Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives

IFAC PapersOnLine 58-7 (2024) 144–149 ScienceDirect Available online at www.sciencedirect.com 2405-8963 Copyright © 2024 The Authors. This is an open access article under the CC BY-NC-ND license . Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2024.08.025 10.1016/j.ifacol.2024.08.025 2405-8963 Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives ⋆ Manuel G. Satu´e ∗J.D. ´ Alvarez ∗∗ Manuel A. Perales ∗∗∗ ∗Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]) ∗∗ Department of Computer Engineering, Automation and Robotics, CIESOL—ceiA3, University of Almer´ıa, 04120 Almer´ıa, Spain (e-mail: [email protected]) ∗∗∗ Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: mperale[email protected]s) Abstract: PI tuning for variable-speed multi-phase drives is a complex task. In this paper, a Pareto analysis is introduced to reveal not previously reported links between figures of merit. The drive used for the study includes a 5-phase induction motor supplied by a voltage source inverter. A finite state predictive method is used for the inner loop (current control). The outer loop (speed control) is governed by a PI. The analysis is done experimentally thus including all sorts of non-idealities not appearing in commonly found models. The experimental results show how the pursuit for better performance is hindered by the existence of links between figures of merit. The importance of the result lies in showing that arbitrary performance enhancements are not possible in a general case. Keywords: Application of power electronics, Digital implementation, Modeling and simulation of power systems, PI control, Tuning 1. INTRODUCTION PI controllers are routinely used in variable speed drives for speed and/or current regulation. The speed control loop (outer loop) is used to maintain the mechanical speed close to its reference value. The current loop has a much faster time scale and is used to produce stator currents that, in turn, produce a torque that drives the speed. In the last years, there has been a trend in academia to replace PI loops with more sophisticated alternatives such as Model Predictive Control (MPC). This trend is a consequence of the relative success attained by MPC in other realms (Berenguel et al., 1998; Ram´ırez-Arias et al., 2005). The case of predictive control of drives is exemplified by the so called Finite State Model Predictive Control (FSMPC), where direct digital control of the Voltage Source Inverter (VSI) is attained. In FSMPC the Pulse Width Modulation (PWM) stage is no longer necessary, thus improving the performance of the system as reported by Lim et al. (2013); Berm´udez et al. (2020). Going back to the speed control loop, the influential works related to PI tuning for Indirect Field Oriented Control (IFOC) date from the 1990s and 2000s (De Wit et al., 1996; Espinosa-Perez and Ortega, 1997; Espinosa-Perez et al., 1998; Bazanella and Reginatto, 2001). These works pro- ⋆This work is part of the Grant TED2021-129558B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. vide methods which utilize some, not very strict, requirements on the knowledge of the time constant of the rotor of the IM to guarantee the global stability. A dynamic model of the current fed IM is used at the basis of these methods. Some variations have been reported making use of harmonic injection (Arahal and Duran, 2009), but, in most cases, the models do not consider practicalities like nonlinearities in the IM dynamics, VSI+modulation behavior and aspects derived from the use of a Digital Signal Processor (DSP) for the PI + IFOC algorithms. In the 20 years that followed, the state of the art has been enriched by some papers expanding the previous works in a variety of ways. Most of the efforts have focused on: sensorless operation, estimation of the rotor time constant, extension of the PI with operating-point based scheduling and/or adaptivity, fault-tolerant capabilities, and tuning methods based on extensive simulation/experimentation. Other technologies have even been proposed to replace the speed PI but, despite all of this, conventional PI control is still reported as competitive due to its simple implementation compared with nonlinear control (Ortega et al., 1993), adaptive control (Marino et al., 1993), robust control (Ding et al., 2000; Alamo et al., 2006) and some fuzzy and/or neural approaches (Masiala et al., 2008; Pham and Le, 2020). The lack of theoretical models to analyze the above cited practical aspects and their influence on complex figures of merit such as the Integral Time-multiplied Absolute Error (ITAE) has led to the use of a experimental approach for the PI tuning. For example, in Shaija and Daniel (2023) metaheuristic optimisation algorithms are used for the optimal PI tuning. Again, MPC Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives ⋆ Manuel G. Satu´e ∗J.D. ´ Alvarez ∗∗ Manuel A. Perales ∗∗∗ ∗Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]) ∗∗ Department of Computer Engineering, Automation and Robotics, CIESOL—ceiA3, University of Almer´ıa, 04120 Almer´ıa, Spain (e-mail: [email protected]) ∗∗∗ Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: mperale[email protected]s) Abstract: PI tuning for variable-speed multi-phase drives is a complex task. In this paper, a Pareto analysis is introduced to reveal not previously reported links between figures of merit. The drive used for the study includes a 5-phase induction motor supplied by a voltage source inverter. A finite state predictive method is used for the inner loop (current control). The outer loop (speed control) is governed by a PI. The analysis is done experimentally thus including all sorts of non-idealities not appearing in commonly found models. The experimental results show how the pursuit for better performance is hindered by the existence of links between figures of merit. The importance of the result lies in showing that arbitrary performance enhancements are not possible in a general case. Keywords: Application of power electronics, Digital implementation, Modeling and simulation of power systems, PI control, Tuning 1. INTRODUCTION PI controllers are routinely used in variable speed drives for speed and/or current regulation. The speed control loop (outer loop) is used to maintain the mechanical speed close to its reference value. The current loop has a much faster time scale and is used to produce stator currents that, in turn, produce a torque that drives the speed. In the last years, there has been a trend in academia to replace PI loops with more sophisticated alternatives such as Model Predictive Control (MPC). This trend is a consequence of the relative success attained by MPC in other realms (Berenguel et al., 1998; Ram´ırez-Arias et al., 2005). The case of predictive control of drives is exemplified by the so called Finite State Model Predictive Control (FSMPC), where direct digital control of the Voltage Source Inverter (VSI) is attained. In FSMPC the Pulse Width Modulation (PWM) stage is no longer necessary, thus improving the performance of the system as reported by Lim et al. (2013); Berm´udez et al. (2020). Going back to the speed control loop, the influential works related to PI tuning for Indirect Field Oriented Control (IFOC) date from the 1990s and 2000s (De Wit et al., 1996; Espinosa-Perez and Ortega, 1997; Espinosa-Perez et al., 1998; Bazanella and Reginatto, 2001). These works pro- ⋆This work is part of the Grant TED2021-129558B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. vide