passivity and dissipativity analysis of a system and its

620 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 62, NO. 2, FEBRUARY 2017

Passivity and Dissipativity Analysis of a Systemand Its Approximation

Meng Xia, Panos J. Antsaklis, Fellow, IEEE , Vijay Gupta, and Feng Zhu

Abstract—In this paper, we consider the following prob-lem: what passivity properties can be inferred for a systemby studying only an approximate mathematical model for it.Our results show that an excess of passivity (whether in theform of input strictly passive, output strictly passive or verystrictly passive) in the approximate model guarantees a cer-tain passivity index for the system, provided that the normof the error between the approximate and the true modelsis sufficiently small in a suitably defined sense. Further,we consider (Q,S,R)-dissipative systems and show that(Q,S,R)-dissipativity has a similar robustness property,even though the supply rates for the system and its approx-imation may be different. These results may be particularlyuseful if either the approximate model is much easier toanalyze, or if the precise system model is unknown. Weillustrate the results by considering particular approxima-tion methods, e.g., model reduction, discretization, quanti-zation, and linearization around an equilibrium point.

Index Terms—Approximation, dissipativity, linearization,linear systems, model reduction, nonlinear systems, pas-sivity, quantization, sampling.

I. INTRODUCTION

TOOLS that can guarantee system-level properties, suchas stability, from the properties of individual components

and their interactions are important to develop a systematicdesign theory for large-scale systems. The classical notionsof passivity, and more generally dissipativity, are relevant inthis respect. Informally, these concepts characterize the energyconsumption of a dynamical system and have a long historyof use in both analysis and synthesis. A useful property ofpassivity is that it is a compositional property for paralleland feedback interconnections. Moreover, under quite generalassumptions, passivity implies stability [1]–[4]. Passivity anddissipativity theory has been used as a powerful analysis anddesign tool in many applications, such as large space structures[5], chemical processes [3], multi-agent systems [6], haptic tele-operation systems [7], and cyber-physical systems [8], [9].

In this paper, we are particularly interested in the passivity ofa system as inferred from studying an approximate model of its

Manuscript received April 15, 2015; revised January 21, 2016 andJanuary 24, 2016; accepted April 12, 2016. Date of publication May 4,2016; date of current version January 26, 2017.This work was supportedin part by the National Science Foundation under Grants CNS-1035655and CNS-1446288, and is gratefully acknowledged. Recommended byAssociate Editor L. Menini.

The authors are with the Department of Electrical Engineering, Uni-versity of Notre Dame, Notre Dame, IN 46556 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2016.2562919

dynamical behavior. In a large scale system, precise knowledgeof the mathematical model will be difficult to obtain. Moreover,even if such a model were obtainable, the classical tradeoffbetween model accuracy and tractability [10], [11] may leadto a simpler model being preferred for analysis. A variety ofapproximation methods can be used for analysis, simulationor control design [12]. While it is known that under someconditions, some of these methods such as linearization [1],[13] and model reduction [14], [15] can be performed so as topreserve passivity, the question of whether passivity of a systemcan be guaranteed if a system with a model close (in a suitablydefined sense) to it is passive still remains open.

To quantify the excess and shortage of passivity propertiesof a system, we use the notion of passivity levels (or indices).Informally speaking, positive passivity levels indicate that asystem is passive and negative passivity levels indicate that thesystem is non-passive. Moreover, the magnitudes of the passiv-ity levels imply “how far” a system is away from being passive.The connections between the passivity levels and conic systemshave been studied in [16] and [17]. Passivity levels have beenused in e.g., [3], [18] to render a non-passive system passivethrough feedback and feedforward. Further, the use of passivitylevels generalizes the passivity theorem for feedback intercon-nected systems. If the plant has negative passivity levels, thenthe controller can be designed to have sufficiently large positivepassivity levels in order to guarantee passivity and stability ofthe feedback interconnection, see e.g., [7], [19]. As a result, itis of great interest to have knowledge of the passivity levels.

The main contribution of this paper is the relationship be-tween passivity levels of two mathematical models, one ofwhich could represent accurately a physical system and theother could represent an approximation. It should be notedthat, in general, the approximate model is different from the“real” plant (or controller). The present paper provides a solidfoundation for obtaining the passivity levels for the “real”plant (or controller) from studying the approximate model. Theapproximate model is assumed to have an excess of passivity,defined through the concept of passivity levels [3]. Informallystated, our main result is of the form that if the error betweenthe system and its approximation is small in a suitably definedsense, the passivity levels for the true system can be guaranteed.We then illustrate our results through approximate models ob-tained using particular approximation methods, such as modelreduction of a higher-order system to obtain a lower-ordermodel [12], [14], sampled-data systems [20], [21], quantization[22], [23] and linearization of a nonlinear system around anequilibrium point [1], [19]. Since there is a rich theory of usingpassivity levels (or indices) to design control laws [3], [4],

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XIA et al.: PASSIVITY AND DISSIPATIVITY ANALYSIS OF A SYSTEM AND ITS APPROXIMATION 621

[24], our results imply that it is possible to use the (hopefullymore tractable) approximate model for control design. It maybe noted that our results may be conservative if a specificapproximation method is used, since they bound passivity forany model that has a bounded difference from the given systemwithout considering any particular characterization.

The closest results to ours in the literature seem to be theline of work that obtains particular approximation methodssuch as model order reduction that preserve passivity [12], [14],[15]. As opposed to these works, we focus on the variationin the passivity levels caused by an arbitrary method used forapproximation. Our objective is not to provide new approxi-mation techniques, but to provide a framework for obtainingpassivity levels when particular approximation methods areconsidered. Another related line of work uses the integralquadratic constraint (IQC) framework for stability analysis of afeedback interconnection of a known system and a perturbation[25]–[28]. We focus on passivity rather than stability directlydue to the fact that beyond guaranteeing stability, passivityalso provides useful properties such as compositionality [3],[29]. Further, our results also provide synthesis tools to designcontrollers so that the system can be rendered passive (andhence stable) [4].

Some preliminary results of this paper have appeared in [30]and [31]. In [30], we focused on model reduction and consid-ered the specific method of positive real truncated balanced re-alization (PR-TBR) for model reduction. In this paper, we haveextended the results to more general approximation methodsthat include as special cases model reduction (not necessarilyPR-TBR), discretization, quantization and linearization aroundan equilibrium point. Further, we have provided new examplesto illustrate the results. While both this paper and [31] considersector bounded quantizers, the main results and the approachesare different. In [31], we focused only on quantization andintroduced an input-output transformation matrix to passivatethe system after quantization. However, in this paper, we derivesufficient conditions for guaranteeing passivity levels for thesystem after more general approximation methods, includingbut not restricted to quantization. Further, this paper does notconsider passivation techniques at all.

The rest of the paper is organized as follows. Section IIprovides background material on passivity and (Q,S,R)-dissipativity. Section III presents the problem statement. Themain results are given in Section IV. In particular, given asystem of interest, we relate the passivity level of the sys-tem to that of its approximate model, when the model isinput strictly passive (Section IV-A), output strictly passive(Section IV-B), very strictly passive (Section IV-C) and(Q,S,R)-dissipative (Section IV-D). We then apply these re-sults to particular approximation methods in Section V. Specif-ically, model reduction of a higher-order system is consideredin Section V-A, discretization is considered in Section V-B,quantization is considered in Section V-C and linearization ofa nonlinear system about an equilibrium point is considered inSection V-D. Numerical examples are provided in Section VI.Concluding remarks are given in Section VII.

Notation: The Euclidean space of dimension m is denotedby R

m. Denote the truncation of a signal u(t) up to time

T (0 ≤ T < ∞) by uT (t). The inner product of truncated

signals uT (t) and yT (t) is denoted by 〈u, y〉T , where 〈u, y〉T Δ=∫ T

0 uT (t)y(t)dt and uT (t) denotes the transpose of u(t). TheL2-induced norm of the signal uT (t) is denoted by ‖uT (t)‖L2

,

where ‖uT (t)‖2L2

Δ=∫ T0 uT (t)u(t)dt. We use the notations u(t)

and u for a signal interchangeably. The H∞ norm of a transferfunction G(s) is denoted by ‖G‖H∞ . For a matrix A ∈ R

n×n,the minimum eigenvalue of A is denoted by λ(A) and themaximum eigenvalue by λ(A). A ≥ 0 denotes that A is positivesemi-definite and A > 0 implies that A is positive definite. Then-dimensional identity matrix is denoted by In×n or simplyI by omitting the dimensions if clear from the context. Thenotation max{a, b} denotes the larger value of a, b ∈ R andmin{a, b} denotes the smaller value of a, b ∈ R. The absolutevalue of a real number a ∈ R is denoted by |a| and the 2-normof a vector x ∈ R

m is denoted by ‖x‖.

II. PRELIMINARIES

A. Definitions

Definition 1 ([1], [32]): Consider a system Σ with input uand output y where u(t) ∈ U ⊂ R

m and y(t) ∈ Y ⊂ Rm. It is

said to be

• passive, if there exists a constant β ≤ 0 such that

〈u, y〉T ≥ β. (1)

• input feedforward passive (IFP), if there exist constants νand β ≤ 0 such that

〈u, y〉T ≥ β + ν〈u, u〉T (2)

we call such a ν an IFP level, denoted as IFP (ν). If ν > 0,then system Σ is said to be input strictly passive (ISP).

• output feedback passive (OFP), if there exist constants ρand β ≤ 0 such that

〈u, y〉T ≥ β + ρ〈y, y〉T (3)

we call such a ρ an OFP level, denoted as OFP (ρ). Ifρ > 0, then system Σ is said to be output strictly passive(OSP).

