Multi-set based model predictive control to explore large freshwater resources

IFAC PapersOnLine 58-2 (2024) 38–43 ScienceDirect Available online at www.sciencedirect.com 2405-8963 Copyright © 2024 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.2024.07.088 10.1016/j.ifacol.2024.07.088 2405-8963 Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Copyright © 2024 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) J.G. Martin et al. / IFAC PapersOnLine 58-2 (2024) 38–43 39 Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Multi-set based model predictive control to explore large freshwater resources J.G. Martin ∗A. Anderson ∗∗ I. S´ anchez ∗∗∗ A. D’ Jorge ∗∗∗ E. Duviella ∗∗∗∗ J.M. Maestre ∗ ∗Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, C/ Camino de los Descubrimientos, s/n., 41092 Sevilla, Spain ∗∗ Department of Mathematics and Statistical Science, University of Idaho, Idaho, United States ∗∗∗ Instituto de Desarrollo Tecnol´ ogico para la Industria Qu´ ımica (INTEC), Consejo Nacional de Investigaciones cient´ ıficas y t´ ecnicas (CONICET), Santa Fe, Argentina ∗∗∗∗ IMT Nord Europe, Univ. Lille, F-59000 Lille, France Abstract: Unmanned surface vehicles can become cutting-edge aquatic laboratories for real-time water quality assessment, evaluating the physical, chemical, and biological profiles of water to detect degradation in freshwater resources. The development of control strategies for surface scanning missions is essential for efficient and effective water resource management. This paper introduces novel exploration methods for analysis experiments. Our approach is founded on an optimizing target-set tracking controller and innovates by adopting a dynamic target-switching method. The target transitions to a further objective once the current target is assured, result in smoother system behavior, as shown by the simulations using the nonlinear model of a commercial surface vehicle tasked with covering a designated area of Heron Lake, located in Villeneuve d’Ascq. The method is implemented in two distinct ways: the first prioritizes passing through all the sets, while the second compromises between passing through and computational cost. Keywords: Unmanned Surface Vehicle, Environmental Monitoring, Water Quality, Coverage Control 1. INTRODUCTION Water scarcity is already a significant issue in several urban populations around the world. Population growth, urbanization, and socioeconomic developments are expected to increase water demand by 50%–80% over the next three decades, and the population facing water scarcity is projected to increase from 933 million in 2016 to at least 1.7 billion people in 2050, with India being the most severely affected country (He et al., 2021; McDonald et al., 2014). In parallel, water quality will deteriorate as well due to urbanization and the growth of cities. Climate change will affect the spatial distribution, air temperature, and rainfall, influencing the dilution of contaminants and strongly impacting the ecological status of freshwater. This dire prognosis has spurred industrial developments and academic interest in technological strategies related to the assessment of water quality; for example, real-time collection and analysis of relevant data to determine the physical, chemical, and biological profile of freshwater resources. City waste affects the water quality of nearby freshwater resources. In this regard, direct in situ measurements with a hand-operated unmanned surface vehicle (USV) were taken in a region of Heron Lake in Villeneuve d’Ascq. One criterion for classifying surface water quality is the water quality index (WQI) presented in Pesce and Wunderlin (2000); S´ anchez et al. (2007). Following such criteria, it can be seen in Fig. 1 that there is a gradient of the WQI from the canal bringing contaminants towards the main part of the lake (where dilution and decontamination occur). Usually, monitoring these parameters exploits highly reliable instrumentation restricted to a few and widely available locations, leaving relevant regions unreachable (Madeo et al., 2020). To overcome these challenges, the use of unmanned surface vehicles (USVs) has been proposed with remarkable results (Sinisterra et al., 2017; Ranjbar et al., 2023). Since sampling of the watersheds should cover all seasons, periodic sampling for the determination of field parameters and laboratory analysis is necessary. Planning the design of proper automatic control strategies