R. Wisniewski, Christoffer Sloth, Manuela L. Bujorianu, Nir Piterman
We consider the safety problem of piecewise-deterministic Markov processes (PDMP). These are systems that have deterministic dynamics and stochastic jumps, where both the time and the destination of the jumps are stochastic. Specifically, we solve a p-safety problem, where we identify the set of initial states from which the probability to reach designated unsafe states is at most 1 - p. Based on the knowledge of the full generator of the PDMP, we are able to develop a system of partial differential equations describing the connection between unsafe and initial states. We then show that by using the moment method, we can translate the infinite-dimensional optimisation problem searching for the largest set of p-safe states to a finite dimensional polynomial optimisation problem. We have implemented this technique on top of GloptiPoly and show how to apply it to a numerical example.
{"title":"Safety Verification of Piecewise-Deterministic Markov Processes","authors":"R. Wisniewski, Christoffer Sloth, Manuela L. Bujorianu, Nir Piterman","doi":"10.1145/2883817.2883836","DOIUrl":"https://doi.org/10.1145/2883817.2883836","url":null,"abstract":"We consider the safety problem of piecewise-deterministic Markov processes (PDMP). These are systems that have deterministic dynamics and stochastic jumps, where both the time and the destination of the jumps are stochastic. Specifically, we solve a p-safety problem, where we identify the set of initial states from which the probability to reach designated unsafe states is at most 1 - p. Based on the knowledge of the full generator of the PDMP, we are able to develop a system of partial differential equations describing the connection between unsafe and initial states. We then show that by using the moment method, we can translate the infinite-dimensional optimisation problem searching for the largest set of p-safe states to a finite dimensional polynomial optimisation problem. We have implemented this technique on top of GloptiPoly and show how to apply it to a numerical example.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"48 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":"131018089","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}
Timothy E. Wang, P. Garoche, P. Roux, Romain Jobredeaux, E. Feron
Robustness analyses play a major role in the synthesis and analysis of controllers. For control systems, robustness is a measure of the maximum tolerable model inaccuracies or perturbations that do not destabilize the system. Analyzing the robustness of a closed-loop system can be performed with multiple approaches: gain and phase margin computation for single-input single-output (SISO) linear systems, mu analysis, IQC computations, etc. However, none of these techniques consider the actual code in their analyses. The approach presented here relies on an invariant computation on the discrete system dynamics. Using semi-definite programming (SDP) solvers, a Lyapunov-based function is synthesized that captures the vector margins of the closed-loop linear system considered. This numerical invariant expressed over the state variables of the system is compatible with code analysis and enables its validation on the code artifact. This automatic analysis extends verification techniques focused on controller implementation, addressing validation of robustness at model and code level. It has been implemented in a tool analyzing discrete SISO systems and generating over-approximations of phase and gain margins. The analysis will be integrated in our toolchain for Simulink and Lustre models autocoding and formal analysis.
{"title":"Formal Analysis of Robustness at Model and Code Level","authors":"Timothy E. Wang, P. Garoche, P. Roux, Romain Jobredeaux, E. Feron","doi":"10.1145/2883817.2883824","DOIUrl":"https://doi.org/10.1145/2883817.2883824","url":null,"abstract":"Robustness analyses play a major role in the synthesis and analysis of controllers. For control systems, robustness is a measure of the maximum tolerable model inaccuracies or perturbations that do not destabilize the system. Analyzing the robustness of a closed-loop system can be performed with multiple approaches: gain and phase margin computation for single-input single-output (SISO) linear systems, mu analysis, IQC computations, etc. However, none of these techniques consider the actual code in their analyses. The approach presented here relies on an invariant computation on the discrete system dynamics. Using semi-definite programming (SDP) solvers, a Lyapunov-based function is synthesized that captures the vector margins of the closed-loop linear system considered. This numerical invariant expressed over the state variables of the system is compatible with code analysis and enables its validation on the code artifact. This automatic analysis extends verification techniques focused on controller implementation, addressing validation of robustness at model and code level. It has been implemented in a tool analyzing discrete SISO systems and generating over-approximations of phase and gain margins. The analysis will be integrated in our toolchain for Simulink and Lustre models autocoding and formal analysis.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"2 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114018663","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}
Timing contracts for embedded controller implementation specify the constraints on the time instants at which certain operations are performed such as sampling, actuation, computation, etc. In this paper, we consider the problem of verifying the stability of embedded control systems under such timing contracts. Reformulating the problem in the framework of impulsive linear systems, we provide theoretical conditions for stability and a verification algorithm based on reachability analysis. In the second part of the paper, given a model of the plant and of the controller we propose an approach to synthesize timing contracts that guarantee stability.