methods which utilize some, not very strict, requirements on the knowledge of the time constant of the rotor of the IM to guarantee the global stability. A dynamic model of the current fed IM is used at the basis of these methods. Some variations have been reported making use of harmonic injection (Arahal and Duran, 2009), but, in most cases, the models do not consider practicalities like nonlinearities in the IM dynamics, VSI+modulation behavior and aspects derived from the use of a Digital Signal Processor (DSP) for the PI + IFOC algorithms. In the 20 years that followed, the state of the art has been enriched by some papers expanding the previous works in a variety of ways. Most of the efforts have focused on: sensorless operation, estimation of the rotor time constant, extension of the PI with operating-point based scheduling and/or adaptivity, fault-tolerant capabilities, and tuning methods based on extensive simulation/experimentation. Other technologies have even been proposed to replace the speed PI but, despite all of this, conventional PI control is still reported as competitive due to its simple implementation compared with nonlinear control (Ortega et al., 1993), adaptive control (Marino et al., 1993), robust control (Ding et al., 2000; Alamo et al., 2006) and some fuzzy and/or neural approaches (Masiala et al., 2008; Pham and Le, 2020). The lack of theoretical models to analyze the above cited practical aspects and their influence on complex figures of merit such as the Integral Time-multiplied Absolute Error (ITAE) has led to the use of a experimental approach for the PI tuning. For example, in Shaija and Daniel (2023) metaheuristic optimisation algorithms are used for the optimal PI tuning. Again, MPC Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives ⋆ Manuel G. Satu´e ∗J.D. ´ Alvarez ∗∗ Manuel A. Perales ∗∗∗ ∗Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]) ∗∗ Department of Computer Engineering, Automation and Robotics, CIESOL—ceiA3, University of Almer´ıa, 04120 Almer´ıa, Spain (e-mail: [email protected]) ∗∗∗ Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: mperale[email protected]s) Abstract: PI tuning for variable-speed multi-phase drives is a complex task. In this paper, a Pareto analysis is introduced to reveal not previously reported links between figures of merit. The drive used for the study includes a 5-phase induction motor supplied by a voltage source inverter. A finite state predictive method is used for the inner loop (current control). The outer loop (speed control) is governed by a PI. The analysis is done experimentally thus including all sorts of non-idealities not appearing in commonly found models. The experimental results show how the pursuit for better performance is hindered by the existence of links between figures of merit. The importance of the result lies in showing that arbitrary performance enhancements are not possible in a general case. Keywords: Application of power electronics, Digital implementation, Modeling and simulation of power systems, PI control, Tuning 1. INTRODUCTION PI controllers are routinely used in variable speed drives for speed and/or current regulation. The speed control loop (outer loop) is used to maintain the mechanical speed close to its reference value. The current loop has a much faster time scale and is used to produce stator currents that, in turn, produce a torque that drives the speed. In the last years, there has been a trend in academia to replace PI loops with more sophisticated alternatives such as Model Predictive Control (MPC). This trend is a consequence of the relative success attained by MPC in other realms (Berenguel et al., 1998; Ram´ırez-Arias et al., 2005). The case of predictive control of drives is exemplified by the so called Finite State Model Predictive Control (FSMPC), where direct digital control of the Voltage Source Inverter (VSI) is attained. In FSMPC the Pulse Width Modulation (PWM) stage is no longer necessary, thus improving the performance of the system as reported by Lim et al. (2013); Berm´udez et al. (2020). Going back to the speed control loop, the influential works related to PI tuning for Indirect Field Oriented Control (IFOC) date from the 1990s and 2000s (De Wit et al., 1996; Espinosa-Perez and Ortega, 1997; Espinosa-Perez et al., 1998; Bazanella and Reginatto, 2001). These works pro- ⋆This work is part of the Grant TED2021-129558B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. vide methods which utilize some, not very strict, requirements on the knowledge of the time constant of the rotor of the IM to guarantee the global stability. A dynamic model of the current fed IM is used at the basis of these methods. Some variations have been reported making use of harmonic injection (Arahal and Duran, 2009), but, in most cases, the models do not consider practicalities like nonlinearities in the IM dynamics, VSI+modulation behavior and aspects derived from the use of a Digital Signal Processor (DSP) for the PI + IFOC algorithms. In the 20 years that followed, the state of the art has been enriched by some papers expanding the previous works in a variety of ways. Most of the efforts have focused on: sensorless operation, estimation of the rotor time constant, extension of the PI with operating-point based scheduling and/or adaptivity, fault-tolerant capabilities, and tuning methods based on extensive simulation/experimentation. Other technologies have even been proposed to replace the speed PI but, despite all of this, conventional PI control is still reported as competitive due to its simple implementation compared with nonlinear control (Ortega et al., 1993), adaptive control (Marino et al., 1993), robust control (Ding et al., 2000; Alamo et al., 2006) and some fuzzy and/or neural approaches (Masiala et al., 2008; Pham and Le, 2020). The lack of theoretical models to analyze the above cited practical aspects and their influence on complex figures of merit such as the Integral Time-multiplied Absolute Error (ITAE) has led to the use of a experimental approach for the PI tuning. For example, in Shaija and Daniel (2023) metaheuristic optimisation algorithms are used for the optimal PI tuning. Again, MPC Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives ⋆ Manuel G. Satu´e ∗J.D. ´ Alvarez ∗∗ Manuel A. Perales ∗∗∗ ∗Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]) ∗∗ Department of Computer Engineering, Automation and Robotics, CIESOL—ceiA3, University of Almer´ıa, 04120 Almer´ıa, Spain (e-mail: [email protected]) ∗∗∗ Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: mperale[email protected]s) Abstract: PI tuning for variable-speed multi-phase drives is a complex task. In this paper, a Pareto analysis is introduced to reveal not previously reported links between figures of merit. The drive used for the study includes a 5-phase induction motor supplied by a voltage source inverter. A finite state predictive method is used for the inner loop (current control). The outer loop (speed control) is governed by a PI. The analysis is done experimentally thus including all sorts of non-idealities not appearing in commonly found models. The experimental results show how the pursuit for better performance is hindered by the existence of links between figures of merit. The importance of the result lies in showing that arbitrary performance enhancements are not possible in a general case. Keywords: Application of power electronics, Digital implementation, Modeling and simulation of power systems, PI control, Tuning 