Dissipativity of system abstractions obtained using approximate input-output simulation
Etika Agarwal, Shravan Sajja, Panos J. Antsaklis, Vijay Gupta
DDISSIPATIVITY OF SYSTEM ABSTRACTIONS OBTAINED USINGAPPROXIMATE INPUT-OUTPUT SIMULATION
ETIKA AGARWAL ∗ , SHRAVAN SAJJA † , PANOS J. ANTSAKLIS ∗ , AND
VIJAY GUPTA ∗ Abstract.
This work focuses on the invariance of important properties between continuousand discrete models of systems which can be useful in the control design of large-scale systems andtheir software implementations. In particular, this paper discusses the relationships between theQSR dissipativity of a continuous state dynamical system and of its abstractions obtained throughapproximate input-output simulation relations. First, conditions to guarantee the dissipativity ofthe continuous system from its abstractions are provided. The reverse problem of determining theQ, S and R dissipativity matrices of the abstract system from that of the continuous system is alsoconsidered. Results characterizing the change in the dissipativity matrices are provided when thesystem abstraction is obtained. Since, under certain conditions, QSR dissipative systems are knownto be stable, the results of this paper can be used to construct stable system abstractions as well. Inthe second part of this paper, we analyze the dissipativity of the approximate feedback compositionof a continuous dynamical system and a discrete controller. We present illustrative examples todemonstrate the results of this paper.
Key words.
Dissipativity, passivity, abstraction, simulation
1. Introduction.
Discrete event and hybrid system models for continuous sys-tems are quite common. For example, such models are useful for sampled and quan-tized systems, and also in software implementations of continuous systems. Further-more, such discrete models can be very useful in the control design of large dynamicalsystems, especially when there are temporal logic performance specifications and ver-ification of safety requirements, e.g., in robotic systems. This is the main motivationbehind studying system properties of interest that are present in both continuous anddiscrete models of the same system.These discrete models can be obtained using abstraction based approaches. Seefor example, [1]-[7]. An abstracted system model approximates a continuous statedynamical system by a system with a pre-order or equivalence relation between thetwo systems. Control design of dynamical systems using abstraction based approachescan be carried out efficiently based on two main factors [8], first, the possibilityof constructing symbolic or purely discrete abstractions of the original system andsecond, the possibility to infer the behavior of the given continuous system based uponits discrete abstraction. The idea of using discrete abstractions for control design ismotivated by the fact that control of a continuous system with a discrete controllerrequires the use of a continuous to discrete and discrete to continuous conversion. Thisset-up can be viewed as an interconnection of a continuous system with a softwaresystem, as shown in Figure 1.Dissipativity based approaches provide attractive alternatives to control analysisand design of such systems. These dissipativity approaches have been used for systemswith control performance described in terms of stability and optimality requirements[9][10] for continuous systems. Dissipativity is an energy based input-output propertyof dynamical systems [11]. A special form of dissipativity is QSR dissipativity whichallows transparent relationships with several important concepts such as passivityand L stability [12]-[14]. Further, QSR dissipativity is preserved over feedback andparallel interconnections; and series interconnections under certain conditions [15]. ∗ Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN ([email protected], [email protected], [email protected]) † IBM Research, Bangalore, India ([email protected])1 a r X i v : . [ c s . S Y ] N ov ence, an added advantage of using control analysis and design techniques based onQSR dissipativity is that they scale well.More recently, [16][17] used dissipativity like concepts for compositional analysis ofinterconnected systems with safety and temporal logic specifications. This opens up anew application area for dissipativity theory where controllers can be designed to meettemporal logic constraints (such as safety and reachability constraints) together withtraditional specifications such as passivity, stability and optimality. As a first steptowards this goal, in this paper, we analyze the dissipativity of system abstractionsand find relationships between the QSR dissipativity, and hence passivity of systemsand their approximately input-output similar abstractions [18][19]. Discrete to continuous Continuous to discreteMeasurement Symbol SymbolControl inputPhysical SystemSoftware Discrete abstraction
Fig. 1 . Finite state approximation of a continuous-time plant interacting with a finite statecontroller (software).
There are several approaches to system abstraction. Different techniques pre-serve different properties of the system such as reachability [3] and compositionality[4] while maintaining equivalence or pre-order in a certain sense between the two sys-tems. One of the popular approaches for abstraction is using the notions of simulation,bisimulation and their approximate versions [3][1] which have been used for both dis-crete and continuous time systems [1][5][6][7]. While these notions are compositionpreserving in some sense [1], approximate simulation and bisimulation are not pre-served for more general input-output interconnections such as cascade and feedback.Tazaki et al [18] proposed a modification of the definitions of approximate simula-tion and bisimulation which, under mild conditions, are interconnection preserving.This means that the abstraction of interconnected system is the interconnection ofsystem abstractions. However, [20] identified that stability is not preserved for simu-lation (or bisimulation) and additional continuity constraints need to be imposed onthe simulation relation. Prabhakar et al [21][22] also proposed some continuity con-straints on simulation relation to preserve variants of stability during abstraction. Analternate approach is offered in the present paper using dissipativity. In this work,we are interested in inferring QSR dissipativity properties of discrete abstractionsfrom that of the corresponding continuous system. These discrete abstractions areobtained using a slight modification of method in [7] and are related to the continuoussystem through approximate input-output simulation relation. The definition of ap-proximate input-output simulation relation used in this paper is a special case of theinterconnection-compatible approximate simulation relation in [18]. QSR dissipativesystems, under mild additional conditions, are stable. This allows us to obtain stableand composable system abstractions without imposing any strict constraints on thesimulation relation. Related work on the passivity of bisimilar systems was done in able 1 Notation
Notation Meaning (cid:107) · (cid:107) infinity norm (cid:107) · (cid:107) Euclidean norm (or induced 2-norm if argument is a matrix) Z +0 set of nonnegative integers R set of real numbers R + set of positive real numbers R +0 set of nonnegative real numbers R n Euclidean space of dimension n P > Q matrix ( P − Q ) is positive definite
2. Preliminaries.
In this section we briefly explain few important notions of ab-stract systems and introduce the properties of dissipativity, passivity and incrementalforward completeness.
The system model we use in this work is that oftransition systems. The following definitions are standard and can be found in [7].
Definition A transition system T = ( X, U, −→ , Y m , H ) consists of: • a set of states X; • a set of inputs U; • a transition map −→ : X × U → X ; • a set of outputs Y ; • and an output map H : X × U → Y . If for any state x ∈ X and u ∈ U there exists at most one state x (cid:48) ∈ X such that x u −−−→ x (cid:48) then the system is deterministic. x (cid:48) is also known as the u -successor of x .If the system is nondeterministic, then for a transition x u −−−→ x (cid:48) the state x (cid:48) maynot be unique. In such a case x (cid:48) belongs to a set of all possible u -successors given by P ost u ( x ) and we will use U ( x ) to denote the set of inputs u ∈ U for which P ost u ( x )is nonempty. Suppose the transition system is equipped with metrics d X : X × X −→ R +0 , d U : U × U −→ R +0 and d Y : Y × Y −→ R +0 representing the “distance” betweentwo elements of state space, input space and output space respectively. This transitionsystem is referred to as a metric transition system . Throughout this work, weassume that the transition system allows for a notion of inner product between inputsand outputs and the distance metric d X , d U and d Y are infinity norms (cid:107) · (cid:107) .Transition systems can be used to describe a large class of dynamical systems.We restrict ourselves to continuous time dynamical systems of the form(1) Σ = ( X, U, Y m , f, h m )where X = R n is the state space; U ⊆ R m : { } ∈ U is the input space; Y m ⊆ R p : { } ∈ Y m is the measured output space; f : X × U → X is a Lipschitz continuous mapdescribing state transition and h m : X × U → Y m is the measured output map. At atany time t ∈ R +0 , the state, input and measured output of Σ are x ( t ) ∈ X , u ( t ) ∈ U , y m ( t ) ∈ Y m and the state and measured output evolve as ˙ x ( t ) = f ( x ( t ) , u ( t )) and y m ( t ) = h m ( x ( t ) , u ( t )). If ξ :] a, b [ −−→ X is a solution of the differential equation˙ x ( t ) = f ( x ( t ) , u ( t )), then we will use ξ ( t, x, u ) to denote a unique point reached at ime t under the input signal u : [0 , t ] → U from an initial condition x ∈ X . Thetransition system associated with Σ is then given by T (Σ) = ( X, U, −→ , Y m , h m )where the state transition map is dictated by the differential equation ˙ x = f ( x, u ).We use notation T (Σ) and T interchangeably in this work. Also, for ease of notation,we will often drop the time index when referring to the state, input and output of Σ.For dissipativity analysis, we consider a separate system output dictated by sys-tem output space Y ⊆ R m and output map h ( x ( t ) , u ( t )) : X × U → Y . The measuredoutput h m ( x ( t ) , u ( t )) can be different from system output h ( x ( t ) , u ( t )). In this work,we consider two different cases (i) when measured output is the system states, and(ii) when measured output is same as the system output. The measured output space Y m and output map h m take values accordingly.We can also define a discrete time system Σ d = ( X d , U d , Y m d , f d , h m d ) where X d , U d , Y m d , f d and h m d are state space, input space, measured output (or measure-ment) space, state transition map and measured output transition map respectively.At any discrete time k the system state x d ( k ) ∈ X d , input u d ( k ) ∈ U d and out-put y m d ( k ) ∈ Y m d evolve in discrete time steps as x d ( k + 1) = f d ( x d ( k ) , u d ( k )) and y m d ( k ) = h m d ( x d ( k ) , u d ( k )) for all k ∈ Z +0 . Similar to continuous time case, systemoutput (used for dissipativity) dictated by system output space Y d ⊆ R m and outputmap h d ( x d ( k ) , u d ( k )) : X d × U d → Y d , can be different from measured output.The following assumptions on system behavior are useful in deriving the mainresults of this paper. Assumption (Incremental forward completeness [7]) The dynamical system Σ is said to be incrementally forward complete if there exist continuous functions α : R +0 × R +0 → R +0 , and α : R +0 × R +0 → R +0 , α ( · , t ) , α ( · , t ) ∈ K ∞ for every t ≥ , such that for any two initial conditions x , x ∈ X , any input trajectories v , v : [0 , t ] ∈ U , and for any t ∈ R +0 the following bound holds: (2) (cid:107) ξ ( t, x , v ) − ξ ( t, x , v ) (cid:107) ≤ α ( (cid:107) x − x (cid:107) , t ) + α ( (cid:107) v − v (cid:107) , t ) where ξ ( t, x i , v i ) is the state trajectory of system with input v i and initial state x i . It should be noted that incremental forward completeness is a weaker conditionthan incremental stability for it does not require the system to be stable. It onlystates that the distance between any two state trajectories is bounded.