• input-feedforward-output-feedback passive (IF-OFP), ifthere exist constants ρ and ν so that

〈u, y〉T ≥ β + ρ〈y, y〉T + ν〈u, u〉T (4)

we call such ρ and ν passivity levels, denoted as IF-OFP(ρ, ν). If both ρ > 0 and ν > 0, then system Σ is said tobe very strictly passive (VSP).1

1Note that VSP is sometimes also referred to as input-output strict passivity(see e.g., [33], [34]). Given ρ > 0, ν > 0 such that (4) is satisfied, we say thatthe system is VSP for (ρ, ν). The concept of VSP may seem restrictive at firstglance; however, a system which is ISP and L2 stable is always VSP, see e.g.,[1], [17], [33].


• finite-gainL2 stable2 (FGS), if there exist constants κ > 0and β ≤ 0 such that

〈y, y〉T ≤ −β + κ2〈u, u〉T . (5)

• (Q,S,R)-dissipative, if there exist Q = QT , R = RT

and S and a constant β ≤ 0, such that

r(u, y)Δ= 〈y,Qy〉T + 2〈y, Su〉T + 〈u,Ru〉T ≥ β. (6)

The function r(u, y) is called the supply rate for Σ.

In all cases, we require that the inequality holds ∀u(t)∈ U ⊂R

m, ∀T ≥ 0 and the corresponding y(t). �Remark 1: We make the following comments.

1) In Definition 1, L2 stability and (Q,S,R)-dissipativityare defined for square systems, i.e., u, y ∈ R

m. We notethat L2 stability and (Q,S,R)-dissipativity can be de-fined for non-square systems, see e.g., [19], [35].

2) It should be noted that a system with IFP (ν) where ν > 0and OFP(ρ) where ρ > 0 may not necessarily imply thatthe system is VSP for (ρ, ν). In fact, it can be shown thata necessary condition for a system to be VSP for (ρ, ν) isgiven by ρν ≤ 1/4 (see e.g., [36]). As a result, for VSP,it may not make sense to define the largest ρ > 0 and thelargest ν > 0 (simultaneously) such that (4) holds ∀u ∈U ⊂ R

m and ∀T ≥ 0.

From (2), it can be seen that ν is a measure of the level ofpassivity for a system. Specifically, if a system Σ is ISP forν > 0, it is also ISP for ν − ε, where 0 ≤ ε < ν. Informally, apositive value of ν can be interpreted as an “excess” of passiv-ity, while a negative value implies a “shortage” of passivity. Asimilar interpretation for ρ can be obtained for an OSP system[3], and for a VSP system [36].

Notice that Definition 1 is an input-output definition. Similardefinitions when a state space model is available are alsopossible, see e.g., [1], [4], [32]. Specifically, given a system

x = f(x, u)

y = h(x, u) (7)

where x ∈ Rn is the system state, u ∈ R

m is the controlinput, and y ∈ R

m is the system output. We can define theconcepts such as passivity possibly through the use of a storagefunction. While we do not use the state space based definitionin this paper, we point out that this representation allows theconsideration of local version of these properties. For instance,if we assume that the functions f and h are real analytic about(x = 0, u = 0) and the pair (x = 0, u = 0) is an equilibriumpoint for system (7), then we can define local passivity ordissipativity in a neighborhood of (x = 0, u = 0) ∈ X × U asin [37] and [38].

Definition 2 ([37], [38]): If any of the properties for system(7) as defined above in Definition 1 hold in a neighborhood of(x=0, u=0)∈X×U , it is called a local property of system (7).

2An equivalent definition for finite-gain L2 stability of Σ is that there existconstants κ1 > 0 and β1 ≤ 0 such that ‖yT ‖L2

≤ κ1‖uT ‖L2− β1 holds

∀u(t) ∈ U , ∀T ≥ 0 and corresponding y(t) (see e.g., [1], [19], [33]).

Fig. 1. Negative feedback interconnection of system H and system G.

B. Feedback Interconnection

Consider the feedback configuration as shown in Fig. 1. Thepassivity theorem states that if both systems H and G are pas-sive, then the feedback interconnected system Σ is also passive.Under mild assumptions, stability of the closed-loop system Σis also guaranteed. The use of passivity levels generalizes thepassivity theorem for feedback interconnected systems in thesense that one of the systems can be allowed to have negativepassivity levels while the other system has positive passivitylevels. The results for feedback interconnection using passivitylevels are summarized in the following two theorems. Formore detailed discussion on guaranteeing passivity and stabilityof feedback interconnected systems using passivity levels, werefer the reader to [4].

Theorem 1 (Theorems 6.1, 6.2, and Lemma 6.8 in [19]):Consider the feedback interconnection of two systems H andG in Fig. 1.

1) If systems H and G are passive, then system Σ is passive.2) If systems H andG are output strictly passive (OSP), then

system Σ is OSP.3) If system H is IF-OFP( ρ1, ν1) and system G is IF-OFP(

ρ2, ν2), where ν1 + ρ2 > 0, ν2 + ρ1 > 0, then system Σis finite gain stable (FGS). �

In Theorem 1, the passivity of both systems G and H isrequired to guarantee that the system Σ is passive. However,it is not necessary for both systems G and H to be passiveto guarantee the finite-gain stability (FGS) of system Σ. Forinstance, if ν1 < 0 and ρ1 < 0 (i.e., system H is not passive),then we require that the system G has passivity levels ν2 >−ρ1 > 0 and ρ2 > −ν1 > 0 in order to guarantee the FGS ofthe closed-loop system Σ.

Remark 2: It should be noted that in Theorem 1, if sys-tem H is locally IF-OFP(ρ1, ν1) and system G is locally IF-OFP(ρ2, ν2), then the closed-loop system is locally finite-gainstable if ν1 + ρ2 > 0 and ν2 + ρ1 > 0, see e.g., [39].

If r2 = 0, the feedback system is given by the mapping r1 →y1. In this case, we have the following less conservative result.

Theorem 2 (Theorem 8 in [18]): Consider the feedbackinterconnection of two systems H and G in Fig. 1. Assumethat r2 = 0.

1) If system H has OFP(ρ) and system G has IFP (ν) whereρ+ ν > 0, then the system r1 → y1 has OFP(ρ+ ν).Further, the system r1 → y1 is finite-gain stable (FGS)with gain κ ≤ 1/(ρ+ ν).

2) If system H has IFP ( ν > 0) and system G has OFP(ρ)where ν + ρ > 0, then the system r1 → y1 has IFP(ρν/(ρ+ ν)).


Fig. 2. Illustration of two systems: system Σ1 with input u ∈ U ⊂ Rm

and output y1, system Σ2 with input u ∈ U ⊂ Rm and output y2 = y1 +

Δy, where u, y1, y2 and Δy are of the same dimensions.

C. Linear Systems

For linear systems, passivity and dissipativity theory is wellestablished, see e.g., [35], [33], [40] and the recent survey[17]. Given an LTI system with transfer function G(s), let itsminimal state-space realization be given by

x =Ax+Buy =Cx+Du (8)

where {A,B} is controllable and {A,C} is observable. If theinitial condition is assumed to be zero, the behavior of system(8) is determined by the corresponding transfer function G(s).The following result is useful to test whether system (8) is(Q,S,R)-dissipative.

Lemma 1 (Lemma 2 in [17] and [41, pp. 140–141]): Sys-tem (8) is (Q,S,R)-dissipative if there exists a P = PT > 0such that

ΠΔ=

[ATP + PA− Q PB − S

(PB − S)T −R

]≤ 0 (9)

where Q, S, R are given by Q = CTQC, S = CTS + CTQD,R = DTQD + (DTS + STD) +R.

Remark 3: This lemma can be used to test whether (8) is

• passive by setting S = (1/2)I,Q = 0, R = 0;• ISP by setting S = (1/2)I,Q = 0, R = −νI;• OSP by setting S = (1/2)I,Q = −ρI,R = 0;• VSP by setting S = (1/2)I,Q = −ρI,R = −νI .

Thus, the linear matrix inequality (9) can be used to find thepassivity levels for the linear system (8).

III. PROBLEM STATEMENT

Consider two system models Σ1 and Σ2 as shown in Fig. 2.One can view Σ1 as the system we are interested in andΣ2 as anapproximation of Σ1. A commonly used measure for judginghow well Σ2 approximates Σ1 is to compare the outputs forthe same excitation function u ∈ U ⊂ R

m [12]. We denote thedifference in the outputs by Δy.3 Note that, in general, Δywill depend on the exact function u. The error may be dueto modeling, linearization, model reduction, or a host of otherreasons. For a good approximation, a reasonable requirementis that the worst case Δy over all control inputs u ∈ U ⊂ R

m

be ‘small’ in terms of a suitably defined norm. More formally,

3In this paper, we focus on system models with additive uncertainty, wherey2 = y1 +Δy. Similar arguments can be developed for system models withmultiplicative uncertainty, where y2 = y1(1 +Δy), see e.g., [3].

with every approximate model, we associate two nonnegativeconstants γ > 0 and ε ≥ 0 (if they exist) such that

〈Δy,Δy〉T ≤ γ2〈u, u〉T + ε, ∀u ∈ U ⊂ Rm and ∀T ≥ 0.

(10)The values of γ and ε reflect how good the approximation is.

Remark 4: Consider the case when U = Rm. Let Σ1 and Σ2

be stable linear systems with zero initial conditions. Further,let Σ1 (resp. Σ2) be characterized by the transfer function G1

(resp. G2) and define ΔG = G1 −G2. If ‖ΔG‖H∞ ≤ γ, wecan guarantee that (10) is satisfied with ε = 0. In this case, γ isan upper bound on the H∞ norm of the difference between thetransfer functions G1 and G2.

