The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. Many algorithms exist for estimating the JSR but not much is known about how to generate an infinite sequence of matrices with an optimal asymptotic growth rate. To the best of our knowledge, the currently known algorithms select a small sequence with large spectral radius using brute force (or branch-and-bound variants) and repeats this sequence infinitely. In this paper we introduce a new approach to this question, using the dual solution of a sum of squares optimization program for JSR approximation. Our algorithm produces an infinite sequence of matrices with an asymptotic growth rate arbitrarily close to the JSR. The algorithm naturally extends to the case where the allowable switching sequences are determined by a graph or finite automaton. Unlike the brute force approach, we provide a guarantee on the closeness of the asymptotic growth rate to the JSR. This, in turn, provides new bounds on the quality of the JSR approximation. We provide numerical examples illustrating the good performance of the algorithm.
{"title":"Generating Unstable Trajectories for Switched Systems via Dual Sum-Of-Squares Techniques","authors":"B. Legat, R. Jungers, P. Parrilo","doi":"10.1145/2883817.2883821","DOIUrl":"https://doi.org/10.1145/2883817.2883821","url":null,"abstract":"The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. Many algorithms exist for estimating the JSR but not much is known about how to generate an infinite sequence of matrices with an optimal asymptotic growth rate. To the best of our knowledge, the currently known algorithms select a small sequence with large spectral radius using brute force (or branch-and-bound variants) and repeats this sequence infinitely. In this paper we introduce a new approach to this question, using the dual solution of a sum of squares optimization program for JSR approximation. Our algorithm produces an infinite sequence of matrices with an asymptotic growth rate arbitrarily close to the JSR. The algorithm naturally extends to the case where the allowable switching sequences are determined by a graph or finite automaton. Unlike the brute force approach, we provide a guarantee on the closeness of the asymptotic growth rate to the JSR. This, in turn, provides new bounds on the quality of the JSR approximation. We provide numerical examples illustrating the good performance of the algorithm.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126643094","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}
Stanley Bak, Sergiy Bogomolov, T. Henzinger, Taylor T. Johnson, P. Prakash
Hybridization methods enable the analysis of hybrid automata with complex, nonlinear dynamics through a sound abstraction process. Complex dynamics are converted to simpler ones with added noise, and then analysis is done using a reachability method for the simpler dynamics. Several such recent approaches advocate that only 'dynamic' hybridization techniques---i.e., those where the dynamics are abstracted on-the-fly during a reachability computation---are effective. In this paper, we demonstrate this is not the case, and create static hybridization methods that are more scalable than earlier approaches. The main insight in our approach is that quick, numeric simulations can be used to guide the process, eliminating the need for an exponential number of hybridization domains. Transitions between domains are generally time-triggered, avoiding accumulated error from geometric intersections. We enhance our static technique by combining time-triggered transitions with occasional space-triggered transitions, and demonstrate the benefits of the combined approach in what we call mixed-triggered hybridization. Finally, error modes are inserted to confirm that the reachable states stay within the hybridized regions. The developed techniques can scale to higher dimensions than previous static approaches, while enabling the parallelization of the main performance bottleneck for many dynamic hybridization approaches: the nonlinear optimization required for sound dynamics abstraction. We implement our method as a model transformation pass in the HYST tool, and perform reachability analysis and evaluation using an unmodified version of SpaceEx on nonlinear models with up to six dimensions.
