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Recent developments on the stability of systems withaperiodic sampling: An overview

Laurentiu Hetel, Christophe Fiter, Hassan Omran, Alexandre Seuret, EmiliaFridman, Jean-Pierre Richard, Silviu Iulian Niculescu

To cite this version:Laurentiu Hetel, Christophe Fiter, Hassan Omran, Alexandre Seuret, Emilia Fridman, et al.. Recentdevelopments on the stability of systems with aperiodic sampling: An overview. Automatica, Elsevier,2017, 76, pp.309 - 335. 10.1016/j.automatica.2016.10.023. hal-01363448

Recentdevelopmentson the stability of systemswith

aperiodic sampling: an overview ⋆

Laurentiu Hetel a,∗ Christophe Fiter a Hassan Omran b Alexandre Seuret c

Emilia Fridman d Jean-Pierre Richard a,e and Silviu Iulian Niculescu f

aCNRS CRISTAL UMR 9189,

Ecole Centrale de Lille / Univ. Lille 1, 59651 Villeneuve d’Ascq, France

bICube lab., UMR 7357 University of Strasbourg/CNRS, France

c CNRS, LAAS, 7 avenue du Colonel Roche, F–31400 Toulouse, France.

d School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel.

eNon-A, INRIA Lille-Nord Europe, France

fLaboratoire des Signaux et Systemes (L2S, UMR CNRS 8506), CNRS - Centrale SUPELEC - Univ. Paris Sud, 3 rue JoliotCurie, 91192 Gif-sur-Yvette, France.

Abstract

This article presents basic concepts and recent research directions about the stability of sampled-data systems with aperiodicsampling. We focus mainly on the stability problem for systems with arbitrary time-varying sampling intervals which has beenaddressed in several areas of research in Control Theory. Systems with aperiodic sampling can be seen as time-delay systems,hybrid systems, Input/Output interconnections, discrete-time systems with time-varying parameters, etc. The goal of thearticle is to provide a structural overview of the progress made on the stability analysis problem. Without being exhaustive,which would be neither possible nor useful, we try to bring together results from diverse communities and present them ina unified manner. For each of the existing approaches, the basic concepts, fundamental results, converse stability theorems(when available), and relations with the other approaches are discussed in detail. Results concerning extensions of Lyapunovand frequency domain methods for systems with aperiodic sampling are recalled, as they allow to derive constructive stabilityconditions. Furthermore, numerical criteria are presented while indicating the sources of conservatism, the problems thatremain open and the possible directions of improvement. At last, some emerging research directions, such as the design ofstabilizing sampling sequences, are briefly discussed.

Key words: networked/embedded control systems, sampled-data control, aperiodic sampling, stability analysis, time-delaysystems, hybrid systems, convex embeddings, Integral Quadratic Constraints.

1 Introduction

The last decade has witnessed an enormous interestin the study of networked and embedded control sys-tems [276,116,107,38]. This interest is mainly due to the

⋆ This work was supported by Region Nord-Pas de Calaisthrough the ARCIR project ESTIREZ, by the ANR ProjectROCC-SYS (agreement ANR-14-CE27-0008) and the CNRSGDRI DelSys.∗ Corresponding author. Tel.: +33 (0)3 20 67 60 13

Email address: laurentiu.hetel@ec-lille.fr(Laurentiu Hetel).

ubiquitous presence of embedded controllers in relevantapplication domains and the growing demand in indus-try on systematic methods to model, analyse and designsystems where sensor and control data are transmittedover a digital communication channel. The study of sys-tems with aperiodic sampling emerged as a modellingabstraction which allows to understand the behaviourof Networked Control Systems (NCS) with samplingjitters, packet drop-outs or fluctuations due to theinter-action between control algorithms and real-timescheduling protocols [276,7,11]. With the emergenceof event-based and self-triggered control techniques[13,10,259,103], the study of aperiodic sampled-data

systems constitutes nowadays a very popular researchtopic in control. In this survey, we focus on questionsarising in the control of systems with arbitrary time-varying sampling intervals. Important practical ques-tions such as the choice of the sampling frequency, theevaluation of necessary computational and energeticresources or the robust control synthesis are mainly re-lated to stability issues. These issues often lead to theproblem of estimating the Maximum Sampling Interval(MSI) for which the stability of a closed-loop sampled-data system is ensured.

The study of aperiodic sampled-data systems has beenaddressed in several areas of research in Control The-ory. Systems with aperiodic sampling can be seen as par-ticular time-delay systems. Sampled-and-hold in controland sensor signals can be modelled using hybrid systemswith impulsive dynamics. Aperiodic sampled-data sys-tems have also been studied in the discrete-time domain.In particular, Linear Time Invariant (LTI) sampled-datasystems with aperiodic sampling have been analysed us-ing discrete-time Linear Parameter Varying (LPV) mod-els. The effect of sampling can be modelled using op-erators and the stability problem can be addressed inthe framework of Input/Output interconnections as typ-ically done in modern Robust Control. While significantadvances on this subject have been presented in the lit-erature, problems related to both the fundamentals ofsuch systems and the derivation of constructive methodsfor stability analysis remain open, even for the case oflinear system. The objective of the article is to presentin a unified and structured manner a collection of signif-icant results on this topic.

The core of the article is dedicated to the analysis of sys-tems with arbitrary varying sampling intervals. We willonly consider the deterministic aspects of the problem.The case when sampling intervals are random variablesgiven by a probability distribution will not be discussedhere. After presenting some generalities and motivationsconcerning sampled-data systems with aperiodic sam-pling (in Section 2), some basic qualitative results arerecalled in Section 3. Section 4 presents the main stabil-ity analysis approaches. At last, in Section 5, we brieflydiscuss some emerging research problems, such as thedesign of stabilizing sampling sequences.We indicate themain challenges, the relations with the arbitrary sam-pling problem and some perspectives on which the cur-rent approaches and tools for aperiodic sampled-datasystems may be useful in the future.

