,

, Monotone transition systems

146 7.2.2 Discrete-time control system as a transition system ,

148 7.3.2 Controller synthesis for monotone transition systems and directed safety specifications ,

,

,

, C

, To prove the second inclusion, it is sufficient to show that C is a safety controller for the transition system T and the safety specification Q S . We have that dom(C) =? dom(C * ) ?? Q S = Q S , where the first equality comes from (i), the second inclusion comes from the fact that C * is a safety controller and the last equality comes from the lower closedness of Q S . Hence, the first condition of Definition B.1 is satisfied. Now let q ? dom(C) and v ? C(q). From construction of the controller C, we have the existence of q ? Q such that q ? Q q , q ? dom(C * ) and v ? C * (q ). Then, we have that ?(q, v) ?? ?(q , v) ?? dom(C * ) = dom(C), where the first inclusion comes from (ii), C, it follows immediately that ? dom(C * ) = dom(C). Let us prove that dom(C * ) = dom(C). Let q ? Q, since q ? (? q) we have that dom(C * ) ? dom(C)

,

, Let C * be the maximal safety controller enforcing the lower closed safety specification Q S ? Q. The following properties hold: (i) dom(C * ) is a lower closed set

, ii) for all q 1 , q 2 ? Q, if q 1 ? Q q 2 then C * (q 2 ) ? C *

, iii) for q ? Q, C * (q) is a lower closed set

, Then, ? dom(C * ) =? dom(C) = dom(C * ), where the last equality comes from (i) in Lemma 7.9. Hence, dom(C * ) is a lower closed set. (ii) Let q 1 , q 2 ? Q with q 1 ? Q q 2, i) We have from (ii) in Lemma 7.9 that dom(C * ) = dom(C)

, Hence, by maximality of C * , we have that v ? C * (q 1 ). Then, C * (q 2 ) ? C * (q 1 ). (iii) Let q ? Q, v ? C * (q) and v ?? v. We have that ?(q, v ) ?? ?(q, v) ?? dom(C * ) = dom(C * ), where the first inclusion comes from the monotonicity of the transition system T , the second inclusion comes from the fact that C * is a safety controller and the last equality comes from the lower closedeness of dom(C * ). Hence, we have ?(q, v ) ? dom(C * ). Then, by maximality of C * , v ? C * (x), Hence, we have that ?(q 1 , v) ?? ?(q 2 , v) ?? dom(C * ) = dom(C * ), where the first inclusion comes from the fact that T is a monotone transition system and the last equality comes from (i)

,

, = InvariantSet(T 1

, Cont(Q s ), vol.1

, 10 return C

, 5) we have that C(q) ? C * (q) for all q ? Cont(Q s ). Let Z i be defined as above, We have from (i) in Proposition 7

, N } such that q ? Z i?1 \ Z i = Z i?1 \ Z i , where the last equality comes from (ii) in Proposition 7.18. Hence, we have that v / ? V i . Then, v ? V i?1 . Moreover, C * is the maximal safety controller, then using the fact that ?(q, v) ? Cont(Q s )

, N } and for any (v i?1 , v i ) ? V i?1 × V i , we have v i ? v i?1 . The synthesis is more efficient. Indeed, for all q ? Z i?1 \ Z i , we have the existence of v ? V i?1 such that ?(q, v) ? Cont(Q s ). Hence, from (iii) in Proposition 7.10 and since V i?2 ?? v, we have that V i?2 ? C * (q). Then, at each step i ? {1, on the inputs ? V satisfies the following property: for all i ? {2

A subset A ? Q S is said to be a controlled invariant if for all q ? A there exists v ? V such that for any (q , y) ? ?(q, v), q ? A. It was shown in [Tab09] that there exists a maximal controlled invariant Cont(Q s ) which is the union of all controlled invariants. The maximal safety controller can be defined as follows: (i) for all q / ? Cont ,

, ii) for all q ? Cont(Q s ), C * (q) = {v ? enab(q) | for (q , y) ? ?(q, v), q ? Cont

, Let us remark that for any safety controller C we have that dom(C) ? Cont(Q s ), while for the maximal safety controller C * , we have dom(C * ) = Cont(Q s )

A survey on explicit model predictive control, Nonlinear model predictive control, pp.345-369, 2009. ,

Compositional refinement for hierarchical hybrid systems, pp.33-48, 2001. ,

Reactive modules. Formal methods in system design, vol.15, pp.7-48, 1999. ,

stability verification, scheduling, and synthesis of cyber-physical systems, 2017. ,

