Passivity-Based Analysis of Sampled and Quantized Control Implementations
PPassivity-Based Analysis of Sampled and Quantized ControlImplementations
Xiangru Xu a , Necmiye Ozay b , Vijay Gupta c a Department of Mechanical Engineering, University of Wisconsin, Madison, WI, 53706, USA b Department of Electrical Engineering & Computer Science, University of Michigan, Ann Arbor, MI, 48109, USA c Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA
Abstract
This paper studies the performance of a continuous controller when implemented on digital devices via sampling and quan-tization, by leveraging passivity analysis. Degradation of passivity indices from a continuous-time control system to its sam-pled, input and output quantized model is studied using a notion of quasi-passivity. Based on that, the passivity propertyof a feedback-connected system where the continuous controller is replaced by its sampled and quantized model is studied,and conditions that ensure the state boundedness of the interconnected system are provided. Additionally, the approximatebisimulation-based control implementation where the controller is replaced by its approximate bisimilar symbolic model whosestates are also quantized is analyzed. Several examples are provided to illustrate the theoretical results.
Key words:
Dissipativity, Passivity Indices, Quantized Control, Approximate Bisimulation, Symbolic Control
Cyber-physical control systems consist of software-basedcontrollers interacting with physical processes (i.e., theplant to be controlled). While the control design can bedone using continuous-time and continuous-space meth-ods, in order to guarantee the desirable operation ofthe closed-loop system, it is important to keep in mindthe restrictions the software implementation imposes. Inparticular, sampling and quantization are prevalent inreal control implementations. Therefore, sampled-dataand quantized control have attracted the attention of re-searchers for decades. However, many problems are stillopen, especially when dynamics of the system are nonlin-ear or when both sampling and quantization are consid-ered [5,3,8,9]. This work presents a unified framework foranalyzing control implementations using tools from pas-sivity theory. In addition to analyzing the effects of sam-pling and quantization of the input and output signalswithin this framework, we also consider quantization of (cid:63)
The material in this paper was partially presented at the54th IEEE Conference on Decision and Control, Dec 15-18,2015, Osaka, Japan. Corresponding author: Xiangru Xu.
Email addresses: [email protected] (Xiangru Xu), [email protected] (Necmiye Ozay), [email protected] (Vijay Gupta). the internal states of the controller. This enables anal-ysis of controllers implemented with finitely many bits,and can be relevant in extremely resource-constrainedsettings like micro-robotics or applications where a com-puter that can do floating-point arithmetic is not feasi-ble, which constitute the motivation for our work.The concept of passivity emerged from the study ofenergy dissipation in circuit analysis, which, roughlyspeaking, means that a system can not generate “inter-nal energy” on its own. In the paper [42], Willems sys-tematized the theory of dissipativity (with passivity asits special case) using the concepts of storage functionand supply rate, where dissipativity is defined as theproperty that the rate of increase of the storage functionis not larger than the supply rate. Since then, the funda-mental connections between dissipativity, stability (ei-ther in the sense of Lyapunov or input/output) and op-timality for control systems have been explored, makingpassivity/dissipativity theory a widely-used tool in con-trol theory [4,20,36]. One important property of passivesystems is the compositionality - the parallel or feedbackconnection of two passive systems remains passive. Thisproperty offers an effective method for analyzing the sta-bility of large-scale, interconnected systems in a com-positional way [26,41]. Several challenging problems incyber-physical systems, such as quantization, time delay
Preprint submitted to Automatica 23 August 2019 a r X i v : . [ m a t h . O C ] A ug nd packet drops, which are induced by the digital de-vices or communication networks, have also been studiedunder the passivity/dissipativity-based framework (see[16,38,10,40,2] and references therein).A general form of dissipativity that has been well-studiedis the QSR-dissipativity whose supply rate is a quadraticfunction of the input and output. A special case of theQSR-dissipativity allows one to use two real numbers,which are termed passivity indices (or passivity levels),to characterize the shortage/excess of passivity for a sys-tem. These two numbers can be either positive or neg-ative, where a negative value implies a shortage of pas-sivity while a positive value implies an excess. Passivityindices are rather useful in the analysis of nonlinear andinterconnected systems; for instance, [14,15] showed thatthe feedback interconnection of two non-passive systemsis finite gain L stable provided that the shortage of pas-sivity of one component can be compensated for by theexcess of passivity of another component, [27] provided acontroller design method for asymptotic stabilization ofa class of nonlinear systems using passivity indices, and[48] and [49] presented conditions on passivity indices ofa closed-loop system in the continuous-time setting andwith the event-triggered mechanism, respectively.It is known that passivity is not preserved under sam-pling or quantization in general (e.g., see [22,2,31]). How-ever, passivity indices, as a quantitative “abstraction”of the passivity property of a system, can be used tostudy how the properties of two related systems (e.g.,a continuous-time system and its sampled or quantizedsystem) differ from each other. Some interesting resultshave been obtained along this direction; for example,[31] analyzed the passivity degradation for a system un-der time sampling with the zero-order hold and the idealsampler, [44] showed that passivity indices of one sys-tem can be inferred from an approximate model of thatsystem when the norm of the model error is small andthe approximate model has an excess of passivity. De-spite these interesting results, passivity analysis for sys-tems that are both sampled and quantized, as well asits control implementations in a feedback loop, are stilllacking, to the best of our knowledge.Symbolic models for dynamical systems provide a uni-fied framework for studying the interactions of softwareand physical phenomena [39]. Such symbolic models aresampled and quantized (in state, input and output),which are used in control software synthesis from high-level specifications [32]. The basic workflow in theseapproaches is to first compute an approximate symbolicmodel of the plant based on (bi)simulation relations,then synthesize a discrete controller for the symbolicplant model, and finally refine that controller and com-pose it with the plant. The constructed controller canbe implemented in software and the closed-loop systemis guaranteed to satisfy the desired high-level specifica-tions. A major limitation of these techniques is the curse of dimensionality: the complexity of the symbolic plantmodel grows exponentially with the dimension of thestate-space of the plant (when quantized uniformly).To tackle this computational problem, compositionalapproaches have been utilized. For instance, [28] pro-posed a compositional symbolic abstraction method fora class of continuous-time nonlinear control systems us-ing the notion of approximate disturbance bisimulationand discussed the related controller synthesis problem,[6] provided a compositional control design method,inspired by small gain theorem and assume-guaranteereasoning, for the feedback composition of two discretesystems with a persistency specification. Passivity,which possesses the nice compositionality property, hasalso been used to tame the computational complexityof symbolic control. For instance, by using simulationfunctions and the dissipativity-based approach, [35,47]investigated compositional abstractions of a networkof continuous-time control systems to a lower dimen-sional, continuous-time interconnected system. How-ever, results that combine the bisimulation-based finitestate abstraction and the passivity/dissipativity-basedapproach are still rare.In this paper, we analyze the passivity property of sam-pled and quantized controller implementations with pas-sivity indices as the main tool. Specifically, the questionwe consider is - suppose that a continuous, dynamic con-troller has been designed to ensure specified passivity in-dices for the closed-loop system, what guarantees on pas-sivity can be inherited if a sampled and quantized con-troller is implemented, and under what conditions theclosed-loop system is (ultimately) bounded? To this end,we first propose the notion of quasi-passivity and thestrong detectability for discrete-time control systems,and give several lemmas that relate ultimate bounded-ness, strong detectability and passivity indices. Then,we explore how passivity degrades from a continuous-time system to its sampled, input and output quantizedmodel, where a set of results quantifying the passiv-ity degradation are derived, which relate the passivityindices, the sampling time and the quantization preci-sion. Based on that, we study the implementation ofthe continuous controller by replacing it with its sam-pled, input and output quantized model, and we pro-vide conditions (on the passivity indices and the strongdetectability of the interconnected components) underwhich the state of the closed-loop system eventually con-verges into a compact set whose size can be made ar-bitrarily small by choosing the quantization precisionsmall enough. Finally, we consider implementing an ap-proximate bisimulation-based symbolic controller whosestates are also quantized, in the closed-loop system, andgive conditions that guarantee the ultimate bounded-ness of the closed-loop system.A preliminary version of the paper appeared in the con-ference publication [46]. In this paper, we extend theresults in [46] in the following important ways: Lemma2, 4 and 5 that relate quasi-passivity, strong detectabil-ity and state boundedness are added, Theorem 1, 2 and3 that present passivity degradation results are derivedusing a relaxed assumption (i.e., Assumption 2), Theo-rem 4 and 5 that present state boundedness results forthe closed-loop system are new, all the complete proofsare included, and a few new examples are provided. Notation and definitions.
