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Observer-based Similarity Output Feedback Control of Cyber-Physical Systems*
Preprints, 5th IFAC Conference on Analysis and Design of Hybrid Preprints, IFAC Conference on Analysis and Design of Hybrid Preprints, 5th Systems 5th Preprints, 5th IFAC IFAC Conference Conference on on Analysis Analysis and and Design Design of of Hybrid Hybrid Systems Available online at www.sciencedirect.com Systems October 14-16, 2015. Georgia Tech, Atlanta, USA Systems October 14-16, 2015. Georgia Tech, Atlanta, USA October October 14-16, 14-16, 2015. 2015. Georgia Georgia Tech, Tech, Atlanta, Atlanta, USA USA
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Observer-based Similarity Output Feedback Observer-based Similarity Output Feedback Observer-based Similarity Output Feedback ⋆⋆ Control of Cyber-Physical Systems Control of Cyber-Physical Systems Control of Cyber-Physical Systems ⋆ Masashi Mizoguchi ∗∗∗ Toshimitsu Ushio ∗∗ ∗∗ ∗∗ Masashi Masashi Mizoguchi Toshimitsu Ushio Masashi Mizoguchi Mizoguchi ∗ Toshimitsu Toshimitsu Ushio Ushio ∗∗ ∗ ∗ Osaka University, Osaka, Japan (e-mail: ∗ University, ∗ Osaka Osaka University, Osaka, Japan (e-mail: [email protected]) Osaka University, Osaka, Osaka, Japan Japan (e-mail: (e-mail: ∗∗ [email protected]) [email protected]) Osaka University, Osaka, Japan (e-mail: [email protected]) ∗∗ ∗∗ Osaka University, Osaka, Japan (e-mail: ∗∗ Osaka University, [email protected]) University, Osaka, Osaka, Japan Japan (e-mail: (e-mail: [email protected]) [email protected]) [email protected]) Abstract: In this paper, we deal with output feedback embedded control of a cyber-physical Abstract: In paper, deal feedback embedded control of cyber-physical Abstract: In this paper, we deal with output feedback embedded control of cyber-physical system. Both plant and awe controller areoutput modeled by state transition systems. the former Abstract: Inathis this paper, we deal with with output feedback embedded control of aaaBut, cyber-physical system. Both a plant and a controller are modeled by state transition systems. But, former system. Both a plant and a controller are modeled by state transition systems. But, the former is infinite while the latter is finite. So, an abstraction of the plant model is used in thethe design of system. Both a plant and a controller are modeled by state transition systems. But, the former is infinite while the latter is finite. So, an abstraction of the plant model is used in the design is infinite infinite whileWe thepropose latter is isan finite. So, an an observer abstraction ofcomputes the plant plant amodel model isallused used in the theabstracted design of of the controller. abstracted thatof set of is possible is while the latter finite. So, abstraction the in design of the propose abstracted observer computes aaa set possible abstracted the controller. We propose an abstracted observer that computes set of all possible abstracted states using theWe sequence injected control inputs that and observed plant. It is shown the controller. controller. We proposeofan an abstracted observer that computesoutputs set of ofofall allthe possible abstracted states using sequence injected control and observed outputs of plant. is states using the sequence of injected control inputs and observed outputs of the plant. It is shown that a simulation theinputs observer the abstracted the statesthere usingisthe the sequence of ofrelation injectedfrom control inputs andto observed outputs plant of the the model. plant. It ItUsing is shown shown that there is a simulation relation from the observer to the abstracted plant model. Using that there is a simulation relation from the observer to the abstracted plant model. Using the observer, we construct a similarity output feedback controller for a given transition system that there is a simulation relation from the observer to the abstracted plant model. Using the the observer, construct a feedback controller aaa given transition system observer, we we desired construct a similarity similarity output feedback controller for given transition system representing behaviors of the output abstracted plant model. It isfor shown using an approximate observer, we construct a similarity output feedback controller for given transition system representing desired behaviors of plant model. is using representing desired behaviors of the abstracted plant model. It is shown using an approximate alternating the controlled plant exhibits abstracted representingrelation desired that behaviors of the the abstracted abstracted plant desired model. It It is shown shown behaviors. using an an approximate approximate alternating relation that the controlled plant exhibits desired abstracted behaviors. alternating relation that the controlled plant exhibits desired abstracted behaviors. alternating relation that the controlled plant exhibits desired abstracted behaviors. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Cyber-physical systems, hybrid systems, observers, output feedback, computer Keywords: Cyber-physical systems, hybrid systems, observers, output feedback, computer Keywords: Cyber-physical systems, hybrid systems, controlled similarity control, abstraction Keywords: systems, Cyber-physical systems, hybrid systems, observers, observers, output output feedback, feedback, computer computer controlled systems, similarity control, abstraction controlled systems, similarity control, abstraction controlled systems, similarity control, abstraction 1. INTRODUCTION On the other hand, Alur et al. (1998) proposed alternating 1. INTRODUCTION On the Alur et (1998) proposed alternating 1. On the other hand, Alur et al. (1998) proposed alternating 1. INTRODUCTION INTRODUCTION (bi)simulation relations analysis composite systems On the other other hand, hand, Alur for et al. al. (1998)of proposed alternating (bi)simulation relations for analysis of composite systems (bi)simulation relations for analysis of composite systems such as multi-agent systems and game automata. The relations for analysis of composite systems An embedded control system is a cyber-physical system (bi)simulation such as multi-agent systems and game automata. The An embedded control system is a cyber-physical system such as multi-agent systems and game automata. The An embedded control system is a cyber-physical system relations are also useful for design symbolic controllers. as multi-agent systems and ofgame automata. The (CPS) consisting of two heterogeneous components: An embedded control system is a cyber-physical systema such relations are also useful for design of symbolic controllers. (CPS) consisting of two heterogeneous components: a relations are also also useful the for design design of symbolic symbolic controllers. (CPS) two aa relations In finite state systems, symbolic control problem can are useful for of controllers. physical componentof a plant to be components: controlled and (CPS) consisting consisting of called two heterogeneous heterogeneous components: In systems, the symbolic problem can component called plant to be controlled and finite state systems, the symbolic control problem can physical component called plant be and be finite solvedstate by using exact (bi)simulation In finite state systems, thealternating symbolic control control problemrelacan aphysical cyber component control physical component called calledaaa acontroller plant to to computing be controlled controlled and In be solved by using exact alternating (bi)simulation relaaa cyber cyber component called a controller computing control be solved by using exact alternating (bi)simulation relaainputs. component called a controller computing control tions. Tabuada (2009) considers a state(bi)simulation feedback controller solved by using exact alternating relaphysicalcalled variables used forcomputing the modelcontrol of the be cyber The component a controller tions. (2009) state inputs. The physical variables used for the model of the tions. Tabuada (2009) considers state feedback controller inputs. physical variables used for model the problem as aconsiders similarityaaa game. tions. Tabuada Tabuada (2009) considers state feedback feedback controller controller physical component real-valued theof cyber inputs. The The physicalare variables usedvectors for the thewhile model of the synthesis synthesis problem as a similarity game. physical component are real-valued vectors while the cyber synthesis problem problem as as aa similarity similarity game. game. physical are vectors while the component has a discrete nature with a finite set. synthesis physical component component are real-valued real-valued vectors while state the cyber cyber In general, there exist disturbances in systems. Moreover, component has discrete nature with finite state set. component aaa discrete with aa set. Thus, these has components dynamical but their component has discretearenature nature with systems a finite finite state state set. In general, there disturbances in Moreover, In general, there exist disturbances in systems. Moreover, in and sensing data are transmitted using In CPSs, general,control there exist exist disturbances in systems. systems. Moreover, Thus, these components are dynamical systems but their Thus, these components are dynamical systems but their modeling is based on different formalisms that have been in Thus, these components are dynamical systems but their CPSs, control and sensing data are transmitted using in communication CPSs, control control and and sensing datadisturbances are transmitted transmitted using a network with or noises. in CPSs, sensing data are using modeling is based on different formalisms that have been modeling based that have developed Overformalisms the past two a aa communication modeling is isindependently. based on on different different formalisms that decades, have been been network with disturbances or noises. communication network with disturbances or noises. The disturbances such as data drop in the network may a communication network with disturbances or noises. developed independently. Over the past two decades, a developed independently. Over two hybrid system has been developed a useful formalismaa The developed independently. Over the the aspast past two decades, decades, such as drop in network The disturbances such as data drop in the network may occurdisturbances asynchronously degrade The disturbances suchand as data data dropcontrol in the the performances. network may may hybrid system has been developed as a useful formalism hybrid system has been developed as a useful formalism for integrating the heterogeneous systems. See occur hybrid system has been developeddynamical as a useful formalism asynchronously and degrade control performances. occur asynchronously and degrade control performances. Thus, robustness of a controller in the presence of the occur asynchronously and degrade control performances. for integrating the heterogeneous dynamical systems. See for integrating the heterogeneous dynamical systems. See Goel, et al. (2012). A major challenging in See the Thus, for integrating the heterogeneous dynamical issue systems. of in the presence of Thus, robustness robustness of aaa controller controller in the presence of the the disturbances is an important issue. See Bloem et al. (2010), Thus, robustness of controller in the presence of the Goel, et al. (2012). A major challenging issue in the Goel, al. A challenging in hybrid system is to develop a symbolic or anissue algorithmic Goel, et et al. (2012). (2012). A major major challenging issue in the the disturbances disturbances is an important issue. See Bloem et al. (2010), is an important issue. See Bloem et al. (2010), and Bloem et al. (2009) for a symbolic approach to the disturbances is an important issue. See Bloem et al. (2010), hybrid system is to develop a symbolic or an algorithmic hybrid is develop an approach to the and the or control using its and hybrid system system is to toverification develop aa symbolic symbolic or an algorithmic algorithmic Bloem al. (2009) for approach to the and Bloem et al. (2009) for symbolic approach to the robustness. Pola Tabuada (2009) proposed and Bloem et et al. and (2009) for aaa symbolic symbolic approach an to apthe approach to the verification and the control using its approach the and control using finite stateto See Tabuada approach toabstraction. the verification verification and the the(2009). controlSimulation using its its robustness. Pola and Tabuada (2009) proposed an aprobustness. Pola and Tabuada (2009) proposed an approximate alternating bisimulation relation and considered robustness. Pola and Tabuada (2009) proposed an apfinite state abstraction. See Tabuada (2009). Simulation finite state abstraction. See Tabuada (2009). Simulation and relations by Milner (1989) proximate finitebisimulation state abstraction. See proposed