N. Davoren, J. M. Logics for hybrid systems. Proceedings of the IEEE, 8(7):985--1010, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of the state space that preserve observations and reachability properties. In addition to reachability, bisimulations of finite transition systems preserve all properties that are expressible in temporal logics such as linear temporal logic (LTL) computational tree logic (CTL) and calculus [9,11]. The notion of bisimulation has been also instrumental in obtaining decidability results for various classes of hybrid systems, by considering finite bisimulations of hybrid systems (see survey [1] In the control community, notions that are similar to bisimulation have been considered in the ....

....logics. In particular, bisimilar transition systems satisfy the same properties expressible in LTL, CTL # , and calculus. In this paper we focus on characterizing bisimilar linear systems, and refer the reader to a very rich literature on the interplay between bisimulations and temporal logic [9,11]. 3 Bisimilar Transition Systems Property preserving partitions of the state space reduce the size of the system by ignoring modeling detail that is irrelevant to the properties of interest. Thus, given a transition system T , and partitions of the state space, we would like to first consider ....

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J. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88(7):985--1010, July 2000.

....motion planning, experiment with them on standard control problems, and extend them to the case of hybrid systems. 1 Introduction and Overview Researchers in the computer science and control theory communities have produced many models for describing the dynamics of hybrid systems (e.g. see [3, 25, 12, 26]) For the purpose of the discussion in this document, we consider a simple illustrative case, in which the constituent continuous state and input spaces (in each mode) are the same. Thus, we have a hybrid system of the form x = f(x, u, q) x J(x, u, q) x, q) D(x, u, q) x J(x, u, ....

J. M. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88:985--1010, July 2000.

....on the continuous evolution of the system within a control mode and on the discrete transitions between control modes that the system might make. A state of the overall system is a pair (control mode, continuous state) For more details on the mathematics and applications of hybrid automata, see [20, 5, 10, 14]. Research and analysis of hybrid automata underlies practical tools such as CHARON and the model checker HYTECH [22] The architecture of a hybrid system in CHARON is expressed as hierarchical agents, a model conceptually similar to hybrid automata and hybrid reactive modules [7] The key ....

....to animation systems. 3. 1 Modal Logic and Properties of Multi agent Animations There are many modal and temporal logics that can be used to reason about hybrid systems, such as CTL, LTL, and the l calculus; significant research has been devoted to the theory and applications of these logics ([14] and [10] are good surveys of recent work) For readers unfamiliar with fundamental modal logic operators, we provide a brief review Modal logics are used to reason about possible worlds and properties of possibility and necessity such as Proposition P is true in all possible worlds or ....

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J. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88:985 1010, July 2000.

....like avoiding an unsafe region of the state space. The process of formal design consists of synthesizing controllers for hybrid systems in order to meet a given specification. Both directions have received large attention in the hybrid systems community, and the reader is referred to [3] 11] [23], 25] 33] 42] 55] and [73] for expositions to much of the research in the field. In this paper, we are interested in the formal analysis of hybrid systems. The formal analysis of large scale, hybrid systems is typically a very difficult process due to the complexity and scale of the ....

....and computation tree logic (CTL) 26] Preserving LTL properties leads to special partitions of the state space given by language equivalence relations, whereas CTL properties are abstracted by bisimulations. A detailed exposition to the use of various logics in hybrid systems can be found in [23]. Similar concepts and constructions, but from a hierarchical control perspective, can be found in [16] and [61] 63] There are immediate obstacles due to undecidability. For example, in [37] it was shown that checking reachability (whether a certain region of the state space can be reached) ....

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J. Davoren and A. Nerode, "Logics for hybrid systems," Proc. IEEE, vol. 88, pp. 985--1010, July 2000.

....from mode q to q # . These sets can be expressed in terms of the non deterministic feedback map c : Q X # 2 Q where H q = x : q # c(q, x) and G qq # = x : q # # c(q, x) The hybrid automata defined here are slightly more restricted than others appearing in this issue, e.g. 17] [18] as there are no jumps or resets in the values of the continuous variables when transitions are taken. These more general models include also a reset map R : Q Q X # 2 X , which decomposes into reset maps R qq # : X # 2 X with dom(R qq # ) G qq # for each (q, q # ) # Q Q. Our models ....

.... is computable, i.e. there is an e#ective way to transform a formula #(x) for F into a formula # # (x) for # 0 (F ) If in the logic in question, the satisfiability problem for nonquantified formulae is decidable, questions such as the inclusion of # 0 (F ) in another set are decidable as well (see [18] for a survey of logic notions) In the simple case where f is constant, i.e. f(x) c, and F is a polyhedral set written as a formula #(x) which is a combination of linear equalities, we get #(x) #t # 0 #(x ct) and quantifier elimination can be performed using elementary linear algebra. ....