The set of non-negative inte-gers and real numbers are denoted as Z ≥ and R ≥ ,respectively. The (cid:96) and (cid:96) ∞ norm of a vector x ∈ R n are denoted as | x | and | x | ∞ , respectively; the (cid:96) ∞ normof a function φ : Z ≥ → R n is denoted as (cid:107) φ (cid:107) :=sup k ∈ Z ≥ | φ [ k ] | . For any A ⊆ R n and µ >
0, [ A ] µ := { a ∈ A | a i = k i µ, k i ∈ Z , i = 1 , , ..., n } . A relation R ⊂ A × B is identified with the map R : A → B , which is definedby b ∈ R ( a ) if and only if ( a, b ) ∈ R . For a set S , the set R ( S ) is defined as R ( S ) = { b ∈ B : ∃ a ∈ S, ( a, b ) ∈ R } .Given a relation R ⊂ A × B , R − denotes the inverse re-lation of R , i.e., R − := { ( b, a ) ∈ B × A : ( a, b ) ∈ R } . Acontinuous function f : R ≥ → R ≥ belongs to class K if it is strictly increasing and f (0) = 0, and f belongs toclass K ∞ if f ∈ K and f ( r ) → ∞ as r → ∞ ; a continu-ous function f : R ≥ × R ≥ → R ≥ belongs to class KL if for each fixed s , function f ( r, s ) ∈ K ∞ with respect to r and for each each fixed r , function f ( r, s ) is decreasingwith respect to s and f ( r, s ) → s → ∞ . A continuous-time control system is a tuple Σ =(
X, U, Y, f, h ) where X ⊆ R n is a set of states, U ⊆ R m is a set of inputs, Y ⊆ R s is a set of out-puts, f : X × U → R n is Lipschitz continuous, and h : X × U → R s is continuous. The state, input andoutput of Σ at time t ∈ R ≥ are denoted by x ( t ), u ( t ), y ( t ), respectively, and their evolution is governed by:˙ x ( t ) = f ( x ( t ) , u ( t )) ,y ( t ) = h ( x ( t ) , u ( t )) , ∀ t ∈ R ≥ . (1)We assume that f (0 ,
0) = 0 and h (0 ,
0) = 0. We alsoassume that given any sufficiently regular control inputsignal u : [0 , T ] → U with T > x ∈ X , there exist a unique state trajectory x anda corresponding output trajectory y defined on [0 , T ]satisfying x (0) = x and Eq. (1). Denote by x ( τ, x , u )the state reached at time τ under the input u from theinitial state x of Σ.A discrete-time control system is a tuple Σ d =( X, U, Y, f d , h d ) where X ⊆ R n is a set of states, U ⊆ R m is a set of inputs, Y ⊆ R s is a set of outputs, f d : X × U → X and h d : X × U → R s are bothcontinuous maps. The state, input and output of Σ d at time step k ∈ Z ≥ are denoted by x [ k ], u [ k ], y [ k ],respectively, and their evolution is governed by: x [ k + 1] = f d ( x [ k ] , u [ k ]) ,y [ k ] = h d ( x [ k ] , u [ k ]) , ∀ k ∈ Z ≥ . (2)The state and output trajectories of the system Σ d arediscrete-time signals satisfying Eq. (2). Definition 1
A transition system is a quintuple T =( Q, L, O, −→ , H ) , where: • Q is a set of states; • L is a set of inputs; • O is a set of outputs; • −→⊂ Q × L × Q is the transition relation; • H : Q × L −→ O is the output function. In the following, we denote an element ( q, (cid:96), p ) ∈−→ ina transition relation by q (cid:96) −−→ p where p, q ∈ Q , (cid:96) ∈ L .Given a continuous-time system Σ = ( R n , U, R s , f, h )and a sampling time τ , we suppose that the con-trol inputs of Σ are piecewise constant, that is, u ( t ) = u (( k − τ ) for any t ∈ [( k − τ, kτ ) , k ∈ Z ≥ .Then, similar to [32], we define a transition system T τ (Σ) = ( X , U , Y , −−→ , H ) associated with the time-sampling of Σ as follows: • X = R n ; • U = U ; • Y = R s ; • p u −−→ q if x ( τ, p, u ) = q where u : [0 , τ ) → { u } , u ∈ U ; • H ( x, u ) = h ( x, u ).We interpret the trajectories of T τ (Σ) in discrete-time,that is, it has an equivalent representation in terms ofa discrete-time control system as in (2), where its state,input and output at time step k ∈ Z ≥ are denotedby x [ k ], u [ k ], y [ k ], respectively. Note that T τ (Σ) can beobtained by putting Σ between a zero-order hold device(D/A) and an uniform sampler (A/D).By further quantizing the state and input spaces of T τ (Σ), we obtain an infinitely countable transitionsystem T τµη (Σ) = ( X , U , Y , −−→ , H ) for some τ, µ, η > • X = [ R n ] η ; • U = [ U ] µ ; • Y = R s ; Note that the sets of state, input and output of T τµη (Σ)are all countable. p u −−→ q if | x ( τ, p, u ) − q | ∞ ≤ η/ u : [0 , τ ) →{ u } , u ∈ U ; • H ( x, u ) = h ( x, u ).Bisimulation is a binary relation between two transitionsystems, which, roughly speaking, requires the two sys-tems match each other’s behavior [39]. In [12], the exactbisimulation was generalized to (cid:15) -approximate bisimu-lation, which allows the states of two transition systemsto be within certain bounds. To further capture the in-put and output behaviors of transition systems, we con-sider the following ( (cid:15), µ ) -approximate bisimulation rela-tion adopted from [19]. Definition 2
Given two transition systems T =( Q , L, O, −−→ , H ) and T = ( Q , L, O, −−→ , H ) where Q ⊆ Q , the sets Q , Q are equipped with the same met-ric d s defined for Q , and the input set L is equipped withthe metric d l , for any (cid:15), µ ∈ R + , a relation R ⊂ Q × Q is said to be an ( (cid:15), µ ) -approximate bisimulation relationbetween T and T , if for any ( q , q ) ∈ R :(i) d s ( q , q ) ≤ (cid:15) ;(ii) q (cid:96) −−→ p implies the existence of (cid:96) ∈ L such that d l ( (cid:96) , (cid:96) ) ≤ µ , q (cid:96) −−→ p and ( p , p ) ∈ R ;(iii) q (cid:96) −−→ p implies the existence of (cid:96) ∈ L such that d l ( (cid:96) , (cid:96) ) ≤ µ , q (cid:96) −−→ p and ( p , p ) ∈ R . If there exists an ( (cid:15), µ )-approximate bisimulation rela-tion R between T and T such that R ( Q ) = Q and R − ( Q ) = Q , T is said to be ( (cid:15), µ )-bisimilar to T ,which is denoted as T ∼ = ( (cid:15),µ ) T . Remark 1
The ( (cid:15), µ ) -approximate bisimulation in Def.2 is different from that in [19]: the finite transition systemin [19] has an observation map (cid:104)·(cid:105) : Q → O but withoutthe output function H , where Q ⊆ Q is not requiredeither. In fact, we can consider the observation map of Σ , T τ (Σ) or T τµη (Σ) above as an identity mapping, whichis the case in [32]. The output function H defined in thispaper is used for the passivity analysis. Definition 3 [1] The continuous-time system Σ in (1) is called incrementally input-to-state stable ( δ -ISS) if itis forward complete and there exist functions β ∈ KL and β ∈ K ∞ such that for any t ∈ R ≥ , any initial state x , x ∈ R n and any input u , v , it holds that | x ( t, x , u ) − x ( t, x , v ) | ≤ β ( | x − x | , t )+ β ( (cid:107) u − v (cid:107) ) . Lemma 1
Consider the continuous-time control system Σ in (1) and any desired precision (cid:15) > . If Σ is δ -ISS sat-isfying | x ( t, x , u ) − x ( t, x , v ) | ≤ β ( | x − x | , t )+ β ( (cid:107) u − v (cid:107) ) and parameters τ, η, µ > satisfy the inequality β ( (cid:15), τ ) + β ( µ ) + η/ ≤ (cid:15), (3) then T τ (Σ) ∼ = ( (cid:15),µ ) T τµη (Σ) . Lemma 1 can be proved following the proof of Theorem5.1 in [32]. The key step is to show (ii) and (iii) in Def-inition 2 by the following fact: for any input (cid:96) ∈ U of T τ (Σ), we can choose input (cid:96) ∈ U of T τµη (Σ) suchthat | (cid:96) − (cid:96) | ∞ ≤ µ , and vice versa. Specifically, we let (cid:96) = Q ( (cid:96) ) where the quantization function Q ( · ) is de-fined entry-wisely as in (20).We call the transition system T τµη (Σ) a symbolic model for Σ where T τ (Σ) ∼ = ( (cid:15),µ ) T τµη (Σ). One nice property ofthis symbolic model is that its evolution can be chosento be deterministic [11], which is appropriate for dis-crete software-based implementation. Particularly, if thestate space and input space of T τµη (Σ) are chosen to bebounded sets, then the resulting T τµη (Σ) will be a finitetransition system. Definition 4 [4] A continuous-time control system Σ as in (1) is called dissipative with respect to a supplyfunction w ( u, y ) if there exists a positive semi-definitestorage function V ( x ) such that the following (integral)dissipation inequality is satisfied for any x ( t ) ∈ R n with t , t ∈ R ≥ , t < t , and any admissible inputs u : V ( x ( t )) − V ( x ( t )) ≤ (cid:90) t t w ( u ( s ) , y ( s )) d s. (4)If V ( x ) is differentiable, an equivalent (differential) formof (4) is ˙ V ( x ( t )) ≤ w ( u ( t ) , y ( t )) , ∀ t ∈ R ≥ . Definition 5 [4] A discrete-time control system Σ d asin (2) is called dissipative with respect to the supply func-tion w ( u, y ) if there exists a positive semi-definite storagefunction V ( x ) such that the following dissipation inequal-ity is satisfied for any x [ k ] ∈ R n with k , k ∈ Z ≥ , k A system Σ (or Σ d ) is called input feed-forward output feedback passive (IF-OFP) with passiv-ity indices ( ν, ρ ) if it is dissipative with respect to thesupply function w ( u, y ) = u (cid:62) y − νu (cid:62) u − ρy (cid:62) y for some ν, ρ ∈ R , denoted as IF-OFP ( ν, ρ ) . ν, ρ reflect the excess or shortage ofpassivity of a system where positive values reflect excessand negative values reflect shortage. A passive systemis IF-OFP(0 , ν, ρ ) with ν > , ρ > ν, ρ are not unique because for a system that is IF-OFP( ν, ρ ),it is also IF-OFP( ν (cid:48) , ρ (cid:48) ) for any ν (cid:48) < ν, ρ (cid:48) < ρ . We will consider the closed-loop configurations shownin Fig. 1, where the feedback connection of T τ ( P ) and˜ T τ (Σ) is denoted as T τ ( P ) × F ˜ T τ (Σ), and the feedbackconnection of T τ ( P ) and ˜ T τµη (Σ) is denoted as T τ ( P ) × F ˜ T τµη (Σ). In these configurations, P and Σ are both con-tinuous models, which can be considered as the origi-nal plant and the controller, respectively; T τ ( P ) (resp. T τ (Σ)) is the time-sampled model of P (resp. Σ), whichconsists of P (resp. Σ), a zero-oder hold device and anuniform sampler; ˜ T τ (Σ) (resp. ˜ T τµη (Σ)) is the modelthat consists of T τ (Σ) (resp. T τµη (Σ)), the input quan-tizer Q and the output quantizer Q . The setup in Fig.1 (a) can be considered as replacing Σ with ˜ T τ (Σ), thesampled and quantized model of Σ; the setup in Fig. 1(b) can be considered as replacing Σ with ˜ T τµη (Σ), theapproximate bisimilar symbolic model of T τ (Σ). We as-sume that 1) the external reference inputs r i , i = 1 , T τ ( P ) and T τ (Σ) are both τ ; 3) all thediscrete-time signals in the feedback loop are synchro-nized.The problems that will be studied are the following: Given the passivity indices of P and Σ and the setup ofFig. 1 (a), what passivity property can be preserved for T τ ( P ) × F ˜ T τ (Σ) , and under what conditions the states of T τ ( P ) × F ˜ T τ (Σ) are bounded? Similarly, given the pas-sivity indices of P and Σ and the setup of Fig. 1 (b), un-der what conditions the states of T τ ( P ) × F ˜ T τµη (Σ) arebounded? We will study the passivity of T τ ( P ) × F ˜ T τ (Σ) and givea boundedness result for it in Section 5, based on thepassivity degradation results in Section 4. After that,we will investigate the state boundedness of T τ ( P ) × F ˜ T τµη (Σ) in Section 6. The main theorems of the paperare summarized in Table 1. In this section, we introduce quasi-passivity, strong de-tectability and some relevant lemmas, which are of in-terest by their own and will be used in later sections. w ++ D/A P A/D T τ (Σ) Q Q − + r u y ˜ u ˜ y r u y ˇ y T τ ( P ) ˜ T τ (Σ) ++ D/A P A/D T τ (Σ) Q Q − + r u y ˜ u ˜ y r u y T τ ( P ) ˜ T τ (Σ) ++1 (a) Feedback connection of T τ ( P ) and ˜ T τ (Σ), denoted as T τ ( P ) × F ˜ T τ (Σ). D/A P A/DA/D Σ D/A Q Q − + r u y ˜ u ˜ y r u y ++ T τ ( P ) T τ (Σ) ˜ T τ (Σ) D/A P A/D T τµη (Σ) Q Q − + r u y ˜ u s ˜ y s r u s y s T τ ( P ) ˜ T τµη (Σ) ++1 (b) Feedback connection of T τ ( P ) and ˜ T τµη (Σ), denoted as T τ ( P ) × F ˜ T τµη (Σ).Fig. 1. Two closed-loop system setups considered. Theorem 1 passivity degradation from Σ to T τ (Σ)Theorem 2 passivity degradation from Σ τ to ˜ T τ (Σ)Theorem 3 passivity inequality of T τ ( P ) × F ˜ T τ (Σ)Theorem 4 state boundedness of T τ ( P ) × F ˜ T τ (Σ)Theorem 5 state boundedness of T τ ( P ) × F ˜ T τµη (Σ) Table 1Summary of the main theorems. Unlike dissipative systems, quasi-dissipative systems(see Def. 1 of [33]) or almost-dissipative systems (seeDef. 2.1 of [7]) allow for “internal energy generation”.The following definition is inspired by [33,7] and Def. 6. Definition 7 The continuous (resp. discrete) time sys-tem Σ in (1) (resp. Σ d in (2) ) is called input feedfor-ward output feedback quasi-passive (IF-OFQP) if it isdissipative with respect to the supply function w ( u, y ) = u (cid:62) y − νu (cid:62) u − ρy (cid:62) y + δ , denoted as IF-OFQP ( ν, ρ, δ ) ,where δ ≥ is a constant. By Def. 7, a discrete-time system Σ d that is IF-OFQP( ν, ρ, δ ) satisfies the following inequality V ( x [ k ]) − V ( x [ k ]) ≤ k − (cid:88) i = k (cid:0) u [ i ] (cid:62) y [ i ] − ν | u [ i ] | − ρ | y [ i ] | + δ (cid:1) (6)5or any k , k ∈ Z ≥ , k < k and any admissible in-put u [ i ] where V ( x ) is a positive semi-definite function.Clearly, IF-OFP is a special case of IF-OFQP with δ = 0.Consider the feedback-connection of two discrete-timesystems Σ , Σ shown in Fig. 2. Define x and x as thestates, and V ( x ) and V ( x ) as the storage functionsof Σ and Σ , respectively. Define r := ( r (cid:62) , r (cid:62) ) (cid:62) asthe external input, and y := ( y (cid:62) , y (cid:62) ) (cid:62) as the overalloutput. The following Lemma shows passivity indices ofthe feedback connection of Σ and Σ . Its proof is similarto that of Theorem 6 in [48] and is thus omitted. Σ Σ − + r r y ++ y u u D/A P A/DA/D Σ D/A Q Q − + r u y ˜ u ˜ y r u y ++ T τ ( P ) T τ (Σ) ˜ T τ (Σ) D/A P A/D T τµη (Σ) Q Q − + r u y ˜ u s ˜ y s r u s y s T τ ( P ) ˜ T τµη (Σ) ++1 Fig. 2. Feedback connection of two discrete-time systems Lemma 2 Consider the feedback connection of twodiscrete-time systems Σ , Σ as shown in Fig. 2. If Σ i ( i = 1 , is IF-OFQP ( ν i , ρ i , δ i ) , then the feedback-connected system is IF-OFQP (ˆ ν, ˆ ρ, ˆ δ ) with respect to theinput r and the output y where ˆ δ = δ + δ and ˆ ν, ˆ ρ canbe chosen as ˆ ν < min { ν , ν } , ˆ ρ ≤ min { ρ − ˆ νν ν − ˆ ν , ρ − ˆ νν ν − ˆ ν } . (7) Definition 8 The discrete-time system (2) is said to be N -step strongly detectable (SD) where N ∈ Z ≥ , if thereexist a constant ϑ ≥ and a radially unbounded, positivedefinite function p : R n → R ≥ such that for any k ∈ Z ≥ , any initial state x [ k ] ∈ R n and any admissibleinput u [ k ] , the following holds: k + N (cid:88) k = k ϑ | u [ k ] | + | y [ k ] | ≥ p ( x [ k ]) . (8)The intuition behind strong detectability is large initialstates must dictate large input-output signals. Similardefinitions were also given in Eq. (38) in [37] and Def.3 in [33]. The strong detectability in Def. 8 implies thezero-state observability given in Def. 6.5 of [20], because u [ k ] = , y [ k ] = , k = k , ..., k + N , implies x [ k ] = . Remark 2 A discrete-time linear system that is ob-servable with an observability index v is v -step SD.Given a system x [ k + 1] = A d x [ k ] + B d u [ k ] , y [ k ] = C d x [ k ] + D d u [ k ] , define U = [ u [ k ] (cid:62) , ..., u [ k + v ] (cid:62) ] (cid:62) , Y = [ y [ k ] (cid:62) , ..., y [ k + v ] (cid:62) ] (cid:62) . The matrix Y can beexpressed as Y = Ox [ k ] + HU where O is the observ-ability matrix and H is a matrix that can be constructedfrom A d , B d , C d , D d . Hence, (cid:80) k + vk = k ϑ | u [ k ] | + | y [ k ] | = ϑU (cid:62) U + x [ k ] (cid:62) O (cid:62) Ox [ k ]+ U (cid:62) H (cid:62) HU +2 x [ k ] (cid:62) O (cid:62) HU .Since rank ( O (cid:62) O ) = rank ( O ) = n , O (cid:62) O (cid:31) . Then itis easy to see that there exist ϑ > and p ( x ) = x (cid:62) M x where M (cid:31) such that (8) holds. Example 1 Consider the double integrator system x [ k + 1] = x [ k ] , x [ k ] = u [ k ] , y [ k ] = x [ k ] + u [ k ] where x , x , y, u ∈ R . It is easy to verify that the sys-tem is not -step SD, but -step SD with ϑ = 2 and p ( x ) = ( x + x ) . The strong detectability of a feedback-connected systemcan be derived by the strong detectability of each sub-system, which is shown by the following Lemma 3. Theproof of this lemma is given in Appendix A. Lemma 3 Consider the feedback connection of twodiscrete-time systems Σ , Σ as shown in Fig. 2. Supposethat Σ is N -step SD that satisfies k + N (cid:88) k = k ϑ | u [ k ] | + | y [ k ] | ≥ p ( x [ k ]) , ∀ k ∈ Z ≥ , (9) and Σ is N -step SD that satisfies k + N (cid:88) k = k ϑ | u [ k ] | + | y [ k ] | ≥ p ( x [ k ]) , ∀ k ∈ Z ≥ , (10) where ϑ , ϑ ≥ and p , p are radially unbounded, pos-itive definite functions, then, the feedback-connected sys-tem is N -step SD with respect to the input r and the out-put y where N = max { N , N } , p ( x ) = (1 − ϑ )( p ( x ) + p ( x )) and ϑ = max { ϑ ϑ + 1 , ϑ ϑ + 1 } . (11)The next lemma shows that a discrete-time system Σ d that is SD and IF-OFQP with ρ > Lemma 4 Suppose that the system Σ d given in (2) is 1) N -step SD satisfying (8) , 2) IF-OFQP ( ν, ρ, δ ) satisfying (6) with ρ > , δ ≥ and a function V ( x ) that is continu-ous, positive semi-definite, radially unbounded. Let λ be anumber such that < λ < ρ , and define η = λ − ν > , η = ρ − λ > . Then, ) for any k ∈ Z ≥ , it holds that x [ k ] ∈ D where D := { z | V ( z ) ≤ ξ } (12) and ξ = c + c , ξ = max { p ( x [0]) , c /η } , c =max z ∈C V ( z ) , c = ( N + 1)[( η + ϑη ) (cid:107) u (cid:107) + δ ] , c = ( N + 1)( η (cid:107) u (cid:107) + δ ) , and C = { z | p ( z ) ≤ ξ } .ii) there exists K ∈ Z ≥ such that x [ k ] ∈ D