Pub Date : 2024-12-31DOI: 10.1109/LCSYS.2024.3524369
Zahra Marvi;Francesco Bullo;Andrew G. Alleyne
In this letter, we address the critical trade-off between safety and performance in control systems by establishing the contractivity of a class of nonlinear systems driven by control barrier function (CBF)-based online feedback optimization. First, we derive a closed-form solution for the control system driven by a CBF-based controller with vector-valued safety constraints. Next, we introduce sufficient design conditions based on the properties of a baseline controller and CBF parameters to ensure both safety and contractivity of the closed-loop system. Under these conditions, we demonstrate the existence of an exponentially stable equilibrium within the safe set and provide an explicit term for the rate of convergence. Building upon these results, we propose a feedback motion planning algorithm that guarantees a global region of attraction within non-convex search areas through a tree of contractive controllers. The contractive nature of our approach ensures robustness against perturbations, making it suitable for dynamic and uncertain environments.
{"title":"Robust and Exponential Stability in Barrier-Certified Systems via Contracting Piecewise Smooth Dynamics","authors":"Zahra Marvi;Francesco Bullo;Andrew G. Alleyne","doi":"10.1109/LCSYS.2024.3524369","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3524369","url":null,"abstract":"In this letter, we address the critical trade-off between safety and performance in control systems by establishing the contractivity of a class of nonlinear systems driven by control barrier function (CBF)-based online feedback optimization. First, we derive a closed-form solution for the control system driven by a CBF-based controller with vector-valued safety constraints. Next, we introduce sufficient design conditions based on the properties of a baseline controller and CBF parameters to ensure both safety and contractivity of the closed-loop system. Under these conditions, we demonstrate the existence of an exponentially stable equilibrium within the safe set and provide an explicit term for the rate of convergence. Building upon these results, we propose a feedback motion planning algorithm that guarantees a global region of attraction within non-convex search areas through a tree of contractive controllers. The contractive nature of our approach ensures robustness against perturbations, making it suitable for dynamic and uncertain environments.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3279-3284"},"PeriodicalIF":2.4,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142938113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-30DOI: 10.1109/LCSYS.2024.3524063
Hang Gu;Chao Shen
Image-based visual servoing (IBVS) applications for autonomous underwater vehicles (AUVs) face significant challenges, including frequent recalibration and lack of constraint handling ability. This letter introduces a novel nonlinear model predictive control (NMPC) approach that integrates the Broyden method for uncalibrated IBVS and incorporates the min-max strategy to tolerate the errors in Jacobian matrix estimation. Our proposed min-max NMPC-IBVS framework estimates the Jacobian matrix online, allowing for continuous adaptation to the underwater environment without the need for prior calibration. This approach significantly enhances computational efficiency and robust control performance, enabling real-time uncalibrated applications. A rigorous proof of recursive feasibility is provided in this letter, ensuring that our NMPC-IBVS method consistently finds feasible optimal solutions that satisfy all constraints over time. Simulation results show that the proposed method is able to respect all design constraints in the AUV IBVS control and achieve robust stability with boosted computational efficiency.
{"title":"Robust NMPC for Uncalibrated IBVS Control of AUVs","authors":"Hang Gu;Chao Shen","doi":"10.1109/LCSYS.2024.3524063","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3524063","url":null,"abstract":"Image-based visual servoing (IBVS) applications for autonomous underwater vehicles (AUVs) face significant challenges, including frequent recalibration and lack of constraint handling ability. This letter introduces a novel nonlinear model predictive control (NMPC) approach that integrates the Broyden method for uncalibrated IBVS and incorporates the min-max strategy to tolerate the errors in Jacobian matrix estimation. Our proposed min-max NMPC-IBVS framework estimates the Jacobian matrix online, allowing for continuous adaptation to the underwater environment without the need for prior calibration. This approach significantly enhances computational efficiency and robust control performance, enabling real-time uncalibrated applications. A rigorous proof of recursive feasibility is provided in this letter, ensuring that our NMPC-IBVS method consistently finds feasible optimal solutions that satisfy all constraints over time. Simulation results show that the proposed method is able to respect all design constraints in the AUV IBVS control and achieve robust stability with boosted computational efficiency.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3237-3242"},"PeriodicalIF":2.4,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142938059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-30DOI: 10.1109/LCSYS.2024.3524056
Jingru Zhu;Cheng Zhao
This letter studies control problems for a class of multi-input multi-output (MIMO) strongly coupled nonlinear uncertain systems with mixed relative degrees one and two. We will design an uncoupled PID controller, where each component of the input vector is determined only by a component of the system output regulation error. Under some suitable conditions on the system nonlinear uncertain functions, we will show that the proposed controller can ensure global stability of the closed-loop system while guaranteeing that the output regulation error converges to zero at an exponential rate. We will also demonstrate that the PID parameters can be chosen from an open and unbounded set constructed based on some prior knowledge regarding the nonlinear functions of the system, and that the designed PID controller exhibits two-sided robustness in terms of the uncertain system structure and the selection of controller parameters. A simulation will be provided to verify the theoretical findings.
{"title":"PID Control of MIMO Nonlinear Uncertain Systems With Low Relative Degrees","authors":"Jingru Zhu;Cheng Zhao","doi":"10.1109/LCSYS.2024.3524056","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3524056","url":null,"abstract":"This letter studies control problems for a class of multi-input multi-output (MIMO) strongly coupled nonlinear uncertain systems with mixed relative degrees one and two. We will design an uncoupled PID controller, where each component of the input vector is determined only by a component of the system output regulation error. Under some suitable conditions on the system nonlinear uncertain functions, we will show that the proposed controller can ensure global stability of the closed-loop system while guaranteeing that the output regulation error converges to zero at an exponential rate. We will also demonstrate that the PID parameters can be chosen from an open and unbounded set constructed based on some prior knowledge regarding the nonlinear functions of the system, and that the designed PID controller exhibits two-sided robustness in terms of the uncertain system structure and the selection of controller parameters. A simulation will be provided to verify the theoretical findings.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3213-3218"},"PeriodicalIF":2.4,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142938056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1109/LCSYS.2024.3523432
Nguyen Thi Lien;Le Van Hien;Nguyen Nhu Thang
This note is concerned with a class of homogeneous cooperative systems with bounded time-varying delays described by the Caputo fractional derivative. We focus on the existence, uniqueness, and Mittag-Leffler stability of positive solutions when the associated vector fields are homogeneous with a degree less than or equal to one. Specifically, the solvability is first exploited through the fixed point theory, leveraging the homogeneity of nonlinear terms. Then, a delay-independent condition for Mittag-Leffler stability is established by utilizing the properties of Mittag-Leffler functions and the comparison principle. Finally, the theoretical results are validated by a given numerical example.
{"title":"Mittag-Leffler Stability of Homogeneous Fractional-Order Systems With Delay","authors":"Nguyen Thi Lien;Le Van Hien;Nguyen Nhu Thang","doi":"10.1109/LCSYS.2024.3523432","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3523432","url":null,"abstract":"This note is concerned with a class of homogeneous cooperative systems with bounded time-varying delays described by the Caputo fractional derivative. We focus on the existence, uniqueness, and Mittag-Leffler stability of positive solutions when the associated vector fields are homogeneous with a degree less than or equal to one. Specifically, the solvability is first exploited through the fixed point theory, leveraging the homogeneity of nonlinear terms. Then, a delay-independent condition for Mittag-Leffler stability is established by utilizing the properties of Mittag-Leffler functions and the comparison principle. Finally, the theoretical results are validated by a given numerical example.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3243-3248"},"PeriodicalIF":2.4,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142938066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1109/LCSYS.2024.3523845
Charlotte Cathcart;Ian Xul Belaustegui;Alessio Franci;Naomi Ehrich Leonard
We present, analyze, and illustrate a first-of-its-kind model of two-dimensional excitable (spiking) dynamics for decision-making over two options. The model, Spiking Nonlinear Opinion Dynamics (S-NOD), provides superior agility, characterized by fast, flexible, and adaptive response to rapid and unpredictable changes in context, environment, or information received about available options. S-NOD derives through the introduction of a single extra term to the previously presented Nonlinear Opinion Dynamics (NOD) for fast and flexible multi-agent decision-making behavior. The extra term is inspired by the fast-positive, slow-negative mixed-feedback structure of excitable systems. The agile behaviors brought about by the new excitable nature of decision-making driven by S-NOD are analyzed in a general setting and illustrated in an application to multi-robot navigation around human movers.