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J.M. Davoren and A. Nerode, "Logics for hybrid systems," Proc. of the IEEE, vol. this, 2000.

....from mode q to q 0 . These sets can be expressed in terms of the non deterministic feedback map c : Q X 2 Q where H q = fx : q 2 c(q; x)g and G qq 0 = fx : q 0 2 c(q; x)g. The hybrid automata de ned here are slightly more restricted than others appearing in this issue, e.g. 17] [18] as there are no jumps or resets in the values of the continuous variables when transitions are taken. These more general models include also a reset map R : Q Q X 2 X , which decomposes into reset maps R qq 0 : X 2 X with dom(R qq 0 ) G qq 0 for each (q; q 0 ) 2 Q Q. Our ....

....Fig. 7. A simple hybrid system with piecewise constant derivatives for which the computation of P does not terminate. the logic in question, the satis ability problem for nonquanti ed formulae is decidable, questions such as the inclusion of 0 (F ) in another set are decidable as well (see [18] for a survey of logic notions) In the simple case where f is constant, i.e. f(x) c, and F is a polyhedral set written as a formula (x) which is a combination of linear equalities, we get (x) 9t 0 (x ct) and quanti er elimination can be performed using elementary linear algebra. ....

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J.M. Davoren and A. Nerode, \Logics for hybrid systems," Proc. of the IEEE, vol. this, 2000.

....like avoiding an unsafe region of the state space. The process of formal design consists of synthesizing controllers for hybrid systems in order to meet a given specification. Both directions have received large attention in the hybrid systems community, and the reader is referred to [3, 11, 23, 25, 33, 42, 55, 73] for expositions to much of the research in the field. In this paper, we are interested in the formal analysis of hybrid systems. The formal analysis of large scale, hybrid systems is typically a very difficult process due to the complexity and scale of the system. This makes the use of ....

....and Computation Tree Logic (CTL) 26] Preserving LTL properties leads to special partitions of the state space given by language equivalence relations, whereas CTL properties are abstracted by bisimulations. A detailed exposition to the use of various logics in hybrid systems can be found in [23]. Similar concepts and constructions, but from a hierarchical control perspective, can be found in [16, 61, 62, 63] There is also a body of work on sufficient abstractions of hybrid systems, for which the reader is referred to [18, 21, 34, 30, 60, 67, 70] There are immediate obstacles due to ....

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J. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE. This issue.

....q to q 0 . These sets can be expressed in terms of the non deterministic feedback map c : Q Theta X 2 Q where H q = fx : q 2 c(q; x)g and G qq 0 = fx : q 0 2 c(q; x)g. The hybrid automata defined here are slightly more restricted than others appearing in this issue, e.g. AHLP00] DN00] as there are no jumps or resets in the values of the continuous variables when transitions are taken. These more general models include also a reset map R : Q Theta Q Theta X 2 X , which decomposes into reset maps R qq 0 : X 2 X with dom(R qq 0 ) G qq 0 for each (q; q 0 ) 2 Q ....

.... computable, i.e. there is an effective way to transform a formula (x) for F into a formula 0 (x) for 0 (F ) If in the logic in question, the satisfiability problem for non quantified formulae is decidable, questions such as the inclusion of 0 (F ) in another set are decidable as well (see [DN00] for a survey of logic notions) In the simple case where f is constant, i.e. f(x) c, and F is a polyhedral set written as a formula (x) which is a combination of linear equalities, we get (x) 9t 0 (x ct) and quantifier elimination can be performed using elementary linear algebra. ....

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J.M. Davoren and A. Nerode, Logics for Hybrid Systems, Proc. of the IEEE, to appear, 2000.

.... with a finite state supervisory controller, linked by an interface consisting of an A D map and a D A map, converting a continuous plant output signal into a discrete controller input signal, and converting a discrete controller output signal into an input to the continuous plant, respectively [16, 12, 7, 5]. Of a particular interest here is the task of controller design; e.g. 7, 16, 9, 4] study classes of control problems in which the continuous plant and D A map are given, and the task is to construct an A D map and a supervisory controller so that the closed loop system fulfills various ....

....B z(#(j) and hence x(t) A y(j) as required. Since x was arbitrary, we conclude that (#, #, B) captures C. 8 Applying DES and discrete behavioural approaches to controller synthesis In tackling the general synthesis problem formulated in Section 5, we can build on previous work in [7, 16, 5], first starting by seeking to find an A D map # and a discrete dynamic specification B spec such that (#, B spec ) captures C spec . The synthesis problem can then be restated purely in terms of the discrete behaviours, BSP## and B spec : find an admissible discrete supervisor with ....

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J.M. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88:985--1010, July 2000.