Assumption Assume that the operator from input u ( t ) to rate of change ofsystem output ˙ y ( t ) has the finite L gain γ , that is (cid:90) τ (cid:107) ˙ y ( t ) (cid:107) d t ≤ γ (cid:90) τ (cid:107) u ( t ) (cid:107) d t for any τ ≥ and admissible input u ( t ) . Assumption 3 is an L gain condition which bounds the rate at which the output y can change with respect to time.We define transition systems T τ (Σ) obtained after sampling Σ, and T τ,µ,η (Σ)obtained after appropriate sampling and quantization of Σ as follows. Definition [7] Let Σ be a dynamical system and the associated transitionsystem be T (Σ) . For any τ > , the sampled transition system T τ (Σ):=( X τ , U τ , u τ −−→ τ ,Y m τ , H m τ ) is defined by: • X τ = X ; • U τ = U ; x τ u τ −−−−→ τ x (cid:48) τ , if there exists a trajectory ξ : [0 , τ ] −−−−→ ξ ( τ, x τ , u ) = x (cid:48) τ where u : [0 , τ ) → u τ , u τ ∈ U τ ; • Y m τ = Y m ; • H m τ ( x τ , u τ ) = h m ( x τ , u τ ) where x τ ∈ X τ , u τ ∈ U τ . T τ (Σ) evolves in discrete time and can be represented as a discrete time system(like Σ d ). As discussed before, for dissipativity analysis, we consider another output(referred to as system output in this work) associated with T τ (Σ). At any discretetime step k , system output is described by the output space Y ⊆ R m and outputfunction h ( x ( k ) , u ( k )) = y ( k ) ∈ Y where x ( k ) ∈ X τ and u ( k ) ∈ U τ . Definition For any incrementally forward complete control system Σ , withsystem states as measurement, i.e., h m ( x, u ) = x , and parameters τ > , η > , µ > and design parameters θ , θ ∈ R + , a countable transition system can be defined as, T τ,µ,η (Σ) := ( X q , U q , u q −−−−→ τ , Y m q , H m q ) , where: • X q = [ X ] η = (cid:8) x ∈ X | x i = k i η, k i ∈ Z , and i = 1 , , . . . , n (cid:9) ; • U q = [ U ] µ = (cid:8) u ∈ U | u i = k i µ, k i ∈ Z , and i = 1 , , . . . , m (cid:9) ; • x q u q −−−−→ τ x (cid:48) q , if (cid:107) ξ ( τ, x q , u q ) − x (cid:48) q (cid:107) ≤ α ( θ , τ ) + α ( θ , τ ) + η/ ; • Y m q = [ X ] η = (cid:8) x ∈ X | x i = k i η, k i ∈ Z , and i = 1 , , . . . , n (cid:9) ; • H m q ( x q , u q ) = x q where x q ∈ X q , u q ∈ U q . α and α are functions from the definition of incremental forward completeness inAssumption . Here, H m q ( x q , u q ) = x q indicates that measured output is same as the systemstates. At any discrete time instant k ∈ Z +0 system output (used for dissipativityanalysis) of T τ,µ,η (Σ) here is h ( x q ( k ) , u q ( k )) where x q ( k ) ∈ X q and u q ( k ) ∈ U q . Notethat this set up is often useful when analyzing systems with state feedback.We also define the sampled and quantized transition system for the case whensystem output is the measurement. Definition For any incrementally forward complete control system Σ , withsystem output as measurement, i.e., h m ( x, u ) = h ( x, u ) and parameters τ > , η > , µ > and design parameters θ , θ ∈ R + , a countable transition system can be definedas T τ,µ,η (Σ) := ( X q , U q , u q −−−−→ τ , Y m q , H m q ) , where: • X q = [ X ] η = (cid:8) x ∈ X | x i = k i η, k i ∈ Z , and i = 1 , , . . . , n (cid:9) ; • U q = [ U ] µ = (cid:8) u ∈ U | u i = k i µ, k i ∈ Z , and i = 1 , , . . . , m (cid:9) ; • x q u q −−−−→ τ x (cid:48) q , if (cid:107) ξ ( τ, x q , u q ) − x (cid:48) q (cid:107) ≤ α ( θ , τ ) + α ( θ , τ ) + η/ ; • Y m q = [ Y ] µ = (cid:8) y ∈ Y | y i = k i µ, k i ∈ Z , and i = 1 , , . . . , m (cid:9) ; • H m q ( x q , u q ) = h q ( x q , u q ) : (cid:107) h q ( x q , u q ) − h ( x q , u ) (cid:107) ≤ µ/ where u : [0 , τ ) → u q , x q ∈ X q , u q ∈ U q .α and α are functions from the definition of incremental forward completeness inAssumption . As compared to Definition 5, since Definition 6 has system output as measure-ment, system output (used for dissipativity analysis) is also quantized and is describedby the output map H m q : X q × U q → Y m q in Definition 6.The transition system T τ,µ,η (Σ) can be countably finite or infinite dependingon the size of state and input spaces. For most practical cases, the system statesand inputs are restricted due to the physical limitations of the system leading to acountably finite T τ,µ,η (Σ). .2. System Relations. Abstraction is an approach to reduce the complexity ofthe description of dynamical systems. One of the popular approaches for abstraction isusing the notion of approximate simulation [1]. Since dissipativity is an input-outputproperty, we talk about a generalized notion of approximate input-output simulation[19].Consider two metric transition systems T and T . The approximate input -output simulation relation can be defined as follows. Definition T is an approximate input-output simulation of T with precision ( (cid:15) u , (cid:15) y ) if there exists an approximate input-output simulation relation R ⊆ X × X such that for all x ∈ X , there exists x ∈ X such that ( x , x ) ∈ R and, for all ( x , x ) ∈ R : for all u ∈ U ( x ) there exists u ∈ U ( x ) such that d U ( u , u ) ≤ (cid:15) u and d Y ( H ( x , u ) , H ( x , u )) ≤ (cid:15) y , for all u ∈ U ( x ) there exists u ∈ U ( x ) such that d U ( u , u ) ≤ (cid:15) u and x u −−−−→ x (cid:48) in T implies the existence of x u −−−−→ x (cid:48) in T such that ( x (cid:48) , x (cid:48) ) ∈ R .This is denoted as T (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T . We can decompose the input and output vectors into two groups, internal andexternal signals as in [18] to facilitate the discussion on system interconnection. It iseasy to see that in this case, Definition 7 becomes a special case of interconnection-compatible approximate simulation in [18] where not only the internal inputs butalso external inputs of the two interconnection-compatible approximately similar sys-tems are required to be close enough to each other. As such, under mild conditions(Theorem 1 in [18]), we can interconnect abstractions (approximate input-output sim-ulation) of systems to obtain an abstraction of interconnection of systems.Similarly, we can also define the notion of approximate input-output alternatingsimulation to take into account the non-deterministic nature of system trajectories.