1. INTRODUCTION PI controllers are routinely used in variable speed drives for speed and/or current regulation. The speed control loop (outer loop) is used to maintain the mechanical speed close to its reference value. The current loop has a much faster time scale and is used to produce stator currents that, in turn, produce a torque that drives the speed. In the last years, there has been a trend in academia to replace PI loops with more sophisticated alternatives such as Model Predictive Control (MPC). This trend is a consequence of the relative success attained by MPC in other realms (Berenguel et al., 1998; Ram´ırez-Arias et al., 2005). The case of predictive control of drives is exemplified by the so called Finite State Model Predictive Control (FSMPC), where direct digital control of the Voltage Source Inverter (VSI) is attained. In FSMPC the Pulse Width Modulation (PWM) stage is no longer necessary, thus improving the performance of the system as reported by Lim et al. (2013); Berm´udez et al. (2020). Going back to the speed control loop, the influential works related to PI tuning for Indirect Field Oriented Control (IFOC) date from the 1990s and 2000s (De Wit et al., 1996; Espinosa-Perez and Ortega, 1997; Espinosa-Perez et al., 1998; Bazanella and Reginatto, 2001). These works pro- ⋆This work is part of the Grant TED2021-129558B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. vide methods which utilize some, not very strict, requirements on the knowledge of the time constant of the rotor of the IM to guarantee the global stability. A dynamic model of the current fed IM is used at the basis of these methods. Some variations have been reported making use of harmonic injection (Arahal and Duran, 2009), but, in most cases, the models do not consider practicalities like nonlinearities in the IM dynamics, VSI+modulation behavior and aspects derived from the use of a Digital Signal Processor (DSP) for the PI + IFOC algorithms. In the 20 years that followed, the state of the art has been enriched by some papers expanding the previous works in a variety of ways. Most of the efforts have focused on: sensorless operation, estimation of the rotor time constant, extension of the PI with operating-point based scheduling and/or adaptivity, fault-tolerant capabilities, and tuning methods based on extensive simulation/experimentation. Other technologies have even been proposed to replace the speed PI but, despite all of this, conventional PI control is still reported as competitive due to its simple implementation compared with nonlinear control (Ortega et al., 1993), adaptive control (Marino et al., 1993), robust control (Ding et al., 2000; Alamo et al., 2006) and some fuzzy and/or neural approaches (Masiala et al., 2008; Pham and Le, 2020). The lack of theoretical models to analyze the above cited practical aspects and their influence on complex figures of merit such as the Integral Time-multiplied Absolute Error (ITAE) has led to the use of a experimental approach for the PI tuning. For example, in Shaija and Daniel (2023) metaheuristic optimisation algorithms are used for the optimal PI tuning. Again, MPC Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives ⋆ Manuel G. Satu´e ∗J.D. ´ Alvarez ∗∗ Manuel A. Perales ∗∗∗ ∗Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]) ∗∗ Department of Computer Engineering, Automation and Robotics, CIESOL—ceiA3, University of Almer´ıa, 04120 Almer´ıa, Spain (e-mail: [email protected]) ∗∗∗ Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: mperale[email protected]s) Abstract: PI tuning for variable-speed multi-phase drives is a complex task. In this paper, a Pareto analysis is introduced to reveal not previously reported links between figures of merit. The drive used for the study includes a 5-phase induction motor supplied by a voltage source inverter. A finite state predictive method is used for the inner loop (current control). The outer loop (speed control) is governed by a PI. The analysis is done experimentally thus including all sorts of non-idealities not appearing in commonly found models. The experimental results show how the pursuit for better performance is hindered by the existence of links between figures of merit. The importance of the result lies in showing that arbitrary performance enhancements are not possible in a general case. Keywords: Application of power electronics, Digital implementation, Modeling and simulation of power systems, PI control, Tuning 1. INTRODUCTION PI controllers are routinely used in variable speed drives for speed and/or current regulation. The speed control loop (outer loop) is used to maintain the mechanical speed close to its reference value. The current loop has a much faster time scale and is used to produce stator currents that, in turn, produce a torque that drives the speed. In the last years, there has been a trend in academia to replace PI loops with more sophisticated alternatives such as Model Predictive Control (MPC). This trend is a consequence of the relative success attained by MPC in other realms (Berenguel et al., 1998; Ram´ırez-Arias et al., 2005). The case of predictive control of drives is exemplified by the so called Finite State Model Predictive Control (FSMPC), where direct digital control of the Voltage Source Inverter (VSI) is attained. In FSMPC the Pulse Width Modulation (PWM) stage is no longer necessary, thus improving the performance of the system as reported by Lim et al. (2013); Berm´udez et al. (2020). Going back to the speed control loop, the influential works related to PI tuning for Indirect Field Oriented Control (IFOC) date from the 1990s and 2000s (De Wit et al., 1996; Espinosa-Perez and Ortega, 1997; Espinosa-Perez et al., 1998; Bazanella and Reginatto, 2001). These works pro- ⋆This work is part of the Grant TED2021-129558B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. vide methods which utilize some, not very strict, requirements on the knowledge of the time constant of the rotor of the IM to guarantee the global stability. A dynamic model of the current fed IM is used at the basis of these methods. Some variations have been reported making use of harmonic injection (Arahal and Duran, 2009), but, in most cases, the models do not consider practicalities like nonlinearities in the IM dynamics, VSI+modulation behavior and aspects derived from the use of a Digital Signal Processor (DSP) for the PI + IFOC algorithms. In the 20 years that followed, the state of the art has been enriched by some papers expanding the previous works in a variety of ways. Most of the efforts have focused on: sensorless operation, estimation of the rotor time constant, extension of the PI with operating-point based scheduling and/or adaptivity, fault-tolerant capabilities, and tuning methods based on extensive simulation/experimentation. Other technologies have even been proposed to replace the speed PI but, despite all of this, conventional PI control is still reported as competitive due to its simple implementation compared with nonlinear control (Ortega et al., 1993), adaptive control (Marino et al., 1993), robust control (Ding et al., 2000; Alamo et al., 2006) and some fuzzy and/or neural approaches (Masiala et al., 2008; Pham and Le, 2020). The lack of theoretical models to analyze the above cited practical aspects and their influence on complex figures of merit such as the Integral Time-multiplied Absolute Error (ITAE) has led to the use of a experimental approach for the PI tuning. For example, in Shaija and Daniel (2023) metaheuristic optimisation