Notations: Throughout the paper, R+ denotes the setλ ∈ R, λ ≥ 0, ‖x‖ represents any norm of the vector xand ‖x‖p , p ∈ N, the p norm of a vector x. For a matrix

M , MT denotes the transpose of M and M⋆, its conju-gate transpose. For square symmetric matrices M, N ,M N (resp. M ≻ N) means that M −N is a positive(resp. definite positive) matrix. ∗, in a symmetric matrix

u(t) = uk

PLANT

y(t)

H

uk

CONTROL

yk = y(tk)

S

TRIGGER

Fig. 1. Classical sampled-data system configuration

represents elements that may be induced by symmetry.‖M‖p , p ∈ N denotes the induced p-norm of a matrix

M . σ (M) denotes the maximum singular value of M .C0(X,Y ), for two metric spaces X and Y , is the set ofcontinuous functions from X to Y . Ln

p (a, b), p ∈ N de-notes the space of functions φ : (a, b) → R

n with norm

‖φ‖Lp=[∫ b

a‖φ(s)‖p ds

] 1p

, and Ln2e[0,∞) is the space of

functions φ : [0,∞) → Rn which are square integrable

on finite intervals.

2 Generalities

2.1 System configuration

In this paper we study the properties of sampled-datasystems consisting of a plant, a digital controller, andappropriate interface elements. A general configurationof such a sampled-data system is illustrated by theblock diagram of Figure 1. In this configuration, y(t) isa continuous-time signal representing the plant output(the plant variables that can be measured). This signalis represented as a function of time t, y : R+ → R

p.

The digital controller is usually implemented as an al-gorithm on an embedded computer. It operates with asampled version of the plant output signal, ykk∈N, ob-tained upon the request of a sampling trigger signal atdiscrete sampling instants tk and using an analog-to-digital converter (the sampler block,S, in Figure 1). Thistrigger may represent a simple clock, as in the classicalperiodic sampling paradigm, or a more complex schedul-ing protocol which may take into account the sensor sig-nal, a memory of its last sampled values, etc. The sam-pling instants are described by a monotone increasingsequence of positive real numbers σ = tkk∈N where

t0 = 0, tk+1 − tk > 0, limk→∞

tk = ∞. (1)

The difference between two consecutive sampling timeshk = tk+1 − tk is called the kth sampling interval. As-suming that the effect of quantizers may be neglected,the sampled version of the plant output is the sequenceykk∈N where yk = y(tk).

In a sampled-data control loop, the digital controllerproduces a sequence of control values ukk∈N using the

2

sampled version of the plant output signal ykk∈N. Thissequence is converted into a continuous-time signal u(t),where u : R+ → R

m (corresponding to the plant input)via a digital-to-analog interface. We consider that thedigital-to-analog interface is a zero-order hold (the holdblock, H , in Figure 1). Furthermore, we assume thatthere is no delay between the sampling instant tk andthe moment the control uk (obtained based on the kth

plant output sample, yk) is effectively implemented atthe plant input. Then the input signal u(t) is a piecewiseconstant signal u(t) = u(tk) = uk, ∀t ∈ [tk, tk+1).

In this survey, we will consider that the plant is modelledby a finite dimensional ordinary differential equation ofthe form

x = F (t, x, u) ,

y = H (t, x, u) ,(2)

where x ∈ Rn represents the plant state-variable. Here

F : R+×Rn×R

m → Rn with F (t, 0, 0) = 0, ∀t ≥ 0, and

H : R+ × Rn × R

m → Rp. It is assumed that for each

constant control and each initial condition (t0, x0) ∈R+ × R

n the function F describing the plant model(2) is such that a unique solution exists for an interval[t0, t0 + ǫ) with ǫ large enough with respect to the max-imum sampling interval. The discrete-time controller isconsidered to be described by an ordinary differenceequation of the form

xck+1 = F c

d (k, xck, yk) ,

uk = Hcd (k, x

ck, yk) ,

(3)

where xck ∈ R

nc is the controller state. Here, F cd : N ×

Rnc × R

p → Rnc and Hc

d : N × Rnc × R

p → Rm. We

will use the denomination sampled-data system for theinterconnection between the continuous-time plant (2)with the discrete-time controller (3) via the relations

yk = y(tk), u(t) = uk, ∀t ∈ [tk, tk+1), ∀k ∈ N, (4)

under a sequence of sampling instants σ = tkk∈N sat-isfying (1).

The different concepts and results will be mostly illus-trated on Linear Time Invariant (LTI) models

x = Ax+Bu, (5)

under a static linear state feedback,

uk = Kxk, k ∈ N, (6)

with xk = x(tk). However, when possible, we will presentextensions to more general nonlinear systems.

2.2 Classical design methods

There are various approaches for the design of a sampled-data controller (3) (see the classical textbooks [16,39]and the tutorial papers [183,182,192,142]).

Emulation. The simplest approach consists in designingfirst a continuous-time controller using classical meth-ods [136,123,140,229].Next, a discrete-time controller ofthe form (3) is obtained by integrating the controller so-lutions over the interval [tk, tk+1).This approach is usu-ally called emulation. Generally, it is difficult to computein a formal manner the exact discrete-time model andapproximations must be used [183,142]. In the LTI case(5) with state feedback (6), the emulation simply meansthat the gain K is set such that the matrix A + BKis Hurwitz and that the plant is driven by the controlu(t) = Kx(tk), ∀t ∈ [tk, tk+1), k ∈ N. While the intu-ition seems to indicate that for sufficiently small sam-pling intervals the obtained sampled-data control givesan approximation of the continuous-time control prob-lem, no guarantee can be given when the sampling in-terval increases, even for constant sampling intervals. Inorder to compensate for the effect of controller discreti-sation, re-design methods may be used [95,189].