Definition [7] T is an approximate input-output alternating simulation of T with precision ( (cid:15) u , (cid:15) y ) if there exists an approximate input-output alternating sim-ulation relation R ⊆ X × X such that for all x ∈ X , there exists x ∈ X suchthat ( x , x ) ∈ R and, for all ( x , x ) ∈ R : for all u ∈ U ( x ) there exists u ∈ U ( x ) such that d U ( u , u ) ≤ (cid:15) u and d Y ( H ( x , u ) , H ( x , u )) ≤ (cid:15) y , for all u ∈ U ( x ) there exists u ∈ U ( x ) such that d U ( u , u ) ≤ (cid:15) u and forevery x (cid:48) ∈ P ost u ( x ) there exists x (cid:48) ∈ P ost u ( x ) such that ( x (cid:48) , x (cid:48) ) ∈ R .This is denoted as T (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T . The two notions of alternating approximate input-output simulation and approximateinput-output simulation coincide in the special case of deterministic systems.
A dynamical system is said to be dissipative if it stores anddissipates energy but does not generate any energy of its own. The notion of energymentioned here is general and is captured using the energy supply rate function. As de-scribed in [11], the supply rate function is a real valued function described on the inputand output space and is locally integrable, i.e., ω : U × Y −→ R , (cid:82) t t | ω ( u ( t ) , y ( t ) | dt < ∞ ∀ t , t ∈ R +0 . Definition [11] A continuous time system Σ with output function y ( t ) = h ( x ( t ) , u ( t )) , is said to be dissipative with respect to the supply rate function ω ( u, y ) , f there exists a nonnegative function V : X → R + , called the storage function, suchthat for all t ≥ t ≥ t ≥ , x ( t ) ∈ X and u ( t ) ∈ U (3) (cid:90) t t ω ( u ( t ) , y ( t )) dt ≥ V ( x ( t )) − V ( x ( t )) holds where x ( t ) is the state at time t resulting from the initial condition x ( t ) andinput function u ( · ) . Definition 9 discusses dissipativity of continuous time system. Similarly, we candefine dissipativity for discrete time system (or sampled transition system) as follows.
Definition [29] A discrete time system Σ d with output function y ( k ) = h ( x ( k ) , u ( k )) , is said to be dissipative with respect to the supply rate function ω ( u, y ) ,if there exists a nonnegative function V : X d → R + , called the storage function, suchthat for all k ≥ k ≥ , x ∈ X d and u ( k ) ∈ U d (4) k − (cid:88) i = k ω ( u ( i ) , y ( i )) ≥ V ( x ( k )) − V ( x ( k )) holds where x ( k ) is state at k resulting from the initial condition x ( k ) and inputfunction u ( · ) . We can obtain different dissipativity structures and system properties dependingon the choice of supply rate function.
Definition (QSR dissipativity and passivity) • A continuous time system Σ is said to be QSR dissipative if it is dissipativewith respect to the supply rate ω ( u, y ) = y T Qy + 2 y T Su + u T Ru . Dissipativityinequality is then given as (5) (cid:90) t t ( y T ( t ) Qy ( t ) + 2 y T ( t ) Su ( t ) + u T ( t ) Ru ( t )) dt ≥ V ( x ( t )) − V ( x ( t )) where Q , S and R are matrices of appropriate dimensions. • Similarly, a discrete time system Σ d is said to be QSR dissipative if it isdissipative with respect to the supply rate ω ( u, y ) = y T Qy + 2 y T Su + u T Ru .Dissipativity inequality is then given as (6) k − (cid:88) i = k ( y T ( i ) Qy ( i ) + 2 y T ( i ) Su ( i ) + u T ( i ) Ru ( i )) ≥ V ( x ( k )) − V ( x ( k )) where Q , S and R are matrices of appropriate dimensions. • A continuous time system Σ (or discrete time system Σ d ) is said to be inputfeed-forward output feedback passive if it is QSR dissipative with Q = − ρ I , S = I , R = − ν I with ρ, ν ≥ . In this work, we refer to
Q, S, and R matrices as dissipativity matrices. A specialcase of QSR dissipativity is input feed-forward output feedback passivity with ν and ρ known as the input and output passivity index. In this work, we use the termspassive and input feed-forward output feedback passive interchangeably. Definition (Quasi-dissipativity [27]) A continuous time system Σ (or dis-crete time system Σ d ) is said to be quasi-dissipative with respect to ω ( u, y ) if thereexists a constant β ≤ such that it is dissipative with respect to the supply rate ω ( u, y ) − β . he boundedness and stability of quasi-dissipative systems is discussed in [28].Similar to dissipativity, quasi-dissipativity also can take different forms dependingon the structure of supply rate function ω ( u, y ). Quasi-QSR-dissipativity and quasi-passivity are two special cases that are of interest to us. They can be defined similar toQSR dissipativity and passivity described earlier. In this work, dissipativity for sys-tems in Definition 5 and Definition 6 is discussed in the context of quasi-dissipativitywhere, the presence of β on the right hand side of (8) indicates the energy generateddue to quantization process. Remark
The definition of dissipativity here is independent of system repre-sentation, i.e., if Σ is dissipative then T (Σ) is also dissipative. Also note that whiletransition system T (Σ) follows dissipativity definition for continuous time system, tra-jectories of transition systems in Definition , Definition and Definition evolvein discrete time and hence follow the dissipativity definition for discrete-time systems.Moreover, [29] showed that for discrete time systems, (4) holds if and only if (7) ω ( u ( k ) , y ( k )) ≥ V ( x ( k + 1)) − V ( x ( k )) for all k ∈ Z +0 , u ( k ) ∈ U and x ( k ) ∈ X . Equivalent condition for quasi-dissipativityis, (8) ω ( u ( k ) , y ( k )) ≥ V ( x ( k + 1)) − V ( x ( k )) + β ∀ k ∈ Z +0 , u ( k ) ∈ U d , x ( k ) ∈ X d .
3. Dissipativity of systems and their abstractions.
In this section we dis-cuss the relationship between the dissipativity properties of a continuous system andits abstraction. We provide two main results. First, we analyze the dissipativity of acontinuous dynamical system when its approximate input-output simulation is QSRdissipative. Secondly, we consider the reverse problem of determining the dissipativityof a system abstraction. We provide conditions under which QSR dissipativity of acontinuous system implies QSR dissipativity of its discrete abstraction obtained usingthe approach in [7].
Con-sider two continuous time systems Σ and Σ and corresponding transition systems T (Σ ) and T (Σ ) . Let u i , y i and x i represent respectively the input, output andstates of T i , i ∈ { , } and both these systems allow a notion of inner product be-tween their inputs and outputs. Suppose T is QSR dissipative with Q , S , R asthe dissipation matrices. If T approximately simulates T then the following theoremgives the conditions under which T is QSR dissipative.
Theorem If T (Σ ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T (Σ ) and T (Σ ) is QSR dissipative, then T (Σ ) is also QSR dissipative with matrices Q , S , R satisfying λ ( Q − Q ) − ζ (cid:107) Q (cid:107) − ζ ≥ λ ( R − R ) − ζ (cid:107) S (cid:107) − ζ (cid:107) R (cid:107) ≥ S = S (9) where ζ , ζ , ζ , ζ ∈ R + are arbitrary non-zero constants, Q , S , R are the dissipa-tivity matrices for T (Σ ) and λ ( · ) represents the smallest eigen value of the matrixin discussion.Proof. See Appendix. lthough the result of Theorem 14 is derived for continuous time systems, it canbe extended to discrete time dynamical systems as well. This result is general inthe sense that it is applicable irrespective of the method which is used to obtain theapproximate input-output simulation. We can also use it to compute upper boundson the passivity indices of transition system T . Corollary If T (Σ ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T (Σ ) and T (Σ ) is passive with passivityindices ρ , ν then T (Σ ) is also passive with passivity indices ρ , ν which satisfythe following ρ (1 + ζ ρ ) ≤ ρ − ζ ν (1 + ζ ν ) ≤ ν − ζ , (10) where ζ , ζ , ζ , ζ ∈ R + are arbitrary non-zero constants such that ρ − ζ > and ν − ζ > , and ρ , ν are the passivity indices for T (Σ ) .Proof. Use Definition 11 in Theorem 14 to obtain this result.The result in Theorem 14 states that if condition (9) is met, then the QSR dissipa-tivity of an approximate input-output simulation of transition system T (Σ ) impliesthe QSR dissipativity of T (Σ ) itself. Note that the reverse of this result is nottrue in general. This can be seen from the definition of approximate input-outputsimulation. For every transition in T (Σ ) there exists a corresponding approximatetransition in T (Σ ). However, T (Σ ) can be a larger system in the sense that theremight be some transitions in T (Σ ) for which there is no corresponding transitionin T (Σ ). Therefore, from Theorem 14, QSR dissipativity of T (Σ ) implies theQSR dissipativity of T (Σ ) and not the other way around in general. In the nextpart of this section, we consider this reverse problem of determining the Q , S and R dissipativity of an approximate input-output simulation from the QSR dissipativityof the original incrementally forward complete continuous system under a particularabstraction technique. Thissection considers the problem of determining dissipativity matrices of the abstractionof a dissipative system. Zamani et al [7] showed that the approximate simulationof incrementally forward complete systems can be computed using time and spacequantization. We make a slight modification to this procedure by introducing an extradesign parameter to obtain finite abstractions which are approximately input-outputsimilar to the original system. We then quantify the change in
QSR dissipativityof the system model under such abstraction. Since in this work the abstraction isobtained with respect to measured output of the system, it makes sense to considertwo different cases, (i) when the measured output of system is same as system statesand, (ii) when measured output is same as the system output.
