Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators

IFAC PapersOnLine 53-2 (2020) 6256–6261 ScienceDirect Available online at www.sciencedirect.com 2405-8963 Copyright © 2020 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.2020.12.1738 10.1016/j.ifacol.2020.12.1738 2405-8963 Copyright © 2020 The Authors. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0 ) Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗ Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], r[email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Eduardo Moya-Lasheras et al. / IFAC PapersOnLine 53-2 (2020) 6256–6261 6257 Copyright © 2020 The Authors. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0 ) Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Model-Free Sliding-Mode Controller for Soft Landing of Reluctance Actuators Eduardo Moya-Lasheras ∗Edgar Ramirez-Laboreo ∗ Carlos Sagues ∗ ∗Departamento de Informatica e Ingenieria de Sistemas (DIIS) and Instituto de Investigacion en Ingenieria de Aragon (I3A), Universidad de Zaragoza, Zaragoza 50018, Spain, (e-mail: [email protected], [email protected], [email protected]) Abstract: Some electromagnetic actuators suffer from high velocity impacts during noncontrolled switching operations, which cause contact bouncing, mechanical wear, and acoustic noise. Soft-landing control strategies aim at minimizing the impact velocities of these devices to improve their performance. This paper presents a sliding-mode controller for soft landing of single-coil reluctance actuators. It is a switching model-free controller, which results in a very simple implementation. A generalized dynamical hybrid model of an actuator is utilized for deriving the robustness condition, based on the Lyapunov theory. Then, the condition is evaluated for a dynamical model, based on a commercial device, and several reference trajectories. Finally, the controller performance is validated through simulations. The effect of the sampling rate on the resulting impact velocities is also analyzed. Keywords: Actuators, Electromagnetic devices, Modeling, Nonlinear control, Robust control, Sliding-mode control, Tracking 1. INTRODUCTION A reluctance actuator is a type of nonlinear electromechanical device which generates a reluctance-based magnetic force to move its armature. Particularly, single-coil actuators are used in an extensive variety of industrial applications because of their fast response, compactness, high energy efficiency, and low cost. Thus, there is a great research interest concerning modeling, identification, estimation, and control of this class of actuators. Regarding the control, one of the main motivations and challenges is to achieve soft landing during switching operations; thus reducing contact bouncing, impact noise and mechanical wear. In the literature, there are several control proposals for reluctance actuators, e.g., based on the backstepping technique (Kahveci and Kolmanovsky, 2010), energy compensation (Yang et al., 2013), a linearization method (Katalenic et al., 2016), or cycle-to-cycle adaptation (Moya-Lasheras et al., 2019), among others. One important drawback of many mass-market singlecoil reluctance actuators is the manufacturing variability among devices from the same ensemble. Moreover, the identification of every unit may impose a prohibitive cost. One major approach to deal with model uncertainties is the sliding-mode control (SMC) theory (Slotine and Li, 1991). There are several works that take this approach. Most commonly, the control law is divided into two terms: This work was partially supported by the Arag´on Regional Government, the Spanish Government, and the European Union, under project RTC-2017-5965-6, project PGC2018-098719-B-I00 (MCIU/ AEI/FEDER, UE), research group DGA-T45 17R, scholarship FPU14/04171, and program FSE Arag´on 2014-2020. an equivalent and a switching control term (Lee et al., 2015; Zhao et al., 2016). Alternatively, Eyabi and Washington (2006) proposed a SMC with only a switching term, which is then approximated to a proportional one. One important aspect that is omitted in these works is the definition of the tracking trajectory, which directly affects the robustness conditions for the SMC. Another important issue is the