Tabuada (2009). Simulation bisimulation relation considered proximate alternating bisimulation relation and considered a symbolicalternating control problem of a nonlinear system with proximate alternating bisimulation relation and and considered and bisimulation relations proposed by Milner (1989) and bisimulation proposed by are notionsrelations that specify the correctness of the aa symbolic and central bisimulation relations proposed by Milner Milner (1989) (1989) control problem of a nonlinear system with symbolic control problem of a nonlinear system with disturbances. Tabuada et al. (2014), Rungger and Tabuada a symbolic control problem of a nonlinear system with are central notions that specify the correctness of the are that specify the of abstraction. The state of an abstracted system based on disturbances. are central central notions notions that specify the correctness correctness of the the Tabuada et al. (2014), Rungger and Tabuada disturbances. Tabuada et al. (2014), Rungger and Tabuada (2014a), Rungger and Tabuada (2014b), and and Rungger and disturbances. Tabuada et al. (2014), Rungger Tabuada abstraction. The state of an abstracted system based on abstraction. The state of an abstracted system based on the relationsThe is astate quotient of the system state set of the abstraction. of anclass abstracted based on (2014a), and (2014b), Rungger and (2014a), Rungger Rungger and Tabuada Tabuada (2014b), and and Rungger and Tabuada (2013) considered an abstraction and refinement (2014a), Rungger and Tabuada (2014b), and Rungger and the relations is a quotient class of the state set of the the relations is a quotient class of the state set of the hybrid system.isThus, an exact discrete of the Tabuada the relations a quotient class of theabstraction state set of (2013) considered an abstraction and refinement Tabuada (2013) considered an abstraction and refinement approach to the design of a controller for a CPS modeled Tabuada (2013) considered an abstraction and refinement hybrid system. Thus, an exact discrete abstraction of the hybrid exact discrete of hybrid system. system Thus, needs an a large of computational system. Thus, an exact amount discrete abstraction abstraction of the the approach the of for approach to the design of controller for CPS modeled by a stateto and discussed themodeled preserapproach totransition the design designsystem, of aaa controller controller for aaa CPS CPS modeled hybrid system needs large amount of computational hybrid needs aa amount of resources and may be an undecidable for a class by hybrid system system needs a large large amountproblem of computational computational a state transition system, and discussed the by a state transition system, and discussed the preservation of input-output dynamical stability (pIODS) of by a state transition system, and discussed the preserpreserresources and may be an undecidable problem for a class resources and may be an undecidable problem for a class of hybrid systems state set ofproblem the hybrid resources and maysince be anthe undecidable for system a class vation of input-output dynamical stability (pIODS) of vation of input-output dynamical stability (pIODS) of the closed system with a state feedback controller that vation of input-output dynamical stability (pIODS) of of hybrid systems since the state set of the hybrid system of hybrid systems since the state set of the hybrid system is See Alursince et al. (2000). Recently, consider an the of infinite. hybrid systems the state set of the to hybrid system closed system with a state feedback controller that the closed system with a state feedback controller that is designed based on the abstracted model of the CPS. the closed system with a state feedback controller that is infinite. See Alur et al. (2000). Recently, to consider an is Alur Recently, to an approximate the hybrid system, the notions is infinite. infinite. See Seeabstraction Alur et et al. al.of(2000). (2000). Recently, to consider consider an is is designed based on the abstracted model of the CPS. designed based on the model of CPS. They show that of pIODS is designed basedthe onpreservation the abstracted abstracted model under of the the model CPS. approximate abstraction of the hybrid system, the notions approximate abstraction of system, notions of an approximate simulation and bisimulation approximate abstraction of the the hybrid hybrid system, the the relation notions They show that the preservation of pIODS under model They show that the preservation of pIODS under model abstraction will be useful in CPS control. However, it was They show that the preservation of pIODS under model of an approximate simulation and bisimulation relation of an approximate simulation and bisimulation relation have proposed. These relations are characterized of an been approximate simulation and bisimulation relation abstraction will be useful in CPS However, it was abstraction will be useful inobserved. CPS control. control. However, it was assumed that all states are In general, all states abstraction will be useful in CPS control. However, it was have been proposed. These relations are characterized have been These are by using Lyapunov-type called a assumed haveinequalities been proposed. proposed. These relations relations functions are characterized characterized that all states are observed. In general, all states assumed that all states are observed. In general, all states of the systems cannot be always observed. Then, we extend assumed that all states are observed. In general, all states by inequalities using Lyapunov-type functions called a by Lyapunov-type functions called simulation and ausing bisimulation function. See Girard andaa of by inequalities inequalities using Lyapunov-type functions called the systems cannot be Then, of the systems cannot be always observed. Then, we extend this output controlobserved. of the system. of theapproach systems to cannot be always always observed. Then, we we extend extend simulation and bisimulation function. See Girard and simulation and bisimulation function. See and Pappas (2007), et al. (2008), Girard al. (2010), simulation and aaaPola bisimulation function. Seeet Girard Girard and this approach to output control of the system. this approach to output control of the system. this approach to output control of the system. Pappas (2007), Pola et al. (2008), Girard et al. (2010), Pappas (2007), Pola et al. (2008), Girard et al. (2010), Tabuada (2008),Pola and et Girard and Pappas Pappas (2007), al. (2008), Girard(2011). et al. (2010), Controller design methods with output information have Tabuada (2008), and Girard and Pappas (2011). Controller design output information have Tabuada (2008), and Girard and Pappas (2011). Controller design methods with output information have Tabuada (2008), and Girard and Pappas (2011). been studied. For methods example, with Tarraf (2012), Tarraf (2014), Controller design methods with output information have ⋆ This work was supported by JSPS KAKENHI No. 15K14007. been studied. For example, Tarraf (2012), Tarraf (2014), ⋆ been studied. For example, Tarraf (2012), Tarraf (2014), ⋆ This work was supported by JSPS KAKENHI No. 15K14007. been studied. For example, Tarraf (2012), Tarraf (2014), This work work was was supported supported by by JSPS JSPS KAKENHI KAKENHI No. No. 15K14007. 15K14007. ⋆ This Copyright IFAC 2015 248 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright IFAC 2015 248 Copyright © IFAC 2015 248 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 248Control. 10.1016/j.ifacol.2015.11.183
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Tarraf et al. (2014), and Fan and Tarraf (2014) proposed the ”certified-by-design” approach, and discussed finite state approximations of plants. Chatterjee et al. (2007) considered the imperfect game winning strategy. Ghaemi and Del Vecchio (2014) used the estimation of a set of possible current states from output information to generate a control input by which a safety specification is achieved. Ehlers and Topcu (2015) proposed an estimator that computes finite approximations of unobserved states. However, these approaches are not based on the (bi)simulation and the alternating simulation. On the other hand, Vu and Takai (2014) considered an observer-based output feedback controller synthesis problem in discrete event systems. See Cassandras and Lafortune (2008) for the design of an observer in discrete event systems. Their approach is based on the exact alternating simulation since the states of the discrete event systems are symbolic. In this paper, we propose a symbolic approach to the design of an output feedback controller with an observer. This observer computes a set of all possible abstracted states using the sequence of injected control inputs and observed outputs of the plant. Using the observer, we construct a similarity output feedback controller for a given transition system representing desired behaviors of the abstracted plant model. It is shown using an approximate alternating relation that the controlled plant exhibits desired abstracted behaviors. The rest of this paper is organized as follows. In Section 2, we define state transition systems, and introduce some relations. Moreover, it is shown that, under a certain condition, we can design a controller for a plant with an abstracted controller. In Section 3, we design an observer. In Section 4, we construct an output feedback controller with the observer. In Section 5, we show an example to demonstrate how the proposed controller works. 2. CYBER-PHYSICAL SYSTEMS 2.1 NOTATIONS We denote the set of natural numbers by N = {0, 1, 2, . . .}, the set of integers by Z, and the set of real numbers by R. R≥0 means the set of all non-negative real numbers. We use |x| to denote the ∞-norm of x ∈ Rn . For a given set A ⊆ Rn , we use [A]η := {x ∈ A | ∃k ∈ Zn : x = 2kη} to denote a uniform grid in A. η will be called an abstraction parameter. For a, b ∈ R with a ≤ b, We denote the closed and half-open intervals in R by [a, b] and [a, b[, respectively. A state transition system is defined by a six tuple (X, X0 , U, r, Y, H) consisting of a set of states X, a set of initial states X0 ⊆ X, a set of inputs U , a transition map r : X × U → 2X , a set of outputs Y , and an output map H : X → Y . Definition 1. A system S is nonblocking if, for any x ∈ X, there exists u ∈ U such that r(x, u) ̸= ∅. 2.2 TRANSITION SYSTEM MODEL A CPS is composed of a cyber component and a physical component. The physical component corresponds to a plant to be controlled while the digital controller is 249
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implemented in the cyber component. We model the physical component as a state transition system S = (X, X0 , U, r, Y, H). Note that the state, the input, and the output of the plant are represented by real-valued variables. On the other hand, the digital controller determines the control input using the discretized model of the plant ˆ , rˆ, X, ˆ I) ˆ with the identity map ˆ X ˆ0, U described by Sˆ = (X, ˆ ˆ ˆ I : X → X. Note that the abstracted model of the plant does not include the information about the outputs since it is used to specify abstracted target behaviors. In this paper, we assume that transition systems S, Sˆ are nonblocking. Definition 2. We consider two transition systems S = ˆ X ˆ0, U ˆ , rˆ, X, ˆ I), ˆ a relation R ⊆ (X, X0 , U, r, Y, H), Sˆ = (X, ˆ ×U ×U ˆ . Let κ, λ ∈ R≥0 , β ∈ [0, 1[ be some X ×X ˆ × U → R≥0 . Let parameters and consider a map d : U ˆ U (x) := ˆ) | (x, x ˆ, u, u ˆ) ∈ R} ⊆ X × X, RX = {(x, x ˆ (ˆ {u | r(x, u) ̸= ∅}, and U x) := {ˆ u | rˆ(ˆ x, u ˆ) ̸= ∅}. A ˆ ×X× parameterized (by ϵ ∈ [κ, ∞[) relation R(ϵ) ⊆ X ˆ U × U is said to be a κ-approximate (β, λ)-contractive alternating simulation relation ((κ, β, λ)-acASR) from Sˆ to S with d(ˆ u, u) if R(ϵ) satisfies R(ϵ) ⊆ R(ϵ′ ) for any ϵ, ϵ′ ∈ [κ, ∞[ with ϵ ≤ ϵ′ , and the following conditions hold for all ϵ ∈ [κ, ∞[ : ˆ 0 , ∃x0 ∈ X0 : (ˆ (1) ∀ˆ x0 ∈ X x0 , x0 ) ∈ RX (κ); ˆ (ˆ u∈U x), ∃u ∈ U (x) : (2) ∀(ˆ x, x) ∈ RX (ϵ), ∀ˆ [(ˆ x, x, u ˆ, u) ∈ R(ϵ)] ∧ [∀x′ ∈ r(x, u), ∃ˆ x′ ∈ rˆ(ˆ x, u ˆ) : ′ ′ (ˆ x , x ) ∈ RX (κ + βϵ + λd(ˆ u, u))].
Especially, if (κ, β, λ) = (0, 0, 0), we can replace an acASR with an alternating simulation relation (ASR). ˆC , X ˆ C0 , U ˆC , rˆC , X ˆ C , IˆC ) with the identity Let SˆC = (X ˆC → X ˆ C be a transition system that repmap IˆC : X ˆ Let resents desired behavior for the abstracted model S. ˆC × X ˆ ×U ˆC × U ˆ be a relation. Then, SC denoted ˆC ⊆ X R by SˆC ×Rˆ C Sˆ = (XC , XC0 , UC , rC , XC , IC ) is their comˆ CX (⊆ X ˆ C × X), ˆ XC0 = (X ˆ C0 × position, where XC = R ˆ ˆ ˆ X0 ) ∩ XC , UC = UC × U , IC : XC → XC is the identity map, and rC : XC × UC → XC is defined by x ,x ˆ), (ˆ u ,u ˆ)) rC ((ˆ C ′ ′c xC , x ˆ ) ∈ XC | x ˆ′C ∈ rˆC (ˆ xC , u ˆC ), x ˆ′ ∈ rˆ(ˆ x, u ˆ)} {(ˆ ˆC , = ˆ, u ˆC , u ˆ) ∈ R if (ˆ xC , x ∅ otherwise.
