provides the conclusions. Notation: Sets R0`and R`(N0`and N`) refers to nonnegative and positive real numbers (integers). For two sets X,YĎRn, the Cartesian product is XˆYfitpx, yq: xPX,yPYu. If tXiuiPNis a finite family of sets indexed by N“t1,...,Nu, then the Cartesian product ŚiPNXiis defined as X1ˆ¨¨¨ˆXN“ tpx1,...,x Nq: x1PX1,...,x NPXNu. The image of a set XĎRn under a linear mapping A:RnÞÑ Rmis given by AXfitAx :xPXu. For sets X,YĎRn, the Minkowski sum is X‘Yfitx`y:xPX,yPYu, and the Pontryagin difference is XaYfitzPRn:Y‘tzuĎXufor YĎX. ||x||arepresents la-norm of the vector xPRnwith aPN`, and ||x||2 Q“xJQx, with Qbeing a weight matrix. 2. PROBLEM SETTINGS This section provides an overview of the dynamics and constraints of the subsystems, defines the coalitional structure given by the cooperation topology of the control network, and also details the coalitions’ dynamics and constraints. 2.1 Subsystem dynamics and constraints The system is composed of N“t1,2,...,Nudynamically coupled subsystems i, whose discrete-time dynamics are: xipk`1q“Aiixipkq`Biiuipkq`wipkq, wipkq“ ř jPMi`Aijxjpkq`Bijujpkq˘`we ipkq,(1) where kPN`is the discrete-time index, xiPRqiand uiPRriare the state and input vectors, respectively, subject to constraints Xiand Ui,wiPRqiis the coupling through states and inputs with neighbors jPMifitjP Nztiu:Aij ‰0_Bij ‰0uwith Aij PRqiˆqj,B ij PRqiˆrj, and the external noise we iis assumed to be bounded by We i. Assumption 1. Xi,We iĂRqiand UiĂRriare compact convex sets that contain the origin in their interiors. 2.2 Control network and cooperation topology Each subsystem iis managed by an agent that has access to local information and can coalesce with its neighbors to improve performance and handle unplanned disturbances such as plug-and-play events. Let us describe the control network by an undirected graph (N,L) with Nagents and LĎNˆNcooperation links. Each link lij “ti, ju“ tj, iu“lji PL, which connects agents iand j, is assumed to provide a bidirectional information flow that can be enabled or disabled by the control scheme. Each enabled link involves a fixed cooperation cost clink PR`. The set of active links at time instant kdefines the cooperation topology Λ ĎL, which is determined by a balance between performance and cooperation costs. Being |L|total number of links, there are 2|L|different topologies that are grouped into a set T“tΛ1,Λ2,...,Λ2|L|u. For convenience, Λ1is the decentralized topology with all links disabled: t0,0,0u, and the centralized topology Λ2|L|denotes full network cooperation: t1,1,1u. The cardinality of Λ is denoted by |Λ|and provides the number of active links in the topology. Definition 2. Given a current topology Λcur, the set of the potential successor topologies Tnew ĎTis defined as: Tnew fitΛPT: distpΛcur,Λqď1uĎT,(2) where distpΛcur,Λqdenotes the Hamming distance between two topologies. E. Masero et al. / IFAC PapersOnLine 56-2 (2023) 7704–7709 7705 Copyright © 2023 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Tube-based coalitional MPC with plug-and-play features ‹ E. Masero ˚,P. R. Baldivieso-Monasterios ˚˚, J. M. Maestre ˚,˚˚˚,P. A. Trodden ˚˚,E. F. Camacho ˚ ˚Dept. of Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Spain (e-mails: {evamasero, pepemaestre, efcamacho}@us.es) ˚˚ Dept. of Automatic Control and Systems Engineering, University of Sheffield, UK (e-mails: {p.baldivieso, p.trodden}@sheffield.ac.uk) ˚˚˚ Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Japan (e-mail: [email protected]) Abstract: This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example. Keywords: Model predictive control, distributed control, multi-agent systems, time-varying systems, robust control, plug-and-play events. 1. INTRODUCTION Model predictive control (MPC) is a control strategy that optimizes control actions in a finite horizon according to an objective function (Maciejowski, 2002). The ability to handle constraints, uncertainties, and multi-variable systems has turned MPC into a successful control strategy, leading to its implementation in distributed multi-agent systems (Maestre and Negenborn, 2014). Distributed MPC techniques involve partitioning the system into subsystems that are managed by local MPC controllers, giving rise to the so-called agents. According to a communication control network, agents share information to improve its local cost while contributing to the global objective. A dynamic control topology that allows enabling and disabling communication links in real-time is the basis for coalitional MPC methods (Fele et al., 2017). Specifically, coalitional control allows local controllers form coalitions to find a balance between performance and communication effort. Depending on the way coalitions are selected, coalitional schemes can be classified into: i) top-down architectures, where a supervisory layer decides the cooperation topology, and ii) bottom-up structures, in which coalitions are formed at the agent level without global knowledge (Baldivieso-Monasterios and Trodden, 2021). The current work follows a top-down coalitional setting of model predictive controllers. ‹This paper was supported by the European Research Council