URL : https://hal.archives-ouvertes.fr/tel-01684705

Scheduling of embedded controllers under timing contracts, Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control, pp.131-140, 2017. ,

URL : https://hal.archives-ouvertes.fr/hal-01540841

Networks of dissipative systems: compositional certification of stability, performance, and safety, 2016. ,

A Lyapunov approach to incremental stability properties, IEEE Transactions on Automatic Control, vol.47, issue.3, pp.410-421, 2002. ,

Forward completeness, unboundedness observability, and their Lyapunov characterizations, Systems and Control Letters, vol.38, issue.4, pp.209-217, 1999. ,

Monotone control systems, IEEE Transactions on automatic control, vol.48, issue.10, pp.1684-1698, 2003. ,

To sample or not to sample: Self-triggered control for nonlinear systems, IEEE Transactions on Automatic Control, vol.55, issue.9, pp.2030-2042, 2010. ,

First steps toward formal controller synthesis for bipedal robots with experimental implementation, Nonlinear Analysis: Hybrid Systems, vol.25, pp.155-173, 2017. ,

Viability theory, 2009. ,

URL : https://hal.archives-ouvertes.fr/inria-00636570

Compositional abstraction of interconnected control systems under dynamic interconnection topology, IEEE Conference on Decision and Control (CDC), pp.3543-3550, 2017. ,

From dissipativity theory to compositional abstractions of interconnected stochastic hybrid systems, IEEE Transactions on Control of Network Systems, 2019. ,

, Transcendental number theory, 1990.

, Contracts for systems design: Theory

, , 2015.

Contracts for systems design: methodology and application cases ,

, , 2015.

Decentralized symbolic control of interconnected systems with application to vehicle platooning, IFAC Proceedings Volumes, vol.46, pp.285-292, 2013. ,

Controlling a class of nonlinear systems on rectangles, IEEE Transactions on Automatic Control, vol.51, issue.11, pp.1749-1759, 2006. ,

Synthesis of reactive (1) designs, Journal of Computer and System Sciences, vol.78, issue.3, pp.911-938, 2012. ,

, Contracts as specifications for dynamical systems in driving variable form, 2018.

Principles of model checking, 2008. ,

Set invariance in control, Automatica, vol.35, issue.11, pp.1747-1767, 1999. ,

Symbolic control design of nonlinear networked control systems, 2014. ,

Design of symbolic controllers for networked control systems, IEEE Transactions on Automatic Control, vol.64, issue.3, pp.1034-1046, 2019. ,

Optimal control of the boost DC-DC converter, IEEE Conference on Decision and Control (CDC), pp.4457-4462, 2005. ,

Formal methods for discrete-time dynamical systems, 2017. ,

A dissipativity approach to safety verification for interconnected systems, IEEE Transactions on Automatic Control, vol.60, issue.6, pp.1722-1727, 2015. ,

Finite abstraction of mixed monotone systems with discrete and continuous inputs, Nonlinear Analysis: Hybrid Systems, vol.23, pp.254-271, 2017. ,

Compositional set invariance in network systems with assume-guarantee contracts, 2018. ,

Safety controller synthesis for switched systems using multi-scale symbolic models, IEEE Conference on Decision and Control and European Control Conference, pp.520-525, 2011. ,

URL : https://hal.archives-ouvertes.fr/hal-00665226

Synthesis of switching controllers using approximately bisimilar multiscale abstractions, Proceedings of the 14th international conference on Hybrid systems: computation and control, pp.191-200, 2011. ,

Assume-guarantee synthesis, International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pp.261-275, 2007. ,

Introduction to discrete event systems, 2009. ,

Distributed synthesis and control of constrained linear systems, American Control Conference, pp.6017-6022, 2012. ,

Modeling cyber-physical systems, Proceedings of the IEEE, vol.100, issue.1, pp.13-28, 2011. ,

An iss small gain theorem for general networks, Mathematics of Control, Signals, and Systems, vol.19, issue.2, pp.93-122, 2007. ,

On compositional symbolic controller synthesis inspired by small-gain theorems, IEEE Conference on Decision and Control (CDC), pp.6133-6138, 2015. ,

Feedback systems: input-output properties, vol.55, 1975. ,

A quantitative approach on assumeguarantee contracts for safety of interconnected systems, European Control Conference, 2019. ,

URL : https://hal.archives-ouvertes.fr/hal-02148745

Compositional transient stability analysis of power systems via the computation of reachable sets, American Control Conference (ACC), pp.2536-2543, 2017. ,