Remark 5: It should be noted that the assumption (10) doesnot require both the systems Σ1 and Σ2 to be stable. However,the “error” between the two systems is required to be stable.As an example, for model reduction for linear systems, it hasbeen shown in [42] norm of the error ‖ΔG‖H∞ even if boththe higher-order system G1 and reduced-order system G2 areunstable. A numerical example is given by considering

G1 =1.01s+ 0.99

(s+ 1)(s− 1)

and its 1st-order approximation (denoted by G2) given by

G2 =1

s− 1.

The “error” system ΔG = G1 −G2 is given by

ΔG =0.01

s+ 1

which is stable and has H∞ norm γ = 0.01.We are now ready to state the problem of interest. Assume

that Σ2 has an excess of passivity. What passivity property forΣ1 can be inferred from that of Σ2? For the case when Σ2 doesnot have an excess of passivity but is (Q2, S2, R2)-dissipative,we may pose the same problem in terms of obtaining condi-tions under which Σ1 is (Q1, S1, R1)-dissipative as well. Theproblem is formulated as follows.

Problem 1: Consider two system models Σ1 and Σ2 asshown in Fig. 2. Suppose that an approximate model Σ2

1) has IFP (ν) where ν > 0; or2) has OFP(ρ) where ρ > 0; or3) is VSP for (ρ, ν) where ρ > 0 and ν > 0; or4) is (Q2, S2, R2)-dissipative.

What passivity or (Q,S,R)-dissipativity properties can be de-rived if the models Σ1 and Σ2 satisfy the relation (10)? �

Our aim is to characterize the passivity levels or the sup-ply rate of the system Σ1 by analyzing the (hopefully moretractable) approximate model Σ2. The obtained passivity levelscan be further used to analyze passivity and stability of feed-back interconnections. As an example, suppose that Σ2 is amathematic model for the physical system Σ1. Assume that thepassivity levels for the model Σ2 are given by ρ2 and ν2. Theanswer from Problem 1 provides the passivity levels forthe system Σ1 which are denoted by ρ1 and ν1. Then, based onTheorem 1, any controller with passivity levels ρc and νc where


ρc > −ν1 and νc > −ρ1 can guarantee the finite-gain stabilityof the closed-loop system.

IV. MAIN RESULTS

We present the results for the cases when the approximatemodel is ISP, OSP, VSP and (Q,S,R)-dissipative. It can beverified that the results are symmetric in Σ1 and Σ2. In otherwords, it does not matter whether we view Σ1 as an approx-imation of Σ2 or Σ2 as an approximation of Σ1. In practice,however, a simpler model is usually used as an approximationof a complex system. The proofs for the theorems are presentedin the Appendix. The proofs for the corollaries are omitted forspace constraints. Complete proofs can be found in [36].

A. Input Strictly Passive Systems

We have the following result that guarantees an ISP levelof Σ1 given the error constraint γ and the input feedforwardpassivity index (IFP) for the approximate model Σ2.

Theorem 3: Consider Σ1 and Σ2 in Fig. 2. Suppose (10) issatisfied for some γ > 0 and ε ≥ 0. If Σ2 has IFP (ν) whereν > 0 and γ < ν, then Σ1 is ISP for ν = ν − γ.

If we are interested merely in determining whether Σ1 ispassive (rather than characterizing the ISP level of Σ1), we canallow γ to be equal to ν.

Corollary 1: Consider Σ1 and Σ2 in Fig. 2. Suppose (10)is satisfied for some γ > 0 and ε ≥ 0. If Σ2 has IFP (ν) whereν > 0 and γ ≤ ν, then Σ1 is passive.

One interpretation of these results is that the IFP (ν) ofΣ2 provides an upper bound on the error γ caused by theapproximation of Σ1 into Σ2. Further, the difference of thesetwo values, ν − γ, provides a lower bound on the IFP of Σ1.It is apparent that there exists a trade-off between how goodthe approximation is (corresponding to the value of γ) and howpassive we can guarantee Σ1 to be (corresponding to the valueof ν − γ).

B. Output Strictly Passive Systems

When the approximate model Σ2 is OSP, we assume alongthe lines of [3] that the inverse4 of Σ2 is L2 stable.

Assumption 1: Consider Σ2 with input u and output y2,where u(t)∈U ⊂R

m and y2(t)∈Y⊂Rm. Assume the inverse

of Σ2 is L2 stable, i.e., there exist η>0 and b≥0, such that

〈u, u〉T ≤ η2〈y2, y2〉T + b, ∀u ∈ U ⊂ Rm and ∀T ≥ 0.

(11)Remark 6: The output feedback passivity index (OFP) of

an LTI system G(s) is defined only if its inverse is causal andstable, i.e., if G(s) is minimum phase and proper [3, pp. 24–32].If the system is not minimum phase, then no feedback gaincan guarantee that the closed loop system remains passive, i.e.,the system cannot be output feedback passive [3, pp. 47–56].Since in this paper, we are interested in generalizing this idea

4Invertibility of linear and nonlinear systems has been studied in e.g., [11],[43], [44].

to possibly nonlinear systems, we impose a similar conditionthrough Assumption 1.

For linear systems, using Assumption 1, the OFP level forG(s) can be shown to be equivalent to the IFP level of the in-verse of G(s), see e.g., [3], [45]. For general nonlinear systems,a sufficient condition for (11) to hold is that Σ2 has IFP ( ν),where ν > 0. This can be seen from the following arguments.The following relation holds ∀u ∈ U ⊂ R

m and ∀T ≥ 0:

1

2ν〈y2, y2〉T − ν

2〈u, u〉T ≥ 〈u, y2〉T − ν〈u, u〉T .

From (2), there exists β ≤ 0 such that 〈u, y2〉T − ν〈u, u〉T ≥β, thus

〈y2, y2〉T ≥ ν2〈u, u〉T + 2βν. (12)

Therefore, condition (11) is satisfied with η2 = 1/ν2, b =(−2β)/ν.

Using Assumption 1, we can characterize an OSP level forΣ1 from the output feedback passivity index (OFP) of Σ2.

Theorem 4: Consider Σ1 and Σ2 in Fig. 2. SupposeAssumption 1 is satisfied for Σ2 and (10) holds for some γ > 0.If Σ2 has OFP(ρ) where ρ > 0 and γ < ρ, then Σ1 is OSP forρ = ρ− γ if

γ2 −(ρ− 2

ρ

)γ +

1

η2− 3 ≥ 0. (13)

If we are interested merely in passivity of Σ1, we have thefollowing result.

Corollary 2: Consider Σ1 and Σ2 in Fig. 2. SupposeAssumption 1 is satisfied for Σ2 and (10) holds for some γ > 0.If Σ2 has OFP(ρ) where ρ > 0 and γη2 ≤ ρ, then Σ1 is passive.

C. Very Strictly Passive Systems

The following result presents the two passivity levels of asystem Σ1 from those of its approximation Σ2.

Theorem 5: Consider Σ1 and Σ2 in Fig. 2. Suppose (10)holds for some γ > 0. Suppose Σ2 is VSP for (ρ, ν) whereρ > 0 and ν > 0 and γ < min{ρ, ν}. Then, Σ1 is VSP for(ρ− γ, ν − γ) if

γ2 −(ρ− 2

ρ

)γ + ν2 − 2 ≥ 0. (14)

The result about passivity of Σ1 follows.Corollary 3: Consider Σ1 and Σ2 in Fig. 2. Suppose (10)

holds for some γ > 0. If Σ2 is VSP for (ρ, ν) where ρ > 0 andν > 0 and ρν2 + ν − γ ≥ 0, then, Σ1 is passive.

Remark 7: To ensure passivity of Σ1 (instead of positivepassivity levels), γ may be larger than the passivity levels ρand ν of Σ2 because for passivity, the only requirement is thatγ ≤ ρν2 + ν.

D. Extension to (Q,S,R)-dissipative Systems

In this section, we extend the results to (Q,S,R)-dissipativesystems that the system may be not passive. The following


result presents a supply rate of a system Σ1 from that of itsapproximation Σ2.

Theorem 6: Consider Σ1 and Σ2 in Fig. 2. Suppose (10)holds for some γ > 0. Let Σ2 be (Q2, S2, R2)-dissipative, thenΣ1 will be (Q1, S1, R1)-dissipative with S1 = S2,5 if thereexists a constant ξ > 0 such that

λ(Q1 −Q2)− ξλ(QT

1 Q1

)≥ 0 (15)

γ2

ξ+ 2max

{0, λ(−Q1)γ

2}

+ 2√λ(ST1 S1

)γ ≤ λ(R1 −R2). (16)

It should be noted that Theorem 6 holds when systems Σ1

and Σ2 represent non-square systems. If this is the case, thenthe matrices Si will be non-square.

With this theorem, the following result is immediate.Proposition 1: Consider Σ1 and Σ2 in Fig. 2. Suppose

(10) holds for some γ > 0 and Σ2 is (Q2, S2, R2)-dissipative.Suppose there exists a ξ > 0 such that (15) and (16) hold. Then,

1) Σ1 is (Q1, S2, R1)-dissipative, where Q1 ≥ Q2 andR1 > R2.

2) If Si = (1/2)I,Qi = −ρiI, Ri = −νiI for i = 1, 2 andρ2 ≤ 0, then ρi and νi satisfy

ρ2 − ρ1 − ξρ21 ≥ 0 (17)

γ2

ξ+ γ ≤ ν2 − ν1. (18)

Proof: We first prove the first item. Because QT1 Q1 ≥ 0,

we have λ(QT1 Q1) ≥ 0. Thus, from (15), we obtain λ(Q1 −

Q2) ≥ 0 and therefore Q1 −Q2 ≥ 0. Further, from (16), wehave λ(R1 −R2) ≥ (γ2/ξ) > 0 and thus λ(R1 −R2) > 0.Therefore, R1 −R2 > 0.