is essential to minimize efforts and simplify motion. Optimization-based strategies for solving motion, pathplanning, and control for USV problems have been gaining attention from the model predictive control (MPC) community (Lindqvist et al., 2020; Manzano et al., 2020; Ranjbar et al., 2023). MPC consists of a receding horizon control strategy, meaning a fixed horizon optimal control problem is solved at every sampling time. From the resulting optimal control sequence, only the first control action is applied to the plant, and the procedure is repeated (Rawlings et al., 2017). A significant advantage of the MPC framework is its ability to take into account (and anticipate) constraints on the inputs and states, such as collision avoidance constraints. Particularly, set-based MPC (Blanchini and Miani, 2015) — in which the guarantee of asymptotic stability is based on set analysis rather than steadystate stabilization — is suitable for controlling uncertain systems where the control objectives are based on reaching regions rather than points, as in this case. Copyright © 2024 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) Fig. 1. Lake in Villeneuve d’Ascq. The values WQI ≥0.7 represent good quality; WQI ≥0.55 medium quality and WQI ≤0.55 bad quality. Using set-based approaches like set-based MPC enhances system robustness, making it more resilient to disturbances and environmental variations, crucial in dynamic environments like aquatic systems (Anderson et al., 2022b). By considering sets rather than individual points, exploration strategies can be efficiently planned, such as grid-based trajectory planning, enabling comprehensive data collection across large water regions (Anderson et al., 2022a). This is vital for sampling and monitoring tasks in aquatic environments, ensuring complete spatial coverage for understanding parameter distribution and evolution. In this study, previous experimental data from (Anderson et al., 2022b) were used to determine the impact on water quality of Lake Villeneuve d’Ascq in northern France. The results indicate possible degradation of water quality, likely due to waste from nearby urban settlements. We present an extension of a previously proposed control strategy by Anderson et al. (2022a) for exploring the surface of the lake using an aquatic laboratory. Our extended control demonstrates improved efficiency via more continuous trajectories and smooth turns without stopping, resulting in more fluid vehicle movement. Importantly, we formalize the control strategy to maintain feasibility. Improvements are computationally demonstrated via USV simulations. 2. PROBLEM STATEMENT In situ measurements using a manually-operated USV were conducted in a region of Heron Lake in Villeneuve d’Ascq and kriging was employed to estimate unexplored areas (Anderson et al., 2022b). Water surface quality classification follows the equation (Pesce and Wunderlin, 2000). WQI =kiCiPi iCi where Cirepresents the normalized parameter value, and Pi denotes the relative weight for each parameter (pH, temperature, dissolved oxygen, turbidity, conductivity, and Redox). The constant kranges from 0.25 to 1, reflecting water quality from highly polluted to apparently good. Maximum weights are assigned to crucial parameters like dissolved oxygen, while minimum weights apply to less significant ones such as temperature and pH (S´ anchez et al., 2007). WQI calculations in Fig. 1 illustrate a gradient in contaminant concentration decreasing towards the main lake area, where dilution and decontamination take place. In the following section, we describe the strategy, proposed on Anderson et al. (2022b), aimed at exploring a large water surface, thereby automating the data collection mission. 3. MULTI-SET TRACKING The USV system can be modeled by the following discrete-time nonlinear system x(k+ 1) = f(x(k),u(k)) (1) where x(k)∈X⊂Rnrepresents the states of the system (position and velocity vectors) and u(k)∈U⊂Rmare the control inputs. The constraint sets Xand Uare compact and convex sets with the origin inside. The dynamic f:X×U→X is a continuous function with f(xs,u s)=0for some feasible point (xs,u s)∈X×U. The exploration region can be partitioned into an ordered grid of target sets Ω=Ω 0, ..., ΩK. The multi-set tracking involves controlling the system to move through the sequence of target sets. This constitutes a multi-tracking strategy based on sets, previously implemented for water quality assessment