Discrete-time design. In this framework, a discrete-time

model of the plant (2) is derived by integration. The ob-tained model represents the evolution of the plant statex(tk) = xk at sampling times 1 . Then, a discrete-timecontroller (3) is designed using the obtained discrete-time model. In the simplest LTI case (5), (6), the evolu-tion of the state between two consecutive sampling in-stants tk and tk+1 is given by

x(t) = Λ(t− tk)x(tk), ∀t ∈ [tk, tk+1], k ∈ N, (7)

with a matrix function Λ defined on R as

Λ(θ) = Ad(θ) +Bd(θ)K = eAθ +

∫ θ

0

eAsdsBK. (8)

Evaluating the closed-loop system’s evolution at t =tk+1 and using the notation hk = tk+1 − tk leads to thelinear difference equation

xk+1 = Λ(hk)xk, ∀k ∈ N (9)

representing the closed-loop system at sampling in-stants. When the sampling interval is constant,

1 Note that generally approximations of the system modelmust be used since the discretized plant model is difficult tocompute formally [181,261]. Even for the case of LTI systemswith constant sampling intervals, the numerical computa-tion of the matrix exponential (or its integral) is subject toapproximations [179].

3

hk = T, ∀k ∈ N, a large variety of discrete-time controldesign methodologies is available in the literature (see[16,39] and the references within). It is well known forthis case that system (9) is asymptotically stable if andonly if the matrix Λ(T ) is Schur. In other words, to de-sign a stabilizing control law (6), the matrix K must beset such as all the eigenvalues of Λ(T ) lay in the openunit disk.

For nonlinear systems with constant sampling intervals,an overview of control design methodologies and relatedissues can be found in [183,182,192,142]. Note that thediscrete-time models such as (9) do not take into con-sideration the inter-sampling behaviour of the system.Relations between the performances of the discrete-timemodel and the performances of the sampled-data loop,can be deduced using the methodology proposed in [196].

Sampled-data design. Infinite dimensional discrete-timemodels which take into account the inter-sampling sys-tem behaviour using signal lifting [19,20,251,257,269]have been proposed in the literature for the case of linearsystems. Specific design methodologies, which are ableto take in consideration continuous-time system perfor-mances, inter-sample ripples and robustness specifica-tions, can be found in the textbook [39] for the case oflinear time invariant systems with periodic sampling.

2.3 Complex phenomena in aperiodic sampling

While in the last fifty years an intensive research hasbeen dedicated to the analysis and design of sampled-data systems under periodic sampling, the study of sys-tems with time-varying sampling intervals is quite un-derdeveloped compared to the periodic conterpart. Thefollowing examples illustrate the rich complexity of phe-nomena that may occur under aperiodic sampling.

Example 1 [274] Consider an LTI sampled-data systemof the form (5),(6) where

A =

[

1 3

2 1

]

, B =

[

1

0.6

]

, K = −[

1 6]

. (10)

For this example, system’s (9) transition matrix Λ(T )is a Schur matrix for any constant sampling interval inT ∈ T = T1, T2, with T1 = 0.18, and T2 = 0.54. Then,in the case of periodic sampling, the sampled-data sys-tem is stable for constant sampling intervals taking val-ues in T . An illustration of the system’s evolution forconstant sampling intervals T1, T2, is given in Figure 2.Clearly, when the sampling interval hk is arbitrarily vary-ing in T , the Schur property of Λ(T ), ∀ T ∈ T , repre-sents a necessary condition for stability of the sampled-data system (1),(5),(6). However, it is not a sufficientone. For example, the sampled-data system with a se-quence of periodically time-varying sampling intervals

0 2 4 6 8 10−30

−20

−10

0

10

t

Constant sampling T1 = 0.18

x

1(t)

x2(t)

u(t)

0 2 4 6 8 10−100

−50

0

50

100

t

Constant sampling T2 = 0.54

x

1(t)

x2(t)

u(t)

Fig. 2. Stability of the system in Example 1 with periodicsampling intervals.

0 2 4 6 8 10−100

−50

0

50

100

t

x

1(t)

x2(t)

u(t)

Fig. 3. Instability of the system in Example 1 with a periodicsampling sequence T1 → T2 → T1 · · · .

hkk∈N = T1, T2, T1, T2, . . . is unstable, as it can beseen in Figure 3. This is due to the fact that the Schurproperty of matrices is not preserved under matrix prod-uct (i.e. the product of two Schur matrices is not neces-sarily Schur). Indeed, the discrete-time system represen-tation over the two sampling instants can be written as

xk+2 = Λ(T2)Λ(T1)xk, ∀k ∈ 2N,

and the transition matrix Λ(T2)Λ(T1) over two samplingintervals T1 and T2, is not Schur. This example showsthe importance of taking into consideration the evolutionof the sampling interval hk when analysing the stabilityof sampled-data systems since variations of the samplinginterval hk may induce instability.

Example 2 [96] Consider now an LTI system with

A =

[

0 1

−2 0.1

]

, B =

[

0

1

]

K =[

1 0]

(11)

Assume that the sampling interval hk is restricted to theset T = T1, T2 with T1 = 2.126 and T2 = 3.950. Thesystem is unstable for both constant sampling intervalsT1 and T2 since for these values system’s (9) transitionmatrix Λ(T ), T ∈ T is not a Schur matrix. However,the product of transition matrices Λ(T1)Λ(T2) has theSchur property. Therefore, the sampled-data system isstable under a periodic evolution of the sampling inter-val hkk∈N = T1, T2, T1, T2, . . .. An example of sys-tem evolution with this particular sampling sequence is

4

provided in Figure 4. In this example the sampling hk

can act on the sampled-data system as a second controlparameter which ensures the system’s stability while thepossible constant sampling configurations are not able toguarantee this property.

0 10 20 30 40 50−1

−0.5

0

0.5

1

t

x

1(t)

x2(t)

u(t)

Fig. 4. Periodic sampling sequence with a stable behaviour.