for all k ≥ K where D := { x | V ( x ) ≤ ξ } (13) and ξ = c + c , ξ = ( c + c ) /η , c is as in1), c = max z ∈C V ( z ) , c is a postive number, and C = { z | p ( z ) ≤ ξ } . Remark 3 Lemma 6.7 in [20] shows the zero-input Lya-punov stability for continuous-time, output strictly pas-sive systems with the zero-state observability assumption.Lemma 4 is a complement to Lemma 6.7 in the discrete-time, quasi-passivity setting.3.3 Input-to-state Practical Stability Definition 9 [25] The discrete-time system Σ d given in (2) is called (globally) input-to-state practically stable (ISpS) if there exist a KL -function β , a K -function β and a positive constant d such that, for any u [ k ] with (cid:107) u (cid:107) < ∞ and any x ∈ R n , it holds that | x [ k ] | ≤ β ( | x | , k ) + β ( max ≤ j ≤ k − | u [ j ] | ) + d, ∀ k ∈ Z ≥ . Definition 10 [25] A continuous function V : R n → R ≥ is called the ISpS-Lyapunov function for Σ d if thereexist d , d , a, b, c, λ > with c ≤ b and a K -function σ , such that the following hold: 1) α ( | x | ) ≤ V ( x ) ≤ α ( | x | ) + d , ∀ x ∈ R n where α ( s ) = as λ , α ( s ) = bs λ ;2) V ( f d ( x, u )) − V ( x ) ≤ − α ( | x | ) + σ ( | u | ) + d , ∀ x ∈ R n , ∀ u ∈ R m , where α ( s ) = cs λ . Clearly, Σ d is input-to-state stable (ISS) when d = 0in Def. 9. In Theorem 2.5 in [25], it was shown thatthe discrete-time ISpS-Lyapunov function is a sufficientcondition for Σ d to be ISpS.The following Lemma 5 shows conditions under which adiscrete-time, quasi-passive system Σ d is ISpS. Its proofis shown in Appendix C. Lemma 5 If there exist λ, a, b, c > , d ≥ such thatthe system Σ d given in (2) is 1) -step SD satisfying (8) with p ( x ) ≥ c | x | λ , 2) IF-OFQP ( ν, ρ, δ ) satisfying (6) with ρ > , δ ≥ and a continuous function V ( x ) thatsatisfies a | x | λ ≤ V ( x ) ≤ b | x | λ + d , then Σ d is ISpS. Lemma 5 establishes the connection between quasi-passivity and ISpS, which enables us to use the results of ISpS to analyze IF-OFQP systems under certain cir-cumstances [17,18,25]. For instance, if V ( x ) in Lemma4 satisfies α ( | x | ) ≤ V ( x ) ≤ α ( | x | ) , ∀ x ∈ R n , where α , α are K ∞ -functions, then the ultimate bound ofΣ d can be derived similar to the proof of Lemma 3.5 in[18]. Specifically, the set D can be given as D = { x | V ( x ) ≤ ξ } (14)where ξ = α − ◦ α − (cid:0) ( η + ϑη ) (cid:107) u (cid:107) + δ (cid:1) , α ∈ K ∞ satisfies α ≤ α ◦ α − , id − α ∈ K , id is the identityfunction, α ( x ) = p ( x ) ∈ K ∞ , and α ∈ K ∞ satisfies id − α ∈ K ∞ . Note that the set D expressed in (13) or(14) can be made arbitrarily small by choosing (cid:107) u (cid:107) and δ small enough. Remark 4 In [13], the input-to-state stability was dis-cussed by a relaxed ISS-Lyapunov function, which satis-fies a weaker decrease condition defined over a boundedtime interval. Because of the N-step strongly detectabilitycondition, the proof of ultimate boundedness in Lemma 4also has to consider the decrease of V every N+1 steps;however, it is different from the proof of [13]. In this section, we study the degradation of passiv-ity indices from a continuous-time system Σ to thetime-sampled system T τ (Σ) and the time-sampled, in-put/output quantized system ˜ T τ (Σ). Configurations ofΣ, T τ (Σ) and ˜ T τ (Σ) are shown in Fig. 3.We make two assumptions on the output function of Σ. Σ u ( t ) y ( t ) Σ D/A A/D u [ k ] y [ k ] T τ (Σ) T τ (Σ) Q Q u [ k ] ˜ y [ k ] ˜ T τ (Σ) D/A P A/DA/D Σ D/A QuantizerQuantizer − r [ k ] u [ k ] y [ k ]1 (a) Σ is a continuous time model. Σ D/A A/D u [ k ] y [ k ] T τ (Σ) T τ (Σ) Q Q u [ k ] ˜ y [ k ] u [ k ] y [ k ] ˜ T τ (Σ) D/A P A/DA/D Σ D/A Q Q − r [ k ] u [ k ] y [ k ] T τ ( P ) T τ (Σ) ˜ T τ ( P ) (b) T τ (Σ) is a time-sampled model of Σ, which consists ofΣ, a ZOH device and an uniform sampler. Σ D/A A/D u [ k ] y [ k ] T τ (Σ) T τ (Σ) Q Q u [ k ] ˜ y [ k ] u [ k ] y [ k ] ˜ T τ (Σ) D/A P A/DA/D Σ D/A Q Q − r [ k ] u [ k ] y [ k ] T τ ( P ) T τ (Σ) ˜ T τ ( P ) (c) ˜ T τ (Σ) is the model that consists of T τ (Σ), the inputquantizer Q and the output quantizer Q .Fig. 3. Three system setups considered. ssumption 1 The output function h of Σ has the ad-ditive form h ( x, u ) = h ( x ) + h ( u ) . Assumption 2 There exist a constant γ > and a func-tion β : R n → R ≥ with β ( ) = 0 such that for any ini-tial state x ∈ R n , h satisfies the following inequalityfor any T > and any admissible u ( t ) : (cid:90) T | ˙ h ( x ( t )) | d t ≤ γ (cid:90) T | u ( t ) | d t + β ( x ) . (15) Remark 5 A gain assumption without considering theinitial state in (15) was made in Theorem 2 of [31] andAssumption 2 of [44]: for any T > and any admissible u ( t ) , it holds that (cid:90) T | ˙ h ( t ) | d t ≤ γ (cid:90) T | u ( t ) | d t. (16) But for the inequality (16) to hold, the function h in[31,44] needs to be independent of u and additionally,the initial state is assumed to be zero (i.e., x = ). Incomparison, Assumption 1 and Assumption 2 are lessrestrictive by considering the additive form and the initialcondition explicitly. Remark 6 If there exist a positive semi-definite func-tion β ( x ) and a positive number γ such that ˙ β ( x ( t )) ≤ γ | u ( t ) | − | ˙ h ( t ) | , ∀ t ∈ R ≥ , (17) then Assumption 2 holds. Particularly, for a stable LTIsystem ˙ x = Ax + Bu with h ( x ) = Cx , if there exist amatrix P (cid:23) and a number γ > such that (cid:32) A (cid:62) P + P A + A (cid:62) C (cid:62) CA P B + A (cid:62) C (cid:62) CBB (cid:62) P + B (cid:62) C (cid:62) CA − γ I + B (cid:62) C (cid:62) CB (cid:33) (cid:22) then one can verify that the function β ( x ) = x T P x sat-isfies (17) , implying that Assumption 2 holds for such asystem. For nonlinear systems whose dynamics are poly-nomial, the sum-of-squares optimization can be used tosearch for β ( x ) that satisfies (17) [44].4.1 Passivity Degradation of T τ (Σ)In this subsection, we consider deriving passivity indicesof T τ (Σ) from those of Σ. As shown in Fig. 3 (a), Σ isa continuous-time system whose input and output are u ( t ) and y ( t ), respectively, and T τ (Σ) is a discrete-timesystem whose input and output are u [ k ] and y [ k ], respec-tively. The following theorem shows quantitatively howpassivity indices degrade from Σ to T τ (Σ). The proof ofthe theorem is given in Appendix D. Theorem 1 Suppose that Σ satisfies Assumption 1 and Assumption 2, and is IF-OFP ( ν, ρ ) with a positive semi-definite storage function V ( x ) . Then T τ (Σ) satisfies thefollowing inequality for any k , k ∈ Z ≥ , k < k andadmissible inputs u ∈ U : ˆ V ( x [ k ]) − ˆ V ( x [ k ]) ≤ δ ( x [ k ]) + k − (cid:88) k = k (cid:0) u [ k ] (cid:62) y [ k ] − ν (cid:48) | u [ k ] | − ρ (cid:48) | y [ k ] | (cid:1) (18) where ˆ V ( x ) = τ V ( x ) and ν (cid:48) = ν − τ γ − τ γ (1 + λ ) | ρ | ,ρ (cid:48) = ρ − | ρ | /λ ,δ ( x [ k ]) = wβ ( x [ k ]) ,w = | ρ | τ (1 + λ ) + 1 γ , (19) with λ > an arbitrary positive number. Although inequality (18) has an additional bias term δ on its right-hand side, we still call (18) the passivityinequality satisfied by Σ d , with some abuse of language.Note that this bias term δ will not affect the summationof the supply function, and δ = 0 when x [0] = . Remark 7 Equation (19) indicates that ν (cid:48) < ν , ρ (cid:48) < ρ ,and when λ is fixed, a smaller sampling time τ can re-sult in a smaller passivity degradation (i.e., larger ν (cid:48) , ρ (cid:48) ),which was also shown in [31,44]. Theorem 1 also general-izes the corresponding results of [31,44] in several ways.Firstly, a more general gain assumption (15) that takesinto account the initial condition is used, which inducesa bias term δ ( x [0]) on the right hand side of (18) . Sec-ondly, the positive number λ in (19) provides additionalflexibility in balancing the passivity indices ν (cid:48) , ρ (cid:48) and theterm δ . Furthermore, to obtain the degraded indices in[44], it was assumed that ν > and an inequality about τ, ν, ρ needs to be satisfied (see Corollary 6 in [44]), butneither of them is required in Theorem 1. Remark 8 The system Σ in Theorem 1 can be nonlinearcontrol system in general. When Σ is a continuous-timeLTI system, dynamics of T τ (Σ) can be given explicitlyand passivity indices of T τ (Σ) can be obtained directly bysolving an LMI (e.g., see Lemma 3 in [21]). Note that inthis case the term δ does not exist. Example 2 Consider the following LTI system: Σ : ˙ x = (cid:20) − . − . . − . (cid:21) x + (cid:20) . . (cid:21) u,y = (cid:20) . − . . . (cid:21) x + (cid:20) . . (cid:21) u, where x ( t ) , u ( t ) , y ( t ) ∈ R . One can verify that Σ is IF- FP (0 . , . with a positive definite storage func-tion V ( x ) = 0 . | x | . Furthermore, Assumption 1 holdsand h satisfies Assumption 2 with γ = 0 . and β ( x ) =0 . | x | . Suppose that the sampling time τ = 0 . . Let-ting λ = 10 in (19) , we obtain ν (cid:48) = 0 . , ρ (cid:48) = 0 . and δ ( x ) = 6 . β ( x ) . By Theorem 1, T τ (Σ) satisfies ˆ V ( x [ k ]) − ˆ V ( x [ k ]) ≤ . | x [ k ] | + k − (cid:88) k = k (cid:0) u [ k ] (cid:62) y [ k ] − . | u [ k ] | − . | y [ k ] | (cid:1) for any k , k ∈ Z ≥ , k < k where ˆ V ( x ) = τ V ( x ) =0 . | x | . By solving the LMI in Lemma 3 of [21], onecan verify that T τ (Σ) is IF-OFP (0 . , . with a pos-itive definite storage function V ( x ) = 0 . | x | , which isdeliberately chosen the same as above. Then, T τ (Σ) sat-isfies the following inequality for any k ∈ Z ≥ : ˆ V ( x [ k + 1]) − ˆ V ( x [ k ]) ≤ u [ k ] (cid:62) y [ k ] − . | u [ k ] | − . | y [ k ] | where ˆ V ( x ) = 0 . | x | . From the two passivity inequal-ities above, one can observe that passivity indices derivedfrom (19) and from solving the LMI are not comparabledirectly.4.2 Passivity Degradation of ˜ T τ (Σ)In this subsection, we consider deriving passivity indicesof ˜ T τ (Σ) from those of T τ (Σ). Recall that the passiv-ity indices of T τ (Σ) not only can be obtained from acontinuous-time system Σ by Theorem 1, in which case T τ (Σ) satisfies (18) with ν (cid:48) , ρ (cid:48) , δ given in (19), but alsomay be obtained directly from the dynamics of T τ (Σ),in which case the bias term δ in (18) does not exist.Note that the system ˜ T τ (Σ) shown in Fig. 3 consistsof T τ (Σ) and two uniform quantizers Q , Q . The (uni-form) quantization function Q ( · ) with a quantizationprecision µ is defined as Q ( s ) := (cid:22) sµ (cid:23) µ, if s ≥ (cid:24) sµ (cid:25) µ, if s < , (20)where (cid:98)·(cid:99) is the floor function (i.e., (cid:98) x (cid:99) is the greatestinteger no larger than x ) and (cid:100)·(cid:101) is the ceiling function(i.e., (cid:100) x (cid:101) is the least integer no less than x ). If s is avector, then Q is implemented entry-wisely.The input of ˜ T τ (Σ), ˜ u [ k ], and the quantized input, u [ k ],are related by u [ k ] = Q (˜ u [ k ]) where the function Q ( · )is given in (20) with a quantization precision µ . It is easy to see that for any k ∈ Z ≥ , | u [ k ] | ≤ | ˜ u [ k ] | and | u [ k ] − ˜ u [ k ] | ≤ √ mµ , which results from the facts that | u [ k ] − ˜ u [ k ] | ≤ √ m | u [ k ] − ˜ u [ k ] | ∞ and | u [ k ] − ˜ u [ k ] | ∞ ≤ µ . The output of ˜ T τ (Σ), ˜ y [ k ], and the output beforequantization, y [ k ], are related by ˜ y [ k ] = Q ( y [ k ]) wherethe function Q ( · ) is given in (20) with a precision µ .It is clear that | y [ k ] − ˜ y [ k ] | ≤ √ mµ for any k ∈ Z ≥ .The following theorem shows how passivity indices de-grade from T τ (Σ) to ˜ T τ (Σ). The proof of the theorem isgiven in Appendix E. Theorem 2 Suppose that T τ (Σ) is IF-OFP ( ν (cid:48) , ρ (cid:48) ) thatsatisfies (18) with a positive semi-definite storage func-tion ˆ V ( x ) . Then ˜ T τ (Σ) satisfies the following passivityinequality for any k , k ∈ Z ≥ , k < k and any admis-sible inputs u ∈ U : ˆ V ( x [ k ]) − ˆ V ( x [ k ]) ≤ δ ( x [ k ]) + k − (cid:88) k = k (cid:16) ˜ u [ k ] (cid:62) ˜ y [ k ] − ˜ ν | ˜ u [ k ] | − ˜ ρ | ˜ y [ k ] | + ˜ δ (cid:17) (21) where δ ( x [ k ]) is given in (19) , ˜ ν = ν (cid:48) − | ν (cid:48) | /λ − / λ , ˜ ρ = ρ (cid:48) − | ρ (cid:48) | /λ − / λ , ˜ δ = [ | ρ (cid:48) | (1 + λ ) + λ ] mµ +[ | ν (cid:48) | (1 + λ ) + λ ] mµ , (22) and λ , · · · , λ > are arbitrary positive numbers. Remark 9 In (21) , the term ˜ δ exists due to the inputand output quantizations of ˜ T τ (Σ) , and the term δ is equalto if δ shown in (18) is equal to (e.g., δ = 0 when ν, ρ are solved by an LMI in Example 2). The followingobservations can also be made from (21) . Firstly, when µ and µ , the quantization precisions, are fixed, thereis a trade-off in choosing λ i (2 ≤ i ≤ : larger λ i resultin larger ˜ ν and ˜ ρ (i.e., more “excess of passivity”), whilesmaller λ i result in smaller ˜ δ (smaller “internal energygeneration”). Secondly, if ν (cid:48) > (resp. ρ (cid:48) > ), thenit is always possible to choose λ , λ (resp. λ , λ ) largeenough such that ˜ ν > (resp. ˜ ρ > ). Thirdly, with fixed λ i , smaller µ , µ (i.e., more precise quantizations) resultin smaller ˜ δ (i.e., less quantization effect), and ˜ δ → when µ , µ → . Therefore, if ν (cid:48) > , ρ (cid:48) > , then itis always possible to choose λ i (2 ≤ i ≤ large enoughand µ , µ small enough such that ˜ ν > , ˜ ρ > and ˜ δ arbitrarily small. Example 3 Consider the system T τ (Σ) in Exam-ple 2 again, which is IF-OFP (0 . , . with astorage function ˆ V ( x ) = 0 . | x | when τ = 0 . .Suppose that µ = µ = 0 . , and let λ i = 20 for i = 2 , , , in (22) . Then, by equation (22) , T τ (Σ) is IF-OFQP (0 . , . , . . That is, ˆ V ( x [ k + 1]) − ˆ V ( x [ k ]) ≤ ˜ u [ k ]˜ y [ k ] − . | ˜ u [ k ] | − . | ˜ y [ k ] | + 0 . for any k ∈ Z ≥ . T τ ( P ) × F ˜ T τ (Σ)In this subsection, we analyze the passivity of the system T τ ( P ) × F ˜ T τ (Σ) shown in Fig. 1 (a).Suppose that P satisfies Assumption 1 and Assump-tion 2 with constant γ and function β , and P is IF-OFP( ν , ρ ) with a continuous, positive definite, radiallyunbounded storage function V . Denote the state, inputand output of T τ ( P ) by x [ k ] ∈ R n , u [ k ] ∈ R m , y [ k ] ∈ R m , respectively. By Theorem 1, T τ ( P ) satisfies the fol-lowing inequality for any k , k ∈ Z ≥ , k < k :1 τ V ( x [ k ]) − τ V ( x [ k ]) ≤ δ ( x [ k ])+ k − (cid:88) k = k (cid:0) u [ k ] (cid:62) y [ k ] − ν (cid:48) | u [ k ] | − ρ (cid:48) | y [ k ] | (cid:1) (23)where ν (cid:48) = ν − τ γ − τ γ (1 + λ ) | ρ | ,ρ (cid:48) = ρ − | ρ | /λ ,δ ( x [ k ]) = w β ( x [ k ]) ,w = | ρ | τ (1 + λ ) + 1 /γ , (24)and λ > γ and function β . Suppose also thatΣ is IF-OFP( ν , ρ ) with a continuous, positive defi-nite, radially unbounded storage function V . Denote thestate, input and output of ˜ T τ (Σ) by x [ k ] ∈ R n , ˜ u [ k ] ∈ R m , ˜ y [ k ] ∈ R m , respectively. By Theorem 1 and The-orem 2, ˜ T τ (Σ) satisfies the following inequality for any k , k ∈ Z ≥ , k < k :1 τ V ( x [ k ]) − τ V ( x [ k ]) ≤ δ ( x [ k ])+ k − (cid:88) k = k (cid:16) ˜ u [ k ] (cid:62) ˜ y [ k ] − ˜ ν | ˜ u [ k ] | − ˜ ρ | ˜ y [ k ] | + ˜ δ (cid:17) (25) where ˜ ν = ν (cid:48) − | ν (cid:48) | /λ − / λ , ˜ ρ = ρ (cid:48) − | ρ (cid:48) | /λ − / λ , ˜ δ = [ | ρ (cid:48) | (1 + λ ) + λ ] mµ +[ | ν (cid:48) | (1 + λ ) + λ ] mµ ,ν (cid:48) = ν − τ γ − τ γ (1 + λ ) | ρ | ,ρ (cid:48) = ρ − | ρ | /λ ,δ ( x [ k ]) = w β ( x [ k ]) ,w = | ρ | τ (1 + λ ) + 1 /γ , (26)and λ , ..., λ are positive real numbers.Define x = (cid:32) x x (cid:33) , r = (cid:32) r r (cid:33) , y = (cid:32) y ˜ y (cid:33) , u = (cid:32) u ˜ u (cid:33) , (27) V ( x ) = 1 τ V ( x ) + 1 τ V ( x ) . (28)The passivity property of the system T τ ( P ) × F ˜ T τ (Σ) inFig. 1 (a) is given by the following theorem whose proofcan be obtained by using Lemma 2. Theorem 3 Suppose that P (resp. Σ ) satisfies Assump-tion 1 and Assumption 2 with constant γ and func-tion β (resp. with constant γ and function β ). Sup-pose also that P (resp. Σ ) is IF-OFP ( ν , ρ ) (resp. IF-OFP ( ν , ρ ) ) with a continuous, positive definite, radi-ally unbounded storage function V (resp. V ). Then thesystem T τ ( P ) × F ˜ T τ (Σ) in Fig. 1 (a) satisfies the fol-lowing passivity inequality for any k , k ∈ Z ≥ , k < k and any admissible input r [ k ] ∈ R m : V ( x [ k ]) − V ( x [ k ]) ≤ δ ( x [ k ]) + k − (cid:88) k = k (cid:0) r [ k ] (cid:62) y [ k ] − ˆ ν | r [ k ] | − ˆ ρ | y [ k ] | + ˜ δ (cid:1) (29) where δ ( x [ k ]) = δ ( x [ k ]) + δ ( x [ k ]) , ˆ ν < min { ν (cid:48) , ˜ ν } , ˆ ρ ≤ min { ρ (cid:48) − ˆ ν ˜ ν ˜ ν − ˆ ν , ˜ ρ − ˆ νν (cid:48) ν (cid:48) − ˆ ν } , (30) and δ , δ , ˜ δ , ν (cid:48) , ρ (cid:48) , ˜ ν , ˜ ρ are given in (24) and (26) . The following corollary shows the passivity inequality T τ ( P ) × F ˜ T τ (Σ) satisfies for a special case when r [ k ] ≡ . Corollary 1 Suppose that conditions in Theorem 3hold. When r [ k ] ≡ , the system T τ ( P ) × F ˜ T τ (Σ) in Fig. (a) satisfies the following inequality V ( x [ k ]) − V ( x [ k ]) ≤ δ ( x [ k ])+ k − (cid:88) k = k (cid:2) − (˜ ν + ρ (cid:48) ) | y [ k ] | − ( ν (cid:48) + ˜ ρ ) | ˜ y [ k ] | + ˜ δ (cid:3) , ∀ k , k ∈ Z ≥ , k < k , (31) where δ ( x [ k ]) = δ ( x [ k ]) + δ ( x [ k ]) , and δ , δ , ˜ δ , ν (cid:48) , ρ (cid:48) , ˜ ν , ˜ ρ are given in (24) and (26) . The particular case when r [ k ] ≡ requires less restric-tive conditions for the closed-loop system to have pos-itive passivity indices. By Remark 9, when r [ k ] ≡ , if ν (cid:48) + ρ (cid:48) > ν (cid:48) + ρ (cid:48) > 0, then it is always possi-ble to choose λ , ..., λ large enough and µ , µ smallenough such that ˜ ν + ρ (cid:48) > ν (cid:48) + ˜ ρ > δ arbi-trarily small. On the contrary, (30) implies that ˆ ρ mightbe negative even if ν (cid:48) , ν (cid:48) , ρ (cid:48) , ρ (cid:48) are all positive. The following theorem shows conditions under whichthe state of T τ ( P ) × F ˜ T τ (Σ) is ultimately bounded. Theproof of the theorem is given in Appendix F. Theorem 4 Consider the system T τ ( P ) × F ˜ T τ (Σ) in Fig. 1 (a). Suppose that 1) T τ ( P ) and ˜ T τ (Σ) sat-isfy passivity inequalities (23) and (25) , respectively;2) T τ ( P ) is N -step SD that satisfies (9) and T τ (Σ) is N -step SD that satisfies (10) ; 3) ˆ ρ shown in (30) can be chosen as ˆ ρ > ; 4) there exist a compact set X ⊂ R n + n and a radially unbounded, positive defi-nite function κ : R n + n → R such that X ⊇ D and η p ( x ) − δ ( x ) ≥ κ ( x ) for all x ∈ X where D := { z | V ( z ) ≤ max { V ( x [0]) , V ( x [1]) , ..., V ( x [ N ]) , d + d + d }} is a compact set, V is defined in (28) , d := ( N +1)[( η + ϑη ) (cid:107) r (cid:107) + ˜ δ ] , d := m (1 − ϑ )( N + 1)( ϑ µ + µ ) , d is a positive real number, d := max z ∈C V ( z ) , C := { z | p ( z ) ≤ d + d + d } , p ( x ) := (1 − ϑ )( p ( x ) + p ( x )) , η := λ − ˆ ν > , η := ˆ ρ − λ > , λ is a number satis-fying < λ < ˆ ρ , ϑ is defined in (11) , δ and ˜ δ are givenin Theorem 3, and N := max { N , N } . Then,i) x [ k ] ∈ D for any k ∈ Z ≥ ;ii) there exists K ∈ Z ≥ such that x [ k ] ∈ D for all k ≥ K where D := { x | V ( x ) ≤ d + d + d } is acompact set. When r [ k ] ≡ , conditions in Theorem 4 can be simpli-fied, as shown by the following corollary. Corollary 2 Consider the system T τ ( P ) × F ˜ T τ (Σ) in Fig. 1 (a) where r [ k ] ≡ . Suppose that condi-tions 1)-2) in Theorem 4 hold, and 3) ˜ ν + ρ (cid:48) > , ν (cid:48) + ˜ ρ > ; 4) there exist a compact set X ⊂ R n + n and a radially unbounded, positive definite func-tion κ : R n + n → R such that X ⊇ D and p ( x ) − δ ( x ) ≥ κ ( x ) for all x ∈ X where D := { z | V ( z ) ≤ max { V ( x [0]) , V ( x [1]) , ..., V ( x [ N ]) , d + d + d }} , V is defined in (28) , d := ( N + 1)˜ δ , d := m ( N +1)( ϑ µ + µ ) , d is a positive real number, d := max z ∈C V ( z ) , C := { z | p ( z ) ≤ d + d + d } and p ( x ) := p ( x ) + p ( x ) , δ and ˜ δ are given in Theorem3, and N := max { N , N } . Then, i) x [ k ] ∈ D for any k ∈ Z ≥ ; ii) there exists K ∈ Z ≥ such that x [ k ] ∈ D for all k ≥ K where D := { x | V ( x ) ≤ d + d + d } isa compact set. The proof of Corollary 2 is given in Appendix G. Corol-lary 2 shows that the state of T τ ( P ) × F ˜ T τ (Σ) with r [ k ] ≡ is ultimately bounded, if the shortage of pas-sivity of one component can be compensated for by theexcess of passivity of another component in the feedbackconnection, and in addition, each component is SD.When the dynamical models of a plant and a controllerare known exactly, it might be easier to analyze theclosed-loop system using these models directly. However,it is almost impossible to obtain an exact model of aplant or a controller. The passivity-based analysis shownabove enables us to compute the passivity degradationfrom the original indices and derive the property of theclosed-loop system (under sampling and quantization)without knowing the exact model (e.g., see [43,34,2]).Therefore, the passivity-based analysis is robust becauseit holds for a family of plants and controllers. Althoughthe ultimate bounds given in Theorem 4 and Corollary 2may not be tight, we point out that the ultimate boundgiven in Theorem 4 will decrease as (cid:107) r (cid:107) , µ , µ decrease,and the ultimate bound given in Corollary 2 can be madearbitrarily small when µ , µ are chosen small enough. Remark 10 In [45], modifying the passivity indices of asystem by simply adding a feedforward loop and/or a feed-back loop was discussed, where the achievable bounds ofthe modified passivity indices were given explicitly. Whencondition 4) in Theorem 4 (or Corollary 2) is not satis-fied, the results in [45] may be used to modify the passivityindices of the feedback-connected system. Remark 11 The emulation-based design is a generalframework for the controller design under time sampling:a controller is designed in the continuous time domain atfirst, and then sampled and implemented using a samplerand hold device so that certain property can be preservedunder the time sampling (see [23,24,29,30] and referencestherein). The notion of ( V, w ) -dissipativity was used tocharacterize the property of the system for preservation.There are two main differences between the emulation-based approach and the method used here: 1) the sam-pling time in the emulation-based approach is an implicitparameter that needs to be chosen sufficiently small butwhose explicit value is hard to compute in general (see[30] for more details), while the sampling time in the re-sults above is an explicit parameter whose effect on the egradation of the passivity indices is explicitly known;2) the input and output quantizations are not consideredin the emulation-based approach. In the next section, wewill show that the symbolic control implementation canbe also studied in our framework. In this section, we study state boundedness of the system T τ ( P ) × F ˜ T τµη (Σ) shown in Fig. 1 (b) where a symboliccontroller T τµη (Σ) approximately bisimilar to Σ is im-plemented. To that end, we introduce an auxiliary con-figuration, denoted by T τ ( P ) × w F ˜ T τ (Σ), as shown in Fig.4 where an external bounded disturbance w is added to˜ y in the system T τ ( P ) × F ˜ T τ (Σ). In what follows, wewill first show the passivity property and state bounded-ness of T τ ( P ) × w F ˜ T τ (Σ) by assuming that w is bounded,and based on that, we will show the state boundednessof T τ ( P ) × F ˜ T τµη (Σ) by showing that there is a partic-ular choice of w that relates the auxiliary configurationand the setup T τ ( P ) × F ˜ T τµη (Σ) of interest. w ++ D/A P A/D T τ (Σ) Q Q − + r u y ˜ u ˇ y r u y ˜ y T τ ( P ) ˜ T τ (Σ) ++ D/A P A/D T τ (Σ) Q Q − + r u y ˜ u ˜ y r u y T τ ( P ) ˜ T τ (Σ) ++1 Fig. 4. Setup of the closed-loop system T τ ( P ) × w F ˜ T τ (Σ),which is T τ ( P ) × F ˜ T τ (Σ) with a bounded disturbance w adding to ˜ y . Denote the states of T τ ( P ), T τ (Σ) and T τµη (Σ) by x [ k ], x [ k ] and x s [ k ], respectively (“s” stands for “symbolic”).Suppose that P (resp. Σ) satisfies Assumption 1 with y ( x , u ) = h ( x ) + h ( u ) (resp. with y ( x , u ) = h ( x ) + h ( u )) and Assumption 2 with constant γ and function β (resp. constant γ and function β ). Sup-pose also that P (resp. Σ) is IF-OFP( ν , ρ ) (resp. IF-OFP( ν , ρ )) with a continuous, positive definite, radi-ally unbounded storage function V (resp. V ). Further-more, suppose that there exist positive numbers (cid:15), µ, η > T τ (Σ) ∼ = ( (cid:15),µ ) T τµη (Σ) holds where µ = µ (recall that µ is the quantization level of the quantizer Q ), and there exists L > z , z ∈ R n , | h ( z ) − h ( z ) | ≤ L | z − z | ∞ . (32)Define x = (cid:32) x x (cid:33) , r = (cid:32) r r (cid:33) , y = (cid:32) y ˇ y (cid:33) , u = (cid:32) u ˜ u (cid:33) for signals depicted in Fig. 4. Next, we show the passivity property of T τ ( P ) × w F ˜ T τ (Σ) by assuming that | w [ k ] | ≤ L(cid:15) + 2 √ mµ , ∀ k ∈ Z ≥ , (33)where µ is the quantization level of quantizer Q . Recallthat for any k ∈ Z ≥ , | u [ k ] | ≤ | ˜ u [ k ] | , | u [ k ] − ˜ u [ k ] | ≤√ mµ and | ˜ y [ k ] − y [ k ] | ≤ √ mµ . In addition, sinceˇ y [ k ] = w [ k ] + ˜ y [ k ], it follows that | ˇ y [ k ] − y [ k ] | ≤| ˇ y [ k ] − ˜ y [ k ] | + | ˜ y [ k ] − y [ k ] | ≤ L(cid:15) + 3 √ mµ , ∀ k ∈ Z ≥ . Similar to Theorem 2, it is easy to show that thefollowing inequality holds for any k , k ∈ Z ≥ , k < k ,and any admissible inputs ˜ u ∈ U :ˆ V ( x [ k ]) − ˆ V ( x [ k ]) ≤ δ + k − (cid:88) k = k (cid:16) ˜ u [ k ] (cid:62) ˇ y [ k ] − ˜ ν | ˜ u [ k ] | − ˜ ρ | ˇ y [ k ] | + ˜ δ (cid:17) (34)where ˆ V ( x ) = τ V ( x ), δ , ˜ ν , ˜ ρ are given by theequations in (26), respectively, and˜ δ =[ | ρ (cid:48) | (1 + λ ) + λ ]( L(cid:15) + 3 √ mµ ) + [ | ν (cid:48) | (1 + λ ) + λ ] mµ , (35)with ν (cid:48) , ρ (cid:48) given in (26).As discussed in Subsection 5.1, ˜ T τ ( P ) satisfies (23) where ν (cid:48) , ρ (cid:48) , δ are given in (24). Similar to Theorem 3, it iseasy to show that T τ ( P ) × w F ˜ T τ (Σ) shown in Fig. 4 sat-isfies inequality (29) for k , k ∈ Z ≥ , k < k and anyadmissible input r [ k ] ∈ R m where V ( x ) = τ V ( x ) + τ V ( x ), ˆ ν, ˆ ρ satisfy (30), δ ( x [0]) = δ ( x [0])+ δ ( x [0]),and δ , δ , ˜ δ , ν (cid:48) , ρ (cid:48) , ˜ ν , ˜ ρ are those in (24) and (34).Now let us consider the SD property of T τ ( P ) × w F ˜ T τ (Σ)similar to the discussion in preceding sections. Supposethat T τ ( P ) and T τ (Σ) are both SD such that inequalities(9) and (10) hold. Note that | ˜ u | ≥ | u | −| ˜ u − u | ≥ | u | − mµ and | ˇ y | ≥ | y | − | ˇ y − y | ≥ | y | − ( L(cid:15) + 3 √ mµ ) . Then, we have ϑ | ˜ u [ k ] | + | ˇ y [ k ] | ≥ ( ϑ | u [ k ] | + | y [ k ] | ) − ϑ mµ − ( L(cid:15) + 3 √ mµ ) ,which implies that (cid:80) k + N k = k ϑ | ˜ u [ k ] | + | ˇ y [ k ] | ≥ p ( x [ k ]) − ( N + 1)[ ϑ mµ + ( L(cid:15) + 3 √ mµ ) ] . Then,similar to the proof of Lemma 3, it is easy to show thatfor any x [ k ] and any r [ k ], we have k + N (cid:88) k = k ϑ | r [ k ] | + | y [ k ] | ≥ p ( x [ k ]) − ( N + 1)[ ϑ mµ + ( L(cid:15) + 3 √ mµ ) ]where N, p, ϑ are those given in Theorem 4.With notations above, we have the following Lemma 612or the system T τ ( P ) × w F ˜ T τ (Σ) shown in Fig. 4. Theproof of Lemma 6 is similar to that of Theorem 4 and isomitted due to the space limitation. Lemma 6 Consider the system T τ ( P ) × w F ˜ T τ (Σ) shownin Fig. 4 where w [ k ] is bounded and satisfies (33) with L given in (32) and (cid:15), µ > . Suppose that inequalities (23) and (34) hold and conditions 2)-4) in Theorem 4hold. Then, there exist compact sets D , D and a con-stant K ∈ Z ≥ such that for any w [ k ] that satisfies (33) , ( x [ k ] , x s [ k ]) ∈ D for k ∈ Z ≥ and ( x [ k ] , x s [ k ]) ∈ D for k ≥ K . Using Lemma 6, we have the following results that pro-vide conditions for the state boundedness of T τ ( P ) × F ˜ T τµη (Σ) shown in Fig. 1 (b). The proof of Theorem 5 isgiven in Appendix H. Theorem 5 Consider the system T τ ( P ) × F ˜ T τµη (Σ) shown in Fig. 1 (b). Suppose that all the conditionsof Lemma 6 hold, and the initial conditions x [0] and x s [0] satisfy | x [0] − x s [0] | ∞ ≤ (cid:15) . Then, there existcompact sets D , D and a constant K ∈ Z ≥ such that ( x [ k ] , x s [ k ]) ∈ D for k ∈ Z ≥ and ( x [ k ] , x s [ k ]) ∈ D for k ≥ K . Corollary 3 Consider the system T τ ( P ) × F ˜ T τµη (Σ) shown in Fig. 1 (b) where r [ k ] ≡ . Suppose that 1) in-equalities (23) and (34) hold, 2) | x [0] − x s [0] | ∞ ≤ (cid:15) , 3)condition 2) of Theorem 4 holds, and 4) conditions 3)-4)in Corollary 2 hold. Then, there exist compact sets D , D and a constant K ∈ Z ≥ such that ( x [ k ] , x s [ k ]) ∈ D for k ∈ Z ≥ and ( x [ k ] , x s [ k ]) ∈ D for k ≥ K . Similar to the discussion in Section 5, the bounds of x , x s can be made arbitrarily small by letting the pa-rameters µ , µ , η, (cid:15) small enough. Furthermore, thebounds of x s can be used to determine how the statespace and input space of T τµη (Σ) can be chosen ascompact sets; the resulting T τµη (Σ) with bounded statespace and input space is a finite transition system thatcan be implemented with finite precision.Above result formalizes the intuition that when a con-troller guaranteeing stability in some robust way is re-placed by its symbolic bisimilar version, the feedback-connected system should be “somewhat stable”. The keychallenge to keep in mind when replacing the controllerin the feedback loop is the fact that the internal signalsdriving the plant changes therefore, within the feedback-loop, symbolic model and the actual controller can bedriven by totally different inputs even if they are ini-tialized with the same initial conditions. The externalbounded signal w we introduce in our analysis capturesthis additional robustness required. Moreover, passivityindices provide us a way to explicitly compute global andultimate bounds on the states of the closed-loop systemwith the symbolic controller implementation. Remark 12 There is a trade-off between the passivitydegradation and the construction of the approximatelybisimilar model T τµη (Σ) : on one hand, a smaller sam-pling time τ can result in a smaller passivity degradation;on the other hand, given a precision (cid:15) , the sampling time τ should be chosen large enough in order to make (3) holds and T τ (Σ) ∼ = ( (cid:15),µ ) T τµη (Σ) . Example 4 Consider the following nonlinear systemthat is adopted from Example 1 of [44]: P : ˙ x = − . x − . x − . x + 0 . u ˙ x = 0 . x − . x + 0 . u y = 0 . x y = 0 . x System P is passive with a storage function V ( x ) = ( x + x ) since ˙ V ≤ u (cid:62) y , which implies P is IF-OFP (0 , . Assumption 1 holds for P since h ( x ) =(0 . x , . x ) (cid:62) , and Assumption 2 holds for h with γ =0 . since (17) can be verified using sum-of-squares op-timization. Consider the configuration of Fig. 1 (b) where P is the plant above and Σ is the controller in Example 2.Suppose that r [ k ] = and choose µ = µ = 0 . . Givena precision (cid:15) = 0 . , parameters τ = 0 . , µ = µ , η = 0 . can be chosen such that T τ (Σ) ∼ = ( (cid:15),µ ) T τµη (Σ) . We canverify that T τ ( P ) is -step SD with ϑ = 0 and p ( x ) =0 . x + 0 . x , and T τ (Σ) is -step SD with ϑ = 0 and p ( x ) = 0 . x − . x x +0 . x . We can also verifythat conditions of Corollary 3 hold, and thus x [ k ] , x s [ k ] are ultimately bounded. To demonstrate the effect of statequantization of T τµη (Σ) , we also consider using quantiza-tion precisions η = 0 . and η = 0 . , both of which alsoguarantee T τ (Σ) ∼ = ( (cid:15),µ ) T τµη (Σ) . In Fig. 5, the trajecto-ries of x [ k ] , x s [ k ] are shown for three different choicesof η where the initial states are all x [0] = ( − . , − (cid:62) , x [0] = (1 . , − . (cid:62) . From these figures, it can be ob-served that the state trajectories are eventually bounded,and the smaller η (i.e., the state quantization precision)is, the smaller the ultimate bound would be. In this paper, we analyzed the passivity degradationfrom a continuous-time control system to its sampled, in-put and output quantized model, and used these resultsto analyze the feedback-connected system when a contin-uous controller is implemented via sampling and quanti-zation. We also considered bisimulation-based symbolicimplementations, where in addition to inputs and out-puts, the internal states of the controller is also quan-tized, therefore the controller has a fully discrete repre-sentation. 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IEEETrans. on Control of Network Systems , 5(3):1003–1015, 2018. [48] F. Zhu, M. Xia, and P. Antsaklis. Passivity analysis andpassivation of feedback systems using passivity indices. In American Control Conference , pages 1833–1838, 2014.[49] F. Zhu, M. Xia, and P. Antsaklis. On passivity analysisand passivation of event-triggered feedback systems usingpassivity indices. IEEE Trans. on Automatic Control ,62(3):1397–1402, 2017. AppendixA Proof of Lemma 3 Clearly 0 ≤ max { ϑ (1 − ϑ ) , ϑ (1 − ϑ ) } ≤ ϑ < p isa radially unbounded, positive definite function. Recallthat r [ k ] = y [ k ] + u [ k ], r [ k ] = u [ k ] − y [ k ]. For any x [ k ] and any r [ k ], k + N (cid:88) k = k ϑ | r [ k ] | + | y [ k ] | = k + N (cid:88) k = k (1 − ϑ )( | y [ k ] | + | y [ k ] | ) + ϑ | u [ k ] | + | u [ k ] | )+ ϑ [( | u [ k ] | | y [ k ] | + 2 u [ k ] (cid:62) y [ k ])]+ ϑ [( | u [ k ] | | y [ k ] | − u [ k ] (cid:62) y [ k ])] ≥ k + N (cid:88) k = k (1 − ϑ )( | y [ k ] | + | y [ k ] | ) + ϑ | u [ k ] | + | u [ k ] | ) ≥ (1 − ϑ ) k + N (cid:88) k = k | y [ k ] | + | y [ k ] | + ϑ | u [ k ] | + ϑ | u [ k ] | ≥ p ( x [ k ]) . This completes the proof. (cid:50) B Proof of Lemma 4 Because νρ ≤ (see [2]), it holds that η > ρ − ν ≥ u (cid:62) y ≤ λ | u | + λ | y | , from (6) we have V ( x [ k + 1]) − V ( x [ k ]) ≤ λ | u [ k ] | + λ | y [ k ] | − ν | u [ k ] | − ρ | y [ k ] | + δ = η | u [ k ] | − η | y [ k ] | + δ ≤ η (cid:107) u (cid:107) + δ. (B.1)Therefore, for any k ∈ Z ≥ , V ( x [ k + N + 1]) − V ( x [ k ]) ≤ k + N (cid:88) k = k (cid:0) η | u [ k ] | − η | y [ k ] | + δ (cid:1) ≤ c − η p ( x [ k ]) , (B.2)where the second inequality is from (8).i). Since V ( x ) is radially unbounded, C , D are com-pact sets and c < ∞ . Clearly, x [0] ∈ D . For k ∈{ , ..., N, N + 1 } , (B.1) implies V ( x [ k ]) ≤ V ( x [0]) + k ( η (cid:107) u (cid:107) + δ ) ≤ c + c . (B.3)15herefore, x [ k ] ∈ D for any k ∈ { , ..., N +1 } . Considernow k ∈ { N + 2 , ..., N + 2 } . If there exists some k ∗ ∈{ N +2 , ..., N +2 } such that x [ k ∗ ] / ∈ D , or equivalently, V ( x [ k ∗ ]) > c + c , (B.4)then by (B.1) we have V ( x [ k ∗ − N − ≥ V ( x [ k ∗ ]) − ( N +1)( η (cid:107) u (cid:107) + δ ) > c , implying that p ( x [ k ∗ − N − >c /η . Then, V ( x [ k ∗ ]) ≤ V ( x [ k ∗ − N − c − η p ( x [ k ∗ − N − ≤ V ( x [ k ∗ − N − ≤ c + c (B.5)where