Anderson et al. (2022b). The control strategy drives the system states into each target set Ωi, then to Ωi+1, and so on, as illustrated in Fig. 2a. In the following section, we propose an extension to this strategy aimed at enhancing the efficiency of the controller. This extension involves switching targets before reaching them, while still ensuring that the system passes through all sets. The strategy involves tracking a target set until the inevitable set (I(Ωi)in Fig. 2b) is reached, which encompasses the target later. This triggers switching to the next target set. The scheme is illustrated in Fig. 2b. 3.1 Proposed strategy: Inevitable-set switching To formalize the multi-set tracking approach, we propose an MPC strategy with the following cost function. Given a fixed prediction horizon N∈Nand the current target set Ωi∈Ω, the cost is defined as: VN(x, Ωi;u)= N−1  j=0 L(x(j),Ωi;u(j)).(2) Here, the current state denoted as x=x(0) and evolves through the system dynamics x(j+ 1) = f(x(j),u(j)) for j=0,1,...,N −1with the input sequence u= {u(0),u(1),...,u(N−1)}. The cost stage function satisfies L(·)≥0and L(x)=0if and only if x∈Ωi. Let (a) Set-after-set: The current target is Ω0until x(k)reaches it, then the target switches to Ω1. F (b) Inevitable set: The current target is Ω0until x(k)reach a point within the set I(Ω0), then the target is switched to Ω1. Fig. 2. Illustration comparing the multi-set tracking strategies, set-after-set vs. inevitable-set switching. 40 J.G. Martin et al. / IFAC PapersOnLine 58-2 (2024) 38–43 XN(Ωi)={x(0) ∈X|x(j)∈X,u(j)∈U,j ∈0,1,...,N −1, x(N)∈Ωi}, (3) be the feasible region that may reach set Ωiin Nsteps. Then, for all x∈XN, the set of control sequences u, satisfying inputs and states constraints at all time steps, including a terminal state constraint x(N)∈Ωiis represented by UN(x). The MPC law at each time instant kis derived from solving the following optimization problem, PN(x, Ωi): V0 N(x, Ωi)=min{VN(x, Ωi;u)|u∈UN(x)} In this equation, Ωiand the current state x∈Xserve as the optimization parameters, while the sequence u∈ UNrepresents a sequence of optimization variables. For the current optimal control sequence given by u0(x) := {u0(0; x),u 0(1; x),...,u 0(N−1; x)}, the optimal trajectory is given by {x0(x)}={x0(0; x),x 0(1; x),...,x 0(N;x)}, with the terminal state x0(N;x)∈Ωi. The control law follows a Receding Horizon Control (RHC) policy κMPC(x)=u0(0; x), where u0(0; x)is the first element of the solution sequence u0(x). Consequently, the closed-loop system under the MPC law is characterized by x(k+ 1) = f(x(k),κ MPC(x(k))), and the optimal cost function defined by V0 N(x, Ωi)=VN(x, Ωi;u0(x)). This MPC formulation was employed through the set-after-set pursuit strategy as outlined in Anderson et al. (2022a). It relies on a objective switch function triggered by the states position, the switch occurs when the system states are within the target set. This provides sufficient (though not necessary) conditions to ensure reaching each objective (thanks to the global asymptotic stability of each). The strategy yields robust yet conservative control, potentially leading to control trajectories with undesired cycles (see simulations Section). Here, we propose extending this strategy for a faster switch of target condition, given by the following concept. Definition 1. (Inevitable set). A set I(Ω) ⊂Xis an inevitable set respect to Ω⊂Xfor System (1) if for every x(k)∈I(Ω), then x(k+ 1) ∈Ωfor all u(k)∈U. Remark 2. The above concept can be recursively extended for k−steps. The k-step inevitable set to Ωcan be defined by Ik(Ω) := I(Ik−1(Ω)) for k≥1, with I0(Ω) := Ω. Now, consider a variable σwhich indicates the current target set, and we define a switched dynamic function σ+(x, σ): R≥0×{0,1,··· ,K}→{0,1,··· ,K}which is used to update the target set in each iteration, defined as σ+(x, σ)=σ:x/∈I(Ωσ) σ+1:otherwise The multi-set tracking MPC is defined in Algorithm 1. Remark 3. To calculate the Inevitable set, the concepts of controllable sets can be utilized (Blanchini and Miani, 2015). For linear systems and convex sets, this computation is straightforward. However, for nonlinear systems, computing the controllable sets entails numerical costs. In the simulations section, we simplify the strategy based on some alternatives for the switching function σ, according to the nonlinear equations of the USV. Algorithm 1: Proposed MPC Data: Function f(·), sets X,U, target sequence Ω, switched