In this section we discuss the dissi-pativity properties of approximate input-output simulation of sampled data systems.The particular class of systems we address here are the ones where measured andactual output of the system are different. For this purpose, we use states as the mea-sured output of T τ (Σ). However, for dissipativity analysis, we use an alternate outputcorresponding to y = h ( x, u ).There have been several approaches to obtain approximate simulation of systems.Most of them concentrate on a restrictive class of systems. [7] introduced a new pro-cedure for construction of abstractions for any non-linear sampled data system which re incrementally forward complete. The approach discussed in [7] provides sufficientconditions in terms of appropriate sampling time and quantization parameters to ob-tain countable transition systems guaranteeing approximate (alternating) simulation.We use this technique to construct approximate input-output (alternating) simula-tion for sampled data systems T τ (Σ) and analyze the dissipativity properties of thusobtained abstract system. Proposition
Consider a control system Σ in (1) whose states are the mea-sured output. Given any desired precision parameters (cid:15) y > , (cid:15) u > , if Σ satis-fies Assumption then for any { τ, θ , θ , η, µ } > satisfying η/ ≤ (cid:15) y ≤ θ and µ/ ≤ (cid:15) u ≤ θ , we have: (11) T τ,µ,η (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T τ (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T τ,µ,η (Σ) where T τ (Σ) and T τ,µ,η (Σ) are defined in Definition and Definition respectively.Proof. Proof follows directly from Theorem 4.1 in [7]. An outline of the proof canbe found in Appendix.We next analyze the
QSR dissipativity of approximately input-output similarsystem T τ,µ,η (Σ) and consider the case where measured output is same as systemstates. Theorem
Consider a dynamical system Σ in (1) whose states are the mea-sured output. Suppose Σ satisfies Assumptions 2 and 3 and it is QSR dissipa-tive with respect to the output function y = h ( x, u ) and a storage function V ( · ) : V ( x ) − V ( x ) ≤ L (cid:107) x − x (cid:107) . Let T τ (Σ) be the transition system corresponding to Σ with a sampling time τ . If the input and state quantization parameters µ and η are chosen such that T τ,µ,η (Σ) in Definition is ( (cid:15) u , (cid:15) y ) - approximately input-outputsimilar to T τ (Σ) , then T τ,µ,η (Σ) is quasi QSR dissipative with matrices Q τ,µ,η , S τ,µ,η , R τ,µ,η satisfying, Q τ,µ,η ≥ Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I S τ,µ,η = SR τ,µ,η ≥ R + τ γ (cid:107) S (cid:107) I + τ γ (cid:107) Q (cid:107) ( τ γ + τ + γ ) I (12) where Q, S, and R are the dissipativity matrices for Σ .Proof. See Appendix.
In the last section, we providedresults for the dissipativity properties of approximate input-output simulation forthe class of systems where measured output is same as the system states. We nowextend these results to the systems where measured output is same as the actualsystem output. To do this, we make an additional assumption on the system outputto be Lipschitz continuous which means that output can not change abruptly. Thedifference from previous section is that system output y = h ( x, u ) is also sampled andquantized here. This can be useful for design of systems with output feedback. Proposition
Consider a control system Σ in (1) whose measured output isthe same as system output. Given any desired precision parameters (cid:15) y > , (cid:15) u > ,if Σ satisfies Assumption and the output function is Lipschitz continuous, i.e., (cid:107) h ( x , u ) − h ( x , u ) (cid:107) ≤ K (cid:107) x − x (cid:107) + K (cid:107) u − u (cid:107) , then for any { τ, θ , θ , η, µ } > satisfying K η/ K + 1) µ/ ≤ (cid:15) y , η/ ≤ θ and µ/ ≤ (cid:15) u ≤ θ , we have: (13) T τ,µ,η (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T τ (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T τ,µ,η (Σ) here T τ (Σ) and T τ,µ,η (Σ) are defined in Definition and Definition respectively.Proof. Proof follows directly from Theorem 4.1 in [7]. An outline of the proof canbe found in Appendix.
Theorem
Consider a dynamical system Σ in (1) whose measured output isthe same as system output. Suppose Σ satisfies Assumptions 2 and 3 and it is QSR dissipative system with respect to the output function y = h ( x, u ) and a storage func-tion V ( · ) : V ( x ) − V ( x ) ≤ L (cid:107) x − x (cid:107) . Let T τ (Σ) be the transition system corre-sponding to Σ with a sampling time τ . If the input and state quantization parameters µ and η are chosen as per Proposition so that T τ,µ,η (Σ) in Definition is ( (cid:15) u , (cid:15) y ) -approximately input-output similar to T τ (Σ) , then T τ,µ,η (Σ) is quasi QSR dissipativewith matrices Q τ,µ,η , S τ,µ,η , R τ,µ,η satisfying, Q τ,µ,η ≥ Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I + ( (cid:107) Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I (cid:107) ) I S τ,µ,η = SR τ,µ,η ≥ R + τ γ (cid:107) Q (cid:107) ( τ γ + τ + γ ) I + (cid:107) S (cid:107) I + ( γ √ mτ µ + γ τ ) I (14) where Q, S, and R are the dissipativity matrices for Σ .Proof. See Appendix.
Remark
20. 1. The inequalities in (9) restrict only the spectral radius ofthe dissipation matrices. This gives some flexibility in choosing the actualstructure of the Q and R matrices.2. Since passivity is a special case of QSR dissipativity, results guaranteeing thepassivity levels for abstract system can be derived from Theorems 17 and 19in a straight forward manner.3. It should be noted that the notion of approximate input-output simulationhere, under mild conditions, is composition preserving [18] in that the inter-connection of two abstractions is the same as the abstraction of interconnec-tion of two systems. Also, passivity is preserved over feedback and parallelinterconnections [11]. Therefore, the composition of passivity of abstractionsis the same as the passivity of abstraction of composition. This is an in-teresting result that can be used to reason about the passivity properties ofabstractions of large scale systems.4. Symbolic models obtained using abstractions have been used for design andanalysis of control systems [8]. However, issues like stability are in generaldifficult to address using symbolic models. Theorems 17 and 19 can be usedto infer dissipativity properties of symbolic models obtained using approx-imate input-output simulation based abstractions. Once the system designis complete, the dissipativity properties of symbolic model can be translatedback to that of the original system using the discrete version of results inTheorem 14. Since, QSR dissipativity implies system stability under cer-tain conditions, apart from offering compositionality, these results on QSRdissipativity can be used to guarantee stability of system abstractions as well.
4. Examples.
In this section, we present two illustrative examples. We describetwo simple linear time invariant (LTI) systems and discuss how the ideas in this papercan be used to obtain a symbolic model and carry out the dissipativity analysis. Forboth these examples, we consider input feed-forward output feedback passivity whichis a special case of
QSR - dissipativity. xample 1. Consider an LTI system Σ : ˙ x = − x + u , with the measured outputdescribed by an identity map, i.e., y m = x . For dissipativity analysis, we consideranother output function y = h ( x, u ) = Cx + Du = x + u . It can be verified thatthis system is input feed-forward output feedback passive with respect to an outputfunction y = x + u and V ( x ) = x T P x = x T (0 . x . The passivity index are(0 . , . X = [ − . , .
2] of the statespace and subset U = [ − . , .
1] of the input space. To construct the symbolic model T τ,µ,η (Σ) of precision (cid:15) u = 0 . , (cid:15) y = 1, choose θ = 1, η = 0 . θ = (cid:15) u = µ = 0 . τ = 0 . µ = 0 . τ = 0 .