under programme Horizon 2020 for project OCONTSOLAR (ref. 789051), by MCIN/AEI/ 10.13039/501100011033 for the project C3PO-R2D2 (Grant PID2020-119476RB-I00), and by the Spanish Ministry of Science and Innovation for University Staff Training (Grant FPU18/04476). One of the main challenges in coalitional control and other distributed settings is dealing with disturbances caused by the dynamic coupling between subsystems. In this regard, robust control techniques are employed when designing controllers. An approach is to consider couplings as bounded additive uncertainties to ensure stability and suitable global performance (Richards and How, 2007). The most conservative fashion for modeling the presence of uncertainties is min-max MPC (Scokaert and Mayne, 1998), which minimizes the control input over the worstcase of the disturbances. The idea of tubes proposed in (Langson et al., 2004) has also become popular for ensuring robust stability for constrained linear systems (Mayne et al., 2005; Trodden and Richards, 2010). Nevertheless, a significant downside is incurred by tightening local constraint sets by margins that can conservatively go beyond the disturbances that subsystems will experience. To reduce conservatism, further tube-based methods have been explored. Rakovi´c et al. (2012) focus on optimizing the diameters of the state and control tubes online, Riverso and Ferrari-Trecate (2012) propose to apply tube-based control twice to exploit the region of attraction of the subsystem for the planned state trajectories of neighbors, Lucia et al. (2015) develop a contract-based scheme in which subsystems share to their neighbors a sequence of the coupling variables that will hold in the future, and Trodden and Maestre (2017) propose to reject dynamic couplings via optimized tubes to outer-bound their disturbance sets more accurately. A similar idea to the latter is here followed, but upgraded to a coalitional setting. Many control approaches are limited in their ability to adapt to physical changes in a system. However, plug-and- play (PnP) control describes the control system’s ability to Ag. NAg. 2Ag. 1 Subs. 1 Subs. 2 Physical Coupling Subs. N Multi-agent system: Control Network Comm. link ... ... c C Coalitions of agents Contr. 1 c 1 plug-in plug-out Plug-and-play subsystems Subs. i Contr. i Contr. N Contr. 2 ··· Fig. 1. Proposed scheme with plug-and-play features for coupled subsystems, where there is a set of coalitions (c1,c 2,...,c C) of local controllers. automatically reconfigure when a new component is added to or removed from the system (Stoustrup, 2009). Multiple interpretations arise around the term plug-and-play. In fault-tolerant control, it is used for automatic recovery of the control objective after a failure (Bodenburg et al., 2014). In micro-grids applications, PnP operations are allowed to adapt control actions in real-time according to varying network conditions (D¨orfler et al., 2014), and are more related to the concept of neutral interactions among the distributed generation units Riverso et al. (2014b), rather than on robustness against subsystem couplings. In this latter context, which is more relevant to the topic of this work, Riverso et al. (2014a) and Lucia et al. (2015) propose control strategies capable of handling PnP subsystems offline, provided that feasibility and stability is not compromised. Otherwise, PnP operations are rejected to ensure overall system’s recursive feasibility and stability. This work presents a tube-based coalitional model predictive control scheme with PnP features for linear multiagent systems (see Fig. 1). We consider that subsystems can join or leave the system in real time, and switching dynamics may be introduced. The formation of coalitions allows for automatic reconfiguration of the control system and avoids the rejection of PnP operations, unlike the studies mentioned above. The set of coalitions is selected in real time to accommodate disturbances and find a trade-off between performance and cooperation burden. Similarly to (Trodden and Maestre, 2017), agents can scale down their constraint sets and share their scaling factors among neighbors to reconfigure the disturbance sets. Our approach goes a step further in exploring an inherent robustness gap to cover disturbances not considered by tubes (e.g., PnP events) and avoid regrouping agents. The robustness gap is created by having each agent employ two different scaling factors: a public one that is implemented and broadcast to the neighborhood, and a private one that contains confidential information. Finally, a case study demonstrates the benefits of this coalitional approach. The paper is structured as follow: §2 introduces the problem setting, §3 describes the tube-based coalitional MPC approach, §4 presents the control algorithm, §5 details PnP operations, §6 reports results on a case study, and §7 provides the conclusions. Notation: Sets R0`and R`(N0`and N`) refers to nonnegative and positive real numbers (integers). For two sets X,YĎRn, the Cartesian product is XˆYfitpx, yq: xPX,yPYu. If tXiuiPNis a finite family of sets indexed by N“t1,...,Nu, then the Cartesian product ŚiPNXiis defined as X1ˆ¨¨¨ˆXN“ tpx1,...,x Nq: x1PX1,...,x NPXNu. The image of a set XĎRn under a linear mapping A:RnÞÑ Rmis given by AXfitAx :xPXu. For sets X,YĎRn, the Minkowski sum is X‘Yfitx`y:xPX,yPYu, and the Pontryagin difference is XaYfitzPRn:Y‘tzuĎXufor YĎX. ||x||arepresents la-norm of the vector xPRnwith aPN`, and ||x||2 Q“xJQx, with Qbeing a weight matrix. 