{"title":"Spiking Nonlinear Opinion Dynamics (S-NOD) for Agile Decision-Making","authors":"Charlotte Cathcart;Ian Xul Belaustegui;Alessio Franci;Naomi Ehrich Leonard","doi":"10.1109/LCSYS.2024.3523845","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3523845","url":null,"abstract":"We present, analyze, and illustrate a first-of-its-kind model of two-dimensional excitable (spiking) dynamics for decision-making over two options. The model, Spiking Nonlinear Opinion Dynamics (S-NOD), provides superior agility, characterized by fast, flexible, and adaptive response to rapid and unpredictable changes in context, environment, or information received about available options. S-NOD derives through the introduction of a single extra term to the previously presented Nonlinear Opinion Dynamics (NOD) for fast and flexible multi-agent decision-making behavior. The extra term is inspired by the fast-positive, slow-negative mixed-feedback structure of excitable systems. The agile behaviors brought about by the new excitable nature of decision-making driven by S-NOD are analyzed in a general setting and illustrated in an application to multi-robot navigation around human movers.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3267-3272"},"PeriodicalIF":2.4,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142938063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1109/LCSYS.2024.3523844
Marcelo Alves Dos Santos;Antonio Ferramosca;Guilherme Vianna Raffo
This letter analyzes the stability properties of a nonlinear Model Predictive Control (MPC) scheme for avoidance. This control approach introduces an extra penalty for avoidance within the nonlinear tracking MPC framework. We demonstrate that, under a mild assumption on the avoidance penalty, the closed-loop system is Input-to-State Stable (ISS) with respect to this penalty. Furthermore, we discuss the conditions under which asymptotic stability can be achieved and present a simplified scheme with relaxed terminal constraints. To illustrate the effectiveness of the proposed strategy, we apply it to the control of a van der Pol oscillator subjected to non-convex constraints.
{"title":"On the Stability of a Nonlinear MPC Scheme for Avoidance","authors":"Marcelo Alves Dos Santos;Antonio Ferramosca;Guilherme Vianna Raffo","doi":"10.1109/LCSYS.2024.3523844","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3523844","url":null,"abstract":"This letter analyzes the stability properties of a nonlinear Model Predictive Control (MPC) scheme for avoidance. This control approach introduces an extra penalty for avoidance within the nonlinear tracking MPC framework. We demonstrate that, under a mild assumption on the avoidance penalty, the closed-loop system is Input-to-State Stable (ISS) with respect to this penalty. Furthermore, we discuss the conditions under which asymptotic stability can be achieved and present a simplified scheme with relaxed terminal constraints. To illustrate the effectiveness of the proposed strategy, we apply it to the control of a van der Pol oscillator subjected to non-convex constraints.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3285-3290"},"PeriodicalIF":2.4,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10817511","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142976168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1109/LCSYS.2024.3523466
Giovanna Amorim;Anastasia Bizyaeva;Alessio Franci;Naomi Ehrich Leonard
We propose and analyze a nonlinear opinion dynamics model for an agent making decisions about a continuous distribution of options in the presence of input. Inspired by perceptual decision-making, we develop new theory for opinion formation in response to inputs about options distributed on the circle. Options on the circle can represent, e.g., the possible directions of perceived objects and resulting heading directions in planar robotic navigation problems. Interactions among options are encoded through a spatially invariant kernel, which we design to ensure that only a small (finite) subset of options can be favored over the continuum. We leverage the spatial invariance of the model linearization to design flexible, distributed opinion-forming behaviors using spatiotemporal frequency domain and bifurcation analysis. We illustrate our model’s versatility with an application to robotic navigation in crowded spaces.