influence of the sampling rate. In general, the sliding accuracy is proportional to the square of the switching delay (Levant, 1993). Still, its effect on the resulting impact velocities needs to be evaluated. This paper presents a robust SMC controller for singlecoil reluctance actuators. It is purely a switching controller, which results in a very simple and computationally inexpensive implementation. Although the resulting controller is model-free—i.e. it does not depend on any model functions or parameters—a dynamic model is required during the design process to guarantee its robustness. The generalized system, which presents both continuous and discrete dynamic behavior, is modeled with a hybrid automaton. A robustness condition is derived, which depends on the system dynamics and the position trajectory. It is then evaluated for a specific dynamic model, based on a commercial solenoid valve, and several trajectories. The first contribution of the paper is the proposal of a switching model-free SMC, which works for every discrete mode of the system. The second contribution is the analysis of the influence of the sampling rate on the impact velocities. 2. SYSTEM DYNAMICS A general single-coil reluctance actuator is represented in Fig. 1. The magnetic core is divided into two parts: a fixed Fig. 1. Schematic representation of a single-coil reluctance actuator. part (stator) and a movable part (mover or armature). The air gap between them is dependent on the position of the mover. There are two types of operations depending on the direction of the movement: in a making operation, the magnetic force is large enough to attract the mover toward the stator; whereas in a breaking operation, the magnetic force is reduced and the passive forces (e.g. elastic or gravity) move the armature in the opposite direction. Moreover, the position of the mover is restricted between a lower and an upper limit. The motion dynamics is given by Newton’s second law, with two forces, ˙v=fv(z,v,φ)= 1 mFpas(z,v)+Fmag(z,φ),(1) where z,v, and ˙vare the position, velocity and acceleration of the mover; Fpas, and Fmag are the passive and magnetic forces; φis the magnetic flux; and mis the moving mass. Note that the dynamic function of vis expressed compactly as fv. The force that can be controlled—albeit indirectly—is Fmag, which is defined as (Ramirez-Laboreo et al., 2016) Fmag =−1 2R g(z)φ2,R g(z)=∂Rg(z) ∂z ,(2) where Rgis the gap reluctance. Note that R g>0, and therefore Fmag ≤0, for all z∈[zmin,z max] (i.e. the magnetic force is always attractive). Then, φcan be related to the current through the coil icoil in terms of the total reluctance, given Amp`ere’s circuital law, Ni coil +ieddy =Rc(φ)+Rg(z)φ, (3) where Nis the number of coil turns, Rcis the core reluctance, and ieddy is the net eddy current through the core. Assuming that the magnetic flux density is constant across the section, ieddy is proportional to the magnetic flux derivative (Ramirez-Laboreo et al., 2019), ieddy =−ke˙ φ. (4) Most commonly, the voltage is treated as the system input u, because it can be directly supplied to the device. The dynamics of the magnetic flux is given by the electrical circuit equation, u=Ri coil +N˙ φ, (5) where Ris the coil resistance. Then, substituting (3) into (5) and solving for ˙ φ, the dynamic function is derived as ˙ φ=fφ(z,φ)+Bφu =−RRg(z)+Rc(φ)φ N2+Rk e +N N2+Rk e u, (6) where the function fφdepends on the position and magnetic flux, and Bφis a constant. ˙z=v ˙v=fv(z,v,φ) ˙ φ=fφ(z,φ)+Bφu ˙z=0 ˙v=0 ˙ φ=fφ(z,φ)+Bφu ˙z=0 ˙v=0 ˙ φ=fφ(z,φ)+Bφu Motion (q= 2 ): Lower limit (q= 1): Upper limit (q= 3): z=zmin ∧v<0⇒v+=0 z=zmax ∧v>0⇒v+=0 fv(z,v,φ)≥0 fv(z,v,φ)≤0 Fig. 2. Hybrid automaton that models the dynamics of reluctance actuators with a limited range of motion. The complete system dynamics can be described through a state-space representation with three state variables (z, v, and φ). As the motion is constrained, zand vmust be static if the mover reaches one of the two limits. Thus, the system is modeled with a hybrid automaton, with three discrete modes, as illustrated in Fig. 2. Each transition is accompanied by its guard condition. There is also a reset function when transitioning to one of the position limits: v+= 0. 