We now prove the second item. Equation (17) is immediatefrom(15). WhenQi=−ρiI , we obtain 2max{0, λ(−Q1)γ

2} =2max{0, ρ1γ2}. Since Q1 ≥ Q2, we have ρ1 ≤ ρ2. Becauseρ2 ≤ 0, we obtain ρ1 ≤ 0. Thus, ρ1γ2 ≤ 0 and from (16), weobtain relation (18). �

In the second case in Proposition 1, system Σ2 has a shortageof passivity when ρ2 < 0. Further, (17) and (18) imply thatthe difference ν2 − ν1 gives an upper bound on the “error” γand the parameter ξ > 0 provides us freedom to trade off thedifferences between ρ2 − ρ1 and ν2 − ν1. {As a special case,we let ξ = γ and ν2 < 0. Then, a pair of passivity levels forsystem Σ1 are given by

ν1 = ν2 − 2γ < 0

ρ1 =2ρ2√

1 + 4γρ2 + 1< 0.

Thus, this result can be used to find the negative passivity levelsof system Σ1 when Σ2 has negative passivity levels.

5Similar arguments can be developed when S1 = S2 does not hold. How-ever, the analysis is more involved.

Remark 8: Although the above development has assumedΣ1 and Σ2 to be continuous-time, similar arguments hold fordiscrete-time systems as well. The definitions of passivity anddissipativity in discrete-time domain can be found in e.g., [17],[45], [46]. The inner product of truncated signals uT (k) and

yT (k) is defined as 〈u, y〉T Δ=∑T

k=0 uT (k)y(k), where T ∈ Z

and 0 ≤ T < ∞. The �2-induced norm of a signal u(k) is

denoted by ‖uT (k)‖�2 , where ‖uT (k)‖2�2Δ=∑T

k=0 uT (k)u(k).

With these changes, we can develop results in discrete-timedomain analogous to the ones above.

Remark 9: So far, we have shown that passivity and dissi-pativity properties of a system can be inferred from its approx-imation, either in continuous-time or discrete-time domain. Wecan also consider the case when Σ1 and Σ2 are hybrid dynam-ical systems, i.e., those described by an interaction betweencontinuous and discrete dynamics, such as switched systems[47]–[51]. Note that Definition 1 applies to switched systemsas well (see also [52]–[55]). Thus, we can use our resultsto investigate passivity/dissipativity properties for a switchedsystem from those of its simplified model.

V. PARTICULAR APPROXIMATION METHODS

We shall now apply the results developed in Section IVto the case when the approximate model is produced usingsome particular approximation methods, e.g., model reduction,discretization, quantization and linearization. It should be notedthat these methods are just particular examples. Our results canbe applied to approximations caused by other factors, such astime delay, parameter uncertainties and saturation, that we donot consider.

A. Model Reduction

Model reduction is an effective approximation techniquewhen dealing with large-scale systems [15], [56]. It can beused to analyze transient stability and to design lower-ordercontrollers, for both linear systems (see e.g., [12], [14]) andnonlinear systems (see e.g., [56], [57]). We will concentrate onthe method of truncated balancing realization (TBR) for modelreduction (see e.g., [14], [15]). Given a passive system, TBRdoes not guarantee passivity of the reduced model except forsome special cases, see e.g., [14], [15].

For the LTI system (8), define its observability gramian byWo and the controllability gramian by Wc, where AWc +WcA

T = −BBT and ATWo +WoA = −CCT . The squareroots of eigenvalues of the product WcWo are called Hankelsingular values and can be used to establish upper bounds onthe error between the transfer functions of the full-order system(denoted by G1) and its reduced-order approximation (denotedby G2). If we denote σi as the ith Hankel singular value (whereσ1 ≥ σ2 ≥ . . . σn ≥ 0, and n is the order of G1), then we have

‖G1 −G2‖H∞≤ 2

n∑i=nr+1

σi (19)


where 0 ≤ nr < n is the order of the reduced-order approxi-mation G2, see e.g., [12]. Obviously, the closer nr is to n, thesmaller the error is.

Using Theorem 3, Theorem 5 and equation (19), we canobtain the following result.

Corollary 4: Consider a stable LTI system G1 with ordern. Let G2 be a reduced order model of G1 with order nr

(0 ≤ nr < n) obtained using TBR. Define γΔ= 2

∑ni=nr+1 σi,

where σi is the ith Hankel singular value of G1 and σ1 ≥ σ2 ≥. . . σn ≥ 0. Then,

1) If G2 has IFP (ν) where ν > 0 and γ < ν, then G1 is ISPfor ν − γ.

2) If G2 is VSP for (ρ, ν) where ρ > 0 and ν > 0 andγ < min{ρ, ν}, then G1 is VSP for (ρ− γ, ν − γ) if γsatisfies the inequality (14).

In the above result, G1 and G2 can exchange places. Given ahigh-order system G1, we can estimate the passivity levels forany reduced-order model G2 from the properties of G1. Thisresult is especially useful when we consider G1 as a high-ordercontroller (with passivity levels ρ1 and ν1) that is designed for aphysical system (with passivity levels ρp and νp). The controllerdesigned for the physical system is of high order when thecontroller is designed e.g., to minimize an LQG performanceor to satisfy an H∞ criterion. To reduce the implementationcost of such a high-order controller, we can use a reduced-ordercontroller G2 instead. Our results can be used to determine theorder of G2 so that stability of the closed-loop system can stillbe guaranteed according to Theorems 1 and 2. As an example,we assume that ρ1 > 0 and ν1 > 0. Then γ in (10) needs tosatisfy

γ < min{ν1 + ρp, ρ1 + νp}

where γ is determined by the order of G2 when G1 is given asshown in Corollary 4.

Remark 10: Given a high-order linear system G1, PR-TBR can be used to guarantee passivity of its reduced-orderapproximation G2 (see e.g., [15], [30]). If we use this methodfor model reduction, Corollary 4 provides a tool to trade off theerror due to this approximation as a function of the variationin the passivity levels for the full-order system G1 and thereduced-order system G2. For more details, we refer the readerto [30], [36].

Remark 11: Transfer functions are considered in Corollary 4.It is expected that tighter results may be possible when extrainformation concerning state-space is provided. For example,consider an LTI system given by (8), where

A = AT ≤ 0, BT = C, D = DT ≥ 0. (20)

An LTI system of the form (8) where (20) holds is calleda relaxation system [15], [35]. Given a relaxation system G1

and its reduced-order approximation G2 through TBR, we canshow that G2 is of relaxation type as well [15]. Further, theIFP for both systems are the same, regardless of the order forthe reduced order system G2 [or equivalently, the value of γin equation (10)]. By comparison with Corollary 4, we can

Fig. 3. Sampled-data System with an ideal sampler and a ZOH device,for which u(t) = ud(k) for kh ≤ t < (k + 1)h, yd(k) = y(kh) for all k ≥0, where h represents the sampling period.

conclude that γ < ν is only a sufficient condition and ν − γis merely a coarse estimate of the IFP of G1.

B. Sampled-Data Systems

Sampled-data systems are common in control because con-tinuous physical systems are typically controlled by digitaldevices, see e.g., [20], [21]. Consider a continuous-time systemΣ1 with input u(t) and output y(t) and a sampled-data systemΣ2 with input ud(k) and output yd(k), see Fig. 3. For standarddiscretization with an ideal sampler and a zero-order hold(ZOH) device, the control inputs for Σ1 and Σ2 are related asu(t) = ud(k) for kh ≤ t < (k + 1)h, where h represents thesampling period and the outputs of the two systems are relatedas yd(k) = y(kh) for all k ≥ 0. It is well known that passivityis not preserved under standard discretization (see e.g., [31],[58], [59]). Passivity degradation under standard discretizationhas been studied in [58] with the following assumption that wealso make.

Assumption 2: Suppose for Σ1, there exists α > 0 such thatfor any T ≥ 0 and all u ∈ U ⊂ R

m

T∫0

‖y(t)‖2 dt ≤ α2

T∫0

‖u(t)‖2 dt. (21)

Finding the constant α in (21) is an interesting problem.Examples on how to find α for linear systems can be foundin [58]. To illustrate how to find α for nonlinear systems, weconsider an example, in which the system Σ1 is given by thefollowing nonlinear equations (with mapping u → y)

x = − 1

2x− 1

2x3 +

1

2u

y =x. (22)

Then, α is given by the L2 gain for the system with mappingu → y, i.e., the system

x = − 1

2x− 1

2x3 +

1

2u

z = − 1

2x− 1

2x3 +

1

2u.

Consider a Lyapunov function V = (1/2)x2 + (1/4)x4

such that

V −(1

4u2 − z2

)= −1

4(x+ x3)

2 ≤ 0.


Thus, system u → y has L2 gain given by 1/2 (see e.g., [19]).Therefore, Assumption 2 is satisfied for the system (22) withα = 1/2.

Next, we investigate how the framework of Section IV can beapplied in this case. We first present a condition that character-izes the approximation induced by sampling.

Proposition 2: Consider Σ1 and Σ2 in Fig. 3 where u ∈U ⊂ R

m. Suppose system Σ1 satisfies Assumption 2. DefineΔy = y − yd, then (10) holds for γ = αh and ε = 0, where his the sampling period.