algorithms are used for the optimal PI tuning. Again, MPC Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives ⋆ Manuel G. Satu´e ∗J.D. ´ Alvarez ∗∗ Manuel A. Perales ∗∗∗ ∗ Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]) ∗∗ Department of Computer Engineering, Automation and Robotics, CIESOL—ceiA3, University of Almer´ıa, 04120 Almer´ıa, Spain (e-mail: [email protected]) ∗∗∗ Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: mper[email protected]) Abstract: PI tuning for variable-speed multi-phase drives is a complex task. In this paper, a Pareto analysis is introduced to reveal not previously reported links between figures of merit. The drive used for the study includes a 5-phase induction motor supplied by a voltage source inverter. A finite state predictive method is used for the inner loop (current control). The outer loop (speed control) is governed by a PI. The analysis is done experimentally thus including all sorts of non-idealities not appearing in commonly found models. The experimental results show how the pursuit for better performance is hindered by the existence of links between figures of merit. The importance of the result lies in showing that arbitrary performance enhancements are not possible in a general case. Keywords: Application of power electronics, Digital implementation, Modeling and simulation of power systems, PI control, Tuning 1. INTRODUCTION PI controllers are routinely used in variable speed drives for speed and/or current regulation. The speed control loop (outer loop) is used to maintain the mechanical speed close to its reference value. The current loop has a much faster time scale and is used to produce stator currents that, in turn, produce a torque that drives the speed. In the last years, there has been a trend in academia to replace PI loops with more sophisticated alternatives such as Model Predictive Control (MPC). This trend is a consequence of the relative success attained by MPC in other realms (Berenguel et al., 1998; Ram´ırez-Arias et al., 2005). The case of predictive control of drives is exemplified by the so called Finite State Model Predictive Control (FSMPC), where direct digital control of the Voltage Source Inverter (VSI) is attained. In FSMPC the Pulse Width Modulation (PWM) stage is no longer necessary, thus improving the performance of the system as reported by Lim et al. (2013); Berm´udez et al. (2020). Going back to the speed control loop, the influential works related to PI tuning for Indirect Field Oriented Control (IFOC) date from the 1990s and 2000s (De Wit et al., 1996; Espinosa-Perez and Ortega, 1997; Espinosa-Perez et al., 1998; Bazanella and Reginatto, 2001). These works pro- ⋆This work is part of the Grant TED2021-129558B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. vide methods which utilize some, not very strict, requirements on the knowledge of the time constant of the rotor of the IM to guarantee the global stability. A dynamic model of the current fed IM is used at the basis of these methods. Some variations have been reported making use of harmonic injection (Arahal and Duran, 2009), but, in most cases, the models do not consider practicalities like nonlinearities in the IM dynamics, VSI+modulation behavior and aspects derived from the use of a Digital Signal Processor (DSP) for the PI + IFOC algorithms. In the 20 years that followed, the state of the art has been enriched by some papers expanding the previous works in a variety of ways. Most of the efforts have focused on: sensorless operation, estimation of the rotor time constant, extension of the PI with operating-point based scheduling and/or adaptivity, fault-tolerant capabilities, and tuning methods based on extensive simulation/experimentation. Other technologies have even been proposed to replace the speed PI but, despite all of this, conventional PI control is still reported as competitive due to its simple implementation compared with nonlinear control (Ortega et al., 1993), adaptive control (Marino et al., 1993), robust control (Ding et al., 2000; Alamo et al., 2006) and some fuzzy and/or neural approaches (Masiala et al., 2008; Pham and Le, 2020). The lack of theoretical models to analyze the above cited practical aspects and their influence on complex figures of merit such as the Integral Time-multiplied Absolute Error (ITAE) has led to the use of a experimental approach for the PI tuning. For example, in Shaija and Daniel (2023) metaheuristic optimisation algorithms are used for the optimal PI tuning. Again, MPC Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives ⋆ Manuel G. Satu´e ∗ J.D. ´ Alvarez ∗∗ Manuel A. Perales ∗∗∗ ∗Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]) ∗∗ Department of Computer Engineering, Automation and Robotics, CIESOL—ceiA3, University of Almer´ıa, 04120 Almer´ıa, Spain (e-mail: [email protected]) ∗∗∗ Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: mperale[email protected]s) Abstract: PI tuning for variable-speed multi-phase drives is a complex task. In this paper, a Pareto analysis is introduced to reveal not previously reported links between figures of merit. The drive used for the study includes a 5-phase induction motor supplied by a voltage source inverter. A finite state predictive method is used for the inner loop (current control). The outer loop (speed control) is governed by a PI. The analysis is done experimentally thus including all sorts of non-idealities not appearing in commonly found models. The experimental results show how the pursuit for better performance is hindered by the existence of links between figures of merit. The importance of the result lies in showing that arbitrary performance enhancements are not possible in a general case. Keywords: Application of power electronics, Digital implementation, Modeling and simulation of power systems, PI control, Tuning 1. INTRODUCTION PI controllers are routinely used in variable speed drives for speed and/or current regulation. The speed control loop (outer loop) is used to maintain the mechanical speed close to its reference value. The current loop has a much faster time scale and is used to produce stator currents that, in turn, produce a torque that drives the speed. In the last years, there has been a trend in academia to replace PI loops with more sophisticated alternatives such as Model Predictive Control (MPC). This trend is a consequence of the relative success attained by MPC in other realms (Berenguel et al., 1998; Ram´ırez-Arias et al., 2005). The case of predictive control of drives is exemplified by the so called Finite State Model Predictive Control (FSMPC), where direct digital control of the Voltage Source Inverter (VSI) is attained. In FSMPC the Pulse Width Modulation (PWM) stage is no longer necessary, thus improving the performance of the system as reported by Lim et al. (2013); Berm´udez et al. (2020). Going back to the speed control loop, the influential works related to PI tuning for Indirect Field Oriented Control (IFOC) date from the 1990s and 2000s (De Wit et al., 1996; Espinosa-Perez and Ortega, 1997; Espinosa-Perez et al., 1998; Bazanella and Reginatto, 2001). These works pro- ⋆This work is part of the Grant TED2021-129558B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. vide methods which utilize some, not very strict, requirements on the knowledge of the time constant of the rotor of the IM to guarantee the global stability. A dynamic model of the current fed IM is used at the basis of these methods. Some variations have been reported making use of harmonic