2.4 Problem set-ups

The core of the survey is dedicated to the robust analysisof sampled-data systems with sampling sequences of theform (1) where the sampling interval hk = tk+1 − tktakes arbitrary values in some set T = [h, h] ⊂ R+. Thisfirst problem set-up may correspond, for example, tothe sampling triggering mechanism from Figure 1 with aclock submitted to jitter [268], or with some schedulingprotocol which is too complex to be modelled explicitly[276,107]. Basically, for the case of LTI models (5) withlinear state feedback (6) under a sampling sequence (1)we will address the robust stability of the closed-loopsystem (12) given below

x(t) = Ax(t) +BKx(tk), ∀t ∈ [tk, tk+1), ∀k ∈ N,

tk+1 = tk + hk, ∀k ∈ N,

t0 = 0, x(t0) = x0 ∈ Rn

(12)as if hk is a time-varying ”perturbation” taking valuesin a bounded set T .

In Section 5 we briefly indicate some basic ideas concern-ing a recently emerging research topic where the sam-pling interval hk plays the role of a control parameterthat may be changed according to the plant state or out-put. This problem set-up corresponds to the design of ascheduling mechanism. For the case of system (12), hk

is considered as an additional input which, by an appro-priate open/closed-loop choice, can ensure the systemstability.

3 Qualitative properties of sampled-data sys-tems

In this section we recall some aspects concerning thequalitative behaviour of sampled-data systems withtime-varying sampling intervals. First, we discuss theexistence of a sufficiently small sampling interval that

preserves asymptotic stability when discretizing acontinuous-time stabilizing controller. Next, we presentqualitative stability results which can be deduced fornonlinear sampled-data systems using linearization.

3.1 Small sampling interval approximations

The choice of sampling intervals is a critical issue in theemulation approach. Intuitively, choosing a sufficientlylarge sampling frequency should preserve the stabilityunder a sampled-data implementation. This conjecturehas been confirmed in [254,105,106] for systems with pe-riodic sampling. For the case of aperiodic sampling, var-ious classes of systems have been treated in the litera-ture [118,34,210]. The case of LTI systems (5) with linearstate feedback (6) has been addressed long ago in [118].

Theorem 1 [118] Consider that system x = (A+BK)x

is exponentially stable. Then there exists a scalar h > 0such that the closed-loop system (12) is ExponentiallyStable for any sequence σ = tkk∈N of the form (1)

satisfying hk = tk+1 − tk < h, ∀k ∈ N.

The proof is based on the existence of a quadratic Lya-punov function for the continuous-time closed-loop sys-tem x = (A + BK)x. An extension to a more generalclass of (input affine) nonlinear systems is given below.

Theorem 2 [118] Let x = 0 be a globally exponentiallystable equilibrium point of system x = f(x) + g(x)u withu = K(x), where f : R

n → Rn, g : R

n → Rn×m,

K : Rn → R

m, and let V : Rn → R+ be a radially

unbounded C1 function such that V (0) = 0, V (x) > 0for all x 6= 0. Assume that the following conditions aresatisfied:

• the functions f(.) and K(.) are globally Lipschitz;• there existsG > 0 such that ‖g(x)‖ ≤ G for all x ∈ R

n;• there exist c1, c2 > 0 such that for any x ∈ R

n

∂V

∂x(f(x) + g(x)K(x)) ≤ −c1 ‖x‖

2,

∥∥∥∥

∂V

∂x

∥∥∥∥≤ c2 ‖x‖ ;

• given a closed set B1 and a bounded open set B2 withB1 ⊂ B2 ⊂ R

n there exists a scalar s(B1, B2) > 0such that, if the initial condition η(0) ∈ B1, then thetrajectory η(t) of η = f(η) satisfies η(t) ∈ B2, for allt ≤ s(B1, B2).

Then there exists a scalar h > 0 such that x = 0 is aGlobally Exponentially Stable Equilibrium point of thesystem

x(t) = f (x(t)) + g (x(t))K (x(tk)) , (13)

t ∈ [tk, tk+1), k ∈ N, for any sequence of samplinginstants σ = tkk∈N of the form (1) satisfying hk =tk+1 − tk < h, ∀k ∈ N.

5

In [118] it was further shown that if the continuous-timecontrol system x = f(x) + g(x)K(x) is asymptoticallystable (instead of exponentially stable), then only prac-tical stability is guaranteed for the sampled-data system(13). A more general case, dealing with the emulation ofdynamical controllers based on Euler discretization wasprovided in [34,118]. An alternative to Theorem 2 canbe found in [210] and concerns the same issue but thedrift f(x) is not required to satisfy any Lipschitz prop-erty. However, the continuous-time control loop shouldensure the exponential decay of a quadratic Lyapunovfunction along its solutions. Furthermore, the result onlystates the practical stability of the sampled-data controlloop. Another extension to a more general class of Net-worked Control Systems can be found in [194]. There itis shown that if a continuous-time controller is designedsuch that it yields input-to-state stability with respectto external disturbances, then the same controller willachieve a semi-global practical input-to-state stabilityproperty when implemented in a sampled-data controlloop via an exact emulation. Qualitative results for theexistence of both a sufficiently small sampling intervaland a stabilizing sampled-data controller can be foundin [133].

3.2 Linear approximations

The study of sampled-data systems with linear modelsand controllers is often easier to address than the nonlin-ear case. For some classes of nonlinear sampled-data sys-tems, local stability can be deduced from the propertiesof a linearized model around the equilibrium [115,119].