Proof: We have the following relation for all kh ≤ t <(k + 1)h and all k ≥ 0:∥∥∥∥∥∥

t∫kh

y(s)ds

∥∥∥∥∥∥ ≤t∫

kh

‖y(s)‖ ds ≤(k+1)h∫kh

‖y(s)‖ ds

and thus the following relation holds:

(k+1)h∫kh

∥∥∥∥∥∥t∫

kh

y(s)ds

∥∥∥∥∥∥2

dt ≤(k+1)h∫kh

⎛⎜⎝ (k+1)h∫kh

‖y(s)‖ ds

⎞⎟⎠2

dt

≤h

⎛⎜⎝ (k+1)h∫kh

‖y(s)‖ ds

⎞⎟⎠2

. (23)

Using Cauchy-Schwarz inequality, we have⎛⎜⎝ (k+1)h∫kh

‖y(s)‖ ds

⎞⎟⎠2

≤ h

(k+1)h∫kh

‖y(s)‖2 ds. (24)

By assumption (21) and setting T = Kh, we obtain

K−1∑k=0

(k+1)h∫kh

‖y(s)‖2 ds ≤ α2

T∫0

‖u(t)‖2 dt.

Together with (23) and (24), we can derive that

K−1∑k=0

(k+1)h∫kh

∥∥∥∥∥∥t∫

kh

y(s)ds

∥∥∥∥∥∥2

dt ≤ h2α2

T∫0

‖u(t)‖2 dt. (25)

Thus, we obtain the following relation from (25):

〈Δy,Δy〉T =

K−1∑k=0

(k+1)h∫kh

‖y(t)− yd(k)‖2 dt

=

K−1∑k=0

(k+1)h∫kh

∥∥∥∥∥∥t∫

kh

y(s)ds

∥∥∥∥∥∥2

dt

≤α2h2〈u, u〉T . (26)

Therefore, (10) is satisfied for γ = αh and ε = 0. This com-pletes the proof. �

The following result is immediate from Theorem 3,Theorem 5 and equation (26).

Corollary 5: Consider a continuous-time system Σ1 and itssampled-data system Σ2 obtained from standard discretizationwith sampling period given by h, as shown in Fig. 3, whereu ∈ U ⊂ R

m. Suppose that Assumption 2 is satisfied.

1) If Σ2 has IFP (ν) where ν > 0 and αh < ν, then Σ1 hasIFP no less than ν − αh. Further, if Σ2 has IFP (ν) whereν > 0 and αh ≤ ν, then Σ1 is passive.

2) If Σ2 is VSP for (ρ, ν) where ρ > 0 and ν > 0 and ρν2 +ν − αh ≥ 0, then Σ1 is passive.

Remark 12:

1) Once again, we note that, Σ1 and Σ2 can exchange placesin the above result. Therefore, if the continuous-timesystem Σ1 has IFP (ν), then the sampled-data system Σ2

has IFP no less than ν − αh.2) If Σ1 has IFP (ν), from αh ≤ ν, we obtain that ν/α

provides an upper bound for the sampling period h forpreserving passivity. When α is large (the system may beoscillatory [58]), we need a small sampling period h toensure passivity.

3) The first part of Corollary 5 were also derived in [58]. Thesecond part of Corollary 5 is different from the results in[58]. If the continuous-time system Σ1 is VSP for (ρ, ν),then one sufficient condition derived in [58] to guaranteethe discrete-time system Σ2 being passive is given by

αh <1

1 + ρν < ν.

Thus, the upper bound on the discretization time intervalh will be larger using the results in the present paperwhere we only require that αh ≤ ρν2 + ν.

C. Quantization of Stable Systems

Control using quantized feedback is an important researchproblem. The effects of quantization have to be considered forinstance, when the control law has to be implemented through acommunication channel, see e.g., [22], [23], [60]. The quantizerwe consider in this paper (see also [31], [60]) is based on thesector bound method and given as

auTu ≤ uTQ(u) ≤ buTu, ∀u ∈ U ⊂ Rm (27)

where Q(u) is the output of the quantizer with input u and0 ≤ a ≤ b < ∞. This kind of quantizer characterizes severalpractical quantizers, such as the uniform quantizer [31] andlogarithmic quantizer [23], [60]. It has been shown in [31] thatthe parameters a, b of the quantizer play an important role inpreserving OSP of a system after quantization.

Consider system Σ1 with input u and output y1, as shown inFig. 4. For simplicity, we assume zero initial conditions and thatsystem Σ1 is finite-gain L2 stable with gain κ. Next, we shall


Fig. 4. Quantized-input, Quantized-output system: Σ1 is finite-gainstable with gain κ and the quantizer Qi satisfies (27) with aiu

Tu ≤uTQi(u) ≤ biu

T u. The control input u and the outputs y1, y2 are ofthe same dimensions.

investigate the passivity properties of the system Σ2 (i.e., thesystem of Σ1 after quantization). We have the following result.

Proposition 3: Consider the two systems in Fig. 4, whereu ∈ U ⊂ R

m, system Σ1 is L2 stable with gain κ > 0. Thequantizer Qi satisfies aiuTu ≤ uTQi(u) ≤ biu

Tu, where 0 ≤ai ≤ bi < ∞. Then, (10) is satisfied for γ

Δ= κ(1 + b1b2) and

ε = 0.Remark 13: By setting a1 = b1 = 1, we have Q1(u) = u,

corresponding to the case only when the output of Σ1 is quan-tized. Likewise, by setting a2 = b2 = 1, we have Q2(u) = u,corresponding to the case only when the input of Σ1 is quan-tized. We do not consider the trivial case when ai = bi = 1 fori = 1, 2 (i.e., if no quantizers are used).

Proof: Denote the input to quantizer Q2 as y, then wehave y2 = Q2(y) and a2y

T y ≤ yTQ2(y) ≤ b2yT y. Therefore,

we obtain QT2 (y)Q2(y) ≤ b22y

T y.Because Σ1 is stable, we have 〈y, y〉T ≤ κ2〈Q1(u),

Q1(u)〉T . Also, from a1uTu ≤ uTQ1(u) ≤ b1u

Tu, we obtainQT

1 (u)Q1(u) ≤ b21uTu. Then, we have

〈Q2(y),Q2(y)〉T ≤ κ2b22b21〈u, u〉T .

Since Σ1 is finite-gain stable with gain κ, from the Cauchy-Schwarz inequality, we can derive that

|〈y1,Q2(y)〉T | ≤ κ2b2b1〈u, u〉T .

From the above relations, we can derive that

〈Δy,Δy〉T Δ= 〈y2 − y1, y2 − y1〉T= 〈Q2(y),Q2(y)〉T + 〈y1, y1〉T −2 〈y1,Q2(y)〉T≤ (1 + b1b2)

2κ2〈u, u〉T .

Therefore, (10) holds for γΔ= κ(1 + b1b2) and ε = 0. This

completes the proof. �We can derive a tighter bound for the error γ in (10) when

the system Σ1 is linear. For notational convenience, denote Has the linear operator which maps the input u of the system Σ1

to the output y1Δ= H[u] of system Σ1.

Proposition 4: Consider the two systems in Fig. 4, wheresystem Σ1 is a linear operator. Assume that the system Σ1

is L2 stable with gain κ > 0. The quantizer Qi satisfiesaiu

Tu ≤ uTQi(u) ≤ biuTu, where 0 ≤ ai ≤ bi < ∞. Define

δiΔ= max{|ai − 1|, |bi − 1|}. Then, (10) is satisfied for γ

Δ=

κ√2(δ21 + δ22b

21) and ε = 0.

Proof: Denote the input to quantizer Q2 as y, thenwe have y = H[Q1(u)] and y2 = Q2(y). Thus, we have thefollowing relations:

‖Δy‖ Δ= ‖y1 − y2‖

= ‖H[u]−Q2 (H [Q1(u)])‖≤‖H[u]−H [Q1(u)]‖+‖Q2(H [Q1(u)])−H [Q1(u)]‖ .

(28)

Since H is a linear operator, we have H[u]−H[Q1(u)] =H[u−Q1(u)]. Further, using aiu

Tu ≤ uTQi(u) ≤ biuTu, we

can obtain that ‖u−Q1(u)‖ ≤ δ1‖u‖ and

‖Q2 (H [Q1(u)])−H [Q1(u)]‖ ≤ δ2 ‖H [Q1(u)]‖ .

Then, from (28), we can obtain that

〈Δy,Δy〉T ≤ 2 〈H [u−Q1(u)] ,H [u−Q1(u)]〉T+ 2δ22 〈H [Q1(u)] ,H [Q1(u)]〉T . (29)

Because the system Σ1 has finite gain κ, we have〈H[Q1(u)],H[Q1(u)]〉T ≤ κ2〈Q1(u),Q1(u)〉T and

〈H [u−Q1(u)] ,H [u−Q1(u)]〉T≤ κ2 〈u−Q1(u), u−Q1(u)〉T≤ κ2δ21〈u, u〉T .

Then, based on the fact that ‖Q1(u)‖2 ≤ b21‖u‖2, from (29), weobtain that

〈Δy,Δy〉T ≤ 2κ2(δ21 + δ22b

21

)〈u, u〉T . (30)

Therefore, (10) is satisfied with γ = κ√2(δ21 + δ22b

21) and

ε = 0. �Remark 14: Consider the case when system Σ1 is linear.

If the quantizers Qi satisfy that ai → 1, bi → 1 (i.e., when thequantized value Qi(u) of u is close enough to the real value

of u), then, we have δiΔ= max{|ai − 1|, |bi − 1|} → 0 and

thus γ = κ√2(δ21 + δ22b

21) → 0. Therefore, the error between

system Σ1 and system Σ2 is sufficiently small.The following result is immediate from Proposition 1 and

Theorem 3.Corollary 6: Consider the two systems in Fig. 4, where

u ∈ U ⊂ Rm, system Σ1 is L2 stable with gain κ > 0. The

quantizer Qi satisfies aiuTu ≤ uTQi(u) ≤ biuTu, where 0 ≤

ai ≤ bi < ∞.