injection (Arahal and Duran, 2009), but, in most cases, the models do not consider practicalities like nonlinearities in the IM dynamics, VSI+modulation behavior and aspects derived from the use of a Digital Signal Processor (DSP) for the PI + IFOC algorithms. In the 20 years that followed, the state of the art has been enriched by some papers expanding the previous works in a variety of ways. Most of the efforts have focused on: sensorless operation, estimation of the rotor time constant, extension of the PI with operating-point based scheduling and/or adaptivity, fault-tolerant capabilities, and tuning methods based on extensive simulation/experimentation. Other technologies have even been proposed to replace the speed PI but, despite all of this, conventional PI control is still reported as competitive due to its simple implementation compared with nonlinear control (Ortega et al., 1993), adaptive control (Marino et al., 1993), robust control (Ding et al., 2000; Alamo et al., 2006) and some fuzzy and/or neural approaches (Masiala et al., 2008; Pham and Le, 2020). The lack of theoretical models to analyze the above cited practical aspects and their influence on complex figures of merit such as the Integral Time-multiplied Absolute Error (ITAE) has led to the use of a experimental approach for the PI tuning. For example, in Shaija and Daniel (2023) metaheuristic optimisation algorithms are used for the optimal PI tuning. Again, MPC Pareto Analysis of PI Tuning in Direct Digital Control of Multi-phase Drives ⋆ Manuel G. Satu´e ∗J.D. ´ Alvarez ∗∗ Manuel A. Perales ∗∗∗ ∗Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain (e-mail: [email protected]) ∗∗ Department of Computer Engineering, Automation and Robotics, CIESOL—ceiA3, University of Almer´ıa, 04120 Almer´ıa, Spain (e-mail: [email protected]) ∗∗∗ Electronic Engineering Department, University of Seville, 41092 Seville, Spain (e-mail: mperale[email protected]s) Abstract: PI tuning for variable-speed multi-phase drives is a complex task. In this paper, a Pareto analysis is introduced to reveal not previously reported links between figures of merit. The drive used for the study includes a 5-phase induction motor supplied by a voltage source inverter. A finite state predictive method is used for the inner loop (current control). The outer loop (speed control) is governed by a PI. The analysis is done experimentally thus including all sorts of non-idealities not appearing in commonly found models. The experimental results show how the pursuit for better performance is hindered by the existence of links between figures of merit. The importance of the result lies in showing that arbitrary performance enhancements are not possible in a general case. Keywords: Application of power electronics, Digital implementation, Modeling and simulation of power systems, PI control, Tuning 1. INTRODUCTION PI controllers are routinely used in variable speed drives for speed and/or current regulation. The speed control loop (outer loop) is used to maintain the mechanical speed close to its reference value. The current loop has a much faster time scale and is used to produce stator currents that, in turn, produce a torque that drives the speed. In the last years, there has been a trend in academia to replace PI loops with more sophisticated alternatives such as Model Predictive Control (MPC). This trend is a consequence of the relative success attained by MPC in other realms (Berenguel et al., 1998; Ram´ırez-Arias et al., 2005). The case of predictive control of drives is exemplified by the so called Finite State Model Predictive Control (FSMPC), where direct digital control of the Voltage Source Inverter (VSI) is attained. In FSMPC the Pulse Width Modulation (PWM) stage is no longer necessary, thus improving the performance of the system as reported by Lim et al. (2013); Berm´udez et al. (2020). Going back to the speed control loop, the influential works related to PI tuning for Indirect Field Oriented Control (IFOC) date from the 1990s and 2000s (De Wit et al., 1996; Espinosa-Perez and Ortega, 1997; Espinosa-Perez et al., 1998; Bazanella and Reginatto, 2001). These works pro- ⋆This work is part of the Grant TED2021-129558B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. vide methods which utilize some, not very strict, requirements on the knowledge of the time constant of the rotor of the IM to guarantee the global stability. A dynamic model of the current fed IM is used at the basis of these methods. Some variations have been reported making use of harmonic injection (Arahal and Duran, 2009), but, in most cases, the models do not consider practicalities like nonlinearities in the IM dynamics, VSI+modulation behavior and aspects derived from the use of a Digital Signal Processor (DSP) for the PI + IFOC algorithms. In the 20 years that followed, the state of the art has been enriched by some papers expanding the previous works in a variety of ways. Most of the efforts have focused on: sensorless operation, estimation of the rotor time constant, extension of the PI with operating-point based scheduling and/or adaptivity, fault-tolerant capabilities, and tuning methods based on extensive simulation/experimentation. Other technologies have even been proposed to replace the speed PI but, despite all of this, conventional PI control is still reported as competitive due to its simple implementation compared with nonlinear control (Ortega et al., 1993), adaptive control (Marino et al., 1993), robust control (Ding et al., 2000; Alamo et al., 2006) and some fuzzy and/or neural approaches (Masiala et al., 2008; Pham and Le, 2020). The lack of theoretical models to analyze the above cited practical aspects and their influence on complex figures of merit such as the Integral Time-multiplied Absolute Error (ITAE) has led to the use of a experimental approach for the PI tuning. For example, in Shaija and Daniel (2023) metaheuristic optimisation algorithms are used for the optimal PI tuning. Again, MPC Copyright © 2024 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) schemes are able to consider more complex models, being this another reason for its proposal (Arahal et al., 2008). Moreover, in recent years, there has been a technological advance in the drives themselves. The motors have moved from 3-phase IMs to multi-phase machines and from twolevel to multi-level VSIs (Rodr´ıguez et al., 2023). The multi-phase case is specially interesting for this study as the overall structure of the controller is IFOC-like. The interaction among phases makes the tuning of the inner loop PIs more complex. Multi-phase drives requires at least 4 PI controllers for the PWM method and just one predictive controller for FSMPC. In particular, the existence (in stator currents of multi-phase IM) of a torque producing plane (α−β) and various harmonic planes (xj− yjwith j≥1) makes the control design more challenging. For instance, modulation schemes for 6 and 5-phase drives have little to do with each other, yet predictive controllers have the same structure for both. So, FSMPC has rendered the problem more tractable but at the cost of higher computational needs (Gon¸calves et al., 2019; Mart´ın et al., 2016). Also, the FSMPC method is becoming popular for its flexibility. For instance, modulation variants that are impossible with PWM can be used with FSMPC (Satu´e et al., 2023; Zhang et al., 2023), the mechanical load characteristics can be considered with ease, and faulttolerance capabilities are better utilized (Yepes et al., 2022). State