Consider the following nonlinear system

x = F (x, u) ,

y = H (x) ,(14)

the discrete-time controller

xck+1 = F c

d (xck, yk) ,

uk = Hcd (x

ck, yk)

(15)

and the interconnection yk = y(tk), u(t) = uk, ∀t ∈[tk, tk+1), ∀k ∈ N, for sampling sequences σ = tkk∈N

as defined in (1). The closed-loop system can be repre-sented by the set of equations

x(t) = f (x(t), xk , xck) , t ∈ [tk, tk+1)

xck+1 = g (xk, x

ck) , k ∈ N,

(16)

where f (x, xk, xc) = F (x,Hc

d (xck, H(xk))) , g (xk, x

ck) =

F cd (x

ck, H(xk)) and xk = x(tk). For f(x, v, w) and

g(v, w) let A = ∂f∂x

∣∣∣0, A0 = ∂f

∂v

∣∣∣0, B = ∂f

∂w

∣∣∣0,

C = ∂g∂w

∣∣∣0, D = ∂g

∂v

∣∣∣0. To system (16) the following

linear model is associated

x(t) = Ax(t) +A0xk +Bxck, t ∈ [tk, tk+1),

xck+1 = Cxc

k +Dxk, k ∈ N.(17)

Integrating the system over a sampling interval and let-

ting zTk =[

xTk xc

kT]

leads to the following linear time-

varying discrete-time system

zk+1 = Ω(hk)zk, ∀k ∈ N, (18)

with

Ω(hk) =

[

eAhk +∫ hk

0eAsdsA0

∫ hk

0eAsdsB

D C

]

. (19)

The following theorem establishes conditions for the sta-bility of the nonlinear system (16) under arbitrary vari-ations of the sampling interval.

Theorem 3 [119] Assume that, for every possible se-quence σ = tkk∈N defined in (1), one has hk = tk+1 −tk ≤ h, and for any k ∈ N, ‖Ω(hk)‖2 < q < 1, where

h and q are constant scalars. Then the equilibrium point[

xT xcT]

= 0 of system (16) is Exponentially Stable.

The nature of the result is in the spirit of the Lyapunov’sfirstmethod [136], as it permits to guarantee the stabilityof the equilibrium of the nonlinear system, by studyingthe stability of its linearization at the origin. In the sameway, it remains qualitative and it does not provide anyestimate of the domain of attraction. However, the resultdoes not require the sampling intervals to be small.

The following theorem uses the linear model (18) in or-der to provide conditions for the stability of the nonlin-ear sampled-data system (16) with a fixed sequence ofsampling instants.

Theorem 4 [119] Let σ = tkk∈N be a sequence defined

in (1) with supk∈Ntk+1 − tk = h < ∞, where h is agiven constant. Assume that either

(i) lim supk→∞ ‖Ω(hk)‖2 < 1, or

(ii) lim supk→∞max |eig(Ωk)| < 1 and every subse-quence of Ω(hk)k∈N converges to a Schur matrix andthe solutions Pk of ΩT (hk)PkΩ(hk) − Pk = −I satisfylim supk→∞ ‖Pk+1 − Pk‖2 < 1, or

(iii) Ω(hk) converges to a Schur matrix.

6

Then the equilibrium point[

xT xcT]

= 0 of system (16)

with the sampling sequence σ is Exponentially Stable.

Note that, in the conditions of Theorem 4, the matrixΩ(hk) is not required to be Schur for all the values ofk ∈ N. One may find particular sequences of samplinginstants σ = tkk∈N satisfying the conditions i) or iii) inTheorem 4 for which the eigenvalues of Ω(hk) are outsidethe unit disk for some values of k. The result is interest-ing when the sampling interval hk can be considered as acontrol parameter, for scheduling the sampling instantsin an appropriate manner. The theorem may be used todetermine scheduling mechanisms with sampling inter-vals larger than in a periodic sampling configuration.

4 Stability analysis under arbitrary time-varying sampling

The previous results are qualitative and prove some niceproperties of sampled-data systems. However, they donot provide any method for estimating the set of sam-pling intervals for which the stability properties are stillguaranteed. In the following, we review some resultswhich provide such an estimation for sampled-data sys-tems with sampling intervals that are arbitrary varying.More formally, over the section, we present results thataddress the following problem:

• Problem A (Arbitrary sampling problem): Con-sider the sampled-data system (1),(2),(3),(4) and abounded subset T ⊂ R+. Determine if the sampled-data system is stable (in some sense) for any arbitrarytime-varying sampling interval hk = tk+1 − tk withvalues in T .

Often the set T is considered of the form T = (0, h]

where h is some positive scalar. The largest value of h forwhich the stability of the closed loop system is ensuredis called Maximum Sampling Interval (MSI).

Several perspectives for addressing Problem A exist.First, we present results that are based on a time-delaymodelling of the sampled-data system (1),(2),(3),(4).Next, we show how the problem can be addressed fromthe point of view of hybrid systems. We continue withapproaches that use the explicit system integration in-between successive sampling instants, such as the onesclassically used in the discrete-time framework. Last, re-sults addressing Problem A from the robust control the-ory point of view are presented.

4.1 Time-delay approach

To the best of our knowledge, this technique was initi-ated in [172,15], and further developed in [66,254,158]

t

t

τ(t

)=

t−

t k

Fig. 5. Sampling seen as a piecewise-continuous time-delay.

and in several other works. For the case of an LTI systemwith sampled-data state feedback (12), we may re-write

u(t) = Kx(tk) = Kx(t− τ(t)),

τ(t) = t− tk, ∀t ∈ [tk, tk+1),(20)

where the delay τ is piecewise-linear, satisfying τ (t) =1 for t 6= tk, and τ(tk) = 0. This delay indicates thetime that has passed since the last sampling instant.An illustration of a typical delay evolution is given inFigure 5. The LTI systemwith sampled-data (12) is thenre-modeled as an LTI system with a time-varying delay

x(t) = Ax(t) +BKx(t− τ(t)), ∀t ≥ 0. (21)

This permits to adapt the tools for the analysis ofsystems with fast varying delays [76,97,225,198]. Thismodel is equivalent to the original sampled-data systemwhen considering that the sampling induced delay hasa known derivative τ (t) = 1, for all t ∈ [tk, tk+1), k ∈ N.

4.1.1 Theoretical foundation

For system (21) it is natural to consider, as a state vari-able, the functional xt(θ) = x(t + θ), ∀θ ∈ [−h, 0],

and, as state space, the set C0([−h, 0

],Rn

)of contin-

uous functions mapping the interval[−h, 0

]into R

n

[68,197,199]. The most popular generalization of the di-rect Lyapunov method for time-delay system has beenproposed by Krasovskii [139]. It uses the existence offunctionals V (t, xt) depending on the state vector xt. Inthe sampled-data case [72,67,150] functionals V (t, xt, xt)depending both on xt and xt (see [137], p.337) are useful.