law σ, initial state x0∈X Result: implicit control law κ(·) 1: x=x0 2: σ=0 3: while σ≤Kdo 4: Update system state x 5: σ=σ+(x, σ) 6: Solve PN(x, Ωσ) 7: Apply u(k)=κMPC(x) 8: end 4. USV SIMULATIONS In this section, we will consider an USV that must explore an area of Heron Lakelocated in Villeneuve d’Ascq, France, depicted in Fig. 1. This artificial lake was created in the 1970s to drain marshy areas and receives rainwater from roads, artificial surfaces, and storm overflows from a mixed drainage system. Water enters from the East and, when the level is high, excess water is pumped out to a nearby river in the west. Natural remediation processes occur within the lake, resulting in a gradient from high biodegradable input at the entrance to the exit (Ivanovsky et al., 2018). The environment is divided into hexagonal sets (Anderson et al., 2022b,a). The USV must take WQI measurements in each cell, so we can consider these as target sets. We are comparing the set-after-set approach from previous works with two new heuristic methods that consider the idea of the “Inevitable” set to change the target set before reaching it. All the simulations in this section have been performed using Matlab®ina3.2GHz Intel Core® i7,16 GB RAM PC. 4.1 USV model CT2MC is a company that has designed freshwater monitoring vessels featuring flat hulls and aerial propulsion. This guarantees contamination-free sampling and inspection missions. The official SPYBOAT® system used here (Fig. 3a) follows standard equipment including sensors and dual actuators for differential heading control. The SPYBOAT® architecture is described in Fig. 3b. A validated model proposed in (Hervagault, 2019) based on system characteristics is crucial for developing model predictive control strategies to automate environment missions. The USV measures turbidity using a Hyperion optical sensor from Valeport 1and temperature, DO, pH, and conductivity using Tripod sensors from AquaLabo. 2 The marine craft moves on an horizontal plane and only surge, sway and yaw are considered. The resulting is a nonlinear model given by the following equations.                  ˙x=ucos(ψ)−vsin(ψ)+wx, ˙y=usin(ψ)+vcos(ψ)+wy, ˙ ψ=r+wψ, ˙u=τu m11 +m22 m11 vr +Xu m11 u, ˙v=−m11 m22 ur +Yv m22 v, ˙r=τr m33 +m22−m11 m33 uv +Nr m33 r (4) 1https://www.valeport.co.uk/content/uploads/2021/05/0901814i-HyperionOptical-Sensors-Operating-Manual.pdf 2https://en.aqualabo.fr/ J.G. Martin et al. / IFAC PapersOnLine 58-2 (2024) 38–43 41 XN(Ωi)={x(0) ∈X|x(j)∈X,u(j)∈U,j ∈0,1,...,N −1, x(N)∈Ωi}, (3) be the feasible region that may reach set Ωiin Nsteps. Then, for all x∈XN, the set of control sequences u, satisfying inputs and states constraints at all time steps, including a terminal state constraint x(N)∈Ωiis represented by UN(x). The MPC law at each time instant kis derived from solving the following optimization problem, PN(x, Ωi): V0 N(x, Ωi)=min{VN(x, Ωi;u)|u∈UN(x)} In this equation, Ωiand the current state x∈Xserve as the optimization parameters, while the sequence u∈ UNrepresents a sequence of optimization variables. For the current optimal control sequence given by u0(x) := {u0(0; x),u 0(1; x),...,u 0(N−1; x)}, the optimal trajectory is given by {x0(x)}={x0(0; x),x 0(1; x),...,x 0(N;x)}, with the terminal state x0(N;x)∈Ωi. The control law follows a Receding Horizon Control (RHC) policy κMPC(x)=u0(0; x), where u0(0; x)is the first element of the solution sequence u0(x). Consequently, the closed-loop system under the MPC law is characterized by x(k+ 1) = f(x(k),κ MPC(x(k))), and the optimal cost function defined by V0 N(x, Ωi)=VN(x, Ωi;u0(x)). This MPC formulation was employed through the set-after-set pursuit strategy as outlined in Anderson et al. (2022a). It relies on a objective switch function triggered by the states position, the switch occurs when the system states are within the target set. This provides sufficient (though not necessary) conditions to ensure reaching each objective (thanks to the global asymptotic stability of each). The strategy yields robust yet conservative control, potentially leading to control trajectories with undesired cycles (see simulations Section). Here, we propose extending this strategy for a faster switch of target condition, given by the following concept. Definition 1. (Inevitable set). A set I(Ω) ⊂Xis an inevitable set respect to Ω⊂Xfor System (1) if for every x(k)∈I(Ω), then x(k+ 1) ∈Ωfor all u(k)∈U. Remark 2. The above concept can be recursively extended