the first inequality is from (B.2), and the thirdinequality is from (B.3). Noting that (B.5) contradictswith (B.4), we conclude that x [ k ] ∈ D for k ∈ { N +2 , ..., N + 2 } . By induction, x [ k ] ∈ D for any k ∈ Z ≥ .ii). We claim that x [ k ] ∈ D for some k ∈ Z ≥ impliesthat x [ k + n ( N + 1)] ∈ D for any n ∈ Z ≥ . Indeed, x [ k ] ∈ D implies that V ( x [ k ]) ≤ c + c . If x [ k + N +1] / ∈D , then V ( x [ k + N + 1]) > c + c . From (B.2) we have V ( x [ k ]) ≥ V ( x [ k + N + 1]) − c > c , which implies that η p ( x [ k ]) > c + c by the definition of c . Then, againfrom (B.2), we have V ( x [ k + N + 1]) ≤ V ( x [ k ]) + c − η p ( x [ k ]) < V ( x [ k ]) ≤ c + c , which contradicts withthe assertion that V ( x [ k + N + 1]) > c + c . Define j s = min { k ∈ Z ≥ | k ≡ s (mod N + 1) , V ( x [ k ]) ∈D } ≤ ∞ for s = 0 , , ..., N where “mod” denotes themodulo. The claim above shows that x [ k ] ∈ D for any k ≥ j s where k ≡ s (mod N + 1). For any k + N +1 < j s where k ≡ s (mod N + 1), k ∈ Z ≥ , (B.2)implies that V ( x [ k + N + 1]) − V ( x [ k ]) ≤ c − η p ( x [ k ]).Since x [ k ] / ∈ D , V ( x [ k ]) > c + c > c , which impliesthat η p ( x [ k ]) > c + c . Therefore, V ( x [ k + N + 1]) − V ( x [ k ]) ≤ − c . Hence, j s ≤ ( V ( x [0] − c − c )) /c < ∞ .Choose K = max { j , j , ..., j N } . Then, x [ k ] ∈ D forany k ≥ K, k ∈ Z ≥ . This completes the proof. (cid:50) C Proof of Lemma 5 The system Σ d is 0-step SD implies that ϑ | u [ k ] | + | y [ k ] | ≥ p ( x [ k ]) , ∀ k ∈ Z ≥ . As in Lemma 4, we choose λ such that 0 < λ < ρ , and define η = λ − ν > η = ρ − λ > 0. Then we have V ( x [ k +1]) − V ( x [ k ]) ≤ η | u [ k ] | − η | y [ k ] | + δ ≤− η p ( x [ k ])+( η + ϑη ) | u [ k ] | + δ ≤− η c | x [ k ] | λ +( η + ϑη ) | u [ k ] | + δ Define K ∞ -functions α ( s ) = as λ , α ( s ) = bs λ , α ( s ) = η cs λ , a K -function σ ( s ) = ( η + ϑη ) s and d = δ .Then V is an ISpS-Lyapunov function of Σ d by Def. 10.The conclusion follows by Theorem 2.5 in [25]. (cid:50) D Proof of Theorem 1 Let the inputs of Σ to be piecewise constant, that is, u ( t ) = u [ k ] for any t ∈ [ kτ, ( k +1) τ ). Since x [ k ] = x ( kτ ),we have | y ( t ) − y [ k ] | = | h ( x ( t )) − h ( x ( kτ )) | ≤ (cid:90) ( k +1) τkτ | ˙ h ( s ) | d s ≤ √ τ (cid:115)(cid:90) ( k +1) τkτ | ˙ h ( s ) | d s, (D.1)where the last inequality is from the Cauchy-Schwarzinequality. For any k , k ∈ Z ≥ , k < k and for anyadmissible u ( t ), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) k τk τ u ( t ) (cid:62) y ( t ) d t − τ k − (cid:88) k = k u [ k ] (cid:62) y [ k ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k − (cid:88) k = k (cid:90) ( k +1) τkτ | u [ k ] || y ( t ) − y [ k ] | d t ≤ τ √ τ k − (cid:88) k = k | u [ k ] | (cid:115)(cid:90) ( k +1) τkτ | ˙ h ( s ) | d s ≤ τ √ τ (cid:118)(cid:117)(cid:117)(cid:116) k − (cid:88) k = k | u [ k ] | (cid:115) γ (cid:90) k τk τ | u ( s ) | d s + β ( x [0]) ≤ τ γ k − (cid:88) k = k | u [ k ] | + τ βγ , where the second inequality is from (D.1), the thirdinequality is from the Cauchy-Schwartz inequality, thefourth inequality is from (15), the fifth inequality is be-cause β ( x [ k ]) > 0. Therefore, (cid:90) k τk τ u ( t ) (cid:62) y ( t )d t ≤ τ k − (cid:88) k = k u [ k ] (cid:62) y [ k ]+ τ γ k − (cid:88) k = k | u [ k ] | + τ βγ . (D.2)It is clear that − ν (cid:90) k τk τ | u ( t ) | d t = − τ ν k − (cid:88) k = k | u [ k ] | . (D.3)Furthermore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) k τk τ y ( t ) (cid:62) y ( t ) d t − τ k − (cid:88) k = k y [ k ] (cid:62) y [ k ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k − (cid:88) k = k (cid:90) ( k +1) τkτ | y ( t ) − y [ k ] | d t + 2 k − (cid:88) k = k (cid:90) ( k +1) τkτ | y [ k ] || y ( t ) − y [ k ] | d t ≤ k − (cid:88) k = k (cid:90) ( k +1) τkτ | y ( t ) − y [ k ] | d t + k − (cid:88) k = k (cid:90) ( k +1) τkτ λ | y ( t ) − y [ k ] | + 1 λ | y [ k ] | d t ≤ τ (1+ λ ) (cid:32) γ (cid:90) k τk τ | u ( t ) | d t + β (cid:33) + τλ k − (cid:88) k = k | y [ k ] | τ γ (1+ λ ) k − (cid:88) k = k | u [ k ] | + τλ k − (cid:88) k = k | y [ k ] | + τ (1+ λ ) β where λ is an arbitrary positive number, the third in-equality is from (15). Therefore, − ρ (cid:90) k τk τ y ( t ) (cid:62) y ( t ) d t ≤ τ γ (1 + λ ) | ρ | k − (cid:88) k = k | u [ k ] | + τ ( | ρ | λ − ρ ) k − (cid:88) k = k | y [ k ] | + τ | ρ | (1 + λ ) β ( x [ k ]) . (D.4)Since V ( x [ k τ ]) = V ( x ( k τ )), V ( x [ k ]) = V ( x ( k τ )), V ( x ( k τ )) − V ( x ( k τ )) ≤ (cid:82) k τk τ u ( t ) (cid:62) y ( t ) − ν | u ( t ) | − ρ | y ( t ) | d t , adding (D.2), (D.3), (D.4) results in (18). (cid:50) E Proof of Theorem 2 For any k ∈ Z ≥ , since | y [ k ] − ˜ y [ k ] | ≤ √ mµ , we have (cid:12)(cid:12) | y [ k ] | − | ˜ y [ k ] | (cid:12)(cid:12) ≤ | ˜ y [ k ] || y [ k ] − ˜ y [ k ] | + | y [ k ] − ˜ y [ k ] | ≤ λ | ˜ y [ k ] | +(1 + λ ) mµ where λ is an arbitrary positive number. Hence, − ρ | y [ k ] | ≤ ( | ρ | λ − ρ ) | ˜ y [ k ] | + | ρ | (1+ λ ) mµ . (E.1)Similarly, since | u [ k ] − ˜ u [ k ] | ≤ √ mµ , we have (cid:12)(cid:12) | u [ k ] | − | ˜ u [ k ] | (cid:12)(cid:12) ≤| u [ k ] − ˜ u [ k ] | +2 | ˜ u [ k ] || u [ k ] − ˜ u [ k ] |≤ λ | ˜ u [ k ] | +(1 + λ ) mµ where λ is an arbitrary positive number. Hence, − ν | u [ k ] | ≤ ( | ν | λ − ν ) | ˜ u [ k ] | + | ν | (1+ λ ) mµ . (E.2)Since | u [ k ] (cid:62) y [ k ] − ˜ u [ k ] (cid:62) ˜ y [ k ] |≤| ˜ u [ k ] || y [ k ] − ˜ y [ k ] | + | ˜ y [ k ] || u [ k ] − ˜ u [ k ] |≤ λ mµ + 14 λ | ˜ u [ k ] | + λ mµ + 14 λ | ˜ y [ k ] | , where λ , λ are arbitrary positive numbers, we have u [ k ] (cid:62) y [ k ] ≤ ˜ u [ k ] (cid:62) ˜ y [ k ] + 14 λ | ˜ u [ k ] | + 14 λ | ˜ y [ k ] | + m ( λ µ + λ µ ) . (E.3)Then the inequality (21) follows immediately from (18),(E.1), (E.2) and (E.3). (cid:50) F Proof of Theorem 4 Since | ˜ u [ k ] − u [ k ] | ≤ √ mµ , | ˜ u [ k ] | ≥ | u [ k ] | −| ˜ u [ k ] − u [ k ] | ≥ | u [ k ] | − mµ ; similarly, | ˜ y [ k ] | ≥ | y [ k ] | − mµ . Hence, k + N (cid:88) k = k ϑ | ˜ u [ k ] | + | ˜ y [ k ] | ≥ p ( x [ k ]) − m ( N + 1)( ϑ µ + µ ) . (F.1)Then, similar to the proof of Lemma 3, it is easy to showthat (cid:80) k + Nk = k ϑ | r [ k ] | + | y [ k ] | ≥ p ( x [ k ]) − d .By Theorem 3, the feedback-connected system satisfiesinequality (29) where V ( x ) is positive definite, radiallyunbounded. Hence, V ( x [ k + N + 1]) − V ( x [ k ]) ≤ δ ( x [ k ])+ δ ( x [ k ])+ k + N (cid:88) k = k (cid:16) η | r [ k ] | − η | y [ k ] | + ˜ δ (cid:17) ≤ d + d − κ ( x [ k ]) . (F.2)Define j s = min { k ∈ Z ≥ | k ≡ s (mod N +1) , V ( x [ k ]) ∈ D } ≤ ∞ for s = 0 , , ..., N − 1. As inpart 2) of the proof of Lemma 4, it can be shown that x [ k ] ∈ D for any k ≥ j s where k ≡ s (mod N + 1),and V ( x [ k + N + 1]) − V ( x [ k ]) ≤ − d < k + N < j s where k ≡ s (mod N + 1), k ∈ Z ≥ . Thenthe conclusion follows immediately. (cid:50) G Proof of Corollary 2 When r [ k ] ≡ , u [ k ] = − ˜ y [ k ] and ˜ u [ k ] = y [ k ]. Recall-ing (9) and (F.1), we have (cid:80) k + N k = k ϑ | ˜ y [ k ] | + | y [ k ] | ≥ p ( x [ k ]), (cid:80) k + N k = k ϑ | y [ k ] | + | ˜ y [ k ] | ≥ p ( x [ k ]) − m ( N + 1)( ϑ µ + µ ) . Since ˜ ν + ρ (cid:48) > ν (cid:48) + ˜ ρ > (cid:80) k + Nk = k (cid:2) (˜ ν + ρ (cid:48) ) | y [ k ] | +( ν (cid:48) + ˜ ρ ) | ˜ y [ k ] | (cid:3) ≥ p ( x [ k ]) − d . By Corollary 1, when r [ k ] ≡ , the feedback-connected system satisfies the inequality shown in (31).Then, V ( x [ k + N + 1]) − V ( x [ k ]) ≤ d + d − κ ( x [ k ]).The following proof is then the same as that of Theorem4 after obtaining (F.2). (cid:50) H Proof of Theorem 5 Consider the system T τ ( P ) × w F ˜ T τ (Σ) shown in Fig. 4where the disturbance w [ k ] is chosen as w [ k ] = ˜ y s [ k ] − ˜ y [ k ] , ∀ k ∈ Z ≥ . (H.1)Because | y [ k ] − y s [ k ] | = | h ( x [ k ]) − h ( x s [ k ]) | ≤ L | x [ k ] − x s [ k ] | ∞ ≤ L(cid:15) by the Lipschitz property of h ,and | ˜ y s [ k ] − y s [ k ] | ≤ √ mµ by the property of the quan-tizer Q , it follows that | w [ k ] | ≤ | y [ k ] − ˜ y [ k ] | + | y [ k ] − y s [ k ] | + | y s [ k ] − ˜ y s [ k ] | ≤ L(cid:15) + 2 √ mµ , ∀ k ∈ Z ≥ . Hence,the particular choice of w [ k ] shown in (H.1) satisfiesassumption (33). Moreover, if both T τ ( P ) × F ˜ T τµη (Σ)and T τ ( P ) × w F ˜ T τ (Σ) are driven by the same r [ k ], then w [ k ] shown in (H.1) ensures that ˇ y [ k ] = ˜ y s [ k ] and u [ k ] = u s [ k ], ∀ k ∈ Z ≥ . By the definition of approxi-mate bisimulation (cf. Def. 2), x [ k ] and x s [ k ] are relatedby | x [ k ] − x s [ k ] | ∞ ≤ (cid:15), ∀ k ∈ Z ≥ . The state bounded-ness of T τ ( P ) × F ˜ T τµη (Σ) can be derived immediatelyby Lemma 6. (cid:50)(cid:50)