estimation methods such as Kalman filtering and observers can also be incorporated with ease in MPC approaches (Bermudez et al., 2020; Cecilia and Costa- Castell´o, 2020), expanding its potential with respect to the PI case. Speed PI tuning can, in theory, make use of an accurate model of the IM. However, some aspects have been resistant to modelling. This is the case of the delay introduced by the modulation technique used (PWM or its variants or even FSMPC). Saturation in some variables (currents, fluxes, etc.) have also not been considered in many cases (Toral et al., 2010). Finally, some figures of merit such as torque ripple are typically not considered in the automatic control community, yet they play a crucial role in applications. In this context there is not a set of equations (or other convenient procedure) to link figures of merit with controller parameters. The proposal of this paper consider a reduced set of indices to assess controllers for the inner loop of drives. These indices summarize the system’s behavior as other indices are related to the set. Then, all possible controller tunings are considered. The Pareto-optimal solutions are then show to pertain to a surface of lower dimension. As a result, any PI tuning must either be dominated (not Pareto-optimal) or lie in the surface. 2. IFOC STRUCTURE USED IN THE EXPERIMENTS In the indirect field-oriented control scheme flux and torque are independently regulated. The flux current set point i∗ dis set to magnetize the motor whereas quadrature current i∗ qis used to manipulate the produced torque. The PI in the velocity feedback loop is responsible for generating i∗ qto drive the mechanical speed control error to zero. i∗ q(t)=kp·e(t)+kit 0 e(τ)dτ (1) where e=ω∗−ωeis the velocity error or difference between the speed set point (ω∗) and the speed measurement (ωe). Once the set-points in d−qcoordinates are known, they are projected to the α−βspace using the Park transformation, obtaining a reference for stator current in α−βplane as I∗ α−β=Di∗ d,i ∗ q⊺, where matrix Dis given by D=cos (θa) sin (θa) −sin (θa) cos (θa)(2) The flux position θais estimated as θa=ωedt where ωe=ωsl +Pω, being Pis the number of pairs of poles of the induction machine, ωthe mechanical speed and ωsl =i∗ q i∗ d 1 ˆτr (3) where ˆτris an estimation of the rotor time constant τr=Lr/Rr. As a result, the set point for stator current tracking i∗(k) has an amplitude I∗=i∗2 d+i∗2 q. Finally, the α−βreferences can be expressed as i∗ α(t)=I∗sin ωet, i∗ β(t)=I∗cos ωet,i∗ x(t)=0,i∗ y(t) = 0, where i∗ xand i∗ yare currents related to losses in the harmonic plane x−y. 2.1 FSMPC Control of a 5-phase IM The inner loop of an IFOC scheme is responsible for producing stator currents in α−βthat follows their references and, at the same time, maintaining x−ycurrents close to zero as they do not produce torque but contribute to losses. This task can be accomplished using the FSMPC strategy. A model of the IM is needed to predict the IM behavior for each possible VSI configuration. Using the vector space decomposition, the 5-phase stator currents are projected to the energy conversion (α−β) plane and harmonic (x−y) plane, resulting in equations relating stator voltages vs(t), stator and rotor currents is(t), ir(t), fluxes Ψs(t), Ψr(t) and rotor electrical angular speed ωr(t) as follows. vαβs(t)=Rsiαβs(t)+pΨαβs(t) 0=Rriαβr(t)+pΨαβr(t)−jωr(t)Ψαβr(t) Ψαβs(t)=Lsiαβs(t)+Lmiαβr(t)=Lsiαβs(t)+Lmiαβr(t) Ψαβr(t)=Lmiαβs(t)+Lriαβr(t)=Lmiαβs(t)+Lriαβr(t) vxys(t)=Rsixys(t)+pΨxys(t) Ψxys(t)=Llsixys(t) (4) where pis the derivative operator, and the following machine parameters are used: resistances Rs,Rr, inductances Ls,Lr, leakage inductance Lls and mutual inductance Lm. Sub-index sstands for stator and rfor rotor. The stator voltages are produced by the VSI according to the gating signals Kjfor VSI legs j=1, ..., 5 accommodated in u=(K1, ..., K5)∈IB 5with IB = {0,1}. The resulting voltages vαβxys are vαβxys =(vαs,v βs,v xs,v ys)=VDCuTM (5) Manuel G. Satué et al. / IFAC PapersOnLine 58-7 (2024) 144–149 145 Copyright © 2024 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) schemes are able to consider more complex models, being this another reason for its proposal (Arahal et al., 2008). Moreover, in recent years, there has been a technological advance in the drives themselves. The motors have moved from 3-phase IMs to multi-phase machines and from twolevel to multi-level VSIs (Rodr´ıguez et al., 2023). The multi-phase case is specially interesting for this study as the overall structure of the controller is IFOC-like. The interaction among phases makes the tuning of the inner loop PIs more complex. Multi-phase drives requires at least 4 PI controllers for the PWM method and just one predictive controller for FSMPC. In particular, the existence (in stator currents of multi-phase IM) of a torque producing plane (α−β) and various harmonic planes (xj− yjwith j≥1) makes the control design more challenging. For instance, modulation schemes for 6 and 5-phase drives have little to do with each other, yet predictive controllers have the same structure for both. So, FSMPC has rendered the problem more tractable but at the cost of higher computational needs (Gon¸calves et al., 2019; Mart´ın et al., 2016). Also, the FSMPC method is becoming popular for its flexibility. For instance, modulation variants that are impossible with PWM can be used with FSMPC (Satu´e et al., 2023; Zhang et al., 2023), the mechanical load characteristics can be considered with ease, and faulttolerance capabilities are better utilized (Yepes et al., 2022). State estimation methods such as Kalman filtering and observers can also be incorporated with ease in MPC approaches (Bermudez et al., 2020; Cecilia and Costa- Castell´o, 2020), expanding its potential with respect to the PI case. Speed PI tuning can, in theory, make use of an accurate model of the IM. However, some aspects have been resistant to modelling. This is the case of the delay introduced by the modulation technique used (PWM or its variants or even FSMPC). Saturation in some variables (currents, fluxes, etc.) have also not been considered in many cases (Toral et al., 2010). Finally, some figures of merit such as torque ripple are typically not considered in the automatic control community, yet they play a crucial role in applications. In this context there is not a set of equations (or other convenient procedure) to link figures of merit with controller parameters. The proposal of this paper consider a reduced set of indices to assess controllers for the inner loop of drives. These indices summarize the system’s behavior as other indices are related to the set. Then, all possible controller tunings are considered. The Pareto-optimal solutions are then show to pertain to a surface of lower dimension. As a result, any PI tuning must either be dominated (not Pareto-optimal) or lie in the surface. 