3. CONTROL DESIGN 3.1 Trajectory planning The first critical aspect of the soft-landing tracking control is the definition of the position trajectory zref , for all time t∈[t0,t f]. For a given operation, the initial position is z0 and the desired final position is zf. Let tland be the intended instant in which the armature reaches the final position. For a perfect soft landing, zref should satisfy zref (tland)=zf,v ref (tland)=0,a ref (tland)=0,(7) where vref (t)= ˙zref (t),a ref (t)=˙vref (t).(8) Equivalently, in order to start the motion smoothly, zref should satisfy zref (ttakeoff )=z0,v ref (ttakeoff )=0,a ref (ttakeoff )=0,(9) where ttakeoff is the take-off instant. During motion (q= 2), the acceleration is determined from (1). Therefore, in order to start moving immediately at t= ttakeoff ,φshould be φtakeoff , such that fv(z0,0,φ takeoff )= 0. Note that there are two symmetrical solutions of φtakeoff (positive and negative), because fvis an even function with respect to φ. Note also that the controller should decrease |φ|if q= 1, or increase it if q= 3, until it reaches φtakeoff . Thus, prior to moving, a static interval is defined, zref (t)=z0,∀t∈[t0,t takeoff ].(10) where ttakeoff −t0should be large enough to let the magnetic flux reach φtakeoff before ttakeoff . In the second interval (from ttakeoff to tland), a position trajectory must be defined to reach zfsmoothly. Thus, zref should be a function of time t, for all t∈[ttakeoff ,t land], 6258 Eduardo Moya-Lasheras et al. / IFAC PapersOnLine 53-2 (2020) 6256–6261 with boundary conditions (9) and (7). In the third interval (from tland to tf), the mover must be kept in the desired final position, so zref (t)=zf,∀t∈[tland,t f].(11) Once a trajectory is defined, its feasibility should be checked. First, the position must be kept inside its bounds, zref (t)∈[zmin,z max],∀t. (12) Secondly, the required magnetic force F∗ mag should be calculated, and ensure that it is always nonpositive, because repelling magnetic forces are not physically possible (see (2)), F∗ mag(t)=ma ref (t)−Fpas(zref (t),v ref (t)) ≤0,∀t. (13) Moreover, magnetic saturation must also be taken into account. Given the saturated value of the magnetic flux φsat, the required magnetic force should also satisfy F∗ mag(t)≥− 1 2R g(zref (t)) φ2 sat,∀t. (14) 3.2 Control for motion dynamics The controller is initially designed based on the dynamic equations of the motion mode (q= 2), which can be expressed compactly as ˙ x=f(x)+Bu, (15) where x=z v φT ,f(x)=v fv(z,v,φ) fφ(z,φ),B=0 0 Bφ.(16) As stated in the introduction, our proposal relies on an SMC. It is assumed that the position z, velocity v, and acceleration acan be obtained either through measurement or estimation. The proposed sliding surface is defined in terms of their errors, s=λ1+d dtλ2+d dt˜z =˜a+(λ1+λ2)˜v+λ1λ2˜z, (17) where λ1and λ2are positive constants; and ˜z,˜v, and ˜a are the position error and its derivatives, ˜z=z−zref ,˜v=v−vref ,˜a=a−aref .(18) To analyze the convergence to the sliding surface s= 0, the following Lyapunov function is defined, V=1 2s2.(19) Thus, to ensure that sconverges to zero in finite time, we impose the following condition, ˙ V=s˙s≤−η|s|,(20) where ηis a strictly positive constant that determines the convergence speed (|˙s|≥η). Then, by deriving (17) and substituting into (20), ˙ V=sj−jref +(λ1+λ2)˜a+λ1λ2˜v,(21) where j=˙ais the jerk and jref =˙aref is the reference jerk. The jerk jcan be derived from fvand the dynamic equation (15) as j=dfv(x) dt=∂fv(x) ∂xf(x)+∂fv(x) ∂xBu. (22) Note that jdepends on u. Thus, the convergence condition (20) can be expressed in terms of the control u, s(fj−jref +ε−Bju)≤−η|s|,(23) where ε=ε(˜v,˜a)=(λ1+λ2)˜a+λ1λ2˜v, (24) fj=fj(x)=∂fv(x) ∂xf(x) =1 m∂Fpas ∂z (z,v)v+∂Fpas ∂v (z,v)a −1 2R g(z)φ2v−R  g(z)φf φ(z,φ),(25) Bj=Bj(x)=−∂fv(x) ∂xB=R gφB φ m.(26) Then, with some manipulations, sgn(s)Bju≥sgn(s)(fj−jref +ε)+η. (27) Note that, assuming R g>0 for all z∈[zmin,z max], it is obtained that sgn(Bj) = sgn(φ). Thus, the control umust satisfy the following condition, sgn(s) sgn(φ)u≥sgn(s)(fj−jref +ε)+η |Bj|.(28) We propose this model-free control, umotion =umax sgn(s) sgn(φ),(29) where umax is a constant that, in order to ensure the convergence to s= 0, must satisfy umax ≥max |fj−jref +ε|+η |Bj|.