for k−steps. The k-step inevitable set to Ωcan be defined by Ik(Ω) := I(Ik−1(Ω)) for k≥1, with I0(Ω) := Ω. Now, consider a variable σwhich indicates the current target set, and we define a switched dynamic function σ+(x, σ): R≥0×{0,1,··· ,K}→{0,1,··· ,K}which is used to update the target set in each iteration, defined as σ+(x, σ)=σ:x/∈I(Ωσ) σ+1:otherwise The multi-set tracking MPC is defined in Algorithm 1. Remark 3. To calculate the Inevitable set, the concepts of controllable sets can be utilized (Blanchini and Miani, 2015). For linear systems and convex sets, this computation is straightforward. However, for nonlinear systems, computing the controllable sets entails numerical costs. In the simulations section, we simplify the strategy based on some alternatives for the switching function σ, according to the nonlinear equations of the USV. Algorithm 1: Proposed MPC Data: Function f(·), sets X,U, target sequence Ω, switched law σ, initial state x0∈X Result: implicit control law κ(·) 1: x=x0 2: σ=0 3: while σ≤Kdo 4: Update system state x 5: σ=σ+(x, σ) 6: Solve PN(x, Ωσ) 7: Apply u(k)=κMPC(x) 8: end 4. USV SIMULATIONS In this section, we will consider an USV that must explore an area of Heron Lakelocated in Villeneuve d’Ascq, France, depicted in Fig. 1. This artificial lake was created in the 1970s to drain marshy areas and receives rainwater from roads, artificial surfaces, and storm overflows from a mixed drainage system. Water enters from the East and, when the level is high, excess water is pumped out to a nearby river in the west. Natural remediation processes occur within the lake, resulting in a gradient from high biodegradable input at the entrance to the exit (Ivanovsky et al., 2018). The environment is divided into hexagonal sets (Anderson et al., 2022b,a). The USV must take WQI measurements in each cell, so we can consider these as target sets. We are comparing the set-after-set approach from previous works with two new heuristic methods that consider the idea of the “Inevitable” set to change the target set before reaching it. All the simulations in this section have been performed using Matlab®ina3.2GHz Intel Core® i7,16 GB RAM PC. 4.1 USV model CT2MC is a company that has designed freshwater monitoring vessels featuring flat hulls and aerial propulsion. This guarantees contamination-free sampling and inspection missions. The official SPYBOAT® system used here (Fig. 3a) follows standard equipment including sensors and dual actuators for differential heading control. The SPYBOAT® architecture is described in Fig. 3b. A validated model proposed in (Hervagault, 2019) based on system characteristics is crucial for developing model predictive control strategies to automate environment missions. The USV measures turbidity using a Hyperion optical sensor from Valeport 1and temperature, DO, pH, and conductivity using Tripod sensors from AquaLabo. 2 The marine craft moves on an horizontal plane and only surge, sway and yaw are considered. The resulting is a nonlinear model given by the following equations.                  ˙x=ucos(ψ)−vsin(ψ)+wx, ˙y=usin(ψ)+vcos(ψ)+wy, ˙ ψ=r+wψ, ˙u=τu m11 +m22 m11 vr +Xu m11 u, ˙v=−m11 m22 ur +Yv m22 v, ˙r=τr m33 +m22−m11 m33 uv +Nr m33 r (4) 1https://www.valeport.co.uk/content/uploads/2021/05/0901814i-HyperionOptical-Sensors-Operating-Manual.pdf 2https://en.aqualabo.fr/ (a) The SPYBOAT® USV. (b) Architecture of SPYBOAT®. Fig. 3. The SPYBOAT® technology is based on the use of an aquatic drone allowing the realization of water sampling and inspections by guaranteeing the non-contamination of the environment. The vector (x, y)denotes position on the surface, and ψrepresents vessel direction, while u,v, and rindicate surge, sway, and yaw velocities, respectively. Note that x,y, and ψrefer to globlal coordinates while u,v, and rrefer to local coordinates of the robot. Inputs are τu=F1+F2and τr=b(F1−F2), where F1and F2are port and starboard thrust forces, and bis half the distance between thrusters. Parameters Xu,Yv, and Nr are linear surge drag coefficient, linear sway drag coefficient from yaw rate, and linear yaw drag moment coefficient, respectively. Gaussian noise wx∼N(µx,σ x),wy∼N(µy,σ y), and wψ∼N(µψ,σ ψ)represent current-induced noise for x,y, and ψ. Known parameters include µx=0.1,µy=0.1,µψ=0, and σx=σy=σψ=0.01. The mass parameters mii encompass added mass contributions representing hydraulic pressure forces and torque due to forced harmonic motion proportional to acceleration. m11 =m+0.05m, m22 =m+0.5(ρπD2L), m33 =m(L2+W2)+0.5(0.1mB2+ρπD2L3) 12 . where mis the actual mass, Lis the effective length (hull’s length in the water), Wis the width, Dis the mean submerged depth, Bis the distance between propellers and ρis the water density. For more detail on the parameters of model (4) see (Hervagault, 2019). 