2. IFOC STRUCTURE USED IN THE EXPERIMENTS In the indirect field-oriented control scheme flux and torque are independently regulated. The flux current set point i∗ dis set to magnetize the motor whereas quadrature current i∗ qis used to manipulate the produced torque. The PI in the velocity feedback loop is responsible for generating i∗ qto drive the mechanical speed control error to zero. i∗ q(t)=kp·e(t)+kit 0 e(τ)dτ (1) where e=ω∗−ωeis the velocity error or difference between the speed set point (ω∗) and the speed measurement (ωe). Once the set-points in d−qcoordinates are known, they are projected to the α−βspace using the Park transformation, obtaining a reference for stator current in α−βplane as I∗ α−β=Di∗ d,i ∗ q⊺, where matrix Dis given by D=cos (θa) sin (θa) −sin (θa) cos (θa)(2) The flux position θais estimated as θa=ωedt where ωe=ωsl +Pω, being Pis the number of pairs of poles of the induction machine, ωthe mechanical speed and ωsl =i∗ q i∗ d 1 ˆτr (3) where ˆτris an estimation of the rotor time constant τr=Lr/Rr. As a result, the set point for stator current tracking i∗(k) has an amplitude I∗=i∗2 d+i∗2 q. Finally, the α−βreferences can be expressed as i∗ α(t)=I∗sin ωet, i∗ β(t)=I∗cos ωet,i∗ x(t)=0,i∗ y(t) = 0, where i∗ xand i∗ yare currents related to losses in the harmonic plane x−y. 2.1 FSMPC Control of a 5-phase IM The inner loop of an IFOC scheme is responsible for producing stator currents in α−βthat follows their references and, at the same time, maintaining x−ycurrents close to zero as they do not produce torque but contribute to losses. This task can be accomplished using the FSMPC strategy. A model of the IM is needed to predict the IM behavior for each possible VSI configuration. Using the vector space decomposition, the 5-phase stator currents are projected to the energy conversion (α−β) plane and harmonic (x−y) plane, resulting in equations relating stator voltages vs(t), stator and rotor currents is(t), ir(t), fluxes Ψs(t), Ψr(t) and rotor electrical angular speed ωr(t) as follows. vαβs(t)=Rsiαβs(t)+pΨαβs(t) 0=Rriαβr(t)+pΨαβr(t)−jωr(t)Ψαβr(t) Ψαβs(t)=Lsiαβs(t)+Lmiαβr(t)=Lsiαβs(t)+Lmiαβr(t) Ψαβr(t)=Lmiαβs(t)+Lriαβr(t)=Lmiαβs(t)+Lriαβr(t) vxys(t)=Rsixys(t)+pΨxys(t) Ψxys(t)=Llsixys(t) (4) where pis the derivative operator, and the following machine parameters are used: resistances Rs,Rr, inductances Ls,Lr, leakage inductance Lls and mutual inductance Lm. Sub-index sstands for stator and rfor rotor. The stator voltages are produced by the VSI according to the gating signals Kjfor VSI legs j=1, ..., 5 accommodated in u=(K1, ..., K5)∈IB 5with IB = {0,1}. The resulting voltages vαβxys are vαβxys =(vαs,v βs,v xs,v ys)=VDCuTM (5) 146 Manuel G. Satué et al. / IFAC PapersOnLine 58-7 (2024) 144–149 Fig. 1. Diagram and photographs of the laboratory setup used in the experiments. where VDC is the DC link voltage, Ta connectivity matrix and Ma coordinate transformation matrix accounting for the spatial distribution of machine windings. TM =     abbbb babbb bbabb bbbab bbbba      ·     dγ c 1γc 2γc 3γc 4 0γs 1γs 2γs 3γs 4 dγ c 2γc 4cϑγc 3 0γs 2γs 4γs 1γs 3 ccccc      (6) The coefficients used are: a=4/5, b=−1/5, c=−b,d= 2/5, γc h= cos hϑ,γs h= sin hϑ and ϑ=2π/5 for the fivephase IM. The actual values of the electrical parameters corresponding to the machine used in the experiments are given in Table 1. At discrete-time k, the control action u(k+1) is computed minimizing a cost function J. The cost function penalizes stator currents tracking error as J=∥i∗(k+2)−ˆ i(k+2)∥2, where ˆ i(k+ 2) are the predicted currents. Note the one sampling period delay due to computations. The developed mechanical torque is derived from the actual values of the direct and quadrature stator currents as T=PL2 m Llr +Lm id·iq.(7) 3. EXPERIMENTS The PI tuning is assessed in this section by means of tests in the experimental setup described below. 3.1 Laboratory Setup The laboratory arrangement depicted in Fig. 1) is used. A five-phase motor with parameters shown in table 1 is Table 1. Parameters of the five-phase IM Parameter Value Unit Stator resistance, Rs12.85 Ω Rotor resistance, Rr4.80 Ω Stator leakage inductance, Lls 79.93 mH Rotor leakage inductance, Llr 79.93 mH Mutual inductance, LM681.7 mH Rotational inertia, Jm0.02 kg m2 Number of pairs of poles, P3- powered by a five-phase VSI constructed from two threephase SEMIKRON SKS 22F modules. A 300V DC power supply is used in the DC-link. The control programs run on a MSK28335 board housing a TMS320F28335 DSP. Hall effect sensors (LH25-NP) are used to measure the stator phase currents. Last, a DC motor sharing the same axis as the IM is used to provide an opposing torque load (TL) for the tests. 3.2 Performance indices The assessment of variable speed drives can be based on a variety of quantitative criteria. Different applications might put more emphasis on some indices over others. In this paper three indices are selected. The proposed indices can be obtained experimentally from a step test. The first one is the percentage overshoot PO, defined for an underdamped system as PO = 100 ·max ω−ω∗ ω∗(8) The second index is the rise time (from 0% to 100%) defined as the time needed to cross the ω∗value for the first time. RT = argmin t≥0 ω∗(t)−ω(t) (9) The final index is the torque ripple, that is defined as QR =    1 N N  k=1 (T∗(k)−T(k))2(10) The set of all three indices will be represented as vector Γ=(PO, RT, QR)=(γ1,γ 2,γ 3) (11) The units of γ1,γ2and γ3are %, s and N ·m respectively. 3.3 Pareto Surface A set of step tests are performed on the 5-phase IM to obtain the figures of merit for various combinations of the PI parameters (kp,k i). The obtained triples Γ for each PI tuning are recorded. Then, a process of elimination is used to exclude the triples that are not Pareto-optimal. The surviving ones are presented in Fig. 2 where a 3D plot (upper right corner) shows the Γ values and three 2D projections show: γ2vs. γ1(upper left), γ3vs. γ1(lower left) and γ3vs. γ2(lower right). The points are presented with color for better visualization. The color has been Table 2. Optimal tunings for different operating points ω∗k0 p·103k0 i ·105γ0 1γ0 2γ0 3Π 500 65 100 3.3 0.17 0.0120 0.0148 375 42 107 5.3 0.17 0.0103 0.0114 250 120 80 7.1 0.11 0.0123 0.0123 125 130 60 12.9 0.09 0.0101 0.0224 linked to γ3to provide a sense of elevation. A red color indicates a high value of γ3and blue color indicate low value of γ3. It is interesting to see that the Γ triples are situated approximately on a cubic Titeica surface (Tzitz´eica, 1908). The surface is determined mathematically by γ1·γ2·γ3= Π (12) where Π=0.0148 for this case. The repercussions of this findings are deep: by PI tuning alone it is impossible to enhance all figures of merit simultaneously past the Pareto front. Since γ1·γ2·γ3is (approximately) a constant, reducing γ1will result in an increase in γ2·γ3. The same argument holds for a reduction in either γ2or γ3. Going back to the tuning problem, it is interesting to notice that a particular point in the Pareto front minimizes the distance to the origin. This particular solution (Γ0) achieves ∥Γ0∥2= min γ2 1+γ2 2+γ2 3(13) This solution is optimal if the control objectives (figures of merit) are equally important. This might no be the case in a practical application. A weighted metric should be used instead. This is easily achieved by applying scale factors to each figure of merit. For the sake of clarity the distance of equation (13) will be used. In this case the solution closest to the origin (denoted as Γ0) is found to be Γ0= (3.3,0.17,0.0120). If one chooses to tune the PI to achieve Γ0, the tuning problem has a definite answer. This is a step forward from various previous works where the existence of limits for tuning is not realized. 