Temporal logic motion planning for dynamic robots, Automatica, vol.45, issue.2, pp.343-352, 2009. ,

Assigning meanings to programs, Program Verification, pp.65-81, 1993. ,

Compositional verification of hybrid systems using simulation relations, 2005. ,

Well-structured transition systems everywhere! Theoretical Computer Science, vol.256, pp.63-92, 2001. ,

Control of switching systems by invariance analysis: applcation to power electronics, 2013. ,

Who is responsible when a self-driving car has an accident? furutism, 2018. ,

Finite bisimulations for switched linear systems, IEEE Transactions on Automatic Control, vol.59, issue.12, pp.3122-3134, 2014. ,

Safety controller synthesis for incrementally stable switched systems using multiscale symbolic models, IEEE Transactions on Automatic Control, vol.61, issue.6, pp.1537-1549, 2016. ,

URL : https://hal.archives-ouvertes.fr/hal-01197426

Controller synthesis for safety and reachability via approximate bisimulation, Automatica, vol.48, issue.5, pp.947-953, 2012. ,

URL : https://hal.archives-ouvertes.fr/hal-00765682

A composition theorem for bisimulation functions, 2013. ,

Approximately bisimilar abstractions of incrementally stable finite or infinite dimensional systems, IEEE Conference on Decision and Control (CDC), pp.824-829, 2014. ,

URL : https://hal.archives-ouvertes.fr/hal-01133212

Approximate simulation relations for hybrid systems. Discrete event dynamic systems, vol.18, pp.163-179, 2008. ,

URL : https://hal.archives-ouvertes.fr/hal-00307035

Feedback linearization of sampleddata systems, IEEE Transactions on Automatic Control, vol.33, issue.9, pp.857-859, 1988. ,

Sparsity-aware finite abstraction, IEEE Conference on Decision and Control (CDC), pp.2366-2371, 2017. ,

Approximation metrics for discrete and continuous systems, IEEE Transactions on Automatic Control, vol.52, issue.5, pp.782-798, 2007. ,

URL : https://hal.archives-ouvertes.fr/hal-00171714

Approximate bisimulation relations for constrained linear systems, Automatica, vol.43, issue.8, pp.1307-1317, 2007. ,

URL : https://hal.archives-ouvertes.fr/hal-00171420

Hierarchical control system design using approximate simulation, Automatica, vol.45, issue.2, pp.566-571, 2009. ,

URL : https://hal.archives-ouvertes.fr/hal-00379793

Approximate bisimulation: A bridge between computer science and control theory, European Journal of Control, vol.17, issue.5-6, pp.568-578, 2011. ,

URL : https://hal.archives-ouvertes.fr/hal-00765660

Approximately bisimilar symbolic models for incrementally stable switched systems, IEEE Transactions on Automatic Control, vol.55, issue.1, pp.116-126, 2010. ,

URL : https://hal.archives-ouvertes.fr/hal-00306990

, Hybrid Dynamical Systems: Modeling, Stability, and Robustness, 2012.

Abstracting partially feedback linearizable systems compositionally, IEEE Control Systems Letters, vol.1, issue.2, pp.227-232, 2017. ,

Ordering by divisibility in abstract algebras, vol.3, pp.326-336, 1952. ,

, Lazy abstraction-based control for reachability, 2018.

Lazy abstraction-based control for safety specifications, IEEE Conference on Decision and Control (CDC), pp.4902-4907, 2018. ,

Multi-layered abstraction-based controller synthesis for continuous-time systems, Proceedings of the 21st International Conference on Hybrid Systems: Computation and Control (part of CPS Week), pp.120-129, 2018. ,

An axiomatic basis for computer programming, Communications of the ACM, vol.12, issue.10, pp.576-580, 1969. ,

Plug and play distributed model predictive control for heavy duty vehicle platooning and interaction with passenger vehicles, IEEE Conference on Decision and Control (CDC), pp.2803-2809, 2018. ,

You assume, we guarantee: Methodology and case studies, International Conference on Computer Aided Verification, pp.440-451, 1998. ,

An assume-guarantee rule for checking simulation, International Conference on Formal Methods in Computer-Aided Design, pp.421-431, 1998. ,

Dimarogonas. A symbolic approach to the self-triggered design for networked control systems, IEEE Control Systems Letters, vol.3, issue.4, pp.1050-1055, 2019. ,