(30) 3.3 Control for hybrid dynamics In the previous section, we have proposed a controller and proved its convergence for the motion dynamics (q= 2). Now, we ensure that it works for the complete hybrid system. For that, we propose a slight modification of the Lyapunov function, V=1 2σ2,(31) where σis a generalization of s. It is defined as σ=fv(z,v,φ)−aref +(λ1+λ2)˜v+λ1λ2˜z. (32) Note that σis equal to sin the case of motion, because a=fv. On the other hand, before the start of motion (t≤ttakeoff ), σ=fv. Therefore, σ= 0 implies that φ=φtakeoff . As a result, if σ= 0, the system behaves as desired both before and after the start of motion. Following the same line of reasoning as in Section 3.2, convergence to σ= 0 requires sgn(σ) sgn(φ)u≥sgn(σ)(fj−jref +ε)+η |Bj|.(33) To keep the controller model-free, the proposal cannot depend on fv. Instead, it should be a function of s. We generalize the proposed control (29), uhybrid =umax sgn(φ) sgn∗(s),(34) Eduardo Moya-Lasheras et al. / IFAC PapersOnLine 53-2 (2020) 6256–6261 6259 with boundary conditions (9) and (7). In the third interval (from tland to tf), the mover must be kept in the desired final position, so zref (t)=zf,∀t∈[tland,t f].(11) Once a trajectory is defined, its feasibility should be checked. First, the position must be kept inside its bounds, zref (t)∈[zmin,z max],∀t. (12) Secondly, the required magnetic force F∗ mag should be calculated, and ensure that it is always nonpositive, because repelling magnetic forces are not physically possible (see (2)), F∗ mag(t)=ma ref (t)−Fpas(zref (t),v ref (t)) ≤0,∀t. (13) Moreover, magnetic saturation must also be taken into account. Given the saturated value of the magnetic flux φsat, the required magnetic force should also satisfy F∗ mag(t)≥− 1 2R g(zref (t)) φ2 sat,∀t. (14) 3.2 Control for motion dynamics The controller is initially designed based on the dynamic equations of the motion mode (q= 2), which can be expressed compactly as ˙ x=f(x)+Bu, (15) where x=z v φT ,f(x)=v fv(z,v,φ) fφ(z,φ),B=0 0 Bφ.(16) As stated in the introduction, our proposal relies on an SMC. It is assumed that the position z, velocity v, and acceleration acan be obtained either through measurement or estimation. The proposed sliding surface is defined in terms of their errors, s=λ1+d dtλ2+d dt˜z =˜a+(λ1+λ2)˜v+λ1λ2˜z, (17) where λ1and λ2are positive constants; and ˜z,˜v, and ˜a are the position error and its derivatives, ˜z=z−zref ,˜v=v−vref ,˜a=a−aref .(18) To analyze the convergence to the sliding surface s= 0, the following Lyapunov function is defined, V=1 2s2.(19) Thus, to ensure that sconverges to zero in finite time, we impose the following condition, ˙ V=s˙s≤−η|s|,(20) where ηis a strictly positive constant that determines the convergence speed (|˙s|≥η). Then, by deriving (17) and substituting into (20), ˙ V=sj−jref +(λ1+λ2)˜a+λ1λ2˜v,(21) where j=˙ais the jerk and jref =˙aref is the reference jerk. The jerk jcan be derived from fvand the dynamic equation (15) as j=dfv(x) dt=∂fv(x) ∂xf(x)+∂fv(x) ∂xBu. (22) Note that jdepends on u. Thus, the convergence condition (20) can be expressed in terms of the control u, s(fj−jref +ε−Bju)≤−η|s|,(23) where ε=ε(˜v,˜a)=(λ1+λ2)˜a+λ1λ2˜v, (24) fj=fj(x)=∂fv(x) ∂xf(x) =1 m∂Fpas ∂z (z,v)v+∂Fpas ∂v (z,v)a −1 2R g(z)φ2v−R  g(z)φf φ(z,φ),(25) Bj=Bj(x)=−∂fv(x) ∂xB=R gφB φ m.(26) Then, with some manipulations, sgn(s)Bju≥sgn(s)(fj−jref +ε)+η. (27) Note that, assuming R g>0 for all z∈[zmin,z max], it is obtained that sgn(Bj) = sgn(φ). Thus, the control umust satisfy the following condition, sgn(s) sgn(φ)u≥sgn(s)(fj−jref +ε)+η |Bj|.(28) We propose this model-free control, umotion =umax sgn(s) sgn(φ),(29) where umax is a constant that, in order to ensure the convergence to s= 0, must satisfy umax ≥max |fj−jref +ε|+η |Bj|.(30) 3.3 Control for hybrid dynamics In the previous section, we have proposed a controller and proved its convergence for the motion dynamics (q= 2). Now, we ensure that it works for the complete hybrid system. For that, we propose a slight modification of the Lyapunov function, V=1 2σ2,(31) where σis a generalization of s. It is defined as σ=fv(z,v,φ)−aref +(λ1+λ2)˜v+λ1λ2˜z. (32) Note that σis equal to sin the case of motion, because a=fv. On the other hand, before the start of motion (t≤ttakeoff ), σ=fv. Therefore, σ= 0 implies that φ=φtakeoff . As a result, if σ= 0, the system behaves as desired both before and after the start of motion. Following the same line of reasoning as in Section 3.2, convergence to σ= 0 requires sgn(σ) sgn(φ)u≥sgn(σ)(fj−jref +ε)+η |Bj|.