{"title":"Scalable Static Hybridization Methods for Analysis of Nonlinear Systems","authors":"Stanley Bak, Sergiy Bogomolov, T. Henzinger, Taylor T. Johnson, P. Prakash","doi":"10.1145/2883817.2883837","DOIUrl":"https://doi.org/10.1145/2883817.2883837","url":null,"abstract":"Hybridization methods enable the analysis of hybrid automata with complex, nonlinear dynamics through a sound abstraction process. Complex dynamics are converted to simpler ones with added noise, and then analysis is done using a reachability method for the simpler dynamics. Several such recent approaches advocate that only 'dynamic' hybridization techniques---i.e., those where the dynamics are abstracted on-the-fly during a reachability computation---are effective. In this paper, we demonstrate this is not the case, and create static hybridization methods that are more scalable than earlier approaches. The main insight in our approach is that quick, numeric simulations can be used to guide the process, eliminating the need for an exponential number of hybridization domains. Transitions between domains are generally time-triggered, avoiding accumulated error from geometric intersections. We enhance our static technique by combining time-triggered transitions with occasional space-triggered transitions, and demonstrate the benefits of the combined approach in what we call mixed-triggered hybridization. Finally, error modes are inserted to confirm that the reachable states stay within the hybridized regions. The developed techniques can scale to higher dimensions than previous static approaches, while enabling the parallelization of the main performance bottleneck for many dynamic hybridization approaches: the nonlinear optimization required for sound dynamics abstraction. We implement our method as a model transformation pass in the HYST tool, and perform reachability analysis and evaluation using an unmodified version of SpaceEx on nonlinear models with up to six dimensions.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116489410","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}
The tool Matlab/Simulink is a numerical simulation environment that is widely used in industry for model-based design. Numerical simulation scales well and can be applied to systems with highly complex dynamics, but it is also inherently incomplete in the sense that critical events or behavior may be overlooked. The application of formal verification techniques to Simulink models could help to overcome this limitation. Set-based verification tools such as SpaceEx use as underlying formalism hybrid automata, which are semantically and structurally different from Simulink models. To address this issue, we are building the tool SL2SX for transforming a subset of the Simulink modeling language into a corresponding SpaceEx model. Our method is designed to preserve the syntactic aspects of a given Simulink diagram: the resulting SpaceEx model shows the same hierarchical structure and preserves the names of components and variables. Placeholders with the correct interface are provided for unsupported Simulink blocks, which can then be translated manually. We illustrate the tool SL2SX and the verification of the transformed models in SpaceEx on two examples provided by the Mathworks example library.
{"title":"SL2SX Translator: From Simulink to SpaceEx Models","authors":"Stefano Minopoli, Goran Frehse","doi":"10.1145/2883817.2883826","DOIUrl":"https://doi.org/10.1145/2883817.2883826","url":null,"abstract":"The tool Matlab/Simulink is a numerical simulation environment that is widely used in industry for model-based design. Numerical simulation scales well and can be applied to systems with highly complex dynamics, but it is also inherently incomplete in the sense that critical events or behavior may be overlooked. The application of formal verification techniques to Simulink models could help to overcome this limitation. Set-based verification tools such as SpaceEx use as underlying formalism hybrid automata, which are semantically and structurally different from Simulink models. To address this issue, we are building the tool SL2SX for transforming a subset of the Simulink modeling language into a corresponding SpaceEx model. Our method is designed to preserve the syntactic aspects of a given Simulink diagram: the resulting SpaceEx model shows the same hierarchical structure and preserves the names of components and variables. Placeholders with the correct interface are provided for unsupported Simulink blocks, which can then be translated manually. We illustrate the tool SL2SX and the verification of the transformed models in SpaceEx on two examples provided by the Mathworks example library.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"330 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132988906","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":"Session details: Methods for Reachability Analysis","authors":"T. Dang","doi":"10.1145/3261112","DOIUrl":"https://doi.org/10.1145/3261112","url":null,"abstract":"","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117299032","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":"Session details: Control Synthesis","authors":"Majid Zamani","doi":"10.1145/3261114","DOIUrl":"https://doi.org/10.1145/3261114","url":null,"abstract":"","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132777951","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}
We introduce SCOTS a software tool for the automatic controller synthesis for nonlinear control systems based on symbolic models, also known as discrete abstractions. The tool accepts a differential equation as the description of a nonlinear control system. It uses a Lipschitz type estimate on the right-hand-side of the differential equation together with a number of discretization parameters to compute a symbolic model that is related with the original control system via a feedback refinement relation. The tool supports the computation of minimal and maximal fixed points and thus natively provides algorithms to synthesize controllers with respect to invariance and reachability specifications. The atomic propositions, which are used to formulate the specifications, are allowed to be defined in terms of finite unions and intersections of polytopes as well as ellipsoids. While the main computations are done in C++, the tool contains a Matlab interface to simulate the closed loop system and to visualize the abstract state space together with the atomic propositions. We illustrate the performance of the tool with two examples from the literature. The tool and all conducted experiments are available at www.hcs.ei.tum.de.