Then, Rungger and Tabuada (2014b) showed the following lemma. ˆ and S. SupLemma 1. We consider three systems SˆC , S, ˆ and a (κ, β, λ)ˆ C from SˆC to S, pose there exists an ASR R acASR R(ϵ) from Sˆ to S with d(ˆ u, u). Then, the following relation RC (ϵ) ⊆ XC × X × UC × U is a (κ, β, λ)-acASR u, u). from SC := SˆC ×Rˆ C Sˆ to S with d(ˆ xC , x ˆ), x, (ˆ uC , u ˆ), u) | [(ˆ x, x, u ˆ, u) ∈ R(ϵ)] RC (ϵ) := {((ˆ (1) ˆ CX ]}. ∧[(ˆ xC , x ˆ) ∈ R
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Rungger and Tabuada (2014b) considered a state feedback embedded controller, where the controlled system is given by the composition of the plant S and the controller SC = SˆC ×Rˆ C Sˆ with a (κ, β, λ)-acASR RC (ϵ) from SC to S. In general, however, all states may not be observed. In the following, we consider an output feedback controller with an observer. 3. OBSERVER In the output feedback control, we use an observer that estimates states of the plant. The observer is designed in such a way that an error between the current state and the estimated state converges to zero. But, the distance between the states is not defined in a state transition system in general. So, in this paper, we use another approach that is used in discrete event systems, that is, the observer lists up all candidates that are reachable from a sequence of a pair of the input and the output. ˇ of a state set X by a We consider an abstracted set X X ˇ map Cell : X → 2 , where each state x ∈ Cell(ˇ x) of ˇ x), where S is abstracted as x ˇ and satisfies H(x) = H(ˇ ˇ x) denotes the output of x H(ˇ ˇ. For simplicity, we assume x) = X and Cell(ˇ x) ∩ Cell(ˇ x′ ) = ∅ for that ∪xˇ∈Xˇ Cell(ˇ ′ ′ ˇ ˇ is a ˇ ̸= x ˇ . Intuitively, the set X any x ˇ, x ˇ ∈ X with x partition of X such that all states abstracted as x ˇ are ˇ x) ∈ Y . We introduce observed as the same output H(ˇ ˇ of the plant ˇ X ˇ 0 , U, rˇ, Y, H) an abstracted system Sˇ = (X, ˇ | Cell(xˇ0 ) ∩ X0 ̸= ∅} and the ˇ 0 = {ˇ x0 ∈ X S, where X ˇ × U → 2Xˇ is defined as follows: for transition map rˇ : X ˇ × U, each (ˇ x, u) ∈ X ˇ | ∃x ∈ Cell(ˇ rˇ(ˇ x, u) = {ˇ x′ ∈ X x) : r(x, u) ∩ Cell(ˇ x′ ) ̸= ∅}. (2) ˆ Note that the abstracted model S is used to specify abstracted desired behaviors of the system S if the state feedback is applied while the abstracted model Sˇ is used to design an observer that estimates a current state of the ˆ ̸= X. ˇ plant. So, in general X Then, by extending the observer design method in discrete event systems, we define the following transition system ˜ , r˜, Y˜ , H) ˜ called an observer of S, where ˜ X ˜0, U S˜ = (X, ˇ ˜ = {˜ ˇ x) = H(ˇ ˇ x′ )}, X x ∈ 2X − {∅} | ∀ˇ x, ∀ˇ x′ ∈ x ˜, H(ˇ ˜ ˇ ˜ x0 } ∈ X | x ˇ0 ∈ X0 }, X0 = {{ˇ ˜ = U, U Y˜ = Y , ˜ :X ˜ → Y˜ is defined by H(˜ ˜ x) = H(ˇ ˇ x) with x H ˇ∈x ˜, and ˜ ×U → 2X˜ satisfies the following two conditions: • r˜ : X ˜ × U, for any (˜ x, u) ∈ X ′ ′′ ˜ x′ ) ̸= H(˜ ˜ x′′ ), · ∀˜ x, x ˜ ∈ r˜(˜ x, u) with x ˜′ ̸= x ˜′′ : H(˜ ′ ′ · ∪xˇ∈˜x rˇ(ˇ x, u) = ∪x˜ ∈˜r(˜x,u) x ˜.
• • • • •
In the following, for simplicity, we consider the accessible ˜ part of S˜ and denote it by S. We will show how the observer works. Definition 3. We consider two systems S = (X, X0 , U, r, ˜ , r˜, Y˜ , H). ˜ A relation R ⊆ X × ˜ X ˜0, U Y ,H) and S˜ = (X, 250
˜ ×U ×U ˜ is said to be a simulation relation (SR) from X ˜ S to S if it satisfies the following conditions: ˜ 0 : (x0 , x x0 ∈ X ˜ 0 ) ∈ RX ; (1) ∀x0 ∈ X0 , ∃˜ ˜ (˜ u∈U x) : (2) ∀(x, x ˜) ∈ RX , ∀u ∈ U (x), ∃˜ [(x, x ˜, u, u ˜) ∈ R] ∧ [∀x′ ∈ r(x, u), ∃˜ x′ ∈ r˜(˜ x, u ˜) : ′ ′ (x , x ˜ ) ∈ RX ]. Then, we have the following lemma. Lemma 2. We consider a plant S = (X, X0 , U, r, Y, H). ˜ , r˜, Y˜ , H) ˜ be the observer of S. Then, ˜ X ˜0, U Let S˜ = (X, ˜ ˜ ×U ×U ˜ is an SR from the following relation R ⊆ X × X ˜ S to S. ˜ = {(x, x R ˜, u, u ˜) | u = u ˜, ∃ˇ x∈x ˜, x ∈ Cell(ˇ x)}.
(3)
˜ . Then, we will show that R ˜ is Proof. Note that U = U ˜ an SR from S to S. ˜ 0 such Consider any x0 ∈ X0 . Then, there exists x ˜0 ∈ X ˜ ˜X . that H(˜ x0 ) = H(x0 ) holds, and we have (x0 , x ˜0 ) ∈ R
˜ X and any u ∈ U (x), we have For any (x, x ˜) ∈ R ˜ (x, x ˜, u, u) ∈ R. Then, we consider any x′ ∈ r(x, u). By ˜ we have x the definition of R, ˇ∈x ˜ satisfying x ∈ Cell(ˇ x). x, u) satisfying By the definition of r˜, there exists x ˜′ ∈ r˜(˜ that there exists x ˇ′ ∈ x ˜′ such that x′ ∈ Cell(ˇ x′ ). Thus, ′ ′ ˜ ˜ ˜ ˜ ) ∈ RX . Therefore, R is an SR from S to S. (x , x Lemma 2 shows that S˜ simulates any behavior of S in terms of containing the current state of S. S˜ estimates the states from the input u ˜ = u and output y. Therefore, the states that are reachable with the same input and output sequences cannot be distinguished. In other words, the observer computes all possible current states using injected inputs and observed outputs. 4. OUTPUT FEEDBACK CONTROLLER
In this section, we design an output feedback controller. Recall that S = (X, X0 , U, r, Y, H) denotes a physical ˆ , rˆ, X, ˆ I) ˆ denotes its ˆ X ˆ0, U plant to be controlled, Sˆ = (X, ˆ ˆ ˆ ˆ C , IˆC ) ˆ abstracted model, and SC = (XC , XC0 , UC , rˆC , X denotes a transition system describing desired behaviors ˆC of the abstracted model. Assume there exists an ASR R ˆ ˆ ˆ from SC to S, and a (κ, β, λ)-acASR R(ϵ) from S to S with d(ˆ u, u). Then, the relation RC (ϵ) ⊆ XC ×X ×UC ×U given by (1) is a (κ, β, λ)-acASR from SC := SˆC ×Rˆ C Sˆ to S with d(ˆ u, u). Since the physical plant S is non-deterministic, we make the following assumption for simplicity. Assumption 3. For any (xC , x, uC ) ∈ XC × X × UC and ϵ ∈ [κ, ∞[, there exists at most one u ∈ U (x) such that (xC , x, uC , u) ∈ RC (ϵ). From Assumption 3 and the definition of RC (ϵ), it is shown that, if (xC , x, uC , u) ∈ RC (ϵ) for some ϵ ∈ [κ, ∞[, then (xC , x, uC , u′ ) ∈ RC (ϵ′ ) implies u = u′ for any ˇ X ˇ 0 , U, rˇ, Y, H) ˇ and ϵ′ > ϵ. Moreover, recall that Sˇ = (X, ˜ ˜ ˜ ˜ S = (X, X0 , U, r˜, Y, H) denote its abstracted system with respect to the output set Y and an observer of S.
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˜ → 2X as follows: for each We define a function Cell∗ : X ˜ x ˜ ∈ X, ∪ Cell∗ (˜ x) = Cell(ˇ x). x ˇ∈˜ x
˜ × UC × U as follows: ˜ We define a relation R(ϵ) ⊆ XC × X ˜ ˜, uC , u) ∈ R(ϵ) if and only if, for any x ∈ Cell∗ (˜ x), (xC , x the following two conditions hold. • (xC , x, uC , u) ∈ RC (ϵ). • For any x′ ∈ r(x, u), there exists x′C ∈ rC (xC , uC ) ˜ C , u)), where such that (x′C , x′ ) ∈ RCX (κ + βϵ + λd(u ˜ u, u) with uC = (ˆ uC , u ˆ). d(uC , u) = d(ˆ In this paper, we introduce the following technical assumption. ˆ x ∈ X, x′ ∈ X, and Assumption 4. For any x ˆ ∈ X, ϵ ∈ [κ, ∞[, if (ˆ x, x) ∈ RX (ϵ) and |x′ − x| ≤ δ, then ′ (ˆ x, x ) ∈ RX (ϵ + δ). Intuitively, R(ϵ) is an error between the plant S and its ˆ Then, R(ϵ) is often defined by the abstracted model S. distance between each state, x and x ˆ. Thus, Assumption 4 is not so restrictive because of the triangle inequality. Then, we have the following lemma. Lemma 5. If the relation RC (ϵ) is a (κ, β, λ)-acASR from ˜ is an SR from S to S, ˜ then SC to S and the relation R ′ ˜ ˜ the relation R(ϵ) is a (κ + κ , β, λ)-acASR from SC to S, where κ′ = sup sup {|x − x′ | | x, x′ ∈ Cell(ˇ x)} . (4) ˇ x ˇ ∈X
Proof. Obviously, we have that, for any ϵ′ ≥ ϵ ≥ κ + κ′ , ˜ ˜ ′ ). R(ϵ) ⊆ R(ϵ
Next, consider any xC0 ∈ XC0 . Since RC (ϵ) is a (κ, β, λ)acASR from SC to S, there exists x0 ∈ X0 such that ˜ 0 such that (xC0 , x0 ) ∈ RCX (κ) holds. We choose x ˜0 ∈ X ˜ ˜ ˜ ˜0 ) ∈ RX holds. Note that R is an SR from S to S, (x0 , x and x ˜0 = {ˇ x0 } such that x0 ∈ Cell(ˇ x0 ). Then, we have ˜ X (κ + κ′ ). ˜0 ) ∈ R (xC0 , x
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the transition depends on the abstraction parameter κ that represents an error of the abstraction for the controller ˜ intuitively. On the other hand, if we use the observer S, the distance between the state of the observer and the abstracted state of SC after the transition depends on not only κ but also κ′ that represents an error of the abstraction for the observer intuitively. We consider a composition S¯C ¯C , X ¯ C0 , U ¯C , r¯C , Y¯ , H ¯ C ), where (X • • • •
=
SC ×R(ϵ) S˜ ˜
=
˜ ¯ C = XC × X, X ¯ C0 = (XC0 × X ˜0) ∩ R ˜ X (κ), X ¯ C = UC × U , U (x′C , x ˜′ ) ∈ r¯C ((xC , x ˜), (uC , u)) if and only if the following conditions hold: · x′C ∈ rC (xC , uC ); · x ˜′ ∈ r˜(˜ x, u); ˜ ˜, uC , u) ∈ R(e(x ˜)), where · (xC , x C, x ˜ X (ϵ)}; ˜) = inf{ϵ ∈ [κ, ∞[ | (xC , x ˜) ∈ R e(xC , x ˜ C , u)). ˜ X (κ + βϵ + λd(u ˜′ ) ∈ R · (x′C , x
Then, we have the following main theorem. Theorem 6. We consider a plant S = (X, X0 , U, r, Y, H), ˆ , rˆ, X, ˆ I), ˆ and a tranˆ X ˆ0, U its abstracted model Sˆ = (X, ˆ ˆ ˆ ˆ ˆ ˆ sition system SC = (XC , XC0 , U , rˆC , XC , IC ) that repreˆ We assume that there exist sents a desired behavior of S. ˆ and a (κ, β, λ)-acASR R(ϵ) ˆ C from SˆC to S, an ASR R from Sˆ to S with d(ˆ u, u). Then, the following relation ¯ ×X ×U ¯ × U is a (κ + κ′ , β, λ)-acASR from S¯C ¯ C (ϵ) ⊆ X R ¯ C , u), u′ ) = d(ˆ to S with d((u u, u′ ), where κ′ is defined by (4). ¯ C (ϵ) = {(xC , x ˜), x, (uC , u), u′ ) | u = u′ , R ˜ ˜ ˜, uC , u) ∈ R(ϵ), (x, x ˜, u, u) ∈ R} (5) (xC , x
˜ ˜ Therefore, R(ϵ) is a (κ + κ′ , β, λ)-acASR from SC to S.