4.2 Set-after-set approach In previous works of Anderson et al. (2022b,a), a set-after-set approach (see Fig. 2a) was proposed, with the results shown in Fig. 4. 3In this strategy, the switched dynamic function σis defined as follows σ+=σ:x/∈Ωσ σ+1:otherwise However, as it can be seen in Fig. 4b, the saw shape of the speed of the USV is undesirable, for it stresses the mechanical actuators and increases the coverage completion time (in this simulation the USV needed 428 seconds to complete the coverage of all sets). 4.3 Inevitable-set (Velocity) approach Let the objectives be identified by Ωi, where i=1,...,n. The velocity vector, which can be modified using the noise 3A video can be seen in https://youtu.be/vnT86giMHr8 (a) Trajectory of the predicted path algorithm. 0 100 200 300 400 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (b) Longitudinal speed F Fig. 4. Trajectory and longitudinal speed of the USV in the set-after-set approach. The time required to complete the coverage was 428 seconds and the mean computation time was 1.22 seconds. mean, indicates the direction in which the USV is moving, so that a line can be drawn on the surface using the position and velocity of the robot to estimate its future range. In case this line intersects the set of objectives Ωi, this set will become Ωi+1, and by repeating the process, we will determine which is the final target cell, i.e., the last set of objectives intersecting this line will become the designated objective. The described process guides the motion of the robot. If the objective is Ωkand the preceding objectives Ωiare “inevitably” reached because the ship must pass through them. The switch law considered can be expressed as σ+=σ:Vxy ∩proj(x,y)Ωσ=∅ σ+1:otherwise where the condition Vxy ∩proj(x,y)Ωσ=∅implies that the effect of current velocities vectors added to the current positions in the discrete system, is not enough to reach set Ωσ. We considered Vxy =(x, y)+α(u+v+r), where αis a control adjustment weight, and proj(x,y)Ωσis the vector projection of Ωσin the surface position dimension (x, y). This algorithm has been tested first in a simulation without noise 4covering all the sets in 282 seconds, and then in a simulation with noise 5covering all the sets in 252 seconds. The trajectory followed in both cases can be seen in Fig. 5a and in Fig. 5c respectively. Note that with this strategy, the computational cost is much lower; however, it does not guarantee that the robot will not skip any sets, especially in the presence of noise. 4.4 Optimal Trajectory Approach The second proposed method consists of calculating the optimal trajectory from the position of the USV to the target set Ωi, considering that [x, y]∈Ωi−Kwith K>0. If the trajectory crosses all the cells from Ωi−Kto Ωi, then the target cell will become Ωi+1. This procedure will be executed iteratively until this condition is no longer met, i.e., the target set is the last of the objective sets in which the computed trajectory crosses all the previous objectives. Then, the computed control actions for the target cell will be applied and this procedure will be repeated in each time step guiding the robot’s movement. The concept of “Inevitable” is similar to the previous algorithm: if the objective is Ωk, the preceding objectives Ωiare inevitably 4A video can be seen in https://youtu.be/4CYdnlDgafk 5A video can be seen in https://youtu.be/2aXI0Zy7xFE 42 J.G. Martin et al. / IFAC PapersOnLine 58-2 (2024) 38–43 (a) Trajectory of the predicted path algorithm without noise. 0 50 100 150 200 250 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (b) Longitudinal speed without noise F (c) Trajectory of the predicted path algorithm with noise. 