Denote byW [−h, 0] the Banach space of absolutely con-

tinuous functions φ : [−h, 0] → Rn with φ ∈ Ln

2 (−h, 0)(the space of square integrable functions) with the norm

‖φ‖W = maxs∈[−h,0]

‖φ(s)‖ +

[∫ 0

−h

∥∥∥φ(s)

∥∥∥

2

ds

] 12

.

7

Theorem 5 (Lyapunov-Krasovskii Theorem)[137] Consider f : R+ × C0[−h, 0] → R

n continuous inboth arguments and locally Lipschitz in the second argu-ment. Assume that f(t, 0) = 0 for all t ∈ R+ and that fmaps R× (bounded sets in C0[−h, 0]) into bounded setsof Rn. Suppose that α, v, w : R+ → R+ are continuousnondecreasing functions, α(s), β(s) and γ(s) are posi-tive for s > 0, lims→∞ α(s) = ∞ and α(0) = β(0) = 0.The trivial solution of

x(t) = f (t, xt)

is Globally Uniformly Asymptotically Stable if thereexists a continuous functional V : R × W [−h, 0] ×

Ln2 (−h, 0) → R+, which is positive-definite, i.e.

α(‖φ(0)‖) ≤ V (t, φ, φ) ≤ β(‖φ‖W ),

for all φ ∈ W [−h, 0], t ∈ R+, and such that its derivativealong the system’s solutions is non-positive

V (t, xt, xt) ≤ −γ(‖xt(0)‖). (22)

The functional V satisfying the conditions of Theorem5 is called a Lyapunov-Krasovskii Functional (LKF). Inthe general case of sampled-data nonlinear systems, theunderlying delay system x = f(t, xt) used in Theorem 5from [137] is described by a function f which is piecewisecontinuous with respect to t. However, the proof of theresult in [137] can be adapted to cover this case. Forthe case of sampled-data systems, in [72] the Lyapunov-KrasovskiiTheoremwas extended to linear systems witha discontinuous sawtooth delay by use of the Barbalatlemma [21]. Another extension to linear sampled-datasystems has been provided in [67], where the LKF isallowed to have discontinuities at sampling times.

4.1.2 Tools and basic steps

The derivation of stability conditions using LKFs usuallyinvolves quite elaborate developments. To give an idea ofthe procedure involved in this approach and to providea glimpse of its technical flavour, we present first thederivation of LMI stability conditions for the case ofLTI systems (12) with the associated time delay model(21). Based on elementary considerations, we expose themain difficulties and the most relevant tools. The basicsteps for deriving constructive stability conditions areillustrated as follows.

Step 1. Propose a candidate Lyapunov-Krasovskii func-tional V .The LKF that is necessary and sufficient for thestability of LTI systems with delay has a rather complexform, even for the case of constant delays [138,71]. In

order to provide constructive stability conditions, sim-plified forms are used, such as

V (xt, xt) = xT (t)Px(t) + h

∫ 0

−h

∫ t

t+θ

xT (s)Rx(s)dsdθ

(23)with P,R ≻ 0, which was considered in [75].

Step 2. Compute the derivative of V . For the functional(23) this leads to

V (xt, xt) = 2xT (t)Px(t) + h2xT (t)Rx(t)

−h

∫ t

t−h

xT (s)Rx(s)ds. (24)

The difficulty now comes from the last integral term,

J(xt, h) = −∫ t

t−h xT (s)Rx(s)ds, which is an impedi-

ment to the analysis of the sign of (24). Such terms arecommon in the derivative of LKFs and they need to betaken into account in an appropriated manner.

Step 3. Over-approximate the integral terms. This pro-cedure is applied in order to replace the integral termsby more simple expressions. Unavoidably, using suchover-approximations introduces some conservatism inthe analysis. A relevant tool is Jensen’s inequality, whichis recalled here.

Lemma 6 (Jensen’s inequality) [96] Given R ≻ 0,θ ≥ 0, and a differentiable function x : [t − θ, t] → R

n,the following inequality holds:

J(xt, θ) = −

∫ t

t−θ

x(s)Rx(s)ds ≤

−1

θ(x(t) − x(t− θ))

TR (x(t)− x(t− θ)) . (25)

For the case presented in (24), splitting the integral intwo terms and applying Jensen’s inequality leads to

d

dtV (xt, xt) ≤ 2xT (t)Px(t) + h2xT (t)Rx(t)

−ζ(t)TR (τ(t)) ζ(t) (26)

where ζ(t) =

[

x(t) − x (t− τ(t))

x (t− τ(t)) − x(t− h)

]

, and

R (τ(t)) =

hτ(t)R 0

0 hh−τ(t)

R

is a matrix depending on the delay value. At this step,sufficient conditions of stability are provided by the neg-ativity of the right-hand side of the inequality (26). How-ever, the obtained conditions need to be checked for all

8

the values of τ(t) ∈ (0, h], due to the dependence on τ(t)of the matrix R(.).

Step 4. Over-approximate the delay dependent terms.The last step consists in over-approximating the ele-ments that depend on τ(t) by simpler expressions whichare constant or depend only on the upper bound h. Forexample, in [72], stability conditions are obtained by

noting that hτ(t) ≥ 1 and h

h−τ(t)≥ 0, which implies that

R (τ(t))

[

R 0

0 0

]

(27)

holds for all delay τ in the interval [0, h]. Of course, thisover-approximation still introduces some conservatismin the analysis. However, for the case of LTI systems(12), over-approximating the terms in (26) leads to

d

dtV (xt, xt) ≤

[

x(t)

x (t− τ(t))

]T

Ψ(P,R)

[

x(t)

x (t− τ(t))

]

with

Ψ(P,R) =

[

PA+ATP P (BK)

(BK)TP 0

]

+h2

[

AT

(BK)T

]

R[

A BK]

−

[

I

−I

]

R[

I −I]

, (28)

from which LMI stability conditions can be derived[75,72] simply by checking the negative definiteness ofΨ(P,R).