2. PROBLEM SETTINGS This section provides an overview of the dynamics and constraints of the subsystems, defines the coalitional structure given by the cooperation topology of the control network, and also details the coalitions’ dynamics and constraints. 2.1 Subsystem dynamics and constraints The system is composed of N“t1,2,...,Nudynamically coupled subsystems i, whose discrete-time dynamics are: xipk`1q“Aiixipkq`Biiuipkq`wipkq, wipkq“ ř jPMi`Aijxjpkq`Bijujpkq˘`we ipkq,(1) where kPN`is the discrete-time index, xiPRqiand uiPRriare the state and input vectors, respectively, subject to constraints Xiand Ui,wiPRqiis the coupling through states and inputs with neighbors jPMifitjP Nztiu:Aij ‰0_Bij ‰0uwith Aij PRqiˆqj,B ij PRqiˆrj, and the external noise we iis assumed to be bounded by We i. Assumption 1. Xi,We iĂRqiand UiĂRriare compact convex sets that contain the origin in their interiors. 2.2 Control network and cooperation topology Each subsystem iis managed by an agent that has access to local information and can coalesce with its neighbors to improve performance and handle unplanned disturbances such as plug-and-play events. Let us describe the control network by an undirected graph (N,L) with Nagents and LĎNˆNcooperation links. Each link lij “ti, ju“ tj, iu“lji PL, which connects agents iand j, is assumed to provide a bidirectional information flow that can be enabled or disabled by the control scheme. Each enabled link involves a fixed cooperation cost clink PR`. The set of active links at time instant kdefines the cooperation topology Λ ĎL, which is determined by a balance between performance and cooperation costs. Being |L|total number of links, there are 2|L|different topologies that are grouped into a set T“tΛ1,Λ2,...,Λ2|L|u. For convenience, Λ1is the decentralized topology with all links disabled: t0,0,0u, and the centralized topology Λ2|L|denotes full network cooperation: t1,1,1u. The cardinality of Λ is denoted by |Λ|and provides the number of active links in the topology. Definition 2. Given a current topology Λcur, the set of the potential successor topologies Tnew ĎTis defined as: Tnew fitΛPT: distpΛcur,Λqď1uĎT,(2) where distpΛcur,Λqdenotes the Hamming distance between two topologies. 7706 E. Masero et al. / IFAC PapersOnLine 56-2 (2023) 7704–7709 2.3 Coalition dynamics and constraints The cooperation topology defines the coalitional structure of the control network as follows. Definition 3. (Coalitional structure). A cooperation topology Λ partitions the system N“t1,...,Nuinto a set of non-overlapping coalitions C“tc1,...,c Cuwith CďN, which cover all subsystems: ŤcPCc“N. Each coalition cPCis a non-empty cluster of subsystems that range from the grand coalition c“Nto a single subsystem c“i. The discrete-time dynamics of each coalition cPCis xcpk`1q“Accxcpkq`Bccucpkq`wcpkq, wcpkq“ ř dPMc`Acdxdpkq`Bcdudpkq˘`we cpkq,(3) where xc“pxiqiPcand uc“puiqiPcare, respectively, aggregate state and input vectors, Acc “rAij si,jPcand Bcc “rBijsi,jPcare state and input matrices, and disturbance term wcrepresents coupling with neighbors plus external noise we c. The neighbors of coalition care gathered in set McfitdPCzc:Acd ‰0_Bcd ‰0u. Assumption 4. State and input constraint sets of a coalition are Xc“ŚiPcXiand Uc“ŚiPcUi, respectively. 3. TUBE-BASED COALITIONAL MPC Given a coalitional structure C, we consider that coalitions can scale down their constraint sets and reconfigure disturbance sets to reduce conservatism of tube-based approaches. The constraint sets of coalition cPCare scaled by factors αcPr0,1sand βcPr0,1sas: Xcpαcq“αcXc,Ucpβcq“βcUc,(4) where the original set of state constraint is Xc“Xcp1q and the set of input constraint is Uc“Ucp1q. Note that states xcand inputs ucsatisfy constraints provided that αcPr0,1sand βcPr0,1s. Given the dynamics (3) and constraints (4) of coalition c, the disturbance wcis constrained by the set: Wcpα, βqfià dPMc`AcdXdpαdq‘BcdUdpβdq˘‘We c,(5) where pα, βqrepresents the dependence of the scaling factors of all neighbors dPMc, and external noise is also assumed to be constrained by set We c. The original disturbance set Wcp1,1qdepends on the original constraint sets Xdp1qand Udp1qfor all dPMc. Definition 5. The set Ωcis robust control invariant (RCI) for the dynamics xcpk`1q“Accxcpkq`Bccucpkq` wcpkqwith constraint sets pXc,Uc,Wcqif ΩcĎXcand there exists a control law uc“µpxcpkqq P Ucsuch that xcpkqPΩcimplies xcpk`1qPΩc,@wcPWc. The control law is said to be invariance-inducing over the set Ωc. Since the size of Ωcdepends on the size of Wcpα, βq, its existence defines a set ΩcpWcqsuch that: ΩcpWcqĎXcpαcq,µ `ΩcpWcq˘ĎUcpβcq.(6) In tube approaches, the existence of invariant sets equips controllers with robustness to handle disturbances from coupling subsystems. For that reason, the following assumption is considered as starting point. Assumption 6. There exists an RCI set ΩcpWcp1,1qq Ď Xcp1qfor all cPCin the decentralized cooperation topology, i.e., for all current and future subsystems. 