{"title":"SCOTS: A Tool for the Synthesis of Symbolic Controllers","authors":"M. Rungger, Majid Zamani","doi":"10.1145/2883817.2883834","DOIUrl":"https://doi.org/10.1145/2883817.2883834","url":null,"abstract":"We introduce SCOTS a software tool for the automatic controller synthesis for nonlinear control systems based on symbolic models, also known as discrete abstractions. The tool accepts a differential equation as the description of a nonlinear control system. It uses a Lipschitz type estimate on the right-hand-side of the differential equation together with a number of discretization parameters to compute a symbolic model that is related with the original control system via a feedback refinement relation. The tool supports the computation of minimal and maximal fixed points and thus natively provides algorithms to synthesize controllers with respect to invariance and reachability specifications. The atomic propositions, which are used to formulate the specifications, are allowed to be defined in terms of finite unions and intersections of polytopes as well as ellipsoids. While the main computations are done in C++, the tool contains a Matlab interface to simulate the closed loop system and to visualize the abstract state space together with the atomic propositions. We illustrate the performance of the tool with two examples from the literature. The tool and all conducted experiments are available at www.hcs.ei.tum.de.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115425086","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":"Session details: Models with Uncertainty","authors":"A. Girard","doi":"10.1145/3261116","DOIUrl":"https://doi.org/10.1145/3261116","url":null,"abstract":"","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133197821","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}
We investigate quantifying the difference between two hybrid dynamical systems under noise and initial-state uncertainty. While the set of traces for these systems is infinite, it is possible to symbolically approximate trace sets using emph{reachpipes} that compute upper and lower bounds on the evolution of the reachable sets with time. We estimate distances between corresponding sets of trajectories of two systems in terms of distances between the reachpipes. In case of two individual traces, the Skorokhod distance has been proposed as a robust and efficient notion of distance which captures both value and timing distortions. In this paper, we extend the computation of the Skorokhod distance to reachpipes, and provide algorithms to compute upper and lower bounds on the distance between two sets of traces. Our algorithms use new geometric insights that are used to compute the worst-case and best-case distances between two polyhedral sets evolving with time.
{"title":"Computing Distances between Reach Flowpipes","authors":"R. Majumdar, Vinayak S. Prabhu","doi":"10.1145/2883817.2883850","DOIUrl":"https://doi.org/10.1145/2883817.2883850","url":null,"abstract":"We investigate quantifying the difference between two hybrid dynamical systems under noise and initial-state uncertainty. While the set of traces for these systems is infinite, it is possible to symbolically approximate trace sets using emph{reachpipes} that compute upper and lower bounds on the evolution of the reachable sets with time. We estimate distances between corresponding sets of trajectories of two systems in terms of distances between the reachpipes. In case of two individual traces, the Skorokhod distance has been proposed as a robust and efficient notion of distance which captures both value and timing distortions. In this paper, we extend the computation of the Skorokhod distance to reachpipes, and provide algorithms to compute upper and lower bounds on the distance between two sets of traces. Our algorithms use new geometric insights that are used to compute the worst-case and best-case distances between two polyhedral sets evolving with time.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124218962","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}
Shromona Ghosh, Dorsa Sadigh, P. Nuzzo, Vasumathi Raman, Alexandre Donzé, A. Sangiovanni-Vincentelli, S. Sastry, S. Seshia
We address the problem of diagnosing and repairing specifications for hybrid systems, formalized in signal temporal logic (STL). Our focus is on automatic synthesis of controllers from specifications using model predictive control. We build on recent approaches that reduce the controller synthesis problem to solving one or more mixed integer linear programs (MILPs), where infeasibility of an MILP usually indicates unrealizability of the controller synthesis problem. Given an infeasible STL synthesis problem, we present algorithms that provide feedback on the reasons for unrealizability, and suggestions for making it realizable. Our algorithms are sound and complete relative to the synthesis algorithm, i.e., they provide a diagnosis that makes the synthesis problem infeasible, and always terminate with a non-trivial specification that is feasible using the chosen synthesis method, when such a solution exists. We demonstrate the effectiveness of our approach on controller synthesis for various cyber-physical systems, including an autonomous driving application and an aircraft electric power system.