2, the control input is piecewise constant signal of duration τ such that {− µ, , µ } = { u − , u , u } = {− . , , . } ∈ U q , and the states of the symbolic system are described by {− η, − η, , η, η } = {− . , − . , , . , . } ∈ X q . η η − η − η u − , u , u u − , u , u u − , u , u u , u u − , u , u u − u u − , u u − , u , u u − , u , u u − , u , u Fig. 2 . Symbolic model for Σ . The transitions between states upon the action of a control input can be calculatedusing the differential equation describing Σ. As seen in Figure 2, the symbolic model T τ,µ,η (Σ) is non-deterministic.The effect of symbolic abstraction on the passivity properties of Σ using The-orem 17. It can be verified that the output y = x + u satisfies Assumption 3 for γ = 1. Hence, using Theorem 17 we can state that T τ,η,µ (Σ) is ( ρ τ,µ,η , ν τ,µ,η )- inputfeedforward output feedback quasi passive where ρ τ,µ,η = ρ − τ (cid:107) ρ (cid:107) ( τ γ + 1) = 0 . ν τ,µ,η = ν − τ γ (cid:107) . (cid:107) − τ γ (cid:107) ρ (cid:107) ( τ γ + τ + γ ) = 0 . (cid:96) T o − ρ τ,µ,η o T o − ν τ,µ,η (cid:96) T (cid:96) ≥ ˆ V ( p ) − ˆ V ( q ) − β is satisfied for all transitions q (cid:96) −−−−→ τ p where p ∈ Post (cid:96) ( q ), o = C q + D (cid:96) , ˆ V ( q ) = τ q T P q and β = Lη τ , i.e., q T F q + (cid:96) T G q + q T G T (cid:96) + (cid:96) T H (cid:96) + τ β − p T P p ≥ F = P − ρ τ,µ,η τ C T C, G = τ C − ρ τ,µ,η τ D T C, H = τ ( D + D T ) − ρ τ,µ,η τ D T D − ν τ,µ,η τ I. Candidate values for ˆ V ( · ) and β are obtained from the proof of Theorem 17.For the symbolic system, we assume that there are M quantized inputs denoted by { (cid:96) , (cid:96) , . . . , (cid:96) M } and there are N quantized states denoted by { q , q , . . . , q N } . All the ransitions in the symbolic system can be represented by q i (cid:96) j −−−−→ τ p ji for i = 1 , . . . N and j = 1 , . . . , M, where p ji represents the next state after time τ with an initial state q i , under the action input (cid:96) j . Hence, passivity verification would entail verificationof the inequality q Ti F q i + (cid:96) Tj G q i + q Ti G T (cid:96) j + (cid:96) Tj H (cid:96) j + τ β −
12 ( p ji ) T P ( p ji ) ≥ i = 1 , . . . N and j = 1 , . . . , M. In order to verify the above inequality for alltransitions in a systematic fashion, welet ¯ q = (cid:2) q , · · · , q N (cid:3) T , ¯ (cid:96) = (cid:2) (cid:96) , · · · , (cid:96) M (cid:3) T andarrange vectors ¯ p = (cid:2) p , · · · , p N (cid:3) T , . . . , ¯ p M = (cid:2) p M , · · · , p MN (cid:3) T together as ¯¯ p = (cid:2) ¯ p , · · · , ¯ p M (cid:3) T . Verification of (17) for i = 1 , . . . N and j = 1 , . . . , M would requireus to verify positivity of M N scalars. All these
M N scalars will be arranged alongthe diagonal of an
M N × M N matrix, and this diagonal matrix would be checkedfor its positive definiteness. This approach allows us to represent all the inequalitiestogether in a compact fashion. This compact representation will be achieved usingthe Kronecker product as given by
P ASSIV E = I M ⊗ (( I N ⊗ ¯ q T )( I N ⊗ F )( I N ⊗ ¯ q )) + (( I N ⊗ ¯ (cid:96) T )( I N ⊗ G )( I N ⊗ ¯ q )) ⊗ I M + (( I N ⊗ ¯ q T )( I N ⊗ G T )( I N ⊗ ¯ (cid:96) )) ⊗ I M + (( I M ⊗ ¯ (cid:96) T )( I M ⊗ H )( I M ⊗ ¯ (cid:96) )) ⊗ I N + I MN ⊗ K(cid:15) y τ − ¯¯ p T ( I MN ⊗ P )¯¯ p ≥ For the nondeterministic cases where ¯¯ p is not unique, we verify (18) for all possiblevalues of ¯¯ p . Performing this test for our numerical example, we obtain the diagonalelements of the PASSIVE matrix for two possible values of ¯¯ p and it can verified thatall the diagonal elements are positive, hence confirming the passivity of the symbolicmodel.This example demonstrates that the results in this paper can be used to avoidlarge computations for determining passivity of discrete abstractions of continuoussystems. Example 2.
In this example we use the results of Theorem 19 to validate the passivityperformance of a plant connected with a controller implemented in software. Considera linear time invariant system,(19) ˙ x = Ax + Bu, y = Cx + DuA = − . . . . − − . . − − − . − − − .
80 0 1 0 . − , B = . . . . . , C = B T , D = (cid:2) . (cid:3) . It can be verified that the system in (19) is passive with passivity indices (0 . , . L gain that bounds the rate of change of output y is γ = CB = 0 .
44. Supposethis system is connected in feedback to a passive LTI controller(20) ˙ z = A c z + B c w, v = C c z + D c wA c = (cid:20) − − − − (cid:21) , B c = (cid:20) . . (cid:21) , C c = (cid:2) (cid:3) , D c = (cid:2) (cid:3) mplemented in software. For a sampling time of 0.2 sec and the state, input andoutput quantization value of 0.1, the passivity indices of this controller are ( ρ c , ν c ) =(0 . , . − . , . − . , . − . , .
1] of the state, input and output space respectively. The controllersymbolic model can be obtained by considering piecewise continuous control inputs { u − , u , u } = {− . , , . } and the symbolic states and outputs described by {− x , − x , x , x , x } = {− . , − . , , . , . } and { y − , y , y } = {− . , , . } re-spectively.As shown in Figure 1, the system (19) is interacting with the software controllerthrough continuous to discrete and discrete to continuous conversion units. Thissystem can be analyzed by considering a discrete abstraction of the continuous plant.Using Theorem 19 for passivity, ρ τ,µ,η = ρ − τ ρ ( τ γ + 1) − | − ρ + τ ρ ( τ γ + 1) | = − . ν τ,µ,η = ν − / − τ γρ ( τ γ + τ + γ ) − ( γ √ mτ µ + γ τ ) = 0 . ρ τ,µ,η < ν τ,µ,η > ρ c > ρ τ,µ,η + ν c >
0, the closed loop system is passive with output passivity index of0.0462.
5. Dissipativity of approximate feedback composition of systems.
Inthis section we discuss the dissipativity property of the approximate feedback com-position of two transition systems as described in [25]. We show that once two tran-sition systems are approximately feedback composable, then QSR dissipativity of oneof those transition systems implies QSR dissipativity of the entire composition.Cyber physical systems can be constructed by interconnecting several individualsubsystems and this process for transition systems can be described using composi-tion operations. It was shown in [25] that approximate feedback composition of twotransition systems can also be used to construct controllers for requirements such assafety and reachability. Approximate feedback composition of two transition systemsis possible for state feedback if there exists an approximate alternating simulationrelation between the two systems that may be a plant and a controller. The idea ofsupervisory control in [25] is that the controller restricts the behavior of the plant byforcing it to simulate the controller. The controller selects an allowable input label,the plant makes any transition having that input label, and the controller makes atransition to maintain alternating simulation relation. This set up works if there isroom for the controller to select its input to properly navigate the plant behavior whilemaintaining the approximate alternating simulation relation. In the original conceptof approximate alternating simulation in [25], the input sets of two systems whichare approximately alternating similar were different. This freedom of nonidenticalinputs of two transition systems along with the other conditions of approximate alter-nating simulation defined in [25] are captured by ( (cid:15) u , (cid:15) y ) approximately input-outputalternating simulation in definition 4, with (cid:15) u (cid:54) = 0. Therefore, we can modify the defi-nition of approximately feedback composable systems in [25] from using approximatealternating similar systems to ( (cid:15) u , (cid:15) y ) approximate input-output alternating similarsystems. This will be clear in the following definition. Definition
A transition system T is said to be ( (cid:15) u , (cid:15) y ) -approximate feed-back composable with system T if there exists an ( (cid:15) u , (cid:15) y ) -approximate input-outputalternating simulation relation R from T to T , that is, T (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T . et T i := ( X i , U i , −−−−→ τ , Y i , H i ), i = { , } be two transition systems with acommon time period τ and common input and output sets equipped with euclideannorm as the metric. Let R be a ( (cid:15) u , (cid:15) y ) - approximate input-output alternatingsimulation relation from T to T . Let us define a feedback relation F ⊂ X × X × U × U defined by all the quadruples ( x , x , u , u ) ∈ X × X × U × U for which( x , x ) ∈ R and conditions 1 and 2 in Definition 8 are met.The feedback composition of T and T with interconnection relation F , denotedby T × ( (cid:15) u ,(cid:15) y ) F T , is the transition system ( X , U , −−−−→ τ , Y , H ) consisting of • X = { ( x , x ) ∈ ( X × X ) d Y ( H ( x , u ) , H ( x , u )) ≤ (cid:15) y } , where d U ( u , u ) ≤ (cid:15) u ; • U = { ( u , u ) | d U ( u , u ) ≤ (cid:15) u , u ∈ U , u ∈ U } ; • ( x , x ) ( u ,u ) −−−−−→ τ ( x (cid:48) , x (cid:48) ) if the following three conditions hold:1. x u −−−−→ τ x (cid:48) in T ;2. x u −−−−→ τ x (cid:48) in T ;3. ( x , x , u , u ) ∈ F ; • Y = Y = Y ; • H ( x , x , u , u ) = ( H ( x , u ) + H ( x , u )).This symmetrical choice of output allows T × ( (cid:15) u ,(cid:15) y ) F T to be commutative. How-ever, we can also choose an output for the composition as H ( x , x , u , u ) = H ( x , u ) or H ( x , x , u , u ) = H ( x , u ).Before analyzing the dissipativity of the feedback composition, we present thefollowing result from [25]. Even though the results in [25] were derived for approximatesimulation relationships they also hold true for approximate input-output simulationrelationships. Proposition
Consider approximate feedback composition of two ( (cid:15) u , (cid:15) y ) ap-proximately input-output similar transition systems T and T , where T (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T .If we define the output of the composition as h ( x , x , u , u ) = ( h ( x , u ) + h ( x ) , u ) then, (a) T × ( (cid:15) u ,(cid:15) y ) F T (cid:22) ( (cid:15) u ,(cid:15) y / IOS T , (b) T × ( (cid:15) u ,(cid:15) y ) F T (cid:22) ( (cid:15) u ,(cid:15) y / IOS T h ( x , x , u , u ) = h ( x , u ) , then T × ( (cid:15) u ,(cid:15) y ) F T (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T h ( x , x , u , u ) = h ( x , u ) , then T × ( (cid:15) u ,(cid:15) y ) F T (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T . Proof.