3.1 Inherent robustness gap For accommodating locally disturbances not covered by tubes without any serious reconfiguration of the control system, we create an extra robustness gap by using two different scaling factors: ‚Public factors αp i,βp iPr0,1s, which are shared with the network and parameterize constraint sets. ‚Private factors αr i,βr iPr0,1s, which are confidential and individually optimized by each controller. At start, all public and private scaling factors are set equal to 1 for all iPN. Then private factors are online optimized by agents to tighten their constraints and the idea is that agents share larger public scaling factors to their neighbors. In this way, the robustness gap is created by setting public factors as outer bounds of private scaling factors: αp i `“"αp ip1´ραq`αr iρα,if αr iďαp i, αr i,if αr iąαp i,(7) βp i `“"βp ip1´ρβq`βr iρβ,if βr iďβp i, βr i,if βr iąβp i,(8) where αp i `and βp i `denote, respectively, the state and input public scaling factors of agent iat instant k`1, and tuning parameters ρα,ρ βPp0,1q. The formation of coalitions is based on the public information of the agents involved, which frozen their public factors pαp i,βp iq. Hence, the coalition scaling factors pαp c,βp cq hold with the corresponding values to be fulfilled: αp cXc“ą iPc αp iXi,β p cUc“ą iPc βp iUi.(9) Remark 7. Public scaling factors of all iPcare frozen while in coalition, unless private factors of any iPc are greater than public factors. In the latter case, factors pαp c,βp cqare recomputed as (9) with the updated factors of those agents iPcwhose αr iąαp iand βr iąβp ito avoid misleading their neighbors about the real disturbances. 3.2 Preliminaries of the tube-based approach Approaches based on tubes are characterized by dealing with a nominal system with no uncertainties: zcpk`1q“Acczcpkq`Bccvcpkq, where vcis the control input obtained by an MPC controller to regulate the nominal state zc, and handling error dynamics: ecpk`1q“Accec`Bcc µc`ecpkq˘`wc, with ecpkq“xcpkq´zcpkqcontained in an invariant set, in this work, the RCI set is ΩcpWcpαp,βpqq. The nominal constraint sets depend on RCI sets as follows: Zcpαp cqfiXcpαp cqaΩcpWcpαp,βpqq, Vcpβp cqfiUcpβp cqaµ`ΩcpWcpαp,βpqq˘,(10) but we can employ an outer bounding of the RCI set to reduce the computational effort of calculating Ωcexplicitly every time the size of Wcpαp,βpqchanges. To this end, a linear programming problem is solved to obtain the RCI set ΩH, as proposed by Rakovi´c et al. (2007): mintϵ:ϕPΦu,(11) where ϕ“pMH,a,b,ϵqand Φ “tϕ:MHPMH,ΩHĎ aX,µpΩHqĎbU,aPr0,1s,bPr0,1s,q xa`qubďϵuwith MH“pM0,...,M H´1q, and weights qxand qu. Here, scaling constants aand btighten, respectively, the state and input constraint sets to guarantee: ΩHĎaXand µpΩHqĎbU(see (Rakovi´c et al., 2007) for more details). Therefore, constraint sets (10) can be substituted by: Zcpαp c,a cq“Xcpαp cqaacXcpαp cq“p1´acqXcpαp cq, Vcpβp c,b cq“Ucpβp cqabcUcpβp cq“p1´bcqUcpβp cq.(12) with acPr0,1sand bcPr0,1s. 3.3 Tube-based coalitional MPC problem The objective of coalition cPCis to lead the state to the origin while fulfilling the constraints. We propose a coalitional MPC scheme based on optimized tubes to manage coalition dynamics (3) with the control policy: ucpkq“µc`ecpkqq ` vo cpzcpkq˘,(13) where µcpecpkqq is the RCI control law, and vo cis the first element of optimal control sequence. The control sequence Vo cpkqfitvo cpkq,...,vo cpk`Np´1qu, the state sequence Zo cpkq“tzo cpkq,...,zo cpk`Npqu, and private factors pαr c,βr cqare obtained by solving the following nominal MPC problem in the prediction horizon Np: min Vcpkq,αr c,βr c Jc`Zcpkq,V cpkq˘`ρααr c`ρββr c,(14) with weights ρα,ρ βPR`, and subject to constraints: zcpk`t`1q“Acczcpk`tq`Bccvcpk`tq zcpkq“˜zcpkq,t“0,...,N p´1, zcpk`tqPαr cZcpαp c,a cq,t“1,...,N p´1, zcpk`NpqPΩf cpαp c,βp c,a c,b cq, vcpk`tqPVcpβp c,b cq,t“0,...,N p´1 αr cPr0,1s,β r cPr0,1s, (15) for all cPC, where ˜zcpkqis the current state value, and the set Ωf cis the terminal state constraint. Assumption 8. Terminal set Ωf cpαp c,βp c,a c,b cqis positively invariant for dynamics zcpk`1q“Acczcpkq`Bccvcpkqunder the control law vc“Kf czc, that is, pAcc `BccKf cqΩf cĎ Ωf cwith Ωf cĎZcpαp c,a cqand Kf cΩf cĎVcpβp c,b cq. The function JcpZcpkq,V cpkqq in problem (14) is: Np´1 ř t“0`}zcpk`tq}2 Qc`}vcpk`tq}2 Rc˘ loooooooooooooooooooomoooooooooooooooooooon lc`zcpk`tq,vcpk`tq˘ `}zcpk`Npq}2 Pc loooooooomoooooooon fc`zcpk`Npq˘ , where stage cost lcp¨q weight the nominal state and input vectors by matrices Qcą0 and Rcą0, respectively, and terminal cost fcp¨q, with Pcą0, is designed such that zcpkqJPczcpkq´zcpk`1qJPczcpk`1qělc`zcpk`1q,Kf czcpk`1q˘, therefore, zcpkqJPczcpkqis a control Lyapunov function. 