.. .. Conclusion,

, Monotone transition systems

. .. Abstractions, 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

.. .. Numerical-example,

.. .. Conclusion,

. ?-dom(c-*-)-=-dom, 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)

Q. , V. , Y. , ?. , and H. ). ,

, 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)

. C(q)-:=-?,

(. Cont, ;. Q-s, Q. , and ?. , = 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

Q. , 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 )

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

Z. Sun and S. Sam-ge, Stability theory of switched dynamical systems, 2011.

A. Saoud, A. Girard, and L. Fribourg, 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

A. Saoud, A. Girard, and L. Fribourg, 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

A. Swikir, A. Girard, and M. Zamani, 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

A. Saoud, M. Jagtap, A. Zamani, and . Girard, 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

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

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

C. Sloth, G. J. Pappas, and R. Wisniewski, Compositional safety analysis using barrier certificates, International Conference on Hybrid Systems: Computation and Control, pp.15-24, 2012.

A. Sangiovanni-vincentelli, W. Damm, and R. Passerone, Taming dr. frankenstein: Contract-based design for cyberphysical systems, European journal of control, vol.18, issue.3, pp.217-238, 2012.

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

A. Swikir and M. Zamani, Compositional abstractions of interconnected discrete-time switched systems, European Control Conference (ECC), 2019.

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

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

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

P. Tabuada, D. Aaron, A. Ames, G. Julius, and . Pappas, Approximate reduction of dynamic systems, Systems & Control Letters, vol.57, issue.7, pp.538-545, 2008.

P. Tendulkar, Mapping and scheduling on multi-core processors using SMT solvers, 2014.
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Y. Tazaki and J. Imura, Bisimilar finite abstractions of interconnected systems, Hybrid Systems: Computation and Control, pp.514-527, 2008.

Y. Tazaki and J. Imura, Discrete-state abstractions of nonlinear systems using multi-resolution quantizer, pp.351-365, 2009.

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

P. Tabuada, J. George, P. Pappas, and . Lima, Compositional abstractions of hybrid control systems. Discrete event dynamic systems, vol.14, pp.203-238, 2004.

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A. Weber, M. Rungger, and G. Reissig, Optimized state space grids for abstractions, IEEE Transactions on Automatic Control, vol.62, issue.11, pp.5816-5821, 2017.

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

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

M. Zamani, A. Abate, and A. Girard, 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

M. Zamani, P. M. Esfahani, R. Majumdar, A. Abate, and J. Lygeros, Symbolic control of stochastic systems via approximately bisimilar finite abstractions, ZEM + 14, vol.59, pp.3135-3150, 2014.

M. Zamani, M. Mazo, and A. Abate, Finite abstractions of networked control systems, IEEE Conference on Decision and Control, pp.95-100, 2014.

M. Zamani, G. Pola, M. Mazo, and P. Tabuada, Symbolic models for nonlinear control systems without stability assumptions, IEEE Transactions on Automatic Control, vol.57, issue.7, pp.1804-1809, 2012.

D. Zonetti, A. Saoud, A. Girard, and L. Fribourg, 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

D. Zonetti, A. Saoud, A. Girard, and L. Fribourg, 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

M. Zamani, I. Tkachev, and A. Abate, Towards scalable synthesis of stochastic control systems, Discrete Event Dynamic Systems, vol.27, issue.2, pp.341-369, 2017.