Proof. Obviously, we have that, for any ϵ′ ≥ ϵ ≥ κ + κ′ , ¯ C (ϵ) ⊆ R ¯ C (ϵ′ ). R ¯ C0 . Note that (xC0 , x Next, consider any (xC0 , x ˜0 ) ∈ X ˜0 ) ∈ ˜ 0 , there exists ˜ R(κ). Then, by the definition of x ˜0 ∈ X ˜ X holds. Thus, we have ˜0 ) ∈ R x0 ∈ X0 such that (x0 , x ′ ¯ ((xC0 , x ˜0 ), x0 ) ∈ RCX (κ + κ ). ¯ CX (ϵ). For any Finally, consider any ((xC , x ˜), x) ∈ R ¯ (uC , u) ∈ UC ((xC , x ˜)), u ∈ U (x) holds. Then, we have ¯ C (ϵ). Note that (xC , x ˜ ˜), x, (uC , u), u) ∈ R ˜) ∈ R(ϵ), ((xC , x ′ ˜ and (x, x ˜) ∈ R. For any x ∈ r(x, u), by the existence ˜ from S to S, ˜ there exists x x, u) such of the SR R ˜′ ∈ r˜(˜ ′ ′ ˜ X holds. From Lemma 5, there exists ˜) ∈ R that (x , x ˜ X (κ + κ′ + βϵ + ˜′ ) ∈ R x′C ∈ rC (xC , uC ) such that (x′C , x ′ ˜ ¯ CX (κ + ˜′ ), x′ ) ∈ R λd(uC , u)) holds. Thus, we have ((xC , x ′ ¯ κ + βϵ + λd((uC , u), u)). ¯ C (ϵ) is a (κ + κ′ , β, λ)-acASR from S¯C to S. Therefore, R
ˆ ̸= X ˇ in general. Both X ˇ and X ˆ are the Note that X abstraction of X, but their partitions are different. If we use the state feedback controller SC , the distance between the state of the plant S and the abstracted state of SC after
Intuitively, Theorem 6 shows that the closed system approximately tracks the behavior of SC . If there is no ”com¯C ((xC , x mon” input allowed in Cell∗ (˜ x), U ˜)) is the empty ¯ set, so that SC is blocking. To compute a nonblocking
Finally, consider any ϵ such that ϵ ≥ κ + κ′ , and any ˜ ˆ C and ˜) ∈ R(ϵ), where xC = (ˆ xC , x ˆ) with x ˆC ∈ X (xC , x ˆ By Assumption 3 and the definition of RC (ϵ), x ˆ ∈ X. for any uC ∈ UC (xC ), there uniquely exists u ∈ U (x) such that (xC , x ˜, uC , u) ∈ RC (ϵ) holds. Consider any x ˜′ ∈ ′ ∗ ′ r˜(˜ x, u). Then, for any x ∈ Cell (˜ x ), there uniquely exists ˇ such that x′ ∈ Cell(ˇ x′ ). Thus, by the definition of x ˇ′ ∈ X r˜, there exist x′′ ∈ Cell(ˇ x′ ) and x ∈ Cell∗ (˜ x) such that x′′ ∈ r(x, u). Then, we have |x′′ − x′ | ≤ κ′ . Since RC (ϵ) is a (κ, β, λ)-acASR, there exists x′C ∈ rC (xC , uC ) such that ˜ C , u)). By Assumption 4, (x′C , x′′ ) ∈ RCX (κ + βϵ + λd(u ˜ C , u)), which we have (ˆ x′ , x′ ) ∈ RCX (κ + κ′ + βϵ + λd(u ′ ′ ′ ˜ C , u)). ˜ ˜ ) ∈ RCX (κ + κ + βϵ + λd(u implies that (ˆ x ,x
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sub-transition system of S¯C , we introduce the following ¯ ¯ ¯ operator F : 2X → 2X : For any W ⊆ X, F (W ) = ¯ : r¯C (¯ {¯ x ∈ W |∃¯ u∈U x, u ¯) ̸= ∅ ∧ r¯C (¯ x, u ¯) ⊆ W }. (6) By the definition of F , the following equation holds: ¯ : [F (Z) ⊆ Z] ∧ [Z ′ ⊆ Z ⇒ F (Z ′ ) ⊆ F (Z)], ∀Z, Z ′ ∈ X which means that F is monotonically decreasing. We ¯ i (i ∈ N): consider the following iterations X ¯C , ¯0 = X (7) X ¯ ¯ (8) Xi+1 = F (Xi ). ¯i ⊇ X ¯ i+1 . Thus, if there Then, for each i ∈ N, we have X ¯ k+1 = F (X ¯ k ), then we have the exists k ∈ N such that X ¯ ∗ of F given by X ¯∗ = X ¯k . supremal fixed point X C C Then, if
¯∗ ∩ X ¯ C0 ̸= ∅ X (9) C holds, we define a transition system ∗ ∗ ∗ ¯ C∗ , X ¯ C0 ¯C , r¯C ¯ C∗ ), = (X ,U , Y, H (10) S¯C where ¯∗ = X ¯∗ ∩ X ¯ C0 , • X C0 C ∗ ¯ ∗ for each x ¯ ∗ and • r¯C (¯ xC , u ¯C ) = r¯C (¯ xC , u ¯C )∩ X ¯C ∈ X C C ¯ u ¯ C ∈ UC , ¯ ∗ (¯ ¯ xC ) for each x ¯∗ . • H ¯C ∈ X C xC ) = HC (¯ C ∗ ¯ Thus, SC is a nonblocking sub-transition system of S¯C . ¯ ∗ (ϵ) be the restriction of R ¯ C (ϵ) to X ¯∗ × X × U ¯ × U. Let R C ¯ ∗ (ϵ) is a (κ + κ′ , β, λ)-acASR Then, it is obvious that R C ∗ ¯ C , u), u′ ). from S¯C to S with d((u Recall that we can always design an observer. The observer estimates the state of the plant by listing up candidates from the control input and the observed output. However, (9) has to be satisfied in order for the controller to always choose an input that is suitable for all candidates. 5. ILLUSTRATIVE EXAMPLE
We transform (13) into the transition system S1 = (X1 , X10 , U1 , r1 , Y1 , H1 ) where X1 = X10 = R2 , U1 = R, Y1 = [[−10, 10]]0.4 , H1 ((x1 , x2 )) = round(0.640x1 − 0.832x2 ), where round(y) means rounding y off to the nearest value in Y1 , and r1 is defined in the obvious way. The control signal v is sent via a wireless connection system where data dropouts may occur. We assume that the dropout does not occur consecutively. The behavior of the dropout is also modeled by a transition system S2 = (X2 , X20 , U2 , r2 , X2 , I2 ), where X2 = {0, 1}, X20 = X2 , U2 = {⊥, ⊤} and I2 (w) = w, the identity map. The dynamics r2 is illustrated in Fig. 1, where state 0 means that the data is received, and state 1 means that it is lost. The plant S is given by the composition of S1 and S2 , that is, S := S1 ×R12 S2 = (X12 = X1 × X2 , X120 , U12 = U1 × U2 , r12 , Y12 = Y1 , H12 = H1 ), where R12 = {((x1 , x2 ), w, v, u2 ) | w = 1 ⇒ v = 0}
and r12 (((x1 , x2 ), w), (v, u2 )) = (r1 ((x1 , x2 ), v), r2 (w, u2 )). ˆ1, X ˆ 10 , U ˆ1 , Next, we construct an abstracted model. Sˆ1 = (X ˆ 1 , Iˆ1 ) is based on the abstraction of S1 . We choose rˆ1 , X ˆ 10 = [[−10, 10]2 ]η , and U ˆ1 = {0, ±0.5, ±1.0, ±1.5}. ˆ X1 = X We set the abstraction parameter η = 0.05. Then, x1 , x ˆ2 ), (x1 , x2 ), vˆ, v) | R1 (ϵ) = {((ˆ
x1 , x ˆ2 )| ≤ ϵ] ∧ [v = vˆ]} [|(x1 , x2 ) − (ˆ
(14)
is a (0.05, 0.5, 0)-acASR from Sˆ1 to S1 . Since S2 is finite, its symbolic model is given by Sˆ2 = S2 . We define the abstracted plant Sˆ := Sˆ1 ×Rˆ 12 Sˆ2 , where ˆ 12 = {((ˆ R x1 , x ˆ2 ), w, ˆ vˆ, u ˆ2 ) | w ˆ = 1 ⇒ vˆ = 0}.
Then, the relation
ˆ12 , u12 ) | [(ˆ x 1 , x1 , u ˆ1 , u1 ) ∈ R1 (ϵ)] R(ϵ) = {(ˆ x12 , x12 , u u2 = u2 ]} ∧[ˆ x2 = x2 ] ∧ [ˆ
is a (0.05, 0.5, 0)-acASR from Sˆ to S.
In this section, we consider an example to show how the proposed output feedback controller works. We consider the following system: [ ] [ ][ ] [1] ξ [k + 1] [k] 1 0 ξ 1 1 = + 1 u[k], ξ2 [k] ξ2 [k + 1] 11 (11) 2 [ ] ξ1 [k] ξ[k] = [ 0 1 ] . ξ2 [k]
Since (11) is not asymptotically stable, we consider a low-level state feedback controller so that there exists an acASR from Sˆ to S. In addition, we introduce the following state [x1 , x2 ]T to enforce contraction property. [ ] [ ][ ] x1 [k] −2.929 −1.953 ξ1 [k] = . (12) x2 [k] ξ2 [k] −2.253 −2.704 Then, (11) is transformed into the following equation: [ ] [ ][ ] [ ] x1 [k + 1] 0.25 0 x1 [k] −3.905 + v[k], x [k + 1] = x2][k] 0 0.5 −3.606 2 [ (13) x1 [k] 0.640 −0.832 . [k] = [ ] ξ 2 x [k] 2
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We can observe only an abstracted position y. Thus, we ˜ X ˜ 0 , U, r˜, Y, H). ˜ It is an introduce an observer S˜ = (X, ˇ important issue how to determine an abstracted set X that is used in the design of the observer. Note that the abstracted output y is given by rounding ξ2 off in such ˇ has to satisfy a way that round(ξ2 ) = y. Recall that X ˇ and x ∈ Cell(ˇ the condition that for any x ˇ ∈ X x), ˇ x′ ) holds. Thus, the set of the abstracted states H(x) = H(ˇ for the observer is based on (ξ1 , ξ2 )-plane. On the other hand, in order to satisfy the contraction property, the set of the abstracted states for the controller is based on (x1 , x2 )plane as shown in Fig. 2.
Fig. 1. Transition system model of data dropouts.