Lazy controller synthesis using three-valued abstractions for safety and reachability specifications, IEEE Conference on Decision and Control (CDC), pp.3567-3572, 2018. ,

Autonomous intelligent cruise control, IEEE Transactions on Vehicular Technology, vol.42, issue.4, pp.657-672, 1993. ,

Approximate equivalence and synchronization of metric transition systems, Systems & Control Letters, vol.58, issue.2, pp.94-101, 2009. ,

Small-gain theorem for iss systems and applications, Mathematics of Control, Signals and Systems, vol.7, issue.2, pp.95-120, 1994. ,

Compositional controller synthesis for vehicular traffic networks, IEEE Conference on Decision and Control (CDC), pp.6165-6171, 2015. ,

A small gain theorem for parametric assume-guarantee contracts, Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control, pp.207-216, 2017. ,

Symbolic control design for monotone systems with directed specifications, Automatica, vol.83, pp.10-19, 2017. ,

Constructing control system abstractions from modular components, Proceedings of the 21st International Conference on Hybrid Systems: Computation and Control (part of CPS Week), pp.137-146, 2018. ,

Symbolic models for incrementally stable singularly perturbed hybrid affine systems, American Control Conference, 2019. ,

URL : https://hal.archives-ouvertes.fr/hal-02043303

Stability verification and timing contract synthesis for linear impulsive systems using reachability analysis, Nonlinear Analysis: Hybrid Systems, vol.25, pp.211-226, 2017. ,

URL : https://hal.archives-ouvertes.fr/hal-01331737

Symbolic models for incrementally stable switched systems with aperiodic time sampling, vol.51, pp.253-258, 2018. ,

URL : https://hal.archives-ouvertes.fr/hal-01760789

Noninear systems, vol.2, pp.5-6, 1996. ,

, Bounding errors due to switching delays in incrementally stable switched systems, 6th IFAC Conference on Analysis and Design of Hybrid Systems ADHS, vol.51, pp.247-252, 2018.

Cyber-physical systems: A perspective at the centennial, Proceedings of the IEEE, vol.100, pp.1287-1308, 2012. ,

Dynamic contracts for distributed temporal logic control of traffic networks, IEEE Conference on Decision and Control (CDC), pp.3640-3645, 2017. ,

Assume-guarantee reasoning for linear dynamical systems, European Control Conference (ECC), pp.5015-5020, 2009. ,

Compositional analysis for linear control systems, Proceedings of the 13th ACM international conference on Hybrid systems: computation and control, pp.21-30 ,

Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Transactions on Automatic control, vol.54, issue.2, pp.308-322, 2009. ,

Mode sequences as symbolic states in abstractions of incrementally stable switched systems, IEEE Conference on Decision and Control (CDC), pp.3225-3230, 2013. ,

Distributed synthesis of state-dependent switching control, International Workshop on Reachability Problems, pp.119-133, 2016. ,

Reachability analysis of linear systems using support functions, Nonlinear Analysis: Hybrid Systems, vol.4, pp.250-262, 2010. ,

Switching in systems and control, ser. systems & control: Foundations & applications, 2003. ,

Approximating the pareto front of multi-criteria optimization problems, International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pp.69-83, 2010. ,

Computing finite abstractions with robustness margins via local reachable set over-approximation, vol.48, pp.1-6, 2015. ,

Introduction to embedded systems: A cyber-physical systems approach, 2016. ,

Compositional (in) finite abstractions for large-scale interconnected stochastic systems, 2018. ,

Reactive controllers for differentially flat systems with temporal logic constraints, IEEE Conference on Decision and Control (CDC), pp.7664-7670, 2012. ,

Compositional abstraction refinement for control synthesis, Nonlinear Analysis: Hybrid Systems, vol.27, pp.437-451, 2018. ,

URL : https://hal.archives-ouvertes.fr/hal-01672059

Pessoa: A tool for embedded controller synthesis, International Conference on Computer Aided Verification, pp.566-569, 2010. ,

Applying'design by contract', Computer, vol.25, issue.10, pp.40-51, 1992. ,

Safety control with performance guarantees of cooperative systems using compositional abstractions, Proceedings of the 5 th IFAC Conference on Analysis and Design of Hyrbid Systems, pp.317-322, 2015. ,

URL : https://hal.archives-ouvertes.fr/hal-01180975

Compositional abstraction and safety synthesis using overlapping symbolic models, IEEE Transactions on Automatic Control, vol.63, issue.6, pp.1835-1841, 2018. ,