(33) To keep the controller model-free, the proposal cannot depend on fv. Instead, it should be a function of s. We generalize the proposed control (29), uhybrid =umax sgn(φ) sgn∗(s),(34) where sgn∗(s)=     −1,if s<0, +1,if s>0, −1,if s=0∧q=1, +1,otherwise. (35) Under the assumption that (30) is satisfied, a sufficient condition for convergence is sgn∗(s) = sgn(σ).(36) Then, convergence is studied in three separate cases. First, if q= 2, the convergence condition is directly guaranteed because s=σ. Secondly, if z=zref =zmax or z=zref =zmin,sis always zero, but σmay be not. Note that aref =˜v=˜z= 0. Then, sgn(σ) = sgn(fv).(37) Note also that fv<0 if q= 1 and fv>0ifq= 3, otherwise the hybrid system would make a transition to q= 2 (see guard conditions in Fig. 2). Therefore, sgn(σ) = sgn(fv)=−1,if q=1, +1,if q=3.(38) Then, given the proposed definition of sgn∗(s), condition (36) holds, so convergence is still guaranteed. Thirdly, we still need to check the convergence of the controller in the case that the position is in one of the limits (q= 2), but the reference is not. In that event, (32) is simplified into σ=fv+s, (39) where s=−aref −(λ1+λ2)vref −λ1λ2(z−zref ).(40) Assuming that the position trajectory is defined smoothly at the start of the movement, condition (36) is satisfied because, when z0=zmin (breaking operation), fv<0and(zref −z0),v ref ,a ref ≥0.(41) Equivalently, when z0=zmax (making operation), fv>0,and (zref −z0),v ref ,a ref ≤0.(42) On the other hand, at the end of movement, if zhas reached the limit but zref not yet, the condition is not necessarily satisfied. This may seem like a limitation but, if the mover has reached the final position prematurely, it is actually preferable to fix it instead of separating it to continue following the trajectory. Thus, expert rules are added to the controller so the mover is kept at zf=zmin (making operation) or zf=zmax (breaking operation), u=umax sgn(φ) if z=zf=zmin, 0,if z=zf=zmax, umax sgn∗(s) sgn(φ),otherwise. (43) 4. ANALYSIS AND DISCUSSION 4.1 Robustness analysis For the given dynamic model, it is impossible to guarantee robustness in general, for any feasible state. As a clear counterexample, setting φ= 0 makes Bj= 0, and umax ≥ ∞(see (30)). Therefore, the robustness must be studied for a given trajectory. To illustrate this, the robustness is analyzed for three different scenarios. Fig. 3. Solenoid valve: schematic representation (left) and photo (right). Table 1. Parameters of the solenoid valve. Param. Value m0.0016 kg ks61.8N/m zs0.019 m c0.8 Ns/m zmin 0m zmax 0.001 m Param. Value N1200 R50 Ω ke1630 Ω−1 Rc,04.41 ×106H−1 φsat 2.6×10−5Wb Then, three position trajectories are defined. Each one of them consists of a making operation, followed by a breaking operation. The motion intervals are defined with a 5th degree polynomial, satisfying the boundary conditions (7), (9). Moreover, for the sake of simplicity, the time intervals are defined in terms of the motion duration (τmov), tland −ttakeoff =τmov,(44) ttakeoff −t0=tf−tland =τmov/4,(45) where τmov is 3, 4 or 5 ms, for each case. The actuator model is particularized to a commercial solenoid valve, depicted in Fig. 3, whose estimated parameters are presented in Table 1. The passive force is generated by the spring and friction. It is modeled as a mass-spring-damper system, Fpas =ks(zs−z)−cv, (46) where ksis the spring constant, zsis the spring resting position, and cis the damping coefficient. Moreover, the core reluctance is given by a parametric expression that takes into account magnetic saturation (Moya-Lasheras et al., 2017), Rc=Rc,0 1−φ/φsat ,(47) where Rc,0is the core reluctance for φ= 0, and φsat is the saturated value of the magnetic flux. The gap reluctance, on the other hand, is highly nonlinear with respect to the position. Instead of a parametric expression, a look-up table is used (see Fig. 4). Its data has been obtained from finite element analysis and experimentation (Ramirez-Laboreo and Sagues, 2018). In Fig. 5, the desired position and its derivatives (zref , vref ,aref ) are displayed. Three additional useful signals are calculated and shown in Fig. 5: the required magnetic force (as described in Section 3.1), the required action u∗, and Bj. The required action is the absolute value of uto be able to track zref in an ideal scenario (no perturbations or errors), u∗=|fj−jref | |Bj|.