{"title":"Verification and Synthesis of Timing Contracts for Embedded Controllers","authors":"M. A. Khatib, A. Girard, T. Dang","doi":"10.1145/2883817.2883827","DOIUrl":"https://doi.org/10.1145/2883817.2883827","url":null,"abstract":"Timing contracts for embedded controller implementation specify the constraints on the time instants at which certain operations are performed such as sampling, actuation, computation, etc. In this paper, we consider the problem of verifying the stability of embedded control systems under such timing contracts. Reformulating the problem in the framework of impulsive linear systems, we provide theoretical conditions for stability and a verification algorithm based on reachability analysis. In the second part of the paper, given a model of the plant and of the controller we propose an approach to synthesize timing contracts that guarantee stability.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"18 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":"126000183","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: Safety and Stability Analysis","authors":"T. Johnson","doi":"10.1145/3261111","DOIUrl":"https://doi.org/10.1145/3261111","url":null,"abstract":"","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"276 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":"126055811","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 hybrid event-based control (HEBC) systems the controller influences a continuous plant $G$ through two input signals with different characteristics. A continuous input is used to attenuate disturbances and to force the plant to follow a reference signal, whereas a discrete-valued input is determined by an event-based component of the controller in order to adjust the operation point of the plant. HEBC systems have typical characteristics of hybrid dynamical systems including state jumps and switching dynamics. This paper analyses HEBC systems with linear components. It derives bounds on the event threshold in order to avoid Zeno behaviour and to guarantee a minimum inter-event time. The main result is a condition under which the closed-loop system is asymptotically stable and has an asymptotic set-point tracking behaviour. An application example illustrates the results.
{"title":"Event-Separation Properties and Asymptotic Behaviour of Hybrid Event-Based Control Systems","authors":"Tobias Noesselt, M. Schultalbers, J. Lunze","doi":"10.1145/2883817.2883835","DOIUrl":"https://doi.org/10.1145/2883817.2883835","url":null,"abstract":"In hybrid event-based control (HEBC) systems the controller influences a continuous plant $G$ through two input signals with different characteristics. A continuous input is used to attenuate disturbances and to force the plant to follow a reference signal, whereas a discrete-valued input is determined by an event-based component of the controller in order to adjust the operation point of the plant. HEBC systems have typical characteristics of hybrid dynamical systems including state jumps and switching dynamics. This paper analyses HEBC systems with linear components. It derives bounds on the event threshold in order to avoid Zeno behaviour and to guarantee a minimum inter-event time. The main result is a condition under which the closed-loop system is asymptotically stable and has an asymptotic set-point tracking behaviour. An application example illustrates the results.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"49 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":"127586738","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 propose tools for the study of robust stabilizability and the design of robustly stabilizing feedback laws for a wide class of hybrid systems given in terms of hybrid inclusions with inputs and disturbances. We introduce notions of robust uniform global stabilizability and stabilization that capture the case when disturbances can be fully rejected, practically rejected, and when they induce a residual set that can be stabilized. Robust control Lyapunov functions are em- ployed to determine when stabilizing static state-feedback laws are available and also to synthesize robustly stabilizing feedback laws with minimum pointwise norm. Sufficient conditions on the data of the hybrid system as well as on the control Lyapunov function are proposed for the said properties to hold. An example illustrates the results throughout the paper.
{"title":"Robust Asymptotic Stabilization of Hybrid Systems using Control Lyapunov Functions","authors":"R. Sanfelice","doi":"10.1145/2883817.2883848","DOIUrl":"https://doi.org/10.1145/2883817.2883848","url":null,"abstract":"We propose tools for the study of robust stabilizability and the design of robustly stabilizing feedback laws for a wide class of hybrid systems given in terms of hybrid inclusions with inputs and disturbances. We introduce notions of robust uniform global stabilizability and stabilization that capture the case when disturbances can be fully rejected, practically rejected, and when they induce a residual set that can be stabilized. Robust control Lyapunov functions are em- ployed to determine when stabilizing static state-feedback laws are available and also to synthesize robustly stabilizing feedback laws with minimum pointwise norm. Sufficient conditions on the data of the hybrid system as well as on the control Lyapunov function are proposed for the said properties to hold. An example illustrates the results throughout the paper.","PeriodicalId":337926,"journal":{"name":"Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control","volume":"99 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":"114237205","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: Temporal Logic Applications","authors":"S. Mitra","doi":"10.1145/3261108","DOIUrl":"https://doi.org/10.1145/3261108","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":"131875280","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: Invited Speaker","authors":"A. Abate","doi":"10.1145/3261115","DOIUrl":"https://doi.org/10.1145/3261115","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":"125191097","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 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}
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