Theorem 7 (adapted from [75,72]) Assume that thereexists P ≻ 0 and R ≻ 0, such that the following linearmatrix inequality Ψ(P,R) ≺ 0 holds with Ψ(P,R) asdefined in (28). Then, the sampled-data system (12) isAsymptotically Stable for all sampling sequences σ =tkk∈N with hk = tk+1 − tk ≤ h.

Theorem 7 is a simplified reformulation of the result in[72], where a more general case of polytopic systems isconsidered using a descriptor model [75,74].

4.1.3 Directions of improvement

The main sources of conservatism in this approach aredue to the choice of the LKF (Step 1 ) and the over-approximation of its derivative (Steps 3 and 4 ).

Step 1. The conservatism related to Step 1 canbe reduced by adding new integral componentsto the LKF. For example the triple integral term

V (xt) = h2

2

∫ 0

−h

∫ 0

θ

∫ t

t+λxT (s)Rx(s)dsdλdθ introduced

by [246] for systems with fast varying delay might alsobe useful in the sampled-data case. For other forms ofLKF see also [68,211,240,9,236]. We point in particu-lar to the results in [67,154,187,232], which have beenspecifically developed for the analysis of sampled-datasystems. There, the proposed LKFs allow for takinginto account the particular sawtooth evolution of thesampling induced delay (τ (t) = 1, ∀t ∈ [tk, tk+1)) whilein the classical fast varying delay approach the delayderivative is assumed to be unknown and arbitrary vary-ing. For example, it has been shown in [67] that the stan-

dard time-independent term∫ 0

−h

∫ t

t+θ xT (s)Rx(s)dsdθ

used in [72] can be advantageously replaced by the term

(tk+1 − t)∫ t

tkxT (s)Rx(s)ds, which provides derivative-

dependent stability conditions. It leads to an LKF ofthe form [67]:

V (t, x(t), xt) =

xT (t)Px(t) + (hk − τ(t))

∫ t

t−τ(t)

xT (s)Rx(s)ds (29)

which improves (23), as the information τ = 1 can beexplicitly taken into account when evaluating its deriva-tive in Step 2. See also [230] for an alternative LMI for-mulation.

Step 3. Concerning the conservatism related to the over-approximation of integral terms (Step 3 ), the accuracyof Jensen’s inequality has been addressed in [94,26,233].Alternatively to the use of Jensen’s inequality, the in-tegral terms may also be over-approximated using ex-tensions of Wirtinger’s inequality [154,233,234,98]. Fur-ther refinements were developed in [145], where the au-thors considered a discretized version of this inequality,in [273], where additional free-weighted matrices are in-troduced or in [214] using auxiliary functions. Other im-provements have been presented in [234,235] based onBessel’s inequality and Legendre polynomials. Finally,following the description given in Step 1, the applicationof the Wirtinger-based inequality to the case of tripleintegral-type of LKFs was considered in [212].

Step 4. The reduction of the conservatism induced bythe over-approximations of delay dependent terms (Step4 ) has been considered by several authors over the lastfew years [100,213,239]. To the best of our knowledge,for the moment, the most accurate over-approximationof delay dependent terms is provided in [213]. This re-sult encompasses several of the existing approximationtechniques proposed in the literature [72,100] and allowsfor an LMI formulation.

4.1.4 A more recent result

Clearly, considering more complex LKFs and more ad-vanced over-approximation techniques increases the

9

complexity of the proposed LMI stability criteria (interms of readability). In the sequel, we present thesimple stability conditions from [67], which take intoaccount some of the presented conservatism reductiontechniques and provide a fair compromise between ac-curacy and complexity.

Theorem 8 [67] Let there exist P ≻ 0, R ≻ 0, P2 andP3 such that the LMI

Φs P − P T2 + (A+BK)TP3

∗ −P3 − P T3 + hR

≺ 0, (30)

Φs P − P T2 + (A+BK)TP3 −hP T

2 A

∗ −P3 − P T3 −hP T

3 A

∗ ∗ −hR

≺ 0, (31)

with Φs = PT2 (A + BK) + (A + BK)TP2, are feasible.

Then system (12) is Exponentially Stable for all samplingsequences σ = tkk∈N with hk = tk+1 − tk ≤ h.

The result takes into account information about the saw-tooth shape of the delay, which is the specificity of thetime-delay model (21) when representing exactly thesampled-data system (12). It can ensure the stabilityfor time-varying delays τ(t) which are longer than anyconstant delay that preserves stability, provided thatτ(t) = 1.

The research on LKFs for sampled-data systems is stilla wide-open domain. Currently, an important effortis dedicated to finding better LKFs and better over-approximations of the derivatives. Note that providingimprovements (in terms of conservatism reduction) atone step usually requires changes at all the others steps.For this reason, the derivation of constructive stabilityconditions may be quite an elaborate analytical processand it is not always very intuitive. However, a notableadvantage of this methodology is the fact that for lin-ear systems the approach can be easily extended tocontrol design [72,248,150] and to the case of systemswith parameter uncertainties [67,232,208,88,216], de-lays [249,164,87,165,231,48] and schedulling protocols[152,153,151]. See also [113,112] for the use of LKFs inthe case of systems with switching control and [70,69]for the control of semilinear 1-D heat equations.