(48) 6260 Eduardo Moya-Lasheras et al. / IFAC PapersOnLine 53-2 (2020) 6256–6261 Fig. 4. Gap reluctance and its derivative with respect to the gap length. Note that, if τmov = 3 ms, F∗ mag is positive in a small interval in the breaking operation (around t/τmov = 2). Thus, this trajectory is infeasible. This can be checked as well in u∗, which tends to infinity as F∗ mag approaches zero. Then, as the motion duration increases, the requirements are less demanding, because vref and aref are reduced. Therefore, the maximum values of u∗are also reduced. A necessary condition for convergence to s= 0 is umax > max(u∗(t)). This condition is sufficient for perfect tracking in the ideal case, in which ε= 0. Otherwise, in general, a sufficient condition for convergence can be derived from (30), umax ≥max(u∗)+max |ε|+η min |Bj|,(49) where εis bounded, assuming that ˜vand ˜aare bounded, max |ε|≤εmax(|˜v|),max(|˜a|).(50) Some assumptions must be made about the bounds of errors ˜vand ˜ato satisfy the previous condition. As an example, the controller constants are set as λ1=λ2= 2000,η= 105.(51) And, for the sake of simplicity, very conservative assumptions are made about the error bounds, |˜v|≤0.2 max(|vref |),|˜a|≤0.2 max(|aref |).(52) Thus, from (49) and (50), the robustness condition is umax ≥39.13 V (if τmov = 4 ms), or umax ≥31.29 V (if τmov = 5 ms). As expected, the condition is less restrictive when the motion duration is increased. Note that, in order for the controller to be robust to modeling disturbances, the model parameters used to derive the robustness criteria (49) should represent the worst-case scenario, assuming the bounds of each model parameter are known. In practice, however, determining the combination of parameters that results in the worse-case scenario may be too cumbersome, due to the immense number of possibilities. Alternatively, a Monte-Carlo evaluation could be performed, permuting all parameters inside their bounds, and then selecting umax such that (49) holds for every case. 4.2 Sampling rate analysis We have proved that robustness can be guaranteed under some reasonable operating conditions. Still, the sampling rate may be a limiting factor, and its influence should be analyzed. Thus, the proposed controller is tested with different sampling periods Ts. As reference, the second position trajectory from Section 4.1 is used (τmov = 4 ms). The controller constants are set as in Section 4.1, with Fig. 5. Simulation results. Note that the time axis is normalized with respect to τmov. umax = 40 V. The dynamic system is simulated using the hybrid automaton from Fig. 2 and the model parameters from Table 1. The impact velocities are calculated for different sampling periods and depicted in Fig. 6, separating the making and breaking operations. With a sampling rate of 100 kHz, the results are very good, specially in the making operation. For larger sampling periods, the results increasingly worsen. Still, with a sampling rate of only 10 kHz, the impact velocities are better than the ones in a non-controlled scenario. For reference, using a square voltage of 40 V and 0 V, the impact velocities are −2.2 and 0.9 m/s, for the making and breaking operations respectively (which are beyond the graph limits). Fig. 7 presents the resulting state variables for three representative sampling periods. With a sampling rate of Eduardo Moya-Lasheras et al. / IFAC PapersOnLine 53-2 (2020) 6256–6261 6261 Fig. 4. Gap reluctance and its derivative with respect to the gap length. Note that, if τmov = 3 ms, F∗ mag is positive in a small interval in the breaking operation (around t/τmov = 2). Thus, this trajectory is infeasible. This can be checked as well in u∗, which tends to infinity as F∗ mag approaches zero. Then, as the motion duration increases, the requirements are less demanding, because vref and aref are reduced. Therefore, the maximum values of u∗are also reduced. A necessary condition for convergence to s= 0 is umax > max(u∗(t)). This condition is sufficient for perfect tracking in the ideal case, in which ε= 0. Otherwise, in general, a sufficient condition for convergence can be derived from (30), umax ≥max(u∗)+max |ε|+η min |Bj|,(49) where εis bounded, assuming that ˜vand ˜aare bounded, max |ε|≤εmax(|˜v|),max(|˜a|).