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Fig. 2. Abstracted states for the observer (blue points), and abstracted states for the controller (red points). 6. CONCLUSION We considered a CPS consisting of a physical plant and an embedded controller, where the controller determines the input using the output of the plant. On the other hand, to estimate the current state, we introduce an observer that lists up all candidates of the current state. A set of abstracted states for the observer is chosen such that, for each abstracted state, its corresponding states in the plant have the same output. Therefore, the set of abstracted states used for the controller is different from that for the observer. By the observer, we estimate a state set in the plant where the current state of the plant exists. Then, we showed that there exists an acASR from the controller to the observer so that the output feedback controller determines a control input for the plant to exhibit a desired abstracted behavior approximately. Rungger and Tabuada (2014b) showed that pIODS is preserved under an acASR. In this paper, we proved that there exists an acASR from the output feedback controller to the plant. Therefore, it is future work to investigate stability of the controlled plant. REFERENCES R. Goebel, R. G. Sanfelice, and A. R. Teel. Hybrid Dynamical Systems Modeling, Stability, and Robustness. Princeton University Press, 2012. P. Tabuada. Verification and Control of Hybrid Systems. A Symbolic Approach. Springer, 2009. R. Alur, T. A. Henzinger, G. Lafferriere, and G. J. Pappas. Discrete abstractions of hybrid systems. In Proceedings of IEEE, 88(7):971-984, 2000. R. Milner. Communication and concurrency. PrenticeHall, 1989. A. Girard and G. J. Pappas. Approximation metrics for discrete and continuous systems. IEEE Transactions on Automatic Control, 52(5):782-798, 2007. G. Pola, A. Girard, and P. Tabuada. Approximately bisimilar symbolic models for nonlinear control systems. Automatica, 44(10):2508-2516, 2008. A. Girard, G. Pola, and P. Tabuada. Approximately bisimilar symbolic models for incrementally stable switched systems. IEEE Transactions on Automatic Control, 55(1):116-126, 2010. P. Tabuada. An approximate simulation approach to symbolic control. IEEE Transactions on Automatic Control, 53(6):1406-1418, 2008. 253
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Observer-based Similarity Output Feedback Observer-based Similarity Output Feedback Observer-based Similarity Output Feedback ⋆⋆ Control of Cyber-Physical Systems Control of Cyber-Physical Systems Control of Cyber-Physical Systems ⋆ Masashi Mizoguchi ∗∗∗ Toshimitsu Ushio ∗∗ ∗∗ ∗∗ Masashi Masashi Mizoguchi Toshimitsu Ushio Masashi Mizoguchi Mizoguchi ∗ Toshimitsu Toshimitsu Ushio Ushio ∗∗ ∗ ∗ Osaka University, Osaka, Japan (e-mail: ∗ University, ∗ Osaka Osaka University, Osaka, Japan (e-mail: [email protected]) Osaka University, Osaka, Osaka, Japan Japan (e-mail: (e-mail: ∗∗ [email protected]) [email protected]) Osaka University, Osaka, Japan (e-mail: [email protected]) ∗∗ ∗∗ Osaka University, Osaka, Japan (e-mail: ∗∗ Osaka University, [email protected]) University, Osaka, Osaka, Japan Japan (e-mail: (e-mail: [email protected]) [email protected]) [email protected]) Abstract: In this paper, we deal with output feedback embedded control of a cyber-physical Abstract: In paper, deal feedback embedded control of cyber-physical Abstract: In this paper, we deal with output feedback embedded control of cyber-physical system. Both plant and awe controller areoutput modeled by state transition systems. the former Abstract: Inathis this paper, we deal with with output feedback embedded control of aaaBut, cyber-physical system. Both a plant and a controller are modeled by state transition systems. But, former system. Both a plant and a controller are modeled by state transition systems. But, the former is infinite while the latter is finite. So, an abstraction of the plant model is used in thethe design of system. Both a plant and a controller are modeled by state transition systems. But, the former is infinite while the latter is finite. So, an abstraction of the plant model is used in the design is infinite infinite whileWe thepropose latter is isan finite. So, an an observer abstraction ofcomputes the plant plant amodel model isallused used in the theabstracted design of of the controller. abstracted thatof set of is possible is while the latter finite. So, abstraction the in design of the propose abstracted observer computes aaa set possible abstracted the controller. We propose an abstracted observer that computes set of all possible abstracted states using theWe sequence injected control inputs that and observed plant. It is shown the controller. controller. We proposeofan an abstracted observer that computesoutputs set of ofofall allthe possible abstracted states using sequence injected control and observed outputs of plant. is states using the sequence of injected control inputs and observed outputs of the plant. It is shown that a simulation theinputs observer the abstracted the statesthere usingisthe the sequence of ofrelation injectedfrom control inputs andto observed outputs plant of the the model. plant. It ItUsing is shown shown that there is a simulation relation from the observer to the abstracted plant model. Using that there is a simulation relation from the observer to the abstracted plant model. Using the observer, we construct a similarity output feedback controller for a given transition system that there is a simulation relation from the observer to the abstracted plant model. Using the the observer, construct a feedback controller aaa given transition system observer, we we desired construct a similarity similarity output feedback controller for given transition system representing behaviors of the output abstracted plant model. It isfor shown using an approximate observer, we construct a similarity output feedback controller for given transition system representing desired behaviors of plant model. is using representing desired behaviors of the abstracted plant model. It is shown using an approximate alternating the controlled plant exhibits abstracted representingrelation desired that behaviors of the the abstracted abstracted plant desired model. It It is shown shown behaviors. using an an approximate approximate alternating relation that the controlled plant exhibits desired abstracted behaviors. alternating relation that the controlled plant exhibits desired abstracted behaviors. alternating relation that the controlled plant exhibits desired abstracted behaviors. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Cyber-physical systems, hybrid systems, observers, output feedback, computer Keywords: Cyber-physical systems, hybrid systems, observers, output feedback, computer Keywords: Cyber-physical systems, hybrid systems, controlled similarity control, abstraction Keywords: systems, Cyber-physical systems, hybrid systems, observers, observers, output output feedback, feedback, computer computer controlled systems, similarity control, abstraction controlled systems, similarity control, abstraction controlled systems, similarity control, abstraction 1. INTRODUCTION On the other hand, Alur et al. (1998) proposed alternating 1. INTRODUCTION On the Alur et (1998) proposed alternating 1. On the other hand, Alur et al. (1998) proposed alternating 1. INTRODUCTION INTRODUCTION (bi)simulation relations analysis composite systems On the other other hand, hand, Alur for et al. al. (1998)of proposed alternating (bi)simulation relations for analysis of composite systems (bi)simulation relations for analysis of composite systems such as multi-agent systems and game automata. The relations for analysis of composite systems An embedded control system is a cyber-physical system (bi)simulation such as multi-agent systems and game automata. The An embedded control system is a cyber-physical system such as multi-agent systems and game automata. The An embedded control system is a cyber-physical system relations are also useful for design symbolic controllers. as multi-agent systems and ofgame automata. The (CPS) consisting of two heterogeneous components: An embedded control system is a cyber-physical systema such relations are also useful for design of symbolic controllers. (CPS) consisting of two heterogeneous components: a relations are also also useful the for design design of symbolic symbolic controllers. (CPS) two aa relations In finite state systems, symbolic control problem can are useful for of controllers. physical componentof a plant to be components: controlled and (CPS) consisting consisting of called two heterogeneous heterogeneous components: In systems, the symbolic problem can component called plant to be controlled and finite state systems, the symbolic control problem can physical component called plant be and be finite solvedstate by using exact (bi)simulation In finite state systems, thealternating symbolic control control problemrelacan aphysical cyber component control physical component called calledaaa acontroller plant to to computing be controlled controlled and In be solved by using exact alternating (bi)simulation relaaa cyber cyber component called a controller computing control be solved by using exact alternating (bi)simulation relaainputs. component called a controller computing control tions. Tabuada (2009) considers a state(bi)simulation feedback controller solved by using exact alternating relaphysicalcalled variables used forcomputing the modelcontrol of the be cyber The component a controller tions. (2009) state inputs. The physical variables used for the model of the tions. Tabuada (2009) considers state feedback controller inputs. physical variables used for model the problem as aconsiders similarityaaa game. tions. Tabuada Tabuada (2009) considers state feedback feedback controller controller physical component real-valued theof cyber inputs. The The physicalare variables usedvectors for the thewhile model of the synthesis synthesis problem as a similarity game. physical component are real-valued vectors while the cyber synthesis problem problem as as aa similarity similarity game. game. physical are vectors while the component has a discrete nature with a finite set. synthesis physical component component are real-valued real-valued vectors while state the cyber cyber In general, there exist disturbances in systems. Moreover, component has discrete nature with finite state set. component aaa discrete with aa set. Thus, these has components dynamical but their component has discretearenature nature with systems a finite finite state state set. In general, there disturbances in Moreover, In general, there exist disturbances in systems. Moreover, in and sensing data are transmitted using In CPSs, general,control there exist exist disturbances in systems. systems. Moreover, Thus, these components are dynamical systems but their Thus, these components are dynamical systems but their modeling is based on different formalisms that have been in Thus, these components are dynamical systems but their CPSs, control and sensing data are transmitted using in communication CPSs, control control and and sensing datadisturbances are transmitted transmitted using a network with or noises. in CPSs, sensing data are using modeling is based on different formalisms that have been modeling based that have developed Overformalisms the past two a aa communication modeling is isindependently. based on on different different formalisms that decades, have been been network with disturbances or noises. communication network with disturbances or noises. The disturbances such as data drop in the network may a communication network with disturbances or noises. developed independently. Over the past two decades, a developed independently. Over two hybrid system has been developed a useful formalismaa The developed independently. Over the the aspast past two decades, decades, such as drop in network The disturbances such as data drop in the network may occurdisturbances asynchronously degrade The disturbances suchand as data data dropcontrol in the the performances. network may may hybrid system has been developed as a useful formalism hybrid system has been developed as a useful formalism for integrating the heterogeneous systems. See occur hybrid system has been developeddynamical as a useful formalism asynchronously and degrade control performances. occur asynchronously and degrade control performances. Thus, robustness of a controller in the presence of the occur asynchronously and degrade control performances. for integrating the heterogeneous dynamical systems. See for integrating the heterogeneous dynamical systems. See Goel, et al. (2012). A major challenging in See the Thus, for integrating the heterogeneous dynamical issue systems. of in the presence of Thus, robustness robustness of aaa controller controller in the presence of the the disturbances is an important issue. See Bloem et al. (2010), Thus, robustness of controller in the presence of the Goel, et al. (2012). A major challenging issue in the Goel, al. A challenging in hybrid system is to develop a symbolic or anissue algorithmic Goel, et et al. (2012). (2012). A major major challenging issue in the the disturbances disturbances is an important issue. See Bloem et al. (2010), is an important issue. See Bloem et al. (2010), and Bloem et al. (2009) for a symbolic approach to the disturbances is an important issue. See Bloem et al. (2010), hybrid system is to develop a symbolic or an algorithmic hybrid is develop an approach to the and the or control using its and hybrid system system is to toverification develop aa symbolic symbolic or an algorithmic algorithmic