{"title":"Spatially-Invariant Opinion Dynamics on the Circle","authors":"Giovanna Amorim;Anastasia Bizyaeva;Alessio Franci;Naomi Ehrich Leonard","doi":"10.1109/LCSYS.2024.3523466","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3523466","url":null,"abstract":"We propose and analyze a nonlinear opinion dynamics model for an agent making decisions about a continuous distribution of options in the presence of input. Inspired by perceptual decision-making, we develop new theory for opinion formation in response to inputs about options distributed on the circle. Options on the circle can represent, e.g., the possible directions of perceived objects and resulting heading directions in planar robotic navigation problems. Interactions among options are encoded through a spatially invariant kernel, which we design to ensure that only a small (finite) subset of options can be favored over the continuum. We leverage the spatial invariance of the model linearization to design flexible, distributed opinion-forming behaviors using spatiotemporal frequency domain and bifurcation analysis. We illustrate our model’s versatility with an application to robotic navigation in crowded spaces.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3231-3236"},"PeriodicalIF":2.4,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142938065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1109/LCSYS.2024.3523584
Ryotaro Shima;Yuji Ito;Tatsuya Miyano
This letter analyzes the contraction property of the nonlinear systems controlled by suboptimal model predictive control (MPC) using the continuation method. We propose a contraction metric that reflects the hierarchical dynamics inherent in the continuation method. We derive a pair of matrix inequalities that elucidate the impact of suboptimality on the contraction of the optimally controlled closed-loop system. A numerical example is presented to verify our contraction analysis. Our results are applicable to other MPCs than stabilization, including economic MPC.
{"title":"Contraction Analysis of Continuation Method for Suboptimal Model Predictive Control","authors":"Ryotaro Shima;Yuji Ito;Tatsuya Miyano","doi":"10.1109/LCSYS.2024.3523584","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3523584","url":null,"abstract":"This letter analyzes the contraction property of the nonlinear systems controlled by suboptimal model predictive control (MPC) using the continuation method. We propose a contraction metric that reflects the hierarchical dynamics inherent in the continuation method. We derive a pair of matrix inequalities that elucidate the impact of suboptimality on the contraction of the optimally controlled closed-loop system. A numerical example is presented to verify our contraction analysis. Our results are applicable to other MPCs than stabilization, including economic MPC.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3261-3266"},"PeriodicalIF":2.4,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10817557","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142938062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1109/LCSYS.2024.3523385
Teo Brandt;Rafael Fierro;Claus Danielson
This letter extends the application of the invariant set motion planner (ISMP) to space vehicles operating in $mathbb {SE}text {(}3text {)} = mathbb {SO}text {(}3text {)} rtimes {mathbb {R}}^{3}$ , considering the quaternion representation of $mathbb {SO}text {(}3text {)}$ . We provide a proof for a collision-free set by extending the concepts of configuration-space bubbles from robotics literature. We derive a constraint admissible positive invariant (CAPI) subset within the configuration-space bubble for a robust linearization of the nonlinear vehicle dynamics. The motion planner constructs a directed graph of position and orientation equilibria covering $mathbb {SE}text {(}3text {)}$ . CAPI sets are constructed to verify that equilibria are connected by a feasible trajectory. Graph search is applied to determine a sequence of reference configurations, starting at an initial position-orientation and terminating at a goal position-orientation. Simulation results are included that demonstrate the safe navigation of a vehicle in the presence of an obstacle. The trajectory is shown to maintain the CAPI conditions and is therefore safe under the nonlinear translational and rotational closed-loop vehicle dynamics.