4. CONTROL ALGORITHM We present a top-down coalitional MPC algorithm based on optimized tubes and an extra robustness margin. The supervisory layer is responsible for selecting the cooperation topology every Tup PN`time instants (and when needed) to improve performance and guarantee recursive feasibility. The algorithm is described as follows: Alg. 1: Supervisory layer Initial data: Xi,Ui,Kf i,H i,@i, Np,ρ α,ρ β,τ α,τ β,clink Start: Λcur “Λcen,zip0q“xip0q,αp i,βp i,α r i,βr i“1,@iPN Inputs: Λcur,α p i,βp i,@iPN.Output: Λnew 1: Given Λcur, measure ˜xcand ˜zcfor all cPC. 2: Calculate Tnew as (2) with Λcur. 3: for each Λnew PTnew do 4: Compute Wcas (5), Ωcby (11), and Ωf c,@cPC. 5: if EΩcfor any cthen 6: Mark Λnew as infeasible, and go to Step 3. 7: end if 8: for each cPCdo 9: Solve (14) with τα,τ β“0 and αr c,βr c“1 to obtain control sequence Ucvia (13) and Γc“řNp t“1plcpxcpk`tq,u cpk`tqq`clink|Λc|. 10: end for 11: Calculate the cost ΓΛ“řcPCΓcfor Λnew. 12: end for 13: if all Λnew PTnew are marked as infeasible then 14: Any cwith EΩcclusters with neighbor dPMc with the largest Wd, and update Λcur. 15: Go to Step 2. 16: else 17: Select topology Λnew PTnew with lowest cost ΓΛ. 18: Sent Λcur to the lower layer (Alg. 2). 19: end if In the lower layer, each cPCimplements Alg. 2 according to the current topology Λcur and public scaling factors: Alg. 2: Lower control layer Initial data: Xc,Uc,H c,Kf c,Np,τα,τ β,σ c“0,@cPC. Inputs: Λcur,x c,α p c,βp c,@cPC Outputs: αp c`,βp c `,x ` c,z` c 1: Compute Wcby (5), Ωcby (11), and Ωf c. 2: Solve (14) to obtain vo cand αr iand βr i,@iPc. 3: Apply ucto get x` cand vo cto obtain z` c. 4: if αr iąαp iand βr iąβp ifor any iPcthen 5: Set αp i `“αr iand βp i `“βr i. 6: Update αp cand βp cby (9). 7: Execute Alg. 1. 8: else if c“tiu, and αr iďαp iand βr iďβp ithen 9: Get αp i `(7), βp i `(8), and active flag σc“1. 10: Share αp c`,βp c `with all cPMc. 11: Update W` cand Ω` c. 12: end if 13: if σc“1, DΩ` c, and x` c´z` cPΩ` c,@cPCthen 14: Set σc“0 and share αp c`,βp c `with all dPMc. 15: else 16: Hold current scaling factors as αp c`and βp c `. 17: end if 5. PLUG-AND-PLAY SUBSYSTEMS We consider that subsystems can join and leave the system in real time, so switching dynamics may be introduced. This fact may force the redesign of the current cooperation topology for feasibility and performance reasons. When a new subsystem i“N`1 is plugged at time instant kplug, the system yields: Npkq“t1,2,...,N `1u. Due to dynamic couplings, the disturbances of neighbors jPNipkq E. Masero et al. / IFAC PapersOnLine 56-2 (2023) 7704–7709 7707 where ϕ“pMH,a,b,ϵqand Φ “tϕ:MHPMH,ΩHĎ aX,µpΩHqĎbU,aPr0,1s,bPr0,1s,q xa`qubďϵuwith MH“pM0,...,M H´1q, and weights qxand qu. Here, scaling constants aand btighten, respectively, the state and input constraint sets to guarantee: ΩHĎaXand µpΩHqĎbU(see (Rakovi´c et al., 2007) for more details). Therefore, constraint sets (10) can be substituted by: Zcpαp c,a cq“Xcpαp cqaacXcpαp cq“p1´acqXcpαp cq, Vcpβp c,b cq“Ucpβp cqabcUcpβp cq“p1´bcqUcpβp cq.(12) with acPr0,1sand bcPr0,1s. 3.3 Tube-based coalitional MPC problem The objective of coalition cPCis to lead the state to the origin while fulfilling the constraints. We propose a coalitional MPC scheme based on optimized tubes to manage coalition dynamics (3) with the control policy: ucpkq“µc`ecpkqq ` vo cpzcpkq˘,(13) where µcpecpkqq is the RCI control law, and vo cis the first element of optimal control sequence. The control sequence Vo cpkqfitvo cpkq,...,vo cpk`Np´1qu, the state sequence Zo cpkq“tzo cpkq,...,zo cpk`Npqu, and private factors pαr c,βr cqare obtained by solving the following nominal MPC problem in the prediction horizon Np: min Vcpkq,αr c,βr c Jc`Zcpkq,V cpkq˘`ρααr c`ρββr c,(14) with weights ρα,ρ βPR`, and subject to constraints: zcpk`t`1q“Acczcpk`tq`Bccvcpk`tq zcpkq“˜zcpkq,t“0,...,N p´1, zcpk`tqPαr cZcpαp c,a cq,t“1,...,N p´1, zcpk`NpqPΩf cpαp c,βp c,a c,b cq, vcpk`tqPVcpβp c,b cq,t“0,...,N p´1 αr cPr0,1s,β r cPr0,1s, (15) for all cPC, where ˜zcpkqis the current state value, and the set Ωf cis the terminal state constraint. Assumption 8. Terminal set Ωf cpαp c,βp c,a c,b cqis positively invariant for dynamics zcpk`1q“Acczcpkq`Bccvcpkqunder the control law vc“Kf czc, that is, pAcc `BccKf cqΩf cĎ Ωf cwith Ωf cĎZcpαp c,a cqand Kf cΩf cĎVcpβp c,b cq. The function JcpZcpkq,V cpkqq in problem (14) is: Np´1 ř t“0`}zcpk`tq}2 Qc`}vcpk`tq}2 Rc˘ loooooooooooooooooooomoooooooooooooooooooon lc`zcpk`tq,vcpk`tq˘ `}zcpk`Npq}2 Pc loooooooomoooooooon fc`zcpk`Npq˘ , where stage cost lcp¨q weight the nominal state and input vectors by matrices Qcą0 and Rcą0, respectively, and terminal cost fcp¨q, with Pcą0, is designed such that zcpkqJPczcpkq´zcpk`1qJPczcpk`1qělc`zcpk`1q,Kf czcpk`1q˘, therefore, zcpkqJPczcpkqis a control Lyapunov function. 4. CONTROL ALGORITHM We present a top-down coalitional MPC algorithm based on optimized tubes and an extra robustness margin. The supervisory layer is responsible for selecting the cooperation topology every Tup PN`time instants (and when needed) to improve performance and guarantee recursive feasibility. The algorithm is described as follows: Alg. 1: Supervisory layer Initial data: Xi,Ui,Kf i,H i,@i, Np,ρ α,ρ β,τ α,τ β,clink Start: Λcur “Λcen,zip0q“xip0q,αp i,βp i,α r i,βr i“1,@iPN Inputs: Λcur,α p i,βp i,@iPN.Output: Λnew 1: Given Λcur, measure ˜xcand ˜zcfor all cPC. 2: Calculate Tnew as (2) with Λcur. 3: for each Λnew PTnew do 4: Compute Wcas (5), Ωcby (11), and Ωf c,@cPC. 5: if EΩcfor any cthen 6: Mark Λnew as infeasible, and go to Step 3. 