We characterize and compute the maximal admissible positively invariant set for asymptotically stable constrained switching linear systems. Motivated by practical problems found, e.g., in obstacle avoidance, power electronics and nonlinear switching systems, in our setting the constraint set is formed by a finite number of polynomial inequalities. First, we observe that the so-called Veronese lifting allows to represent the constraint set as a polyhedral set. Next, by exploiting the fact that the lifted system dynamics remains linear, we establish a method based on reachability computations to characterize and compute the maximal admissible invariant set, which coincides with the domain of attraction when the system is asymptotically stable. After developing the necessary theoretical background, we propose algorithmic procedures for its exact computation, based on linear or semidefinite programs. The approach is illustrated in several numerical examples.
{"title":"Computing the Domain of Attraction of Switching Systems Subject to Non-Convex Constraints","authors":"N. Athanasopoulos, R. Jungers","doi":"10.1145/2883817.2883823","DOIUrl":"https://doi.org/10.1145/2883817.2883823","url":null,"abstract":"We characterize and compute the maximal admissible positively invariant set for asymptotically stable constrained switching linear systems. Motivated by practical problems found, e.g., in obstacle avoidance, power electronics and nonlinear switching systems, in our setting the constraint set is formed by a finite number of polynomial inequalities. First, we observe that the so-called Veronese lifting allows to represent the constraint set as a polyhedral set. Next, by exploiting the fact that the lifted system dynamics remains linear, we establish a method based on reachability computations to characterize and compute the maximal admissible invariant set, which coincides with the domain of attraction when the system is asymptotically stable. After developing the necessary theoretical background, we propose algorithmic procedures for its exact computation, based on linear or semidefinite programs. The approach is illustrated in several numerical examples.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130229787","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}
In this paper, we design a supervisor to prevent vehicle collisions at intersections. An intersection is modeled as an area containing multiple conflict points where vehicle paths cross in the future. At every time step, the supervisor determines whether there will be more than one vehicle in the vicinity of a conflict point at the same time. If there is, then an impending collision is detected, and the supervisor overrides the drivers to avoid collision. A major challenge in the design of a supervisor as opposed to an autonomous vehicle controller is to verify whether future collisions will occur based on the current drivers choices. This verification problem is particularly hard due to the large number of vehicles often involved in intersection collision, to the multitude of conflict points, and to the vehicles dynamics. In order to solve the verification problem, we translate the problem to a job-shop scheduling problem that yields equivalent answers. The job-shop scheduling problem can, in turn, be transformed into a mixed-integer linear program when the vehicle dynamics are first-order dynamics, and can thus be solved by using a commercial solver.
{"title":"Semi-autonomous Intersection Collision Avoidance through Job-shop Scheduling","authors":"Heejin Ahn, D. Vecchio","doi":"10.1145/2883817.2883830","DOIUrl":"https://doi.org/10.1145/2883817.2883830","url":null,"abstract":"In this paper, we design a supervisor to prevent vehicle collisions at intersections. An intersection is modeled as an area containing multiple conflict points where vehicle paths cross in the future. At every time step, the supervisor determines whether there will be more than one vehicle in the vicinity of a conflict point at the same time. If there is, then an impending collision is detected, and the supervisor overrides the drivers to avoid collision. A major challenge in the design of a supervisor as opposed to an autonomous vehicle controller is to verify whether future collisions will occur based on the current drivers choices. This verification problem is particularly hard due to the large number of vehicles often involved in intersection collision, to the multitude of conflict points, and to the vehicles dynamics. In order to solve the verification problem, we translate the problem to a job-shop scheduling problem that yields equivalent answers. The job-shop scheduling problem can, in turn, be transformed into a mixed-integer linear program when the vehicle dynamics are first-order dynamics, and can thus be solved by using a commercial solver.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116687799","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}
{"title":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","authors":"","doi":"10.1145/2883817","DOIUrl":"https://doi.org/10.1145/2883817","url":null,"abstract":"","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122541309","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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