{"title":"Diagnosis and Repair for Synthesis from Signal Temporal Logic Specifications","authors":"Shromona Ghosh, Dorsa Sadigh, P. Nuzzo, Vasumathi Raman, Alexandre Donzé, A. Sangiovanni-Vincentelli, S. Sastry, S. Seshia","doi":"10.1145/2883817.2883847","DOIUrl":"https://doi.org/10.1145/2883817.2883847","url":null,"abstract":"We address the problem of diagnosing and repairing specifications for hybrid systems, formalized in signal temporal logic (STL). Our focus is on automatic synthesis of controllers from specifications using model predictive control. We build on recent approaches that reduce the controller synthesis problem to solving one or more mixed integer linear programs (MILPs), where infeasibility of an MILP usually indicates unrealizability of the controller synthesis problem. Given an infeasible STL synthesis problem, we present algorithms that provide feedback on the reasons for unrealizability, and suggestions for making it realizable. Our algorithms are sound and complete relative to the synthesis algorithm, i.e., they provide a diagnosis that makes the synthesis problem infeasible, and always terminate with a non-trivial specification that is feasible using the chosen synthesis method, when such a solution exists. We demonstrate the effectiveness of our approach on controller synthesis for various cyber-physical systems, including an autonomous driving application and an aircraft electric power system.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116806059","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}
Alena Rodionova, E. Bartocci, D. Ničković, R. Grosu
We show that metric temporal logic (MTL) the extension of linear temporal logic to real time, can be viewed as linear time-invariant filtering, by interpreting addition, multiplication, and their neutral elements, over the idempotent dioid (max,min,0,1). Moreover, by interpreting these operators over the field of reals (+,x,0,1), one can associate various quantitative semantics to a metric-temporal-logic formula, depending on the filter's kernel used: square, rounded-square, Gaussian, low-pass, band-pass, or high-pass. This remarkable connection between filtering and metric temporal logic allows us to freely navigate between the two, and to regard signal-feature detection as logical inference. To the best of our knowledge, this connection has not been established before. We prove that our qualitative, filtering semantics is identical to the classical MTL semantics. We also provide a quantitative semantics for MTL, which measures the normalized, maximum number of times a formula is satisfied within its associated kernel, by a given signal. We show that this semantics is sound, in the sense that, if its measure is 0, then the formula is not satisfied, and it is satisfied otherwise. We have implemented both of our semantics in Matlab, and illustrate their properties on various formulas and signals, by plotting their computed measures.
{"title":"Temporal Logic as Filtering","authors":"Alena Rodionova, E. Bartocci, D. Ničković, R. Grosu","doi":"10.1145/2883817.2883839","DOIUrl":"https://doi.org/10.1145/2883817.2883839","url":null,"abstract":"We show that metric temporal logic (MTL) the extension of linear temporal logic to real time, can be viewed as linear time-invariant filtering, by interpreting addition, multiplication, and their neutral elements, over the idempotent dioid (max,min,0,1). Moreover, by interpreting these operators over the field of reals (+,x,0,1), one can associate various quantitative semantics to a metric-temporal-logic formula, depending on the filter's kernel used: square, rounded-square, Gaussian, low-pass, band-pass, or high-pass. This remarkable connection between filtering and metric temporal logic allows us to freely navigate between the two, and to regard signal-feature detection as logical inference. To the best of our knowledge, this connection has not been established before. We prove that our qualitative, filtering semantics is identical to the classical MTL semantics. We also provide a quantitative semantics for MTL, which measures the normalized, maximum number of times a formula is satisfied within its associated kernel, by a given signal. We show that this semantics is sound, in the sense that, if its measure is 0, then the formula is not satisfied, and it is satisfied otherwise. We have implemented both of our semantics in Matlab, and illustrate their properties on various formulas and signals, by plotting their computed measures.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"48 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113954414","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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