7: end if 8: for each cPCdo 9: Solve (14) with τα,τ β“0 and αr c,βr c“1 to obtain control sequence Ucvia (13) and Γc“řNp t“1plcpxcpk`tq,u cpk`tqq`clink|Λc|. 10: end for 11: Calculate the cost ΓΛ“řcPCΓcfor Λnew. 12: end for 13: if all Λnew PTnew are marked as infeasible then 14: Any cwith EΩcclusters with neighbor dPMc with the largest Wd, and update Λcur. 15: Go to Step 2. 16: else 17: Select topology Λnew PTnew with lowest cost ΓΛ. 18: Sent Λcur to the lower layer (Alg. 2). 19: end if In the lower layer, each cPCimplements Alg. 2 according to the current topology Λcur and public scaling factors: Alg. 2: Lower control layer Initial data: Xc,Uc,H c,Kf c,Np,τα,τ β,σ c“0,@cPC. Inputs: Λcur,x c,α p c,βp c,@cPC Outputs: αp c`,βp c `,x ` c,z` c 1: Compute Wcby (5), Ωcby (11), and Ωf c. 2: Solve (14) to obtain vo cand αr iand βr i,@iPc. 3: Apply ucto get x` cand vo cto obtain z` c. 4: if αr iąαp iand βr iąβp ifor any iPcthen 5: Set αp i `“αr iand βp i `“βr i. 6: Update αp cand βp cby (9). 7: Execute Alg. 1. 8: else if c“tiu, and αr iďαp iand βr iďβp ithen 9: Get αp i `(7), βp i `(8), and active flag σc“1. 10: Share αp c`,βp c `with all cPMc. 11: Update W` cand Ω` c. 12: end if 13: if σc“1, DΩ` c, and x` c´z` cPΩ` c,@cPCthen 14: Set σc“0 and share αp c`,βp c `with all dPMc. 15: else 16: Hold current scaling factors as αp c`and βp c `. 17: end if 5. PLUG-AND-PLAY SUBSYSTEMS We consider that subsystems can join and leave the system in real time, so switching dynamics may be introduced. This fact may force the redesign of the current cooperation topology for feasibility and performance reasons. When a new subsystem i“N`1 is plugged at time instant kplug, the system yields: Npkq“t1,2,...,N `1u. Due to dynamic couplings, the disturbances of neighbors jPNipkq 7708 E. Masero et al. / IFAC PapersOnLine 56-2 (2023) 7704–7709 m1m2m3m4m5 h12 h23 h34 h45 κ12 κ23 κ34 κ45 F2 F1F3F4F5 r2 r1r3r4r5 Fig. 2. System compound of an array of coupled trucks. grow, and recursive feasibility may be lost. To prevent that, plug-ins are only executed when the supervisory layer is triggered, and cooperation topology is updated according to Alg. 1 at the infinitesimal instant after the plugin, k` plug. Conversely, when a subsystem iis disconnected from the system, it results in Npkq“t1,2,...,i´1,i` 1,...,Nu. Since disturbances of neighbors dPMidecrease, Alg. 1 could be executed to select another topology that improves performance. Otherwise, the current topology could be held, which is computationally less expensive. 6. ILLUSTRATIVE EXAMPLE We use the system presented in (Trodden and Maestre, 2017), where four trucks are coupled by springs and dampers with their neighbors, as depicted in Fig. 2. At time instant kplug, a fifth truck is connected to the system. Each truck iPNpkqhas a state xi, which is composed of position displacement riand velocity vi, and follows continuous-time dynamics: „9ri 9viȷ loomoon 9xi “» – 01 ´1 miÿ jPMi κij ´1 miÿ jPMi hijfi fl„ri viȷ`„0 100ȷui`wi, where each truck’s controller applies a horizontal force Fi“uir0,100sJwith uibeing the control input, and the disturbance wiis defined as: wi“ř jPNi » – 00 1 miÿ jPMi κij 1 miÿ jPMi hijfi fl„rj vjȷ`we i, where the external noise |we i|ďr2.5e ´3,2.5e ´3sJ.A discrete-time model with sample time Ts“0.2sthat approximates the continuous-time model is employed to simulate and control each truck i. As for model parameters in simulations, masses [kg]: m1,m 3“3, m2,m 4“2, and m5“5; spring constants [N/m]: κ12 “0.5, κ23 “0.7, κ34 “1, and κ45 “0.8; and damping factors [Ns/m]: h12 “0.3, h23 “0.4, h34 “0.5, and h45 “0.2. The objective is to lead trucks from initial states x1p0q“ r1.5,0sJ,x2p0q “ r´0.5,0sJ,x3p0q“r1,0sJ,x4p0q“ r´1,0sJ, and x5pkplugq“r1,0sJto their equilibrium positions, considering constraints |ri|ď4 m, |vi|ď1 m/s, and |ui|ď0.5 N/kg, and handling a plug-in truck at kplug “16 s. The weighting matrices for each truck are Qi“Iand Ri“100, and are aggregated as Qc“ diagpQiqiPcand Rc“diagpRiqiPc. The LQR terminal controller Kf c“diagpKf iqiPc, where Kf 1“ r´0.0365,´0.0460s, Kf 2“ r´0.0334,´0.0443s,Kf 3“ r´0.0335,´0.0446s,Kf 4“ r´0.0306,´0.0441s,Kf 5“ r´0.0363,´0.0463s, and the terminal weight matrix Pc“diagpPiqiPc, where P1“„4.3327 ´2.7765 ´2.7765 3.9817 ȷ,P 2“„4.2137 ´2.7240 ´2.7240 3.9148 ȷ,P 3“„4.2203 ´2.7763 ´2.7163 3.9196 ȷ, P4“„4.1235 ´2.6833 ´2.6833 3.8755 ȷ,P 5“„4.3305 ´2.7743 ´2.7743 3.9825 ȷ. Fig. 3. Formation of coalitions. 