{"title":"Safe Vehicle Motion Planning Using Constraint Admissible Positive Invariant Sets on SE(3)","authors":"Teo Brandt;Rafael Fierro;Claus Danielson","doi":"10.1109/LCSYS.2024.3523385","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3523385","url":null,"abstract":"This letter extends the application of the invariant set motion planner (ISMP) to space vehicles operating in <inline-formula> <tex-math>$mathbb {SE}text {(}3text {)} = mathbb {SO}text {(}3text {)} rtimes {mathbb {R}}^{3}$ </tex-math></inline-formula>, considering the quaternion representation of <inline-formula> <tex-math>$mathbb {SO}text {(}3text {)}$ </tex-math></inline-formula>. We provide a proof for a collision-free set by extending the concepts of configuration-space bubbles from robotics literature. We derive a constraint admissible positive invariant (CAPI) subset within the configuration-space bubble for a robust linearization of the nonlinear vehicle dynamics. The motion planner constructs a directed graph of position and orientation equilibria covering <inline-formula> <tex-math>$mathbb {SE}text {(}3text {)}$ </tex-math></inline-formula>. CAPI sets are constructed to verify that equilibria are connected by a feasible trajectory. Graph search is applied to determine a sequence of reference configurations, starting at an initial position-orientation and terminating at a goal position-orientation. Simulation results are included that demonstrate the safe navigation of a vehicle in the presence of an obstacle. The trajectory is shown to maintain the CAPI conditions and is therefore safe under the nonlinear translational and rotational closed-loop vehicle dynamics.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3255-3260"},"PeriodicalIF":2.4,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142975928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-26DOI: 10.1109/LCSYS.2024.3522594
Xu Shang;Jorge Cortés;Yang Zheng
Koopman operator theory and Willems’ fundamental lemma both can provide (approximated) data-driven linear representation for nonlinear systems. However, choosing lifting functions for the Koopman operator is challenging, and the quality of the data-driven model from Willems’ fundamental lemma has no guarantee for general nonlinear systems. In this letter, we extend Willems’ fundamental lemma for a class of nonlinear systems that admit a Koopman linear embedding. We first characterize the relationship between the trajectory space of a nonlinear system and that of its Koopman linear embedding. We then prove that the trajectory space of Koopman linear embedding can be formed by a linear combination of rich-enough trajectories from the nonlinear system. Combining these two results leads to a data-driven representation of the nonlinear system, which bypasses the need for the lifting functions and thus eliminates the associated bias errors. Our results illustrate that both the width (more trajectories) and depth (longer trajectories) of the trajectory library are important to ensure the accuracy of the data-driven model.
{"title":"Willems’ Fundamental Lemma for Nonlinear Systems With Koopman Linear Embedding","authors":"Xu Shang;Jorge Cortés;Yang Zheng","doi":"10.1109/LCSYS.2024.3522594","DOIUrl":"https://doi.org/10.1109/LCSYS.2024.3522594","url":null,"abstract":"Koopman operator theory and Willems’ fundamental lemma both can provide (approximated) data-driven linear representation for nonlinear systems. However, choosing lifting functions for the Koopman operator is challenging, and the quality of the data-driven model from Willems’ fundamental lemma has no guarantee for general nonlinear systems. In this letter, we extend Willems’ fundamental lemma for a class of nonlinear systems that admit a Koopman linear embedding. We first characterize the relationship between the trajectory space of a nonlinear system and that of its Koopman linear embedding. We then prove that the trajectory space of Koopman linear embedding can be formed by a linear combination of rich-enough trajectories from the nonlinear system. Combining these two results leads to a data-driven representation of the nonlinear system, which bypasses the need for the lifting functions and thus eliminates the associated bias errors. Our results illustrate that both the width (more trajectories) and depth (longer trajectories) of the trajectory library are important to ensure the accuracy of the data-driven model.","PeriodicalId":37235,"journal":{"name":"IEEE Control Systems Letters","volume":"8 ","pages":"3135-3140"},"PeriodicalIF":2.4,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142962894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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