Bloem al. (2009) for approach to the and Bloem et al. (2009) for symbolic approach to the robustness. Pola Tabuada (2009) proposed and Bloem et et al. and (2009) for aaa symbolic symbolic approach an to apthe approach to the verification and the control using its approach the and control using finite stateto See Tabuada approach toabstraction. the verification verification and the the(2009). controlSimulation using its its robustness. Pola and Tabuada (2009) proposed an aprobustness. Pola and Tabuada (2009) proposed an approximate alternating bisimulation relation and considered robustness. Pola and Tabuada (2009) proposed an apfinite state abstraction. See Tabuada (2009). Simulation finite state abstraction. See Tabuada (2009). Simulation and relations by Milner (1989) proximate finitebisimulation state abstraction. See proposed Tabuada (2009). Simulation bisimulation relation considered proximate alternating bisimulation relation and considered a symbolicalternating control problem of a nonlinear system with proximate alternating bisimulation relation and and considered and bisimulation relations proposed by Milner (1989) and bisimulation proposed by are notionsrelations that specify the correctness of the aa symbolic and central bisimulation relations proposed by Milner Milner (1989) (1989) control problem of a nonlinear system with symbolic control problem of a nonlinear system with disturbances. Tabuada et al. (2014), Rungger and Tabuada a symbolic control problem of a nonlinear system with are central notions that specify the correctness of the are that specify the of abstraction. The state of an abstracted system based on disturbances. are central central notions notions that specify the correctness correctness of the the Tabuada et al. (2014), Rungger and Tabuada disturbances. Tabuada et al. (2014), Rungger and Tabuada (2014a), Rungger and Tabuada (2014b), and and Rungger and disturbances. Tabuada et al. (2014), Rungger Tabuada abstraction. The state of an abstracted system based on abstraction. The state of an abstracted system based on the relationsThe is astate quotient of the system state set of the abstraction. of anclass abstracted based on (2014a), and (2014b), Rungger and (2014a), Rungger Rungger and Tabuada Tabuada (2014b), and and Rungger and Tabuada (2013) considered an abstraction and refinement (2014a), Rungger and Tabuada (2014b), and Rungger and the relations is a quotient class of the state set of the the relations is a quotient class of the state set of the hybrid system.isThus, an exact discrete of the Tabuada the relations a quotient class of theabstraction state set of (2013) considered an abstraction and refinement Tabuada (2013) considered an abstraction and refinement approach to the design of a controller for a CPS modeled Tabuada (2013) considered an abstraction and refinement hybrid system. Thus, an exact discrete abstraction of the hybrid exact discrete of hybrid system. system Thus, needs an a large of computational system. Thus, an exact amount discrete abstraction abstraction of the the approach the of for approach to the design of controller for CPS modeled by a stateto and discussed themodeled preserapproach totransition the design designsystem, of aaa controller controller for aaa CPS CPS modeled hybrid system needs large amount of computational hybrid needs aa amount of resources and may be an undecidable for a class by hybrid system system needs a large large amountproblem of computational computational a state transition system, and discussed the by a state transition system, and discussed the preservation of input-output dynamical stability (pIODS) of by a state transition system, and discussed the preserpreserresources and may be an undecidable problem for a class resources and may be an undecidable problem for a class of hybrid systems state set ofproblem the hybrid resources and maysince be anthe undecidable for system a class vation of input-output dynamical stability (pIODS) of vation of input-output dynamical stability (pIODS) of the closed system with a state feedback controller that vation of input-output dynamical stability (pIODS) of of hybrid systems since the state set of the hybrid system of hybrid systems since the state set of the hybrid system is See Alursince et al. (2000). Recently, consider an the of infinite. hybrid systems the state set of the to hybrid system closed system with a state feedback controller that the closed system with a state feedback controller that is designed based on the abstracted model of the CPS. the closed system with a state feedback controller that is infinite. See Alur et al. (2000). Recently, to consider an is Alur Recently, to an approximate the hybrid system, the notions is infinite. infinite. See Seeabstraction Alur et et al. al.of(2000). (2000). Recently, to consider consider an is is designed based on the abstracted model of the CPS. designed based on the model of CPS. They show that of pIODS is designed basedthe onpreservation the abstracted abstracted model under of the the model CPS. approximate abstraction of the hybrid system, the notions approximate abstraction of system, notions of an approximate simulation and bisimulation approximate abstraction of the the hybrid hybrid system, the the relation notions They show that the preservation of pIODS under model They show that the preservation of pIODS under model abstraction will be useful in CPS control. However, it was They show that the preservation of pIODS under model of an approximate simulation and bisimulation relation of an approximate simulation and bisimulation relation have proposed. These relations are characterized of an been approximate simulation and bisimulation relation abstraction will be useful in CPS However, it was abstraction will be useful inobserved. CPS control. control. However, it was assumed that all states are In general, all states abstraction will be useful in CPS control. However, it was have been proposed. These relations are characterized have been These are by using Lyapunov-type called a assumed haveinequalities been proposed. proposed. These relations relations functions are characterized characterized that all states are observed. In general, all states assumed that all states are observed. In general, all states of the systems cannot be always observed. Then, we extend assumed that all states are observed. In general, all states by inequalities using Lyapunov-type functions called a by Lyapunov-type functions called simulation and ausing bisimulation function. See Girard andaa of by inequalities inequalities using Lyapunov-type functions called the systems cannot be Then, of the systems cannot be always observed. Then, we extend this output controlobserved. of the system. of theapproach systems to cannot be always always observed. Then, we we extend extend simulation and bisimulation function. See Girard and simulation and bisimulation function. See and Pappas (2007), et al. (2008), Girard al. (2010), simulation and aaaPola bisimulation function. Seeet Girard Girard and this approach to output control of the system. this approach to output control of the system. this approach to output control of the system. Pappas (2007), Pola et al. (2008), Girard et al. (2010), Pappas (2007), Pola et al. (2008), Girard et al. (2010), Tabuada (2008),Pola and et Girard and Pappas Pappas (2007), al. (2008), Girard(2011). et al. (2010), Controller design methods with output information have Tabuada (2008), and Girard and Pappas (2011). Controller design output information have Tabuada (2008), and Girard and Pappas (2011). Controller design methods with output information have Tabuada (2008), and Girard and Pappas (2011). been studied. For methods example, with Tarraf (2012), Tarraf (2014), Controller design methods with output information have ⋆ This work was supported by JSPS KAKENHI No. 15K14007. been studied. For example, Tarraf (2012), Tarraf (2014), ⋆ been studied. For example, Tarraf (2012), Tarraf (2014), ⋆ This work was supported by JSPS KAKENHI No. 15K14007. been studied. For example, Tarraf (2012), Tarraf (2014), This work work was was supported supported by by JSPS JSPS KAKENHI KAKENHI No. No. 15K14007. 15K14007. ⋆ This Copyright IFAC 2015 248 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright IFAC 2015 248 Copyright © IFAC 2015 248 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 248Control. 10.1016/j.ifacol.2015.11.183
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Tarraf et al. (2014), and Fan and Tarraf (2014) proposed the ”certified-by-design” approach, and discussed finite state approximations of plants. Chatterjee et al. (2007) considered the imperfect game winning strategy. Ghaemi and Del Vecchio (2014) used the estimation of a set of possible current states from output information to generate a control input by which a safety specification is achieved. Ehlers and Topcu (2015) proposed an estimator that computes finite approximations of unobserved states. However, these approaches are not based on the (bi)simulation and the alternating simulation. On the other hand, Vu and Takai (2014) considered an observer-based output feedback controller synthesis problem in discrete event systems. See Cassandras and Lafortune (2008) for the design of an observer in discrete event systems. Their approach is based on the exact alternating simulation since the states of the discrete event systems are symbolic. In this paper, we propose a symbolic approach to the design of an output feedback controller with an observer. This observer computes a set of all possible abstracted states using the sequence of injected control inputs and observed outputs of the plant. Using the observer, we construct a similarity output feedback controller for a given transition system representing desired behaviors of the abstracted plant model. It is shown using an approximate alternating relation that the controlled plant exhibits desired abstracted behaviors. The rest of this paper is organized as follows. In Section 2, we define state transition systems, and introduce some relations. Moreover, it is shown that, under a certain condition, we can design a controller for a plant with an abstracted controller. In Section 3, we design an observer. In Section 4, we construct an output feedback controller with the observer. In Section 5, we show an example to demonstrate how the proposed controller works. 2. CYBER-PHYSICAL SYSTEMS 2.1 NOTATIONS We denote the set of natural numbers by N = {0, 1, 2, . . .}, the set of integers by Z, and the set of real numbers by R. R≥0 means the set of all non-negative real numbers. We use |x| to denote the ∞-norm of x ∈ Rn . For a given set A ⊆ Rn , we use [A]η := {x ∈ A | ∃k ∈ Zn : x = 2kη} to denote a uniform grid in A. η will be called an abstraction parameter. For a, b ∈ R with a ≤ b, We denote the closed and half-open intervals in R by [a, b] and [a, b[, respectively. A state transition system is defined by a six tuple (X, X0 , U, r, Y, H) consisting of a set of states X, a set of initial states X0 ⊆ X, a set of inputs U , a transition map r : X × U → 2X , a set of outputs Y , and an output map H : X → Y . Definition 1. A system S is nonblocking if, for any x ∈ X, there exists u ∈ U such that r(x, u) ̸= ∅. 2.2 TRANSITION SYSTEM MODEL A CPS is composed of a cyber component and a physical component. The physical component corresponds to a plant to be controlled while the digital controller is 249
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implemented in the cyber component. We model the physical component as a state transition system S = (X, X0 , U, r, Y, H). Note that the state, the input, and the output of the plant are represented by real-valued variables. On the other hand, the digital controller determines the control input using the discretized model of the plant ˆ , rˆ, X, ˆ I) ˆ with the identity map ˆ X ˆ0, U described by Sˆ = (X, ˆ ˆ ˆ I : X → X. Note that the abstracted model of the plant does not include the information about the outputs since it is used to specify abstracted target behaviors. In this paper, we assume that transition systems S, Sˆ are nonblocking. Definition 2. We consider two transition systems S = ˆ X ˆ0, U ˆ , rˆ, X, ˆ I), ˆ a relation R ⊆ (X, X0 , U, r, Y, H), Sˆ = (X, ˆ ×U ×U ˆ . Let κ, λ ∈ R≥0 , β ∈ [0, 1[ be some X ×X ˆ × U → R≥0 . Let parameters and consider a map d : U ˆ U (x) := ˆ) | (x, x ˆ, u, u ˆ) ∈ R} ⊆ X × X, RX = {(x, x ˆ (ˆ {u | r(x, u) ̸= ∅}, and U x) := {ˆ u | rˆ(ˆ x, u ˆ) ̸= ∅}. A ˆ ×X× parameterized (by ϵ ∈ [κ, ∞[) relation R(ϵ) ⊆ X ˆ U × U is said to be a κ-approximate (β, λ)-contractive alternating simulation relation ((κ, β, λ)-acASR) from Sˆ to S with d(ˆ u, u) if R(ϵ) satisfies R(ϵ) ⊆ R(ϵ′ ) for any ϵ, ϵ′ ∈ [κ, ∞[ with ϵ ≤ ϵ′ , and the following conditions hold for all ϵ ∈ [κ, ∞[ : ˆ 0 , ∃x0 ∈ X0 : (ˆ (1) ∀ˆ x0 ∈ X x0 , x0 ) ∈ RX (κ); ˆ (ˆ u∈U x), ∃u ∈ U (x) : (2) ∀(ˆ x, x) ∈ RX (ϵ), ∀ˆ [(ˆ x, x, u ˆ, u) ∈ R(ϵ)] ∧ [∀x′ ∈ r(x, u), ∃ˆ x′ ∈ rˆ(ˆ x, u ˆ) : ′ ′ (ˆ x , x ) ∈ RX (κ + βϵ + λd(ˆ u, u))].
Especially, if (κ, β, λ) = (0, 0, 0), we can replace an acASR with an alternating simulation relation (ASR). ˆC , X ˆ C0 , U ˆC , rˆC , X ˆ C , IˆC ) with the identity Let SˆC = (X ˆC → X ˆ C be a transition system that repmap IˆC : X ˆ Let resents desired behavior for the abstracted model S. ˆC × X ˆ ×U ˆC × U ˆ be a relation. Then, SC denoted ˆC ⊆ X R by SˆC ×Rˆ C Sˆ = (XC , XC0 , UC , rC , XC , IC ) is their comˆ CX (⊆ X ˆ C × X), ˆ XC0 = (X ˆ C0 × position, where XC = R ˆ ˆ ˆ X0 ) ∩ XC , UC = UC × U , IC : XC → XC is the identity map, and rC : XC × UC → XC is defined by x ,x ˆ), (ˆ u ,u ˆ)) rC ((ˆ C ′ ′c xC , x ˆ ) ∈ XC | x ˆ′C ∈ rˆC (ˆ xC , u ˆC ), x ˆ′ ∈ rˆ(ˆ x, u ˆ)} {(ˆ ˆC , = ˆ, u ˆC , u ˆ) ∈ R if (ˆ xC , x ∅ otherwise.
Then, Rungger and Tabuada (2014b) showed the following lemma. ˆ and S. SupLemma 1. We consider three systems SˆC , S, ˆ and a (κ, β, λ)ˆ C from SˆC to S, pose there exists an ASR R acASR R(ϵ) from Sˆ to S with d(ˆ u, u). Then, the following relation RC (ϵ) ⊆ XC × X × UC × U is a (κ, β, λ)-acASR u, u). from SC := SˆC ×Rˆ C Sˆ to S with d(ˆ xC , x ˆ), x, (ˆ uC , u ˆ), u) | [(ˆ x, x, u ˆ, u) ∈ R(ϵ)] RC (ϵ) := {((ˆ (1) ˆ CX ]}. ∧[(ˆ xC , x ˆ) ∈ R
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Rungger and Tabuada (2014b) considered a state feedback embedded controller, where the controlled system is given by the composition of the plant S and the controller SC = SˆC ×Rˆ C Sˆ with a (κ, β, λ)-acASR RC (ϵ) from SC to S. In general, however, all states may not be observed. In the following, we consider an output feedback controller with an observer. 3. OBSERVER In the output feedback control, we use an observer that estimates states of the plant. The observer is designed in such a way that an error between the current state and the estimated state converges to zero. But, the distance between the states is not defined in a state transition system in general. So, in this paper, we use another approach that is used in discrete event systems, that is, the observer lists up all candidates that are reachable from a sequence of a pair of the input and the output. ˇ of a state set X by a We consider an abstracted set X X ˇ map Cell : X → 2 , where each state x ∈ Cell(ˇ x) of ˇ x), where S is abstracted as x ˇ and satisfies H(x) = H(ˇ ˇ x) denotes the output of x H(ˇ ˇ. For simplicity, we assume x) = X and Cell(ˇ x) ∩ Cell(ˇ x′ ) = ∅ for that ∪xˇ∈Xˇ Cell(ˇ ′ ′ ˇ ˇ is a ˇ ̸= x ˇ . Intuitively, the set X any x ˇ, x ˇ ∈ X with x partition of X such that all states abstracted as x ˇ are ˇ x) ∈ Y . We introduce observed as the same output H(ˇ ˇ of the plant ˇ X ˇ 0 , U, rˇ, Y, H) an abstracted system Sˇ = (X, ˇ | Cell(xˇ0 ) ∩ X0 ̸= ∅} and the ˇ 0 = {ˇ x0 ∈ X S, where X ˇ × U → 2Xˇ is defined as follows: for transition map rˇ : X ˇ × U, each (ˇ x, u) ∈ X ˇ | ∃x ∈ Cell(ˇ rˇ(ˇ x, u) = {ˇ x′ ∈ X x) : r(x, u) ∩ Cell(ˇ x′ ) ̸= ∅}. (2) ˆ Note that the abstracted model S is used to specify abstracted desired behaviors of the system S if the state feedback is applied while the abstracted model Sˇ is used to design an observer that estimates a current state of the ˆ ̸= X. ˇ plant. So, in general X Then, by extending the observer design method in discrete event systems, we define the following transition system ˜ , r˜, Y˜ , H) ˜ called an observer of S, where ˜ X ˜0, U S˜ = (X, ˇ ˜ = {˜ ˇ x) = H(ˇ ˇ x′ )}, X x ∈ 2X − {∅} | ∀ˇ x, ∀ˇ x′ ∈ x ˜, H(ˇ ˜ ˇ ˜ x0 } ∈ X | x ˇ0 ∈ X0 }, X0 = {{ˇ ˜ = U, U Y˜ = Y , ˜ :X ˜ → Y˜ is defined by H(˜ ˜ x) = H(ˇ ˇ x) with x H ˇ∈x ˜, and ˜ ×U → 2X˜ satisfies the following two conditions: • r˜ : X ˜ × U, for any (˜ x, u) ∈ X ′ ′′ ˜ x′ ) ̸= H(˜ ˜ x′′ ), · ∀˜ x, x ˜ ∈ r˜(˜ x, u) with x ˜′ ̸= x ˜′′ : H(˜ ′ ′ · ∪xˇ∈˜x rˇ(ˇ x, u) = ∪x˜ ∈˜r(˜x,u) x ˜.
• • • • •
In the following, for simplicity, we consider the accessible ˜ part of S˜ and denote it by S. We will show how the observer works. Definition 3. We consider two systems S = (X, X0 , U, r, ˜ , r˜, Y˜ , H). ˜ A relation R ⊆ X × ˜ X ˜0, U Y ,H) and S˜ = (X, 250
˜ ×U ×U ˜ is said to be a simulation relation (SR) from X ˜ S to S if it satisfies the following conditions: ˜ 0 : (x0 , x x0 ∈ X ˜ 0 ) ∈ RX ; (1) ∀x0 ∈ X0 , ∃˜ ˜ (˜ u∈U x) : (2) ∀(x, x ˜) ∈ RX , ∀u ∈ U (x), ∃˜ [(x, x ˜, u, u ˜) ∈ R] ∧ [∀x′ ∈ r(x, u), ∃˜ x′ ∈ r˜(˜ x, u ˜) : ′ ′ (x , x ˜ ) ∈ RX ]. Then, we have the following lemma. Lemma 2. We consider a plant S = (X, X0 , U, r, Y, H). ˜ , r˜, Y˜ , H) ˜ be the observer of S. Then, ˜ X ˜0, U Let S˜ = (X, ˜ ˜ ×U ×U ˜ is an SR from the following relation R ⊆ X × X ˜ S to S. ˜ = {(x, x R ˜, u, u ˜) | u = u ˜, ∃ˇ x∈x ˜, x ∈ Cell(ˇ x)}.
(3)
˜ . Then, we will show that R ˜ is Proof. Note that U = U ˜ an SR from S to S. ˜ 0 such Consider any x0 ∈ X0 . Then, there exists x ˜0 ∈ X ˜ ˜X . that H(˜ x0 ) = H(x0 ) holds, and we have (x0 , x ˜0 ) ∈ R
˜ X and any u ∈ U (x), we have For any (x, x ˜) ∈ R ˜ (x, x ˜, u, u) ∈ R. Then, we consider any x′ ∈ r(x, u). By ˜ we have x the definition of R, ˇ∈x ˜ satisfying x ∈ Cell(ˇ x). x, u) satisfying By the definition of r˜, there exists x ˜′ ∈ r˜(˜ that there exists x ˇ′ ∈ x ˜′ such that x′ ∈ Cell(ˇ x′ ). Thus, ′ ′ ˜ ˜ ˜ ˜ ) ∈ RX . Therefore, R is an SR from S to S. (x , x Lemma 2 shows that S˜ simulates any behavior of S in terms of containing the current state of S. S˜ estimates the states from the input u ˜ = u and output y. Therefore, the states that are reachable with the same input and output sequences cannot be distinguished. In other words, the observer computes all possible current states using injected inputs and observed outputs. 4. OUTPUT FEEDBACK CONTROLLER
In this section, we design an output feedback controller. Recall that S = (X, X0 , U, r, Y, H) denotes a physical ˆ , rˆ, X, ˆ I) ˆ denotes its ˆ X ˆ0, U plant to be controlled, Sˆ = (X, ˆ ˆ ˆ ˆ C , IˆC ) ˆ abstracted model, and SC = (XC , XC0 , UC , rˆC , X denotes a transition system describing desired behaviors ˆC of the abstracted model. Assume there exists an ASR R ˆ ˆ ˆ from SC to S, and a (κ, β, λ)-acASR R(ϵ) from S to S with d(ˆ u, u). Then, the relation RC (ϵ) ⊆ XC ×X ×UC ×U given by (1) is a (κ, β, λ)-acASR from SC := SˆC ×Rˆ C Sˆ to S with d(ˆ u, u). Since the physical plant S is non-deterministic, we make the following assumption for simplicity. Assumption 3. For any (xC , x, uC ) ∈ XC × X × UC and ϵ ∈ [κ, ∞[, there exists at most one u ∈ U (x) such that (xC , x, uC , u) ∈ RC (ϵ). From Assumption 3 and the definition of RC (ϵ), it is shown that, if (xC , x, uC , u) ∈ RC (ϵ) for some ϵ ∈ [κ, ∞[, then (xC , x, uC , u′ ) ∈ RC (ϵ′ ) implies u = u′ for any ˇ X ˇ 0 , U, rˇ, Y, H) ˇ and ϵ′ > ϵ. Moreover, recall that Sˇ = (X, ˜ ˜ ˜ ˜ S = (X, X0 , U, r˜, Y, H) denote its abstracted system with respect to the output set Y and an observer of S.
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˜ → 2X as follows: for each We define a function Cell∗ : X ˜ x ˜ ∈ X, ∪ Cell∗ (˜ x) = Cell(ˇ x). x ˇ∈˜ x
˜ × UC × U as follows: ˜ We define a relation R(ϵ) ⊆ XC × X ˜ ˜, uC , u) ∈ R(ϵ) if and only if, for any x ∈ Cell∗ (˜ x), (xC , x the following two conditions hold. • (xC , x, uC , u) ∈ RC (ϵ). • For any x′ ∈ r(x, u), there exists x′C ∈ rC (xC , uC ) ˜ C , u)), where such that (x′C , x′ ) ∈ RCX (κ + βϵ + λd(u ˜ u, u) with uC = (ˆ uC , u ˆ). d(uC , u) = d(ˆ In this paper, we introduce the following technical assumption. ˆ x ∈ X, x′ ∈ X, and Assumption 4. For any x ˆ ∈ X, ϵ ∈ [κ, ∞[, if (ˆ x, x) ∈ RX (ϵ) and |x′ − x| ≤ δ, then ′ (ˆ x, x ) ∈ RX (ϵ + δ). Intuitively, R(ϵ) is an error between the plant S and its ˆ Then, R(ϵ) is often defined by the abstracted model S. distance between each state, x and x ˆ. Thus, Assumption 4 is not so restrictive because of the triangle inequality. Then, we have the following lemma. Lemma 5. If the relation RC (ϵ) is a (κ, β, λ)-acASR from ˜ is an SR from S to S, ˜ then SC to S and the relation R ′ ˜ ˜ the relation R(ϵ) is a (κ + κ , β, λ)-acASR from SC to S, where κ′ = sup sup {|x − x′ | | x, x′ ∈ Cell(ˇ x)} . (4) ˇ x ˇ ∈X
Proof. Obviously, we have that, for any ϵ′ ≥ ϵ ≥ κ + κ′ , ˜ ˜ ′ ). R(ϵ) ⊆ R(ϵ
Next, consider any xC0 ∈ XC0 . Since RC (ϵ) is a (κ, β, λ)acASR from SC to S, there exists x0 ∈ X0 such that ˜ 0 such that (xC0 , x0 ) ∈ RCX (κ) holds. We choose x ˜0 ∈ X ˜ ˜ ˜ ˜0 ) ∈ RX holds. Note that R is an SR from S to S, (x0 , x and x ˜0 = {ˇ x0 } such that x0 ∈ Cell(ˇ x0 ). Then, we have ˜ X (κ + κ′ ). ˜0 ) ∈ R (xC0 , x
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the transition depends on the abstraction parameter κ that represents an error of the abstraction for the controller ˜ intuitively. On the other hand, if we use the observer S, the distance between the state of the observer and the abstracted state of SC after the transition depends on not only κ but also κ′ that represents an error of the abstraction for the observer intuitively. We consider a composition S¯C ¯C , X ¯ C0 , U ¯C , r¯C , Y¯ , H ¯ C ), where (X • • • •
=
SC ×R(ϵ) S˜ ˜
=
˜ ¯ C = XC × X, X ¯ C0 = (XC0 × X ˜0) ∩ R ˜ X (κ), X ¯ C = UC × U , U (x′C , x ˜′ ) ∈ r¯C ((xC , x ˜), (uC , u)) if and only if the following conditions hold: · x′C ∈ rC (xC , uC ); · x ˜′ ∈ r˜(˜ x, u); ˜ ˜, uC , u) ∈ R(e(x ˜)), where · (xC , x C, x ˜ X (ϵ)}; ˜) = inf{ϵ ∈ [κ, ∞[ | (xC , x ˜) ∈ R e(xC , x ˜ C , u)). ˜ X (κ + βϵ + λd(u ˜′ ) ∈ R · (x′C , x
Then, we have the following main theorem. Theorem 6. We consider a plant S = (X, X0 , U, r, Y, H), ˆ , rˆ, X, ˆ I), ˆ and a tranˆ X ˆ0, U its abstracted model Sˆ = (X, ˆ ˆ ˆ ˆ ˆ ˆ sition system SC = (XC , XC0 , U , rˆC , XC , IC ) that repreˆ We assume that there exist sents a desired behavior of S. ˆ and a (κ, β, λ)-acASR R(ϵ) ˆ C from SˆC to S, an ASR R from Sˆ to S with d(ˆ u, u). Then, the following relation ¯ ×X ×U ¯ × U is a (κ + κ′ , β, λ)-acASR from S¯C ¯ C (ϵ) ⊆ X R ¯ C , u), u′ ) = d(ˆ to S with d((u u, u′ ), where κ′ is defined by (4). ¯ C (ϵ) = {(xC , x ˜), x, (uC , u), u′ ) | u = u′ , R ˜ ˜ ˜, uC , u) ∈ R(ϵ), (x, x ˜, u, u) ∈ R} (5) (xC , x
˜ ˜ Therefore, R(ϵ) is a (κ + κ′ , β, λ)-acASR from SC to S.