1) If Σ1 has IFP (ν) where ν > 0, then Σ2 has an IFP levelν − κ(1 + b1b2).

2) If Σ1 is linear and κ√2(δ21 + δ22b

21) < ν (where δi

Δ=

max{|ai − 1|, |bi − 1|}), then Σ2 is ISP for ν −κ√2(δ21 + δ22b

21).

It has been shown that passivity may not be preserved afterquantization, see e.g., [31]. However, a passivity level of systemΣ2 is desired in many cases, especially when system Σ2 as a


subsystem to be interconnected with another system throughfeedback configurations as in Theorem 2. Corollary 6 presentsa lower bound for the passivity index of the system afterquantization.

D. Linearization of Nonlinear Systems

The results so far hold for the case when u ∈ U ⊂ Rm and

u ∈ U = Rm. We now consider a case when u ∈ U ⊂ R

m

and the state x ∈ X ⊂ Rn. In this case, we need to make the

additional assumption that the state x remains in the set X afterapplication of the input u ∈ U . This is a standard assumption inmany different streams, e.g., [1], [39], [37], [61], [62]. Specifi-cally, we consider linearization of a nonlinear system around anequilibrium point. The result shows that if the linearized systemis passive, then “local” passivity for the nonlinear system canbe guaranteed in a neighborhood of the equilibrium point underwhich linearization is done.

Consider the following nonlinear system Σ1 (with initialstate x1(t0) = 0 for simplicity):

x1 = f(x1) + g(x1)u

y1 = ζ(x1) + J(x1)u (31)

where x1 ∈ X ⊂ Rn and u ∈ U ⊂ R

m, f , g, ζ and J aresmooth mappings of appropriate dimensions and f(0) =0, ζ(0) = 0 without loss of generality. We assume that thepair (x1 = 0, u = 0) is an equilibrium point for the nonlinearsystem (31). Define

AΔ=

∂f

∂x1|x1=0, B

Δ= g(0)

CΔ=

∂ζ

∂x1|x1=0, D

Δ= J(0). (32)

With (32), the linearized system Σ2 about the equilibrium point(x1 = 0, u = 0) is given by

x2 =Ax2 +Bu

y2 =Cx2 +Du. (33)

The linearized model Σ2 is accurate up to the first order and iscalled first-order approximation of Σ1 [10], [11].

Proposition 5: Consider a nonlinear system Σ1 given by(31), where f , g, ζ and J are real analytic at x1 = 0. Let Σ2

be the linearization of Σ1 given by (33) with (32). Assume

that A is a Hurwitz matrix. Define ΔyΔ= y2 − y1. Then, in

a neighborhood of the equilibrium point (x1 = 0, u = 0) ∈X × U , there exist a constant γ > 0, such that

‖ΔyT‖L2≤ γ‖uT‖L2

. (34)

Proof: First, consider the following system with input uand output y = x1:

x1 = f(x1) + g(x1)u

y =x1. (35)

It can be seen that x1 satisfies (31). Since the matrix A isHurwitz, the linearization of system (35) has finite L2 gain (seee.g., Corollary 5.2 in [19, pp. 201–205]). Then, in some neigh-borhood of (x1 = 0, u = 0), the nonlinear system (35) also hasfinite L2 gain (see e.g., Corollary 8.3.4 in [1, pp. 211–215]).Therefore, there exists d>0 such that for any u∈L2e, wecan find a sufficiently small neighborhood of (x1=0, u=0)such that

‖x1T ‖L2< d‖uT ‖L2

.

Define vΔ=

[x1

u

], we can further obtain that

‖vT ‖L2<√d2 + 1‖uT‖L2

. (36)

Next, Taylor series expansions for f , g, ζ and J about x1 = 0can be obtained as

f(x1) =Ax1 + F (x1), ζ(x1) = Cx1 +H(x1)

g(x1) =B +G(x1), J(x1) = D +M(x1) (37)

where F (x), H(x), G(x) and M(x) contain higher-order termscorresponding to f(x), ζ(x), g(x) and J(x), respectively.According to Taylor’s Theorem (see e.g., [63]), there exist con-stants li > 0 (i = 1, 2, 3, 4) and r > 0, such that the followingrelations hold when ‖x1‖ ≤ r:

F (x1) ≤ l1‖x1‖2, H(x1) ≤ l3‖x1‖2

G(x1) ≤ l2‖x1‖ ≤ l2r, M(x1) ≤ l4‖x1‖ ≤ l4r.

Define eΔ= x2 − x1. Together with (31)–(33), we can obtain the

following “error” system with input v and output Δy:

e =F(e, v)

Δy =H(e, v) (38)

where F(e, v) = Ae− F (x1)−G(x1)u and H(e, v) = Ce−H(x1)−M(x1)u. We will show that system (38) is finite-gainL2 stable according to Theorem 5.1 in [19, pp. 201–205].

• Since A is Hurwitz, then there exists P > 0 such that (seee.g., Theorem 8.2 in [64, pp. 66–70])

ATP + PA = −I.

Therefore, e = 0 is an exponentially stable equilibriumpoint of system (38).

• When ‖x1‖ ≤ r, the function F (e, v) satisfies

‖F(e, v)−F(e, 0)‖ = ‖F (x1)−G(x1)u‖≤ l1‖x1‖2 + l2‖x1‖‖u‖≤ l1r‖x1‖+ l2r‖u‖≤ (l1 + l2)r‖v‖.

• Similarly, when ‖x1‖ ≤ r, he function H(e, v) satisfies

‖H(e, v)‖ ≤‖C‖‖e‖+ ‖H(x1) +M(x1)u‖≤‖C‖‖e‖+ l3‖x1‖2 + l4‖x1‖‖u‖≤‖C‖‖e‖+ (l3 + l4)r‖v‖.


Therefore, there exists rv > 0 such that for each vT ∈ L2e with‖v‖ ≤ rv , the output Δy satisfies

‖ΔyT ‖L2≤ k‖vT‖L2

where k > 0 is given by

k = (l3 + l4)r +‖C‖λ(P )‖P‖(l1 + l2)

λ(P )r. (39)

Together with (36), we obtain that in some neighborhood of(x1 = 0, u = 0), the following relation holds:

‖ΔyT‖L2≤ γ‖uT‖L2

where γ = k√d2 + 1 > 0. This completes the proof. �

We consider the case when Σ2 is VSP. If Σ2 is observable,then VSP implies it is also asymptotically stable, see e.g., [3],[19]. Using Theorem 5, Corollary 3 and equation (34), weobtain the following result.

Corollary 7: Consider Σ1 and Σ2 in Fig. 2, where Σ1 isgiven by (31) and Σ2 is linearization of Σ1 given by (33) with(32). Suppose the linearized model Σ2 is observable and VSPfor (ρ, ν) where ρ > 0 and ν > 0. Then, in a neighborhood ofthe equilibrium point (0, 0), there exist a constant γ > 0 suchthat (34) holds. Further, we have the following results:

1) If γ ≤ ρν2 + ν, then Σ1 is locally passive.2) If γ < min{ρ, ν} and γ2 − (ρ− (2/ρ))γ + ν2 − 2 ≥ 0,

then Σ1 is locally VSP for (ρ− γ, ν − γ).

Remark 15:

1) The value of γ is determined by the radius of the ballaround x1 = 0 that is under consideration. As x1(t) → 0,γ → 0 and the difference between passivity levels of thetwo systems ((ρ, ν) for Σ2 and (ρ− γ, ν − γ) for Σ1

within the neighborhood of (x1 = 0, u = 0)) tends tozero as well.

2) It is worthwhile to mention that the results hold true formore general nonlinear models (which are not affine incontrol) and in discrete-time setting as well. More de-velopments of the relations of passivity and dissipativityproperties between a nonlinear system and its linearizedmodel can be found in [38].

VI. NUMERICAL EXAMPLES

In this section, we provide a few numerical examples toillustrate the theoretical results developed in this paper. Ingeneral, Σ1 represents the system we are interested in and Σ2 isan approximation of Σ1. We assume zero initial conditions forsimplicity so that β = 0 in Definition 1. Note that the examplesfor approximation of high-order linear systems by using modelreduction method can be found in [30] and [36].

Example 1: Consider the following nonlinear system repre-sented by the state model:

x1 = − x1 − x31 − x2 + u

x2 = x1 − x32

y = x1. (40)

Fig. 5. Sampled-data system of a continuous-time dynamic system withan ideal sampler (A/D) and a ZOH device (D/A). In Example 1, thecontinuous-time system is given by (40).

Fig. 6. Example 1 with input u(k) = (1/3) cos(2k + (π/6)) + (1/2).

The system model (40) has been used in [19, pp. 233-237]to represent an RLC network with nonlinear resistors. Witha storage function given by V (x) = (1/2)(x2

1 + x22), we can

obtain

uT y − V ≥ x41 + x4

2 ≥ 0.

Thus, the nonlinear system (40) is passive, i.e., the passivitylevels for system (40) satisfy ρ2 = 0 and ν2 = 0.

Consider the discrete-time model of the continuous-timesystem (40) by using standard discretization as shown in Fig. 5.Then according to Proposition 2, (10) is satisfied with γ = αh,where h is the sampling period and α is defined in (21). We canapply Proposition 1 to estimate the supply rate of the sampled-data system with input u(k) and output y(k). For instance,take ρ1 = 0, then (17) is satisfied with any ξ > 0. We takeξ = γ, then from (18), we obtain that ν1 ≤ −2γ < 0. Thus, thesampled-data system u → y has passivity level ν1 = −2γ.