URL : https://hal.archives-ouvertes.fr/hal-01620532

Communication and concurrency, vol.84, 1989. ,

An introduction to motion planning under multirate digital control, IEEE Conference on Decision and Control, pp.1780-1785, 1992. ,

Issues on nonlinear digital control, European Journal of Control, vol.7, issue.2-3, pp.160-177, 2001. ,

Smart buildings in the smart grid: Contract-based design of an integrated energy management system, Cyber Physical Systems Approach to Smart Electric Power Grid, pp.103-132, 2015. ,

Constrained model predictive control: Stability and optimality, Automatica, vol.36, issue.6, pp.789-814, 2000. ,

Compositional synthesis of finite state abstractions, IEEE Transactions on Automatic Control, vol.64, issue.6, pp.2629-2636, 2018. ,

Approximately bisimilar symbolic models for digital control systems, Computer Aided Verification, pp.362-377, 2012. ,

Correct-by-construction adaptive cruise control: Two approaches, IEEE Transactions on Control Systems Technology, vol.24, issue.4, pp.816-823, 2014. ,

Synthesis of separable controlled invariant sets for modular local control design, American Control Conference, pp.5656-5663, 2016. ,

Methodology for the design of analog integrated interfaces using contracts, IEEE Sensors Journal, vol.12, issue.12, pp.3329-3345, 2012. ,

Compositional construction of abstractions via relaxed small-gain conditions part ii: discrete case, European Control Conference (ECC), pp.1-4, 2018. ,

Compositional construction of abstractions via relaxed small-gain conditions part i: continuous case, European Control Conference (ECC), vol.2, pp.1-25, 2013. ,

Bisimilar linear systems, Automatica, vol.39, issue.12, pp.2035-2047, 2003. ,

Concurrency and automata on infinite sequences, Theoretical computer science, pp.167-183, 1981. ,

Control of cyberphysical-systems with logic specifications: A formal methods approach, Annual Reviews in Control, 2019. ,

Approximately bisimilar symbolic models for nonlinear control systems, Automatica, vol.44, issue.10, pp.2508-2516, 2008. ,

URL : https://hal.archives-ouvertes.fr/hal-00379755

The temporal logic of programs, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977), pp.46-57 ,

, IEEE, 1977.

Decentralized supervisory control of networks of nonlinear control systems, IEEE Transactions on Automatic Control, vol.63, issue.9, pp.2803-2817, 2018. ,

Symbolic models for networks of control systems, IEEE Transactions on Automatic Control, vol.61, issue.11, pp.3663-3668, 2016. ,

Symbolic models for timevarying time-delay systems via alternating approximate bisimulation, International Journal of Robust and Nonlinear Control, vol.25, issue.14, pp.2328-2347, 2015. ,

Symbolic models for nonlinear time-delay systems using approximate bisimulations, Systems & Control Letters, vol.59, issue.6, pp.365-373, 2010. ,

Symbolic models for nonlinear control systems: Alternating approximate bisimulations, SIAM Journal on Control and Optimization, vol.48, issue.2, pp.719-733, 2009. ,

A power consensus algorithm for dc microgrids, Automatica, vol.89, pp.364-375, 2018. ,

Computing abstractions of nonlinear systems, IEEE Transactions on Automatic Control, vol.56, issue.11, pp.2583-2598, 2011. ,

Plugand-play decentralized model predictive control for linear systems, IEEE Transactions on Automatic Control, vol.58, issue.10, pp.2608-2614, 2013. ,

Practical set invariance for decentralized discrete time systems, IEEE Conference on Decision and Control, pp.3283-3288, 2010. ,

Model predictive control: Theory and design, 2009. ,

Computing reachable sets for uncertain nonlinear monotone systems, Nonlinear Analysis: Hybrid Systems, vol.4, pp.263-278, 2010. ,

URL : https://hal.archives-ouvertes.fr/hal-00629909

Feedback refinement relations for the synthesis of symbolic controllers, IEEE Transactions on Automatic Control, vol.62, issue.4, pp.1781-1796, 2017. ,

Scots: A tool for the synthesis of symbolic controllers, Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control, pp.99-104, 2016. ,

Compositional construction of approximate abstractions of interconnected control systems, IEEE Transactions on Control of Network Systems, vol.5, issue.1, pp.116-127, 2018. ,

Safety control of monotone systems with bounded uncertainties, IEEE Conference on Decision and Control (CDC), pp.4874-4879, 2016. ,