This is a direct consequence of Proposition 11.8 in [25].Based on Proposition 22 and Theorem 14 we obtain the following result.
Theorem
Let T :=( X , U , −−−−→ τ , Y , H ) and T :=( X , U , −−−−→ τ , Y , H ) be two transition systems with a common time period τ and common input and outputsets equipped with euclidean norm as the metric. Let T be ( (cid:15) u , (cid:15) y ) - approximatelyinput-output alternatingly similar to T , T (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T . If T is QSR - dissipative withrespect to an output function h ( x , u ) where x ∈ X and u ∈ U , then T × ( (cid:15) u ,(cid:15) y ) F T s also QSR dissipative w.r.t. ( h ( x , u ) + h ( x , u )) where ( x , x , u , u ) ∈ F ,and the dissipation matrices Q , S , R of the composed system satisfy λ ( Q − Q ) − ζ (cid:107) Q (cid:107) − ζ ≥ λ ( R − R ) − ζ (cid:107) S (cid:107) − ζ (cid:107) R (cid:107) ≥ S = S (21) where ζ , ζ , ζ , ζ ∈ R + are arbitrary non-zero constants, Q , S , R are the dissi-pativity matrices for T . λ ( · ) and λ ( · ) represent the smallest and largest eigen valuesof the concerned matrix.Also, T × ( (cid:15) u ,(cid:15) y ) F T is QSR -dissipative w.r.t. h ( x , u ) where x ∈ X such that ( x , x , u , u ) ∈ X × U with the dissipation matrices Q , S , R as in (21).Proof. See Appendix.This result states that once two transition systems are approximately feedbackcomposable, then QSR dissipativity of one of those transition systems implies QSRdissipativity of the composed transition system. Since passivity is a special case ofQSR dissipativity, Theorem 23 can be applied to design discrete supervisory con-trollers for plants, while preserving the passive nature of the interconnection.The continuous-time system should be preceded by a sample and hold element toconvert the common quantized input symbol into a piecewise constant input. Underthis framework, the interconnected system is passive w.r.t. outputs(i) h ( x , u ), where x is a discrete state of the controller and(ii) ( h ( x , u ) + h ( x , u )) where x is the discrete plant state and x is thediscrete controller state.Once the interconnection is passive, we can guarantee the stability of this intercon-nection [11]. Further discussion on supervisory control using passivity is a subject ofour future work.
6. Conclusion.
In this paper we characterized the dissipativity properties of asystem and its abstracted model. First we presented results to compute the
Q, S ,and R dissipativity matrices of transition system from those of its approximate input-output similar abstraction. We also provided conditions determining the dissipativitymatrices of an abstract system from those of the corresponding incrementally forwardcomplete continuous system. Abstraction was obtained by an approximate input-output simulation of the continuous system. Further, we considered the approximatefeedback composition of a continuous QSR dissipative system with a finite state tran-sition system that may be a symbolic controller. We showed that if one of the com-ponents in this interconnection is QSR dissipative, then the approximate feedbackcomposition is also QSR dissipative.We can also use the results presented here to design symbolic controllers to pas-sivate and hence stabilize a dynamical system. Although a lot of work has been doneto design continuous passivating controllers for a variety of dynamical systems, it ischallenging to design such continuous controllers when other non-traditional controlconstraints, for example expressed in terms of temporal logic, need to be met alongwith passivity specifications. In such cases, the importance of supervisory or discretecontrollers is more apparent. As a future work, we will concentrate on designing con-trollers that passivate the system along with ensuring other system specifications suchas safety and reachability. Appendix. roof of Theorem 14. Proof.
Suppose at any time t , the input, output and state of transition sys-tem T i (Σ i ) be u i ( t ) , y i ( t ) and x i ( t ) respectively. Define the supply rate function ω ( u ( t ) , y ( t )) = y T ( t ) Q y ( t ) + 2 y T ( t ) S u ( t ) + u T ( t ) R u ( t ) and storage function V ( · ). Since T is QSR dissipative with dissipativity matrices Q , S and R , r = (cid:82) t t ω ( u ( t ) , y ( t )) dt ≥ V ( x ( t )) − V ( x ( t )) . Consider another quadratic supply ratefunction ω ( u ( t ) , y ( t )) = y T ( t ) Q y ( t ) + 2 y T ( t ) S u ( t ) + u T ( t ) R u ( t ) . Let this bethe quadratic supply rate for transition system T . Define r = (cid:82) t t ω ( u ( t ) , y ( t )) dt . T is ( (cid:15) u , (cid:15) y ) - approximately input-output similar to T with an approximateinput-output simulation relation R . Hence we can always find a transition x u −−−−−→ t x (cid:48) in T for every transition x u −−−−−→ t x (cid:48) in T such that d U ( u , u ) = (cid:107) u − u (cid:107) = (cid:107) − ∆ u (cid:107) ≤ (cid:15) u and d Y ( h ( x , u ) , h ( x , u )) = (cid:107) y − y (cid:107) = (cid:107) − ∆ y (cid:107) ≤ (cid:15) y with( x , x ) ∈ R and ( x (cid:48) , x (cid:48) ) ∈ R . For ease of representation, we drop the time indexfrom input and output signals. r = (cid:90) t t (( y − ∆ y ) T Q ( y − ∆ y ) + 2( y − ∆ y ) T S ( u − ∆ u ) + ( u − ∆ u ) T R ( u − ∆ u )) dtr − r = (cid:90) t t ( y T ( Q − Q ) y + 2 y T ( S − S ) u + u T ( R − R ) u − y T Q ∆ y − y T S u − y T S ∆ u − u T R ∆ u + 2∆ y T S ∆ u + ∆ u T R ∆ u + ∆ y T Q ∆ y ) dt (22) 2 (cid:104) y , Q ∆ y (cid:105) t ≤ (cid:15) y ζ + ζ (cid:107) Q (cid:107) (cid:104) y , y (cid:105) t , (cid:104) ∆ y, S u (cid:105) t ≤ (cid:15) y ζ + ζ (cid:107) S (cid:107) (cid:104) u , u (cid:105) t (cid:104) y , S ∆ u (cid:105) t ≤ (cid:15) u ζ (cid:107) S (cid:107) + ζ (cid:104) y , y (cid:105) t , (cid:104) u , R ∆ u (cid:105) ≤ (cid:15) u ζ + ζ (cid:107) R (cid:107) (cid:104) u , u (cid:105) t (cid:104) ∆ u, R ∆ u (cid:105) t ≥ λ ( R ) (cid:104) ∆ u, ∆ u (cid:105) t , (cid:104) ∆ y, Q ∆ y (cid:105) t ≥ λ ( Q ) (cid:104) ∆ y, ∆ y (cid:105) t (23) where λ ( · ) is the smallest eigen value of matrix under consideration and notation (cid:104) y, y (cid:105) t = (cid:82) t t y T ( t ) y ( t ) dt .Substituting the set of equations (23) in (22), we can show that if (9) is satisfiedthen then r > β , where β = r − (cid:15) y ζ − (cid:15) y ζ − (cid:15) u ζ (cid:107) S (cid:107) − (cid:15) u ζ − max { , λ ( − Q ) } (cid:15) y − max { , λ ( − R ) } (cid:15) u − max { , − (cid:104) ∆ y, S ∆ u (cid:105) t } . Since T is QSR dissipative, as perTheorem 3.1.11 of [30], the available storage S a ( x ( t )) = sup u ( · ) ,t ≥ t − r ( t ) < ∞ ,x ( t ) = x ∀ x ∈ X . In particular, this is true for any x : ( x , x ) ∈ R . Clearly,available storage S a ( x ) = sup u ( · ) ,t ≥ − r ( t ) < ∞ , x ( t ) = x ∀ x ∈ X . Therefore,from Theorem 3.1.11 of [30], T is QSR dissipative.
