Abstract A nonsmooth optimization control (NOC) based on a sandwich model with hysteresis is proposed to control a micropositioning system (MPS) with a piezoelectric actuator (PEA). In this control scheme, the hysteresis phenomenon inherent in the PEA is described by a Duhem submodel embedded between two linear dynamic submodels that describe the behavior of the drive amplifier and the flexible hinge with load, respectively, thus constituting a sandwich model with hysteresis. Based on this model, a nonsmooth predictor for sandwich systems with hysteresis is constructed. To avoid the complicated online search for the optimal value of the generalized gradient at a nonsmooth point, the method of the so-called weighted estimation of generalized gradient is proposed. In order to compensate for the model error caused by model uncertainty, a model error compensator (MEC) is integrated into the online optimization control strategy. Afterwards, the stability of the control system is analyzed based on Lyapunov’s theory. Finally, the proposed NOC-MEC method is verified on an MPS with a PEA, and the corresponding experimental results are presented.
{"title":"Nonsmooth Optimization Control Based on a Sandwich Model with Hysteresis for Piezo–Positioning Systems","authors":"Sen Yang, Yonghong Tan, Ruili Dong, Qingyuan Tan","doi":"10.34768/amcs-2023-0033","DOIUrl":"https://doi.org/10.34768/amcs-2023-0033","url":null,"abstract":"Abstract A nonsmooth optimization control (NOC) based on a sandwich model with hysteresis is proposed to control a micropositioning system (MPS) with a piezoelectric actuator (PEA). In this control scheme, the hysteresis phenomenon inherent in the PEA is described by a Duhem submodel embedded between two linear dynamic submodels that describe the behavior of the drive amplifier and the flexible hinge with load, respectively, thus constituting a sandwich model with hysteresis. Based on this model, a nonsmooth predictor for sandwich systems with hysteresis is constructed. To avoid the complicated online search for the optimal value of the generalized gradient at a nonsmooth point, the method of the so-called weighted estimation of generalized gradient is proposed. In order to compensate for the model error caused by model uncertainty, a model error compensator (MEC) is integrated into the online optimization control strategy. Afterwards, the stability of the control system is analyzed based on Lyapunov’s theory. Finally, the proposed NOC-MEC method is verified on an MPS with a PEA, and the corresponding experimental results are presented.","PeriodicalId":502322,"journal":{"name":"International Journal of Applied Mathematics and Computer Science","volume":"22 1","pages":"449 - 461"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract An event-triggered adaptive control algorithm is proposed for cooperative tracking control of high-order nonlinear multi-agent systems (MASs) with prescribed performance and full-state constraints. The algorithm combines dynamic surface technology and the backstepping recursive design method, with radial basis function neural networks (RBFNNs) used to approximate the unknown nonlinearity. The barrier Lyapunov function and finite-time stability theory are employed to prove that all agent states are semi-globally uniform and ultimately bounded, with the tracking error converging to a bounded neighborhood of zero in a finite time. Numerical simulations are provided to demonstrate the effectiveness of the proposed control scheme.
{"title":"Event–Triggered Cooperative Control for High–Order Nonlinear Multi–Agent Systems with Finite–Time Consensus","authors":"Shiyin Gong, Meirong Zheng, Jing Hu, Anguo Zhang","doi":"10.34768/amcs-2023-0032","DOIUrl":"https://doi.org/10.34768/amcs-2023-0032","url":null,"abstract":"Abstract An event-triggered adaptive control algorithm is proposed for cooperative tracking control of high-order nonlinear multi-agent systems (MASs) with prescribed performance and full-state constraints. The algorithm combines dynamic surface technology and the backstepping recursive design method, with radial basis function neural networks (RBFNNs) used to approximate the unknown nonlinearity. The barrier Lyapunov function and finite-time stability theory are employed to prove that all agent states are semi-globally uniform and ultimately bounded, with the tracking error converging to a bounded neighborhood of zero in a finite time. Numerical simulations are provided to demonstrate the effectiveness of the proposed control scheme.","PeriodicalId":502322,"journal":{"name":"International Journal of Applied Mathematics and Computer Science","volume":"100 1","pages":"439 - 448"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139346937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals [0,T ] have a turnpike structure in the following sense: For large T the contribution to the objective functional that comes from the subinterval [T/2,T ], i.e., from the second half of the time interval [0,T ], is at most of the order 1/T . More generally, the result holds for subintervals of the form [rT,T ], where r ∈ (0, 1/2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in T with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.
{"title":"Optimal Control Problems without Terminal Constraints: The Turnpike Property with Interior Decay","authors":"M. Gugat, Martin Lazar","doi":"10.34768/amcs-2023-0031","DOIUrl":"https://doi.org/10.34768/amcs-2023-0031","url":null,"abstract":"Abstract We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals [0,T ] have a turnpike structure in the following sense: For large T the contribution to the objective functional that comes from the subinterval [T/2,T ], i.e., from the second half of the time interval [0,T ], is at most of the order 1/T . More generally, the result holds for subintervals of the form [rT,T ], where r ∈ (0, 1/2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in T with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.","PeriodicalId":502322,"journal":{"name":"International Journal of Applied Mathematics and Computer Science","volume":"29 1","pages":"429 - 438"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139343872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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