Proof. Obviously, we have that, for any ϵ′ ≥ ϵ ≥ κ + κ′ , ¯ C (ϵ) ⊆ R ¯ C (ϵ′ ). R ¯ C0 . Note that (xC0 , x Next, consider any (xC0 , x ˜0 ) ∈ X ˜0 ) ∈ ˜ 0 , there exists ˜ R(κ). Then, by the definition of x ˜0 ∈ X ˜ X holds. Thus, we have ˜0 ) ∈ R x0 ∈ X0 such that (x0 , x ′ ¯ ((xC0 , x ˜0 ), x0 ) ∈ RCX (κ + κ ). ¯ CX (ϵ). For any Finally, consider any ((xC , x ˜), x) ∈ R ¯ (uC , u) ∈ UC ((xC , x ˜)), u ∈ U (x) holds. Then, we have ¯ C (ϵ). Note that (xC , x ˜ ˜), x, (uC , u), u) ∈ R ˜) ∈ R(ϵ), ((xC , x ′ ˜ and (x, x ˜) ∈ R. For any x ∈ r(x, u), by the existence ˜ from S to S, ˜ there exists x x, u) such of the SR R ˜′ ∈ r˜(˜ ′ ′ ˜ X holds. From Lemma 5, there exists ˜) ∈ R that (x , x ˜ X (κ + κ′ + βϵ + ˜′ ) ∈ R x′C ∈ rC (xC , uC ) such that (x′C , x ′ ˜ ¯ CX (κ + ˜′ ), x′ ) ∈ R λd(uC , u)) holds. Thus, we have ((xC , x ′ ¯ κ + βϵ + λd((uC , u), u)). ¯ C (ϵ) is a (κ + κ′ , β, λ)-acASR from S¯C to S. Therefore, R
ˆ ̸= X ˇ in general. Both X ˇ and X ˆ are the Note that X abstraction of X, but their partitions are different. If we use the state feedback controller SC , the distance between the state of the plant S and the abstracted state of SC after
Intuitively, Theorem 6 shows that the closed system approximately tracks the behavior of SC . If there is no ”com¯C ((xC , x mon” input allowed in Cell∗ (˜ x), U ˜)) is the empty ¯ set, so that SC is blocking. To compute a nonblocking
Finally, consider any ϵ such that ϵ ≥ κ + κ′ , and any ˜ ˆ C and ˜) ∈ R(ϵ), where xC = (ˆ xC , x ˆ) with x ˆC ∈ X (xC , x ˆ By Assumption 3 and the definition of RC (ϵ), x ˆ ∈ X. for any uC ∈ UC (xC ), there uniquely exists u ∈ U (x) such that (xC , x ˜, uC , u) ∈ RC (ϵ) holds. Consider any x ˜′ ∈ ′ ∗ ′ r˜(˜ x, u). Then, for any x ∈ Cell (˜ x ), there uniquely exists ˇ such that x′ ∈ Cell(ˇ x′ ). Thus, by the definition of x ˇ′ ∈ X r˜, there exist x′′ ∈ Cell(ˇ x′ ) and x ∈ Cell∗ (˜ x) such that x′′ ∈ r(x, u). Then, we have |x′′ − x′ | ≤ κ′ . Since RC (ϵ) is a (κ, β, λ)-acASR, there exists x′C ∈ rC (xC , uC ) such that ˜ C , u)). By Assumption 4, (x′C , x′′ ) ∈ RCX (κ + βϵ + λd(u ˜ C , u)), which we have (ˆ x′ , x′ ) ∈ RCX (κ + κ′ + βϵ + λd(u ′ ′ ′ ˜ C , u)). ˜ ˜ ) ∈ RCX (κ + κ + βϵ + λd(u implies that (ˆ x ,x
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sub-transition system of S¯C , we introduce the following ¯ ¯ ¯ operator F : 2X → 2X : For any W ⊆ X, F (W ) = ¯ : r¯C (¯ {¯ x ∈ W |∃¯ u∈U x, u ¯) ̸= ∅ ∧ r¯C (¯ x, u ¯) ⊆ W }. (6) By the definition of F , the following equation holds: ¯ : [F (Z) ⊆ Z] ∧ [Z ′ ⊆ Z ⇒ F (Z ′ ) ⊆ F (Z)], ∀Z, Z ′ ∈ X which means that F is monotonically decreasing. We ¯ i (i ∈ N): consider the following iterations X ¯C , ¯0 = X (7) X ¯ ¯ (8) Xi+1 = F (Xi ). ¯i ⊇ X ¯ i+1 . Thus, if there Then, for each i ∈ N, we have X ¯ k+1 = F (X ¯ k ), then we have the exists k ∈ N such that X ¯ ∗ of F given by X ¯∗ = X ¯k . supremal fixed point X C C Then, if
¯∗ ∩ X ¯ C0 ̸= ∅ X (9) C holds, we define a transition system ∗ ∗ ∗ ¯ C∗ , X ¯ C0 ¯C , r¯C ¯ C∗ ), = (X ,U , Y, H (10) S¯C where ¯∗ = X ¯∗ ∩ X ¯ C0 , • X C0 C ∗ ¯ ∗ for each x ¯ ∗ and • r¯C (¯ xC , u ¯C ) = r¯C (¯ xC , u ¯C )∩ X ¯C ∈ X C C ¯ u ¯ C ∈ UC , ¯ ∗ (¯ ¯ xC ) for each x ¯∗ . • H ¯C ∈ X C xC ) = HC (¯ C ∗ ¯ Thus, SC is a nonblocking sub-transition system of S¯C . ¯ ∗ (ϵ) be the restriction of R ¯ C (ϵ) to X ¯∗ × X × U ¯ × U. Let R C ¯ ∗ (ϵ) is a (κ + κ′ , β, λ)-acASR Then, it is obvious that R C ∗ ¯ C , u), u′ ). from S¯C to S with d((u Recall that we can always design an observer. The observer estimates the state of the plant by listing up candidates from the control input and the observed output. However, (9) has to be satisfied in order for the controller to always choose an input that is suitable for all candidates. 5. ILLUSTRATIVE EXAMPLE
We transform (13) into the transition system S1 = (X1 , X10 , U1 , r1 , Y1 , H1 ) where X1 = X10 = R2 , U1 = R, Y1 = [[−10, 10]]0.4 , H1 ((x1 , x2 )) = round(0.640x1 − 0.832x2 ), where round(y) means rounding y off to the nearest value in Y1 , and r1 is defined in the obvious way. The control signal v is sent via a wireless connection system where data dropouts may occur. We assume that the dropout does not occur consecutively. The behavior of the dropout is also modeled by a transition system S2 = (X2 , X20 , U2 , r2 , X2 , I2 ), where X2 = {0, 1}, X20 = X2 , U2 = {⊥, ⊤} and I2 (w) = w, the identity map. The dynamics r2 is illustrated in Fig. 1, where state 0 means that the data is received, and state 1 means that it is lost. The plant S is given by the composition of S1 and S2 , that is, S := S1 ×R12 S2 = (X12 = X1 × X2 , X120 , U12 = U1 × U2 , r12 , Y12 = Y1 , H12 = H1 ), where R12 = {((x1 , x2 ), w, v, u2 ) | w = 1 ⇒ v = 0}
and r12 (((x1 , x2 ), w), (v, u2 )) = (r1 ((x1 , x2 ), v), r2 (w, u2 )). ˆ1, X ˆ 10 , U ˆ1 , Next, we construct an abstracted model. Sˆ1 = (X ˆ 1 , Iˆ1 ) is based on the abstraction of S1 . We choose rˆ1 , X ˆ 10 = [[−10, 10]2 ]η , and U ˆ1 = {0, ±0.5, ±1.0, ±1.5}. ˆ X1 = X We set the abstraction parameter η = 0.05. Then, x1 , x ˆ2 ), (x1 , x2 ), vˆ, v) | R1 (ϵ) = {((ˆ
x1 , x ˆ2 )| ≤ ϵ] ∧ [v = vˆ]} [|(x1 , x2 ) − (ˆ
(14)
is a (0.05, 0.5, 0)-acASR from Sˆ1 to S1 . Since S2 is finite, its symbolic model is given by Sˆ2 = S2 . We define the abstracted plant Sˆ := Sˆ1 ×Rˆ 12 Sˆ2 , where ˆ 12 = {((ˆ R x1 , x ˆ2 ), w, ˆ vˆ, u ˆ2 ) | w ˆ = 1 ⇒ vˆ = 0}.
Then, the relation
ˆ12 , u12 ) | [(ˆ x 1 , x1 , u ˆ1 , u1 ) ∈ R1 (ϵ)] R(ϵ) = {(ˆ x12 , x12 , u u2 = u2 ]} ∧[ˆ x2 = x2 ] ∧ [ˆ
is a (0.05, 0.5, 0)-acASR from Sˆ to S.
In this section, we consider an example to show how the proposed output feedback controller works. We consider the following system: [ ] [ ][ ] [1] ξ [k + 1] [k] 1 0 ξ 1 1 = + 1 u[k], ξ2 [k] ξ2 [k + 1] 11 (11) 2 [ ] ξ1 [k] ξ[k] = [ 0 1 ] . ξ2 [k]
Since (11) is not asymptotically stable, we consider a low-level state feedback controller so that there exists an acASR from Sˆ to S. In addition, we introduce the following state [x1 , x2 ]T to enforce contraction property. [ ] [ ][ ] x1 [k] −2.929 −1.953 ξ1 [k] = . (12) x2 [k] ξ2 [k] −2.253 −2.704 Then, (11) is transformed into the following equation: [ ] [ ][ ] [ ] x1 [k + 1] 0.25 0 x1 [k] −3.905 + v[k], x [k + 1] = x2][k] 0 0.5 −3.606 2 [ (13) x1 [k] 0.640 −0.832 . [k] = [ ] ξ 2 x [k] 2
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We can observe only an abstracted position y. Thus, we ˜ X ˜ 0 , U, r˜, Y, H). ˜ It is an introduce an observer S˜ = (X, ˇ important issue how to determine an abstracted set X that is used in the design of the observer. Note that the abstracted output y is given by rounding ξ2 off in such ˇ has to satisfy a way that round(ξ2 ) = y. Recall that X ˇ and x ∈ Cell(ˇ the condition that for any x ˇ ∈ X x), ˇ x′ ) holds. Thus, the set of the abstracted states H(x) = H(ˇ for the observer is based on (ξ1 , ξ2 )-plane. On the other hand, in order to satisfy the contraction property, the set of the abstracted states for the controller is based on (x1 , x2 )plane as shown in Fig. 2.
Fig. 1. Transition system model of data dropouts.
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Fig. 2. Abstracted states for the observer (blue points), and abstracted states for the controller (red points). 6. CONCLUSION We considered a CPS consisting of a physical plant and an embedded controller, where the controller determines the input using the output of the plant. On the other hand, to estimate the current state, we introduce an observer that lists up all candidates of the current state. A set of abstracted states for the observer is chosen such that, for each abstracted state, its corresponding states in the plant have the same output. Therefore, the set of abstracted states used for the controller is different from that for the observer. By the observer, we estimate a state set in the plant where the current state of the plant exists. Then, we showed that there exists an acASR from the controller to the observer so that the output feedback controller determines a control input for the plant to exhibit a desired abstracted behavior approximately. Rungger and Tabuada (2014b) showed that pIODS is preserved under an acASR. In this paper, we proved that there exists an acASR from the output feedback controller to the plant. Therefore, it is future work to investigate stability of the controlled plant. REFERENCES R. Goebel, R. G. Sanfelice, and A. R. Teel. Hybrid Dynamical Systems Modeling, Stability, and Robustness. Princeton University Press, 2012. P. Tabuada. Verification and Control of Hybrid Systems. A Symbolic Approach. Springer, 2009. R. Alur, T. A. Henzinger, G. Lafferriere, and G. J. Pappas. Discrete abstractions of hybrid systems. In Proceedings of IEEE, 88(7):971-984, 2000. R. Milner. Communication and concurrency. PrenticeHall, 1989. A. Girard and G. J. Pappas. Approximation metrics for discrete and continuous systems. IEEE Transactions on Automatic Control, 52(5):782-798, 2007. G. Pola, A. Girard, and P. Tabuada. Approximately bisimilar symbolic models for nonlinear control systems. Automatica, 44(10):2508-2516, 2008. A. Girard, G. Pola, and P. Tabuada. Approximately bisimilar symbolic models for incrementally stable switched systems. IEEE Transactions on Automatic Control, 55(1):116-126, 2010. P. Tabuada. An approximate simulation approach to symbolic control. IEEE Transactions on Automatic Control, 53(6):1406-1418, 2008. 253
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