0246810 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 1 2 3 4 5 Time step k Agent i kplug Unavailable agent Available agent Agents in coalition Table 1. Comparison of costs. Methods Jperf Jcoop Jtotal Jtotal[%] Cen. MPC 28.47 14.50 42.97 — Coal. MPC 28.84 2.00 30.84 28.24 % 6.1 Simulated results We consider simulations of length Nsim “40, a prediction horizon Np“10, and a cost per link clink “0.1. We also define the supervisory layer period Tup “5, parameter ρα“0.1, and weight τα“5e ´5. In Fig. 3, we present the formation of coalitions starting from the centralized topology. Since the supervisory layer decides the new topology every Tup “5 instants, the coalitional structure is C“ tt1u,t2,3,4uu for kă5, and C“ tt1u,t2u,t3,4uu for 6 ďkă11. Afterwards, agents work independently until i“5 is introduced, causing the disturbance of its neighbor to increase. As a result, agent 4 forms a coalition with agent 3. This coalition is disbanded when disturbances decrease enough as a consequence of shrinking constraints sets of neighbors. In Fig. 4, we show the evolution of the public and private scaling factors of all agents. When agents 3 and 4 break their coalition at k“21 (see Fig. 3), factor αr 4ąαp 4. At next instant k“22, it is updated αp 4“αr 4, and Alg. 1 is triggered to hold recursive feasibility. By chance, instant k“22 coincides with the normal moment of execution of the supervisory layer. Afterwards, all agents work individually until the end. In Fig. 5, we illustrate the nominal and real state trajectories of agents and their tubes, which can shrink and grow due to changes of topology and factors αp ifor all iPNpkq. For example, tube of agent 3 generally shrinks but, at instants k“11 and k“21, its disturbances increase due to its coalition breakups (see Fig. 3), so its tube grows to handle more uncertainty. Finally, Table 1 shows a comparison of the performance, cooperation, and total cost for all MPC schemes: Jperf “ Nsim ÿ k“1 }xpkq}2 Q`}upkq}2 R,J coop “ Nsim ÿ k“1 clink|Λpkq|. As shown, centralized MPC provides the lowest performance cost at the expense of a high cooperation cost, while coalitional MPC yields a total cost (Jtotal “Jperf ` Jcoop) reduction of 28.24 %. We report that decentralized MPC cannot be implemented since it becomes infeasible. Therefore, coalitional MPC provides a suitable control framework to deal with plug-and-play operations online, while reducing the total cost. 7. CONCLUSIONS We propose a tube-based coalitional MPC method with plug-and-play (PnP) features for linear multi-agent systems. Agents exchange information about the size of their 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 1 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 2 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 3 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 4 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 5 αp iαr i Fig. 4. Evolution of scaling factors in coalitional MPC. 010 20 30 40 0 1 ´1 ´0.5 0 Time step k r1 v1 010 20 30 40 ´1 0 1 ´0.5 0 0.5 Time step k r2 v2 010 20 30 40 ´1 0 1 ´1 ´0.5 0 Time step k r3 v3 010 20 30 40 ´1 0 1 ´0.5 0 0.5 1 Time step k r4 v4 010 20 30 40 0 0.5 1 ´0.8 ´0.6 ´0.4 ´0.2 0 Time step k r5 v5 Tube Coal. MPC ziCoal. MPC xiCoal. MPC xiCen. MPC Fig. 5. Evolution of tubes cross sections for each agent. state and input constraints to reconfigure their disturbance sets and reduce conservatism. Moreover, agents can form coalitions to improve their performance and handle PnP operations. As shown in simulations, our coalitional MPC outperforms centralized MPC in total cost and has a degree of flexibility for handling PnP using a distributed setting. Since tubes depend on disturbances, they shrink when states come close to the origin. Future work will include a bottom-up implementation of our proposal and a rigorous analysis of the recursive feasibility and stability. REFERENCES Baldivieso-Monasterios, P.R. and Trodden, P.A. (2021). Coalitional predictive control: Consensus-based coalition forming with robust regulation. Automatica, 125. Bodenburg, S., Niemann, S., and Lunze, J. (2014). Experimental evaluation of a fault-tolerant plug-and-play controller. In 13th European Control Conference (ECC), 1945–1950. IEEE. D¨orfler, F., Simpson-Porco, J.W., and Bullo, F. (2014). Plug-and-play control and optimization in microgrids. In 53rd Conference on Decision and Control (CDC), 211–216. IEEE. Fele, F., Maestre, J.M., and Camacho, E.F. (2017). Coalitional control: Cooperative game theory and control. IEEE Control Systems Magazine, 37(1), 53–69. Langson, W., Chryssochoos, I., Rakovi´c, S., and Mayne, D.Q. (2004). Robust model predictive control using tubes. Automatica, 40(1), 125–133. Lucia, S., K¨ogel, M., and Findeisen, R. (2015). Contractbased predictive control of distributed systems with plug and play capabilities. IFAC-PapersOnLine, 48(23). Maciejowski, J.M. (2002). Predictive control: with constraints. Pearson Education. Maestre, J.M. and Negenborn, R.R. (2014). Distributed model predictive control made easy, volume 69. Springer. Mayne, D.Q., Seron, M.M., and Rakovi´c, S. (2005). Robust model predictive control of constrained linear systems with bounded disturbances. Automatica, 41(2). Rakovi´c, S.V., Kerrigan, E.C., Mayne, D.Q., and Kouramas, K.I. (2007). Optimized robust control invariance for linear discrete-time systems: Theoretical foundations. Automatica, 43(5), 831–841. Rakovi´c, S.V., Kouvaritakis, B., Findeisen, R., and Cannon, M. (2012). Homothetic tube model predictive control. Automatica, 48(8), 1631–1638. Richards, A. and How, J.P. (2007). Robust distributed model predictive control. International Journal of Control, 80(9), 1517–1531. Riverso, S., Farina, M., and Ferrari-Trecate, G. (2014a). Plug-and-play model predictive control based on robust control invariant sets. Automatica, 50(8), 