3.4 Influence of the operating regime So far, just an operating point has been analyzed. The method is now extended to other values of ω∗to check to what extent the Pareto surface and optimal tuning are affected. Fig. 3 presents the Pareto front and its projections for ω∗= 250 (rpm). It can be seen that the Pareto-optimal solutions still lie (approximately) in a Titeica surface. However, for this operating point, the figures of merit have different values. This can be seen comparing Fig. 2 with Fig. 3 and also in table 2. In said table, various speed references are considered. For each one the PI tuning that yields Γ0is provided along the actual values of γ0 iand the Π parameter. It can be seen that the optimal tuning is different for each operating point. This support the adoption of scheduling solutions such as in Arahal et al. (2019). In most cases, however, a single PI tuning is uses. The particular tuning can be found by trial and error aiming at Table 3. Comparison of results in different operating points ω∗Tuning γ0 1γ0 2γ0 3 500 T1 3.92 0.177 0.0281 500 T2 3.98 0.168 0.0283 375 T1 5.25 0.152 0.0290 375 T2 5.34 0.141 0.0295 250 T1 7.37 0.131 0.0289 250 T2 7.34 0.119 0.0293 125 T1 13.77 0.128 0.0277 125 T2 13.83 0.109 0.0284 improving performance near the nominal operating point. This of course has the effect of diminished performance far from said nominal point. This is illustrated by table 3 where two PI tunings are compared in terms of the figures of merit for each speed regime. Tuning T1 corresponds to the optimal tuning for 500 rpm (kp= 65 ·103,ki= 100 · 105), whereas T2 uses as PI parameters the mean value over the various speed regimes, producing (kp= 89 · 103,ki= 86 ·105). It can be seen that the results are not so different and the degradation with respect to the optimal case might be acceptable in many applications. This provides support for the usual engineering practice used for tuning. However, for high end applications, the results of this paper show that PI tuning has limits and that something else is needed to provide high performance in all operating points. This is a powerful result that can not be arrived at without the Pareto analysis. 4. CONCLUSION Assessment of controllers is presented in many works by means of a few cases. The results of this paper show that such procedure is insufficient because two important issues are kept in the dark. First, the trade-offs between figures of merit, and second the relationship between optimal tuning and operating regimes. Without a complete Pareto analysis covering all operating regimes the conclusions are necessarily baseless. ACKNOWLEDGEMENTS This work is part of the Grant TED2021-129558B- C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. Also, the authors want to thank the support provided by project I+D+i/PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/ by “ERDF A way of making Europe”. REFERENCES Alamo, T., Normey-Rico, J.E., Arahal, M.R., Limon, D., and Camacho, E.F. (2006). Introducing linear matrix inequalities in a control course. IFAC Proceedings Volumes, 39(6), 205–210. Arahal, M.R., Cirre, C.M., and Berenguel, M. (2008). Serial grey-box model of a stratified thermal tank for hierarchical control of a solar plant. Solar Energy, 82(5), 441–451. Arahal, M.R., Kowal, G., Barrero, F., and del Mar Castilla, M. (2019). Cost function optimization for multi-phase induction machines predictive control. Manuel G. Satué et al. / IFAC PapersOnLine 58-7 (2024) 144–149 147 Table 2. Optimal tunings for different operating points ω∗k0 p·103k0 i ·105γ0 1γ0 2γ0 3Π 500 65 100 3.3 0.17 0.0120 0.0148 375 42 107 5.3 0.17 0.0103 0.0114 250 120 80 7.1 0.11 0.0123 0.0123 125 130 60 12.9 0.09 0.0101 0.0224 linked to γ3to provide a sense of elevation. A red color indicates a high value of γ3and blue color indicate low value of γ3. It is interesting to see that the Γ triples are situated approximately on a cubic Titeica surface (Tzitz´eica, 1908). The surface is determined mathematically by γ1·γ2·γ3=Π (12) where Π=0.0148 for this case. The repercussions of this findings are deep: by PI tuning alone it is impossible to enhance all figures of merit simultaneously past the Pareto front. Since γ1·γ2·γ3is (approximately) a constant, reducing γ1will result in an increase in γ2·γ3. The same argument holds for a reduction in either γ2or γ3. Going back to the tuning problem, it is interesting to notice that a particular point in the Pareto front minimizes the distance to the origin. This particular solution (Γ0) achieves ∥Γ0∥2= min γ2 1+γ2 2+γ2 3(13) This solution is optimal if the control objectives (figures of merit) are equally important. This might no be the case in a practical application. A weighted metric should be used instead. This is easily achieved by applying scale factors to each figure of merit. For the sake of clarity the distance of equation (13) will be used. In this case the solution closest to the origin (denoted as Γ0) is found to be Γ0= (3.3,0.17,0.0120). If one chooses to tune the PI to achieve Γ0, the tuning problem has a definite answer. This is a step forward from various previous works where the existence of limits for tuning is not realized. 3.4 Influence of the operating regime So far, just an operating point has been analyzed. The method is now extended to other values of ω∗to check to what extent the Pareto surface and optimal tuning are affected. Fig. 3 presents the Pareto front and its projections for ω∗= 250 (rpm). It can be seen that the Pareto-optimal solutions still lie (approximately) in a Titeica surface. However, for this operating point, the figures of merit have different values. This can be seen comparing Fig. 2 with Fig. 3 and also in table 2. In said table, various speed references are considered. For each one the PI tuning that yields Γ0is provided along the actual values of γ0 iand the Π parameter. It can be seen that the optimal tuning is different for each operating point. This support the adoption of scheduling solutions such as in Arahal et al. (2019). In most cases, however, a single PI tuning is uses. The particular tuning can be found by trial and error aiming at Table 3. Comparison of results in different operating points ω∗Tuning γ0 1γ0 2γ0 3 500 T1 3.92 0.177 0.0281 500 T2 3.98 0.168 0.0283 375 T1 5.25 0.152 0.0290 375 T2 5.34 0.141 0.0295 250 T1 7.37 0.131 0.0289 250 T2 7.34 0.119 0.0293 125 T1 13.77 0.128 0.0277 125 T2 13.83 0.109 0.0284 improving performance near the nominal operating point. This of course has the effect of diminished performance far from said nominal point. This is illustrated by table 3 where two PI tunings are compared in terms of the figures of merit for each speed regime. Tuning T1 corresponds to the optimal tuning for 500 rpm (kp= 65 ·103,ki= 100 · 105), whereas T2 uses as PI parameters the mean value over the various speed regimes, producing (kp= 89 · 103,ki= 86 ·105). It can be seen that the results are not so different and the degradation with respect to the optimal case might be acceptable in many applications. This provides support for the usual engineering practice used for tuning. However, for high end applications, the results of this paper show that PI tuning has limits and that something else is needed to provide high performance in all operating points. This is a powerful result that can not be arrived at without the Pareto analysis. 4. CONCLUSION Assessment of controllers is presented in many works by means of a few cases. The results of this paper show that such procedure is insufficient because two important issues are kept in the dark. First, the trade-offs between figures of merit, and second the relationship between optimal tuning and operating regimes. Without a complete Pareto analysis covering all operating regimes the conclusions are necessarily baseless. ACKNOWLEDGEMENTS This work is part of the Grant TED2021-129558B- C22 funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. 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