0 50 100 150 200 250 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (d) Longitudinal speed with noise F Fig. 5. Trajectory and longitudinal speed of the USV during the trajectory in the Velocity-Line Algorithm with and without noise. The time required to complete the coverage was 282 seconds in the case without noise and 252 seconds in the case with noise. The mean computation time was 0.63 seconds per step in the case without noise and 0.6seconds per step in the case with noise. reached because the boat must pass through them. The switch law considered can be expressed as σ+= max σ(Pxy ∩proj(x,y)Ωσ=∅) where the condition Pxy ∩proj(x,y)Ωσ=∅implies that the furthest set in the sequence that the prediction reaches is the set Ωσ. We considered Pxy as the secuence of predicted positions for the system, and proj(x,y)Ωσis the vector projection of Ωσ in the surface position dimension (x, y). This algorithm has been tested first in a simulation without noise 6covering all the sets in 228 seconds, and then in a simulation with noise 7covering all the sets in 195 seconds. The trajectory followed in both cases can be seen in Fig. 6a and in Fig. 6c respectively. Also, the longitudinal speed in both cases can be seen in Fig. 6b and in Fig. 6d respectively. Note that the saw shape of the speed has been significantly reduced and the mean speed is significantly higher at the cost of higher computation burden. 5. CONCLUSIONS In this work, we have extended the strategy developed in Anderson et al. (2022b,a) by adding a method to switch the set Ωi in such a way that it is ensured that the system passes through all of them. This method is based on the inevitable set of the system, which is the set of states that the system cannot avoid passing through given its current state. A numerical assessment of the proposed controller and a comparison concerning previously presented strategies have been included using the model of a commercially available aquatic 6A video can be seen in https://youtu.be/DSN3WHn0GyE 7A video can be seen in https://youtu.be/qUpq1IBdrrw (a) Trajectory of the predicted path algorithm without noise. 0 50 100 150 200 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (b) Longitudinal speed without noise F (c) Trajectory of the predicted path algorithm with noise. 0 50 100 150 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (d) Longitudinal speed with noise F Fig. 6. Trajectory and longitudinal speed of the USV during the trajectory in the predicted path algorithm with and without noise. The time required to complete the coverage was 228 seconds in the case without noise and 195 seconds in the case with noise. The mean computation time was 3.13 seconds per step in the case without noise and 3.04 seconds per step in the case with noise. drone in a real-life scenario, namely the Heron Lake. The numerical experimentation results show improved performance and smoother behavior of the vehicle, which is typically desired in robotic vehicle applications. More precisely, both the total time to cover the area and the average computational cost per iteration can be seen in Table 1, which clearly shows that the velocity-line algorithm offers a good compromise between computational cost and efficiency. Also, note that both proposed algorithms significantly decrease the coverage time from 428 seconds of the set-after-set approach. More precisely, the predicted path algorithm has decreased it by 54.44% and the velocity-line algorithm has decreased it by 41.12%. Also, not that the accelerations, and thus, the required energy is significantly lower in the proposed algorithm. Finally, note the main difference between the predicted path approach and the velocity-line approach is that the second one obtains a larger mean velocity but the first one optimize the trajectory. Future works will include enhanced methods for the computation of the inevitable sets as well as the computation of the set sequences to be traversed. Additionally, another potential area for future work is the inclusion of forbidden sets, which would represent obstacles or constraints within the system. REFERENCES Anderson, A., Martin, J., Bouraqadi, N., Etienne, L., Fabresse, L., Langueh, K., Lozenguez, G., Rajaoarisoa, L., Maestre, J., and Duviella, E. (2022a). Map meshing impact on the efficiency of nonlinear set-based model predictive control for water quality assessment. IFAC-PapersOnLine, 55(33), 105– 110. Anderson, A., Martin, J., Mougin, J., Bouraqadi, N., Duviella, E., Etienne, L., Fabresse, L., Langueh, K., Lozenguez, G., Alary, C., et al. (2022b). Water quality map extraction from J.G. Martin et al. / IFAC PapersOnLine 58-2 (2024) 38–43 43 (a) Trajectory of the predicted path algorithm without noise. 0 50 100 150 200 250 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (b) Longitudinal speed without noise F (c) Trajectory of the predicted path algorithm with noise. 