Proof of Proposition 16.
Proof.
We first show that T τ (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T τ,µ,η (Σ). Let us define a relation R ⊂ X τ × X q such that ( x τ , x q ) ∈ R if (cid:107) x τ − x q (cid:107) ≤ η/
2. Note that for any x τ ∈ X τ and u τ ∈ U τ , there always exist x q ∈ X q and u q ∈ U q such that (cid:107) x τ − x q (cid:107) ≤ η/ ≤ (cid:15) y and (cid:107) u τ − u q (cid:107) ≤ µ/ ≤ (cid:15) u . This is possible because of the specific quantization whichallows x τ to be within η/ x q and u τ to be within µ/ u q . Fromthe definitions of output functions H m τ ( x τ , u τ ) = x τ and H m q ( x q , u q ) = x q , we have (cid:107) H m τ ( x τ , u τ ) − H m q ( x q , u τ ) (cid:107) = (cid:107) x τ − x q (cid:107) ≤ (cid:15) y , hence condition (i) of Definition 7 issatisfied. ow if we consider the transition x τ u τ −−−−→ τ x (cid:48) τ in the transition system T τ (Σ) ,then the distance between x (cid:48) τ and ξ ( τ, x q , u q ) can estimated based on the incrementalforward complete property of Σ, (cid:107) x (cid:48) τ − ξ ( τ, x q , u q ) (cid:107) ≤ α ( η, τ ) + α ( µ, τ ) ≤ α ( (cid:15) y , τ ) + α ( (cid:15) u , τ )As mentioned earlier, due to the particular structure of quantization, for any x (cid:48) τ ∈ X τ there always exists x (cid:48) q ∈ X q such that(24) (cid:107) x (cid:48) τ − x (cid:48) q (cid:107) ≤ η/
2. From the triangular inequality we have (cid:107) ξ ( τ, x q , u q ) − x (cid:48) q (cid:107) ≤ (cid:107) ξ ( τ, x q , u q ) − x (cid:48) τ (cid:107) + (cid:107) x (cid:48) τ − x (cid:48) q (cid:107)≤ α ( (cid:15) y , τ ) + α ( (cid:15) u , τ ) + η/ ≤ α ( θ , τ ) + α ( θ , τ ) + η/ T τ,µ,η (Σ) implies the existence of x q u q −−−→ x (cid:48) q in T τ,µ,η (Σ). Therefore, from inequality (24) we conclude that ( x (cid:48) τ , x (cid:48) q ) ∈ R and condi-tion (ii) in Definition 7 holds. Thus, T τ (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T τ,µ,η (Σ).Along similar lines we can prove that T τ,µ,η (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T τ (Σ). The key is todefine R ⊆ X q × X τ such that ( x q , x τ ) ∈ R if (cid:107) x q − x τ (cid:107) = 0 and notice that for every x q ∈ X q , we can choose x τ = x q , x τ ∈ X τ which satisfies condition (i) of definition 4(i.e., (cid:107) x τ − x q (cid:107) = 0 < (cid:15) y ). This is possible because X q ⊆ X τ .For every u q ∈ U q , choose u τ = u q , u τ ∈ U τ (this satisfies (cid:107) u τ − u q (cid:107) = 0 < (cid:15) u ). Con-sider the unique transition x τ u τ −−−−→ τ x (cid:48) τ = ξ ( τ, x τ , u τ ) ∈ Post u τ ( x τ ). The distancebetween x (cid:48) τ and ξ ( τ, x q , u q ) can be bounded using the incrementally forward completeproperty of Σ, i.e.,(25) (cid:107) x (cid:48) τ − ξ ( τ, x q , u q ) (cid:107) ≤ α (0 , τ ) + α (0 , τ )Proceeding same as the proof of T τ (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T τ,µ,η (Σ), we can show that for every x (cid:48) τ ∈ Post u τ ( x τ ) there exists x (cid:48) q ∈ Post u q ( x q ) such that ( x (cid:48) q , x (cid:48) τ ∈ R ). This iscondition (ii) of Definition 8. Thus, T τ,µ,η (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T τ (Σ). Proof of Theorem 17.
Proof.
At any discrete time k, the state, input and output of T τ,µ,η in Defini-tion 5 are x q , u q and h ( x q , u q ). Since the original continuous time system Σ is QSRdissipative, inequality (5) holds for any input and all t > t >
0. Therefore, it willbe valid even if we substitute t = kτ, t = ( k + 1) τ and u ( t ) = u q ∈ U q ⊂ U for kτ ≤ t ≤ ( k + 1) τ where τ is the sampling time and x ( t ) = x q ∈ X q ⊂ X . Underthese conditions, system output of Σ is h ( x ( t ) , u q ). For simplicity of notation werepresent h ( x q , u q ) as y ( kτ ) and h ( x ( t ) , u q ) as y ( t ). Dissipativity inequality becomes, (cid:90) ( k +1) τkτ ( y ( t ) T Qy ( t ) + y ( t ) T Su q + u Tq Ru q ) dt ≥ V ( ξ ( τ, x q , u q )) − V ( x q )(26) ounds for (cid:82) ( k + ) τ k τ y ( t ) T Qy ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( k +1) τkτ ( y ( t ) T Qy ( t ) − y ( kτ ) T Qy ( kτ )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( k +1) τkτ (cid:90) tkτ dds ( y T ( s ) Qy ( s )) ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) ( k +1) τkτ (cid:90) tkτ (cid:107) Q T y ( s ) (cid:107) (cid:107) ˙ y ( s ) (cid:107) ds dt ≤ (cid:90) ( k +1) τkτ (cid:90) τkτ (cid:107) Q T y ( s ) (cid:107) (cid:107) ˙ y ( s ) (cid:107) ds dt ≤ τ (cid:107) Q (cid:107) ( (cid:90) ( k +1) τkτ y ( t ) T y ( t ) dt + γ τ u Tq u q )(27) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( k +1) τkτ ( y ( t ) T y ( t ) − y ( kτ ) T y ( kτ )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) ( k +1) τkτ ( (cid:107) y ( t ) − y ( kτ ) (cid:107) + 2 (cid:107) y ( kτ ) (cid:107) (cid:107) y ( t ) − y ( kτ ) (cid:107) ) dt Using (cid:107) y ( t ) − y ( kτ ) (cid:107) ≤ √ τ (cid:113)(cid:82) ( k +1) τkτ (cid:107) y ( s ) (cid:107) ds ≤ τ γ (cid:107) u q (cid:107) in the above equation, (cid:90) ( k +1) τkτ y ( t ) T y ( t ) dt ≤ ( τ γ + τ γ ) (cid:90) ( k +1) τkτ u Tq u q dt + ( τ γ + 1) (cid:90) ( k +1) τkτ y ( kτ ) T y ( kτ ) dt. This can be used in (27) to compute bounds on the first term of (26), (28) (cid:90) ( k +1) τkτ y ( t ) T Qy ( t ) dt ≤ y ( kτ ) T ( τ Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I ) y ( kτ )+ u Tq ( τ (cid:107) Q (cid:107) ( τ γ + γ ) + τ γ (cid:107) Q (cid:107) ) I u q Bounds for (cid:82) ( k + ) τ k τ y ( t ) T Su q dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( k +1) τkτ y ( t ) T Su q dt − (cid:90) ( k +1) τkτ y ( kτ ) T Su q dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) Su q (cid:107) (cid:90) ( k +1) τkτ (cid:90) tkτ (cid:107) ˙ y ( s ) (cid:107) dsdt ≤ τ √ τ (cid:107) Su q (cid:107) (cid:115)(cid:90) τ (cid:107) ˙ y ( s ) (cid:107) ds ≤ (cid:90) ( k +1) τkτ u Tq τ γ (cid:107) S (cid:107) u q dt (29) ⇒ (cid:90) ( k +1) τkτ y ( t ) T Su q dt ≤ u Tq τ γ (cid:107) S (cid:107) u q + τ y ( kτ ) T Su q Bounds for V ( ξ ( τ, x q , u q )) : Now we consider a transition x q u q −−−−→ τ x (cid:48) q in T τ,µ,η (Σ)and by Definition of T τ,µ,η (Σ) we have (cid:107) ξ ( τ, x q , u q ) − x (cid:48) q (cid:107) ≤ α ( θ , τ )+ α ( θ , τ )+ η/ V ( ξ ( τ, x q , u q )) ≥ V ( x (cid:48) q ) − L ( (cid:107) x (cid:48) q − ξ ( τ, x q , u q ) (cid:107) ) ≥ V ( x (cid:48) q ) − L ( α ( θ , τ )+ α ( θ , τ )+ η/ Proof of Theorem Proposition 18.