To verify this, we implement the system model in MATLAB/Simulink with sampling period h = 0.1. One can verify thatthe assumption (21) is satisfied with α = 2.5 for this example.As shown in the top plot of Fig. 6, at some time constants k,the value of u(k)y(k) < 0, where the control input is given byu(k) = (1/3) cos(2k + (π/6)) + (1/2). This implies that thediscrete-time model is not passive. Further, with a feedforwardcontroller given by a constant gain 2γ (see e.g., [3, pp. 24-29]),the sampled-data system can become passive, as shown in thebottom plot of Fig. 6. It should be noted that one can verify theresults by considering other control inputs u(k). Finally, if weconsider a feedback interconnection as in Theorem 2, where the


Fig. 7. System (41) after quantization using a logarithmic quantizer Q.

sampled-data system represents the controllerG in the feedbackloop, then the closed-loop system is guaranteed to be OSP if theplant H has an OFP level ρ > 2γ. �

Example 2: Consider the following discrete-time system:

x(k + 1) =Ax(k) +Bu(k)

y(k) =Cx(k) +Du(k)

where the system matrices {A,B,C,D} are given by

A =

[0 0.20.5 0

], B =

[0.10

], C =

[0 0.2

], D = 2.

(41)It can be verified by solving the LMI in Lemma 1 thatsystem (41) is OSP with passivity level ρ2 = 0.49. Further,Assumption 1 is satisfied with η = 0.5.

Consider the system (41) after quantization with input u(k)and y(k), as shown in Fig. 7. The logarithmic quantizer Q isdefined as follows (see e.g., [23], [60]):

Q(ε) =

⎧⎪⎨⎪⎩�jμ, if �jμ

1+δ ≤ ε< �jμ1−δ , j=0,±1,±2, . . .

0, if ε = 0

−Q(−ε), if ε < 0

where μ > 0 is a scaling parameter, 0 < � < 1 implies thequantization density and δ = (1 − �)/(1 + �). Note that asmall � implies coarse quantization and a large � implies densequantization.

The system after quantization u(k) → y(k) may not be OSPwith the same passivity level ρ2 = 0.49 as the system (41).According to Corollary 6, assumption (10) is satisfied withγ = κδ

√2(2 + δ2 + 2δ), where κ = 2 is the H∞ norm of the

system (41). If � = 0.8, then condition (10) is satisfied. Then,according to Theorem 4, the system after quantization u(k) →y(k) is OSP with passivity level ρ2 − γ = 0.15.

To verify this, we implement the system model in MATLAB/Simulink with the quantizer parameters μ = 1 and � = 0.8.As shown in the top plot of Fig. 8, at some time constants k,the value of uT (k)y(k)− ρ2y

T (k)y(k) < 0, where the controlinput is given by u(k) = sin(0.2k − (π/3)). This implies thatthe system after quantization u(k) → y(k) does not have theOFP level ρ2. As shown in the bottom plot of Fig. 8, forall k ≥ 0, we have uT (k)y(k)− (ρ2 − γ)yT (k)y(k) ≥ 0. Thisimplies that the system after quantization u(k) → y(k) have anOFP level ρ2 − γ. It should be noted that one can verify theresults by considering other control inputs u(k). Further, if weconsider a feedback interconnection as in Theorem 2, wherethe system after quantization represents the plant H , then theclosed-loop system is guaranteed to be finite-gain stable if thecontroller G has an IFP level ν > γ − ρ2. �

Fig. 8. Example 2 with input u(k) = sin(0.2k − (π/3)).

Example 3: Consider the following nonlinear system

x1 = − x21 + x2

x2 = − x1 − x2 + (x1 + 1)u

y = x1 + 2x2 + (2x2 + 1)u. (42)

The linearized model of the nonlinear system (42) around theorigin is given by (8) with

A =

[0 1−1 −1

], B =

[01

], C = [1 2], D = 1.

By solving the LMI (9) in Lemma 1, we can obtain that P =[1 1/2

1/2 1

]> 0 is a solution such that the linearized model

is VSP with passivity levels ρ2 = 0.1 and ν2 = 0.4.If we apply V (x) = xTPx as a locally defined storage

function for the nonlinear system, then we obtain

V − uT y ≤ −x21

(1

2− |2x1| − |x2| − 2|u|

)− u2 (1− |2x2|)− x2

2

(1

2− |u|

).

To guarantee local passivity of the nonlinear system (42), weconsider (x, u) ∈ X × U , where the set X is defined as

X Δ=

{x ∈ R

2 : |x1(t)| ≤1

16, |x2(t)| ≤

1

16, ∀ t ≥ 0

}and the set U is defined as

U Δ=

{u ∈ R : |u(t)| ≤ 1

8, such that x ∈ X , ∀ t ≥ 0

}.

We have the following inequality holds for (x, u) ∈ X × U :

V − uT y ≤ − 1

16

(x21 + x2

2

)− 1

2u2 ≤ 0.


Fig. 9. Example 3 with input u(t) = 0.1 exp(−0.8t).

To obtain the local passivity levels for the nonlinear system(42), we consider X and U defined above. It can be verifiedthat (10) is satisfied with γ = 0.094. Further, with ρ2 = 0.1 andν2 = 0.4, the following inequality holds:

γ2 − (ρ2 − 2/ρ2)γ + ν22 − 2 = 0.04 > 0.

Then according to Corollary 7, the nonlinear system (42) islocally VSP with passivity levels ρ1 and ν1, where ρ1 = ρ2 −γ = 0.006 and ν1 = ν2 − γ = 0.306.

To verify this, we implement the system model in MATLAB.As shown in the top plot of Fig. 9, the states satisfy|x1(t)| ≤ 1/16 and |x2(t)| ≤ 1/16 with the control inputu(t) = 0.1 exp(−0.8t). As shown in bottom plot of Fig. 9,uT y − ρ1y

T y − ν1uTu ≥ 0 for all t ≥ 0. This implies that ρ1

and ν1 are valid local passivity levels for the nonlinear system(42). It should be noted that one can verify the results byconsidering other control inputs u(t). Further, if we consider afeedback interconnection as in Theorem 1, where the nonlinearsystem (42) represents the plant H and the control objectiveis to minimize an LQG performance. In order to use the LQGtechniques for linear systems, we can design an LQG controllerG with passivity levels ρc and νc for the linearized modelof system (42). Within a neighborhood around the origin, theclosed-loop system is guaranteed to be stable if ρc + ν1 > 0and νc + ρ1 > 0. �

VII. CONCLUSION

In this paper, we established conditions under which thepassivity properties of a system can be obtained by analyzingits approximation. The approximate model is assumed to beinput/output/very strictly passive and the general result statesthat if the error between the system and its approximation issmall, the original system has a guaranteed passivity level aswell. The analysis is extended to the case when the approx-imation is (Q,S,R)-dissipative (not necessarily passive). Thepassivity levels obtained for the system can further be used

to analyze the passivity and stability of feedback intercon-nections when the system is in the loop. The results may beinterpreted as robustness properties of passivity with respect tomodel uncertainties. To illustrate the results, we study particularapproximation methods such as reduced-order approximation,discretization, quantization and linearization around an equilib-rium point.

APPENDIX

Proof of Theorem 3: Consider systems Σ1 and Σ2

with an arbitrary input u, so that the corresponding outputsare y1 and y2, where y2 = y1 +Δy. First we note that for anyγ > 0, |uTΔy| ≤ (1/2γ)ΔyTΔy + (γ/2)uTu, thus, we havethe following relation:

|〈u,Δy〉T | ≤T∫

0

|uTΔy|dt ≤ 1

2γ〈Δy,Δy〉T +

γ

2〈u, u〉T .

(43)Thus, from assumption (10), we obtain

|〈u,Δy〉T | ≤γ

2〈u, u〉T +

1

2γ

(γ2〈u, u〉T + ε

)= γ〈u, u〉T +

ε

2γ. (44)

Now for the system Σ2, we have

〈u, y2〉T − ν〈u, u〉T = 〈u, y1〉T − ν〈u, u〉T + 〈u,Δy〉T≤〈u, y1〉T − ν〈u, u〉T + |〈u,Δy〉T |

≤ 〈u, y1〉T − (ν − γ)〈u, u〉T +ε

2γ.

By assumption, Σ2 is ISP for a given ν > 0. Thus, for a givenconstant β ≤ 0

〈u, y2〉T − ν〈u, u〉T ≥ β.

By defining ν = ν − γ, β = β − (ε/2γ) ≤ 0, we obtain for thesystem Σ1

〈u, y1〉T − ν〈u, u〉T +ε

2γ≥ β

or equivalently 〈u, y1〉T − ν〈u, u〉T ≥ β for a constant β ≤ 0.Now note that u and T are arbitrary. Therefore, if γ < ν, thenΣ1 is ISP for ν > 0. �

Proof of Theorem 4: System Σ2 has OFP(ρ), then thereexists a constant β ≤ 0 such that 〈u, y2〉T − ρ〈y2, y2〉T ≥ β.From the relation that uTy2 ≤ (1/2ρ)uTu+ (ρ/2)yT2 y2, weobtain

〈u, y2〉T ≤ 1

2ρ〈u, u〉T +

ρ

2〈y2, y2〉T .