(50) Some assumptions must be made about the bounds of errors ˜vand ˜ato satisfy the previous condition. As an example, the controller constants are set as λ1=λ2= 2000,η= 105.(51) And, for the sake of simplicity, very conservative assumptions are made about the error bounds, |˜v|≤0.2 max(|vref |),|˜a|≤0.2 max(|aref |).(52) Thus, from (49) and (50), the robustness condition is umax ≥39.13 V (if τmov = 4 ms), or umax ≥31.29 V (if τmov = 5 ms). As expected, the condition is less restrictive when the motion duration is increased. Note that, in order for the controller to be robust to modeling disturbances, the model parameters used to derive the robustness criteria (49) should represent the worst-case scenario, assuming the bounds of each model parameter are known. In practice, however, determining the combination of parameters that results in the worse-case scenario may be too cumbersome, due to the immense number of possibilities. Alternatively, a Monte-Carlo evaluation could be performed, permuting all parameters inside their bounds, and then selecting umax such that (49) holds for every case. 4.2 Sampling rate analysis We have proved that robustness can be guaranteed under some reasonable operating conditions. Still, the sampling rate may be a limiting factor, and its influence should be analyzed. Thus, the proposed controller is tested with different sampling periods Ts. As reference, the second position trajectory from Section 4.1 is used (τmov = 4 ms). The controller constants are set as in Section 4.1, with Fig. 5. Simulation results. Note that the time axis is normalized with respect to τmov. umax = 40 V. The dynamic system is simulated using the hybrid automaton from Fig. 2 and the model parameters from Table 1. The impact velocities are calculated for different sampling periods and depicted in Fig. 6, separating the making and breaking operations. With a sampling rate of 100 kHz, the results are very good, specially in the making operation. For larger sampling periods, the results increasingly worsen. Still, with a sampling rate of only 10 kHz, the impact velocities are better than the ones in a non-controlled scenario. For reference, using a square voltage of 40 V and 0 V, the impact velocities are −2.2 and 0.9 m/s, for the making and breaking operations respectively (which are beyond the graph limits). Fig. 7 presents the resulting state variables for three representative sampling periods. With a sampling rate of Fig. 6. Impact velocities in making (left) and breaking (right) operations, as functions of the sampling period. Fig. 7. Simulated state variables using the controller with three different sampling periods Ts. 1 MHz, the tracking is almost perfect. With a sampling rate of 100 kHz, there is a slight error in the position (almost imperceptible in the graphic), but the impact velocities are appreciably larger. Still, the performance is very good. With a sampling rate of 10 kHz, the results are much worse. The high ripple of the magnetic flux is filtered, but leads to significant tracking errors. Even though the position errors may seem small, the velocity errors and, more importantly, the impact velocities are much larger than in the other cases. 5. CONCLUSIONS We have addressed the soft-landing control of single-coil reluctance actuators, presenting a sliding-mode controller which does not use any information about the dynamic system. We have also derived the convergence criteria, based on a generalized dynamical model. This controller requires to know the position and its derivatives, as well as the sign of φ. Alternatively, the current through the coil can be restricted to nonnegative values. That way, the magnetic flux is always nonnegative, simplifying the control. 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