Proof.
We first show that T τ (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T τ,µ,η (Σ). Consider a relation R ⊂ X τ × X q such that ( x τ , x q ) ∈ R if (cid:107) x τ − x q (cid:107) ≤ η/
2. Note that for any x τ ∈ X τ and u τ ∈ U τ , there always exist x q ∈ X q and u q ∈ U q such that (cid:107) x τ − x q (cid:107) ≤ η/ (cid:107) u τ − u q (cid:107) ≤ µ/ ≤ (cid:15) u . This is possible because of the specific quantization which llows x τ to be within η/ x q and u τ to be within µ/ u q .From the definitions of output functions we have, (cid:107) h τ ( x τ , u τ ) − h q ( x q , u q ) (cid:107) = (cid:107) h ( x τ , u τ ) − h ( x q , u q ) + h ( x q , u q ) − h q ( x q , u q ) (cid:107)≤ K (cid:107) x τ − x q (cid:107) + K (cid:107) u τ − u q (cid:107) + (cid:107) h ( x q , u q ) − h q ( x q , u q ) (cid:107)≤ K η/ K µ/ µ/ ≤ (cid:15) y (31)Hence condition (i) of Definition 7 is satisfied. Along the lines of proof of Proposi-tion 16, it can be shown that condition (ii) of Definition 7 is also satisfied, proving T τ (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T τ,µ,η (Σ).Similar to the steps described above and proof of Proposition 16, we can showthat T τ,η,µ (Σ) (cid:22) ( (cid:15) u ,(cid:15) y ) IOAS T τ (Σ). Proof of Theorem Theorem 19.
Proof.
At any discrete time k, the state, input and output of T τ,µ,η in Defini-tion 6 are x q , u q and h q ( x q , u q ). Since the original continuous time system Σ is QSRdissipative, inequality (5) holds for any input and all t > t >
0. Therefore, it willbe valid even if we substitute t = kτ, t = ( k + 1) τ and u ( t ) = u q ∈ U q ⊂ U for kτ ≤ t ≤ ( k + 1) τ where τ is the sampling time and x ( t ) = x q ∈ X q ⊂ X . Underthese conditions, system output of Σ is h ( x ( t ) , u q ).For simplicity of notation we represent h ( x q , u q ) = y ( kτ ), h ( x ( t ) , u q ) = y ( t ), h q ( x q , u q ) = ˆ y ( kτ ) and ∆ y = y ( kτ ) − ˆ y ( kτ ). Dissipativity inequality becomes, (cid:90) ( k +1) τkτ ( y ( t ) T Qy ( t ) + y ( t ) T Su q + u Tq Ru q ) dt ≥ V ( ξ ( τ, x q , u q )) − V ( x q )(32)In order to prove the dissipativity of the approximate input-output similar system T τ,µ,η , we find bounds on each term of equation (32). Bounds for (cid:82) ( k + ) τ k τ y ( t ) T Qy ( t ) dt : From proof of Theorem 17 (cid:90) ( k +1) τkτ y ( t ) T Qy ( t ) dt ≤ y ( kτ ) T ( Qτ + τ (cid:107) Q (cid:107) ( τ γ + 1) I ) y ( kτ ) + u Tq ( τ (cid:107) Q (cid:107) ( τ γ + γ ) + τ γ (cid:107) Q (cid:107) ) I u q ≤ ˆ y ( kτ ) T ( Qτ + τ (cid:107) Q (cid:107) ( τ γ + 1) I )ˆ y ( kτ ) + 2ˆ y ( kτ ) T ( Qτ + τ (cid:107) Q (cid:107) ( τ γ + 1) I )∆ y + ∆ y T ( Qτ + τ (cid:107) Q (cid:107) ( τ γ + 1) I )∆ y + u Tq ( τ (cid:107) Q (cid:107) ( τ γ + γ ) + τ γ (cid:107) Q (cid:107) ) I u q . (33)2ˆ y ( kτ ) T ( Qτ + τ (cid:107) Q (cid:107) ( τ γ + 1) I )∆ y ≤ τ ∆ y T ∆ y + τ (cid:107) Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I (cid:107) ˆ y ( kτ ) T ˆ y ( kτ ) ≤ τ m µ τ (cid:107) Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I (cid:107) ˆ y ( kτ ) T ˆ y ( kτ )(34) ∆ y T ( Qτ + τ (cid:107) Q (cid:107) ( τ γ + 1) I )∆ y ≤ ¯ λ ( Qτ + τ (cid:107) Q (cid:107) ( τ γ + 1) I )∆ y T ∆ y ≤ mτ µ ¯ λ ( Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I )(35) where ¯ λ ( · ) is the largest eigen value of matrix under consideration. Using (34) and(35) in (33), (cid:90) ( k +1) τkτ y ( t ) T Qy ( t ) dt ≤ ˆ y ( kτ ) T ( Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I + ( (cid:107) Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I (cid:107) ) I ) τ ˆ y ( kτ )+ u Tq ( τ (cid:107) Q (cid:107) ( τ γ + γ ) + τ γ (cid:107) Q (cid:107) ) I u q + 14 mτ µ (1 + ¯ λ ( Q + τ (cid:107) Q (cid:107) ( τ γ + 1) I )) . (36) ounds for 2 (cid:82) ( k + ) τ k τ y ( t ) T Su q dt (cid:90) ( k +1) τkτ ( y ( t ) T Su q dt − ˆ y (0) T Su q ) dt ≤ (cid:90) ( k +1) τkτ (( y ( t ) − ˆ y ( kτ )) T ( y ( t ) − ˆ y ( kτ )) + ( Su q ) T ( Su q )) dt ≤ (cid:90) ( k +1) τkτ ( y ( t ) − ˆ y ( kτ )) T ( y ( t ) − ˆ y ( kτ )) dt + τ (cid:107) S (cid:107) u Tq u q (37) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( k +1) τkτ ( y ( t ) − ˆ y ( kτ )) T ( y ( t ) − ˆ y ( kτ )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) ( k +1) τkτ (cid:107) y ( t ) − ˆ y ( kτ ) (cid:107) dt ≤ (cid:90) ( k +1) τkτ ( (cid:107) y ( t ) − y ( kτ ) (cid:107) + (cid:107) y ( kτ ) − ˆ y ( kτ ) (cid:107) ) dt ≤ τ ( (cid:115) γ (cid:90) ( k +1) τkτ (cid:107) u q (cid:107) dt + √ mµ/ ≤ ( γ τ + γτ √ mτ µ ) u Tq u q + µτ ( mµ + γ √ mτ )4 . Using this in (37), (cid:90) ( k +1) τkτ y ( t ) T Su q dt ≤ τ ˆ y ( kτ ) T Su q + ( γ τ + γτ √ mτ µ + τ (cid:107) S (cid:107) ) u Tq u q + µτ ( mµ + γ √ mτ )4 . (38) Similar to the proof of Theorem 14, V ( ξ ( τ, x q , u q )) can also be bounded. We can usethis bound on V ( ξ ( τ, x q , u q )), (36) and (38) to bound the terms in (32) and rearrangethe resulting equation to get (14). Proof of Theorem 23.
Proof.
Output of T is h ( x , u ) and we consider two possible outputs of T = T × ( (cid:15) u ,(cid:15) y ) F T . From the definition of approximate feedback composition and Propo-sition 22, possible relations between T and T are given by Case 1: h ( x , x , u , u ) = 12 ( h ( x , u ) + h ( x , u )) ⇒ T (cid:22) ( (cid:15) u ,(cid:15) y / IOS T Case 2: h ( x , x , u , u ) = h ( x , u ) ⇒ T (cid:22) ( (cid:15) u ,(cid:15) y ) IOS T . Using Theorem 14 for both these cases gives the result.
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