Stability theory of switched dynamical systems, 2011. ,

Contract based design of symbolic controllers for interconnected multiperiodic sampled-data systems, IEEE Conference on Decision and Control (CDC), pp.773-779, 2018. ,

URL : https://hal.archives-ouvertes.fr/hal-01857389

On the composition of discrete and continuous-time assume-guarantee contracts for invariance, European Control Conference (ECC), pp.435-440, 2018. ,

URL : https://hal.archives-ouvertes.fr/hal-01712710

From dissipativity theory to compositional synthesis of symbolic models, Indian Control Conference (ICC), pp.30-35, 2018. ,

URL : https://hal.archives-ouvertes.fr/hal-01620525

Compositional abstraction-based synthesis for cascade discrete-time control systems, 6th IFAC Conference on Analysis and Design of Hybrid Systems ADHS, vol.51, pp.13-18, 2018. ,

URL : https://hal.archives-ouvertes.fr/hal-01761180

Interdependence quantification for compositional control synthesis with an application in vehicle safety systems, IEEE Conference on Decision and Control (CDC), pp.5700-5707, 2016. ,

Input to state stability: Basic concepts and results, Nonlinear and optimal control theory, pp.163-220, 2008. ,

Compositional safety analysis using barrier certificates, International Conference on Hybrid Systems: Computation and Control, pp.15-24, 2012. ,

Taming dr. frankenstein: Contract-based design for cyberphysical systems, European journal of control, vol.18, issue.3, pp.217-238, 2012. ,

Compositional synthesis of finite abstractions for networks of systems: A small-gain approach, 2018. ,

Compositional abstractions of interconnected discrete-time switched systems, European Control Conference (ECC), 2019. ,

Compositional synthesis of symbolic models for networks of switched systems, IEEE Control Systems Letters, vol.3, issue.4, pp.1056-1061, 2019. ,

An approximate simulation approach to symbolic control, IEEE Transactions on Automatic Control, vol.53, issue.6, pp.1406-1418, 2008. ,

Verification and control of hybrid systems: a symbolic approach, 2009. ,

Approximate reduction of dynamic systems, Systems & Control Letters, vol.57, issue.7, pp.538-545, 2008. ,

Mapping and scheduling on multi-core processors using SMT solvers, 2014. ,

URL : https://hal.archives-ouvertes.fr/tel-01087271

Bisimilar finite abstractions of interconnected systems, Hybrid Systems: Computation and Control, pp.514-527, 2008. ,

Discrete-state abstractions of nonlinear systems using multi-resolution quantizer, pp.351-365, 2009. ,

Linear time logic control of discrete-time linear systems, IEEE Transactions on Automatic Control, vol.51, issue.12, pp.1862-1877, 2006. ,

Compositional abstractions of hybrid control systems. Discrete event dynamic systems, vol.14, pp.203-238, 2004. ,

Equivalence of dynamical systems by bisimulation, IEEE transactions on automatic control, vol.49, issue.12, pp.2160-2172, 2004. ,

Optimized state space grids for abstractions, IEEE Transactions on Automatic Control, vol.62, issue.11, pp.5816-5821, 2017. ,

Approximately bisimilar symbolic models for randomly switched stochastic systems, Systems & Control Letters, vol.69, pp.38-46, 2014. ,

Compositional abstraction for networks of control systems: A dissipativity approach, IEEE Transactions on, 2017. ,

Symbolic models for stochastic switched systems: A discretization and a discretization-free approach, Automatica, vol.55, pp.183-196, 2015. ,

URL : https://hal.archives-ouvertes.fr/hal-01170568

Symbolic control of stochastic systems via approximately bisimilar finite abstractions, ZEM + 14, vol.59, pp.3135-3150, 2014. ,

Finite abstractions of networked control systems, IEEE Conference on Decision and Control, pp.95-100, 2014. ,

Symbolic models for nonlinear control systems without stability assumptions, IEEE Transactions on Automatic Control, vol.57, issue.7, pp.1804-1809, 2012. ,

A symbolic approach to voltage stability and power sharing in time-varying DC microgrids, European Control Conference, 2019. ,

URL : https://hal.archives-ouvertes.fr/hal-02070070

Decentralized monotonicity-based voltage control of dc microgrids with zip loads. 8th IFAC Workshop on Distributed Estimation and Control in Networked Systems, 2019. ,

URL : https://hal.archives-ouvertes.fr/hal-02208665

Towards scalable synthesis of stochastic control systems, Discrete Event Dynamic Systems, vol.27, issue.2, pp.341-369, 2017. ,