0 50 100 150 200 250 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (d) Longitudinal speed with noise F Fig. 5. Trajectory and longitudinal speed of the USV during the trajectory in the Velocity-Line Algorithm with and without noise. The time required to complete the coverage was 282 seconds in the case without noise and 252 seconds in the case with noise. The mean computation time was 0.63 seconds per step in the case without noise and 0.6seconds per step in the case with noise. reached because the boat must pass through them. The switch law considered can be expressed as σ+= max σ(Pxy ∩proj(x,y)Ωσ=∅) where the condition Pxy ∩proj(x,y)Ωσ=∅implies that the furthest set in the sequence that the prediction reaches is the set Ωσ. We considered Pxy as the secuence of predicted positions for the system, and proj(x,y)Ωσis the vector projection of Ωσ in the surface position dimension (x, y). This algorithm has been tested first in a simulation without noise 6covering all the sets in 228 seconds, and then in a simulation with noise 7covering all the sets in 195 seconds. The trajectory followed in both cases can be seen in Fig. 6a and in Fig. 6c respectively. Also, the longitudinal speed in both cases can be seen in Fig. 6b and in Fig. 6d respectively. Note that the saw shape of the speed has been significantly reduced and the mean speed is significantly higher at the cost of higher computation burden. 5. CONCLUSIONS In this work, we have extended the strategy developed in Anderson et al. (2022b,a) by adding a method to switch the set Ωi in such a way that it is ensured that the system passes through all of them. This method is based on the inevitable set of the system, which is the set of states that the system cannot avoid passing through given its current state. A numerical assessment of the proposed controller and a comparison concerning previously presented strategies have been included using the model of a commercially available aquatic 6A video can be seen in https://youtu.be/DSN3WHn0GyE 7A video can be seen in https://youtu.be/qUpq1IBdrrw (a) Trajectory of the predicted path algorithm without noise. 0 50 100 150 200 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (b) Longitudinal speed without noise F (c) Trajectory of the predicted path algorithm with noise. 0 50 100 150 Steps 0 0.5 1 1.5 2 2.5 [m/s] u (d) Longitudinal speed with noise F Fig. 6. Trajectory and longitudinal speed of the USV during the trajectory in the predicted path algorithm with and without noise. The time required to complete the coverage was 228 seconds in the case without noise and 195 seconds in the case with noise. The mean computation time was 3.13 seconds per step in the case without noise and 3.04 seconds per step in the case with noise. drone in a real-life scenario, namely the Heron Lake. The numerical experimentation results show improved performance and smoother behavior of the vehicle, which is typically desired in robotic vehicle applications. More precisely, both the total time to cover the area and the average computational cost per iteration can be seen in Table 1, which clearly shows that the velocity-line algorithm offers a good compromise between computational cost and efficiency. Also, note that both proposed algorithms significantly decrease the coverage time from 428 seconds of the set-after-set approach. More precisely, the predicted path algorithm has decreased it by 54.44% and the velocity-line algorithm has decreased it by 41.12%. Also, not that the accelerations, and thus, the required energy is significantly lower in the proposed algorithm. Finally, note the main difference between the predicted path approach and the velocity-line approach is that the second one obtains a larger mean velocity but the first one optimize the trajectory. Future works will include enhanced methods for the computation of the inevitable sets as well as the computation of the set sequences to be traversed. Additionally, another potential area for future work is the inclusion of forbidden sets, which would represent obstacles or constraints within the system. REFERENCES Anderson, A., Martin, J., Bouraqadi, N., Etienne, L., Fabresse, L., Langueh, K., Lozenguez, G., Rajaoarisoa, L., Maestre, J., and Duviella, E. (2022a). Map meshing impact on the efficiency of nonlinear set-based model predictive control for water quality assessment. IFAC-PapersOnLine, 55(33), 105– 110. 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