Thus, we have β ≤ (1/2ρ)〈u, u〉T − (ρ/2)〈y2, y2〉T . We canfurther obtain

〈y2, y2〉T ≤ 1

ρ2〈u, u〉T − 2β

ρ. (45)


From Cauchy-Schwarz inequality, (10), (45) and the fact β ≤ 0,we obtain

|〈y2,Δy〉T | ≤√〈Δy,Δy〉T

√〈y2, y2〉T

≤ γ

ρ

√〈u, u〉T +

ε

γ2

√〈u, u〉T − 2βρ

≤ γ

ρ

(〈u, u〉T +

ε

γ2− 2βρ

)=

γ

ρ〈u, u〉T +

ε

γρ− 2βγ. (46)

Together with (44), if we define aΔ= ρ− γ > 0, we obtain

ΦΔ= γ〈y2, y2〉T − 〈u,Δy〉T + 2a〈Δy, y2〉T − a〈Δy,Δy〉T≥ γ〈y2, y2〉T −|〈u,Δy〉T |−2a |〈Δy, y2〉T | − a〈Δy,Δy〉T

≥ γ〈y2, y2〉T −(γ + 2a

γ

ρ+ aγ2

)〈u, u〉T − ε

2γ

− 2a

(ε

γρ− 2βγ

)− aε.

Using Assumption 1, we obtain 〈y2, y2〉T ≥ (1/η2)〈u, u〉T −(b/η2) and thus

γ〈y2, y2〉T −(γ + 2a

γ

ρ+ aγ2

)〈u, u〉T

≥[1

η2−(1 + 2a

1

ρ+ aγ

)]γ〈u, u〉T − γb

η2.

Thus, if (13) is satisfied, we have Φ ≥ β, where β =−(ε/2γ)− 2a((ε/γρ)− (2β/γ))− aε− (γb/η2) ≤ 0.

From the fact that 〈u, y1〉T − (ρ−γ)〈y1, y1〉T = 〈u, y2〉T −ρ〈y2, y2〉T +Φ, defining β = β + β ≤ 0, we have for Σ1 that

〈u, y1〉T − (ρ− γ)〈y1, y1〉T ≥ β.

Note that u and T are arbitrary. Therefore, for γ < ρ, Σ1 is OSPfor ρ = ρ− γ. �

Proof of Theorem 5: Since Σ2 is VSP for (ρ, ν), then Σ2

is ISP for ν > 0 and OSP for ρ > 0. Thus

〈y2, y2〉T ≥ ν2〈u, u〉T + 2βν. (47)

Together with (10), (44), and (46), if we define aΔ= ρ− γ >

0 and ψΔ= 2a〈y2,Δy〉T − 〈u,Δy〉T − a〈Δy,Δy〉T , then we

obtain

|ψ| ≤ |〈u,Δy〉T |+ 2a |〈y2,Δy〉T |+ a〈Δy,Δy〉T

≤(γ + 2a

γ

ρ+ aγ2

)〈u, u〉T +

ε

2γ

+ 2a

(ε

γρ− 2βγ

)+ aε.

Thus, the following relation holds by substituting (47):

γ〈u, u〉T + γ〈y2, y2〉T + ψ

≥ γ(1 + ν2)〈u, u〉T + 2βνγ − |ψ|

≥ γ

(ν2 − 2a

ρ− aγ

)〈u, u〉T − ε

2γ− 2a

(ε

ργ− 2βγ

)− aε+ 2βνγ.

If (14) is satisfied, we obtain ν2 − (2a/ρ)− aγ ≥ 0. Thus,we have

γ〈u, u〉T + γ〈y2, y2〉T + ψ ≥ − ε

2γ

− 2a

(ε

γρ− 2βγ

)− aε+ 2βνγ.

For Σ1 with input u and output y1 = y2 −Δy, we have

〈u, y1〉T − (ν − γ)〈u, u〉T − (ρ− γ)〈y1, y1〉T= 〈u, y2〉T − ν〈u, u〉T − ρ〈y2, y2〉T+ γ〈u, u〉T + γ〈y2, y2〉T + ψ.

Σ2 is assumed to be VSP for (ρ, ν) and therefore there exists aconstant β ≤ 0 such that

〈u, y2〉T − ν〈u, u〉T − ρ〈y2, y2〉T ≥ β.

Defining βΔ=β−(ε/2γ)−2a((ε/γρ)−2βγ)−aε+2βνγ≤0,

we obtain for Σ1

〈u, y1〉T − (ν − γ)〈u, u〉T − (ρ− γ)〈y1, y1〉T ≥ β.

Note u and T are arbitrary. Thus, if γ < ρ and γ < ν, Σ1 isVSP for (ρ− γ, ν − γ). �

Proof of Theorem 6: For notational convenience, we

define cΔ= 2max{0, λ(−Q1)γ

2}, λ1Δ=√λ(ST

1 S1) ≥ 0 and

λ2Δ= λ(QT

1 Q1) ≥ 0. From Cauchy-Schwarz inequality and(10), we obtain

|〈S1u,Δy〉T | ≤√〈S1u, S1u〉T

√〈Δy,Δy〉T

≤√λ(ST1 S1

)〈u, u〉T

√γ2〈u, u〉T + ε

≤λ1γ〈u, u〉T +ελ1

γ. (48)

We have the following relation for any ξ > 0 from (10):

2〈Q1y2,Δy〉T ≤ 1

ξ〈Δy,Δy〉T + ξ〈Q1y2, Q1y2〉T

≤ γ2

ξ〈u, u〉T + ξλ2〈y2, y2〉T +

ε

ξ. (49)

Denote the supply rate for Σi as ri(u, yi)Δ= 〈yi, Qiyi〉T +

2〈yi, Siu〉T + 〈u,Riu〉T . Since Σ2 is (Q2, S2, R2)-dissipative,


r2 ≥ β2. Also, we can derive that

r1 = r2 + 〈y2, (Q1 −Q2)y2〉T + 〈u, (R1 −R2)u〉T− 2〈y2, Q1Δy〉T + 〈Δy,Q1Δy〉T − 2〈Δy, S1u〉T .

Note that 〈Q1y2,Δy〉T = 〈y2, Q1Δy〉T because Q1 is sym-metric. When Q1 −Q2 ≥ 0 and R1 −R2 > 0, together with(48) and (49), we obtain

r1 ≥ (λ(Q1 −Q2)− ξλ2) 〈y2, y2〉T

+

(λ(R1 −R2)−

γ2

ξ− 2λ1γ

)〈u, u〉T

+ β2 −ε

ξ− 2

ελ1

γ+ 〈Δy,Q1Δy〉T . (50)

Defining β1Δ= β2 − (ε/ξ)− 2(ελ1/γ) ≤ 0, from (15), we have

r1≥β1 + 〈Δy,Q1Δy〉T +

(λ(R1−R2)−

γ2

ξ−2λ1γ

)〈u, u〉T .

Two cases are possible. If Q1 > 0, then c = 0 and〈Δy,Q1Δy〉T ≥ 0. Thus, from (16), we obtain r1 ≥ β1. IfQ1 ≤ 0, we have c = λ(−Q1)γ

2 and from (10)

〈Δy,Q1Δy〉T ≥ −λ(−Q1)γ2〈u, u〉T − ελ(−Q1).

Again, from (16), we obtain r1 ≥ β1 − ελ(−Q1). Thus, therealways exits a β ≤ 0 such that r1 ≥ β for both cases. ThereforeΣ1 is (Q1, S1, R1)-dissipative. �

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Meng Xia received the B.S. degree in appliedmathematics in 2008 and the M.S. degree incontrol science and engineering in 2010, bothfrom the Harbin Institute of Technology, China.She received the Ph.D. degree in electrical en-gineering from the University of Notre Dame,Notre Dame, IN, USA, in 2015.

She is currently an Autonomous ControlSystem Engineer at the MathWorks, Natick,MA. Her research interests include cyber-physical systems, optimization algorithms and

networked control systems.

Panos J. Antsaklis (S’74–M’76–SM’86–F’91)is the Brosey Professor of Electrical Engineer-ing at the University of Notre Dame. He is agraduate of the National Technical University ofAthens, Greece, and holds the M.S. and Ph.D.degrees from Brown University.

His research addresses problems of controland automation and examines ways to designcontrol systems that will exhibit high degreeof autonomy. His recent research focuses oncyber-physical Systems and addresses prob-

lems in the interdisciplinary research area of control, computing andcommunication networks, and on hybrid and discrete event dynamicalsystems. He had co-authored two research monographs on discreteevent systems, two graduate textbooks on Linear Systems and has co-edited six books on Intelligent Autonomous Control, Hybrid Systems andNetworked Embedded Control Systems. He is the Editor-in-Chief of theIEEE TRANSACTIONS ON AUTOMATIC CONTROL.

Dr. Antsaklis is an IFAC and AAAS Fellow, the 2006 recipient of theEngineering Alumni Medal of Brown University and a 2012 honorarydoctorate recipient from the University of Lorraine, France.

Vijay Gupta received the B. Tech degree fromthe Indian Institute of Technology, Delhi and theM.S. and Ph.D. degrees from the California Insti-tute of Technology, all in electrical engineering.

He is an Associate Professor in the De-partment of Electrical Engineering at the Uni-versity of Notre Dame, Notre Dame, IN, USA.His research interests include cyber-physicalsystems, distributed estimation, detection andcontrol, and, in general, the interaction of com-munication, computation and control.

Dr. Gupta received the NSF CAREER award in 2009, and the DonaldP. Eckman award from the American Automatic Control Council in 2013.

Feng Zhu received the Ph.D. degree in elec-trical engineering from the University of NotreDame, Notre Dame, IN, USA, in 2014, the M.S.degrees in electrical engineering and appliedmathematics from the University of Notre Dame,and the B.S. degree from the Harbin Institute ofTechnology, China, in 2008.

His research interests are hybrid systems,discrete event dynamical systems and optimalcontrols. He is currently a Data Scientist atMicrosoft, Redmond, WA.

passivity and dissipativity analysis of a system and its

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