....classes are quite distinct, a key technical tool for both cases are metric tolerance relations, which are put to use in di erent ways. This paper also demonstrates the exibility and adaptability of our novel methodology for hybrid controller synthesis based on modal logic, rst developed in [6, 7]. For our purposes, modal logic is best viewed as a formalism for reasoning about sets of states and operators on sets arising from relations on the state space. Considered as a family of logics, modal logic includes the temporal logics more commonly used in formal veri cation of hybrid systems. ....

....must traverse some minimum spatial distance when ful lling the event sequence speci cation. 4 Modal Logics for Hybrid Systems This section sets out only the bare details of modal logics and their application to hybrid systems. For a more substantial account, the reader is referred to [7] and also to [6] The handbook chapter [18] gives a broader introduction to the family of modal and temporal logics. A modal signature is a pair (Rel; Prp) where Rel is an alphabet of atomic relation labels, and Prp is an alphabet of atomic propositions. The set L(Rel; Prp) of modal formulas ....

J.M. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88:985-1010, July 2000.

....must traverse some minimum spatial distance when fulfilling the event sequence specification. 4 Modal logics for hybrid systems This section sets out only the bare details of modal logics and their application to hybrid systems. For a more substantial account, the reader is referred to [7] and also to [6] The handbook chapter [16] gives a broader introduction to the family of modal and temporal logics. For our purposes, modal logic is best viewed as a formalism for reasoning about sets of states and operators on sets arising from relations on the state space. A modal signature is ....

J.M. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88:985--1010, July 2000.

....and the use of deductive proof systems for such logics, where one seeks to give a formal deduction of a specification formula from a theory (set of formulas) already known to be true of the system model. A comprehensive tutorial and survey of various logics for hybrid systems is o#ered in [12]. The present paper combines ideas from both control theory and computer science to develop a logic based approach to the design and synthesis of hybrid control systems. The plant, or system to be controlled, consists of a finite number of continuous systems x = F c (x) over a common state space ....

....discrete transition (q, q # ) we continue with this established notation. This hybrid control configuration is an extension of the switching controller framework of [6] and is closely related to the DES supervisory control framework of [18] the latter relationship is developed in more detail in [12]. Figure 1: Controller designated regions of plant state space We study a quite general class of hybrid control problems. The problem is framed as the task of constructing a controller for a given plant, so that the resulting closed loop hybrid automaton is guaranteed to satisfy a given list of ....

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J.M. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88, July 2000.

....manufacturing [19, 2] Formal reasoning about discrete dynamics is well studied in the analysis and verification of computer hardware and software using temporal and modal logics; the challenge in the hybrid setting is to also incorporate continuous dynamics into a common logical framework. As in [8], the key and elementary move is to consider relations R a # S S as a generic way of representing continuous as well as discrete dynamics. Consider a state space S # R n equipped with the standard Euclidean metric topology TE , and consider a di#erential equation d dt x(t) F (x(t) ....

.... the real valued variables (possibly the identity map) Hybrid trajectories are then formed by the alternating composition of flow and reset relations, and a rich array of properties of hybrid trajectories can be encoded as formulae of PDL, the propositional calculus L, and topological extensions [8]. While topological semantics for S4 like modalities have been used in a number of recent studies of bimodal logics, our work is the first to directly address continuity properties of relations in a bimodal framework. Earlier work by Davoren [5] and independent work by Kremer and Mints [14] uses ....

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J.M. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88, July 2000.

....logics has been highly successful for discrete systems, where the component transition relations have a homogeneous character, the poly modal exogenous approach has a clear attraction in the case of hybrid systems, 3 whose components are necessarily heterogenous in nature. In other papers [14, 17], we show how the modal calculus provides a substantial and usable logical foundation for the formal analysis of hybrid systems. In the modal framework, hybrid trajectories can be simply described by alternating compositions of continuous evolution and discrete reset relations, and in these ....

....M, # # # # i# J#K M # # J#K M # . For sentences # # S (#, #) the denotation J#K M # is independent of variable assignments #, so written J#K M . Thus M # i# J#K M = S. A more detailed review of the calculus and the theory of approximations of fixed points can be found in [33, 19, 14, 17]. Formal verification by model checking can only be applied to sentences # # S (#, #) since the task is to recursively compute the denotation J#K M and thus determine whether M #. Model checking of L sentences is decidable for LTS models M which have a finite bisimulation quotient (so ....

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J. M. Davoren and A. Nerode. Logics for hybrid systems. In Antsaklis and van Schuppen [9]. To appear.

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N. Davoren, J. M. Logics for hybrid systems. Proceedings of the IEEE, 8(7):985--1010, 2000.

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J. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE. This issue.

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J. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88, July 2000.

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J. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE, 88:985--1010, 2000.

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