2179–2186. Riverso, S. and Ferrari-Trecate, G. (2012). Tube-based distributed control of linear constrained systems. Automatica, 48(11), 2860–2865. Riverso, S., Sarzo, F., and Ferrari-Trecate, G. (2014b). Plug-and-play voltage and frequency control of islanded microgrids with meshed topology. IEEE Trans. on Smart Grid, 6(3), 1176–1184. Scokaert, P.O. and Mayne, D.Q. (1998). Min-max feedback model predictive control for constrained linear systems. IEEE Trans. on Automatic Control, 43(8), 1136–1142. Stoustrup, J. (2009). Plug & play control: Control technology towards new challenges. European Journal of Control, 15(3-4), 311–330. Trodden, P.A. and Maestre, J.M. (2017). Distributed predictive control with minimization of mutual disturbances. Automatica, 77, 31–43. Trodden, P.A. and Richards, A. (2010). Distributed model predictive control of linear systems with persistent disturbances. International Journal of Control, 83(8), 1653–1663. E. Masero et al. / IFAC PapersOnLine 56-2 (2023) 7704–7709 7709 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 1 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 2 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 3 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 4 010 20 30 40 0 0.2 0.4 0.6 0.8 1 Time step k Agent 5 αp iαr i Fig. 4. Evolution of scaling factors in coalitional MPC. 010 20 30 40 0 1 ´1 ´0.5 0 Time step k r1 v1 010 20 30 40 ´1 0 1 ´0.5 0 0.5 Time step k r2 v2 010 20 30 40 ´1 0 1 ´1 ´0.5 0 Time step k r3 v3 010 20 30 40 ´1 0 1 ´0.5 0 0.5 1 Time step k r4 v4 010 20 30 40 0 0.5 1 ´0.8 ´0.6 ´0.4 ´0.2 0 Time step k r5 v5 Tube Coal. MPC ziCoal. MPC xiCoal. MPC xiCen. MPC Fig. 5. Evolution of tubes cross sections for each agent. state and input constraints to reconfigure their disturbance sets and reduce conservatism. Moreover, agents can form coalitions to improve their performance and handle PnP operations. As shown in simulations, our coalitional MPC outperforms centralized MPC in total cost and has a degree of flexibility for handling PnP using a distributed setting. Since tubes depend on disturbances, they shrink when states come close to the origin. Future work will include a bottom-up implementation of our proposal and a rigorous analysis of the recursive feasibility and stability. REFERENCES Baldivieso-Monasterios, P.R. and Trodden, P.A. (2021). Coalitional predictive control: Consensus-based coalition forming with robust regulation. Automatica, 125. Bodenburg, S., Niemann, S., and Lunze, J. (2014). Experimental evaluation of a fault-tolerant plug-and-play controller. In 13th European Control Conference (ECC), 1945–1950. IEEE. D¨orfler, F., Simpson-Porco, J.W., and Bullo, F. (2014). Plug-and-play control and optimization in microgrids. In 53rd Conference on Decision and Control (CDC), 211–216. IEEE. Fele, F., Maestre, J.M., and Camacho, E.F. (2017). Coalitional control: Cooperative game theory and control. IEEE Control Systems Magazine, 37(1), 53–69. Langson, W., Chryssochoos, I., Rakovi´c, S., and Mayne, D.Q. (2004). Robust model predictive control using tubes. Automatica, 40(1), 125–133. Lucia, S., K¨ogel, M., and Findeisen, R. (2015). Contractbased predictive control of distributed systems with plug and play capabilities. IFAC-PapersOnLine, 48(23). Maciejowski, J.M. (2002). Predictive control: with constraints. Pearson Education. Maestre, J.M. and Negenborn, R.R. (2014). Distributed model predictive control made easy, volume 69. Springer. Mayne, D.Q., Seron, M.M., and Rakovi´c, S. (2005). Robust model predictive control of constrained linear systems with bounded disturbances. Automatica, 41(2). Rakovi´c, S.V., Kerrigan, E.C., Mayne, D.Q., and Kouramas, K.I. (2007). Optimized robust control invariance for linear discrete-time systems: Theoretical foundations. Automatica, 43(5), 831–841. Rakovi´c, S.V., Kouvaritakis, B., Findeisen, R., and Cannon, M. (2012). Homothetic tube model predictive control. Automatica, 48(8), 1631–1638. Richards, A. and How, J.P. (2007). Robust distributed model predictive control. International Journal of Control, 80(9), 1517–1531. Riverso, S., Farina, M., and Ferrari-Trecate, G. (2014a). Plug-and-play model predictive control based on robust control invariant sets. Automatica, 50(8), 2179–2186. Riverso, S. and Ferrari-Trecate, G. (2012). Tube-based distributed control of linear constrained systems. Automatica, 48(11), 2860–2865. Riverso, S., Sarzo, F., and Ferrari-Trecate, G. (2014b). Plug-and-play voltage and frequency control of islanded microgrids with meshed topology. IEEE Trans. on Smart Grid, 6(3), 1176–1184. Scokaert, P.O. and Mayne, D.Q. (1998). Min-max feedback model predictive control for constrained linear systems. IEEE Trans. on Automatic Control, 43(8), 1136–1142. Stoustrup, J. (2009). Plug & play control: Control technology towards new challenges. European Journal of Control, 15(3-4), 311–330. Trodden, P.A. and Maestre, J.M. (2017). Distributed predictive control with minimization of mutual disturbances. Automatica, 77, 31–43. Trodden, P.A. and Richards, A. (2010). Distributed model predictive control of linear systems with persistent disturbances. International Journal of Control, 83(8), 1653–1663.