4.1.5 An extension to nonlinear systems

Concerning nonlinear systems, [163] has extended theideas in [72] for the case of control affine non-autonomoussystems. Consider the nonlinear system:

x(t) = f(t, x(t)) + g(t, x(t))u(t), (32)

with the state x(t) ∈ Rn and the input u(t) ∈ R

m, andwith functions f , g that are locally Lipschitz with respect

to x and piecewise continuous in t. Assume that a C1

controller u(t) = K(t, x) is designed in order to make thesystem (32) Globally Uniformly Asymptotically Stable.Moreover, assume that there exist a C1 positive definiteand radially unbounded function V , and a continuouspositive definite function W such that:

−[∂V

∂t(t, x) +

∂V

∂x(f(t, x) + g(t, x)K(t, x))

]

≥ W (x),

(33)for all t ≥ t0 and x ∈ R

n. Also, consider K(t, 0) = 0for all t ∈ R. Hence, V is a strict Lyapunov function forx = f(t, x) + g(t, x)K(t, x), and one can fix class K∞

functions α1 and α2 such that α1(‖x‖2) ≤ V (t, x) ≤α2(‖x‖2), for all t ≥ t0 and x ∈ R

n. Define the function

ρ(t, x) =∂K

∂t(t, x)+

∂K

∂x

(

f(t, x)+g(t, x)K(t, x))

. (34)

Theorem 9 (adapted from [163]) Suppose that there ex-ist constants c1, c2, c3 and c4 such that:

∥∥∥∥

∂K

∂x(t, x)g(t, x)

∥∥∥∥

2

2

≤ c1,

∥∥∥∥

∂V

∂x(t, x)g(t, x)

∥∥∥∥

2

2

≤ c2,

‖ρ(t, x)‖22 ≤ c3W (x),∥∥∥∥

∂V

∂x(t, x)g(t, x)K(t, x)

∥∥∥∥2

≤ c4(V (t, x) + 1),

hold for all t ≥ t0 and x ∈ Rn. Consider the system (32)

in closed-loop with: u(t) = K(tk, x(tk)), t ∈ [tk, tk+1),σ = tkk∈N as defined in (1) and hk = tk+1−tk ∈ [h, h],∀k ∈ N. Then, the closed-loop system is Globally Uni-

formly Asymptotically Stable if h ≤ (4c1 + 8c2c3)−1/2 .

The stability is proven by means of a Lyapunov func-tional of the form

U(t, xt) = V (t, x(t)) +ǫ

h

∫ 0

−h

∫ t

t+θ

‖Ψ(s, xs)‖22 dsdθ,

where Ψ(t, xt) =∂K∂t (t, xs(0)) +

∂K∂x (t, xt(0)) xt(0) and

ǫ > 0. This functional is reminiscent of the form (29)used in [72] to study LTI systems. However, differentlyfrom the LTI case, it is far more complex to determinehow conservative the result is.

4.1.6 Further reading

Aside from the Lyapunov-Krasovskii method, the sta-bility of sampled-data systems can also be analysedusing the method proposed by Razumikhin [224]. Con-nections between Razumikhin’s method and the ISSnonlinear small gain theorem [243] have been estab-lished in [253]. This relation has been used in [254] inorder to show the preservation of ISS properties undersufficiently fast sampling for nonlinear systems with

10

an emulated sampled-data controller. Razumikhin’smethod has been used in [61] for the case of LTIsampled-data systems. In [132], the Razumikhin methodis explored for nonlinear sampled-data systems on thebasis of vector Lyapunov-Razumikhin functions. Formore general extensions to the control design problemsee [134], concerning the case of nonlinear feed-forwardsystems and [135], for nonlinear sampled-data systemwith input delays. At last, we would like to mentionthe Input/Output approach for the analysis of time-delay systems [78,96,130], which makes use of classicalrobust control tools [278,168]. The application of theInput/Output approach for the case of sampled-datasystems has been discussed in [174,154]. The approachwas further developed by [81,204,203,207,206,40] with-out passing through the time-delay system model. Itwill be presented in more detail in Section 4.4.

4.2 Hybrid system approach

Due to the existence of both continuous and discrete dy-namics, it is quite natural to model sampled-data sys-tems as hybrid dynamical systems [91,92,99]. The firstmentions to sampled-data systems as hybrid dynamicalsystems date back to the middle of the ’80s [184]. Lateron, in the ’90s, the use of hybrid models has been de-veloped for linear sampled-data systems with uniformand multi-rate sampling as an interesting approach forthe H∞ and H2 control problems [126,247,256]. The ap-proach has also been developed for nonlinear sampled-data systems in [115,270]. For systems with aperiodicsampling, impulsive models had been used starting with[256,56,170]. Recently, more general hybrid models havebeen proposed in the context of Networked ControlledSystems by [193,195]. A solid theoretic foundation hasbeen established for hybrid systems in the frameworkproposed by [91,92] and it proves to be very useful in theanalysis of sampled-data systems.

In this section we will present some basic hybrid mod-els encountered in the analysis of sampled-data systems.The extensions of the Lyapunov stability theory for hy-brid systems will be introduced together with construc-tive numerical and analytic stability analysis criteria.

4.2.1 Impulsive models for sampled-data systems

Consider the case of LTI sampled-data systems with lin-ear state feedback, as in system (12). Let x denote apiecewise constant signal representing the most recentstate measurement of the plant available at the con-troller, x(t) = x(tk), for all t ∈ [tk, tk+1), k ∈ N. Usingthe augmented system state χ(t) = [xT (t), xT (t)]T ∈R

nχ with nχ = 2n, the dynamics of the LTI sampled-data system (12) can be written under the form

χ(t) = Fχ(t), t 6= tk, k ∈ N,

χ(tk) = Jχ(t−k ), k ∈ N,(35)

with

χ(t−) = limθ↑t

χ(θ), F =

[

A BK

0 0

]

, J =

[

I 0

I 0

]

. (36)

Similar models can be determined by considering an aug-mented state vector χ including the most recent con-trol value implemented at the plant u(t) = u(tk), thesampling error e(t) = x(t) − x(t), the actuation erroreu(t) = u(t) − u(t), etc. Models of the form (35),(36)fit into the framework of impulsive dynamical systems[173,99,143,17] (sometimes also called discontinuous dy-namical systems or simply jump systems). More generalnonlinear sampled-data systems lead to impulsive sys-tems of the form [187,193]

χ(t) = Fk(t, χ(t)), t 6= tk, k ∈ N, (37a)

χ(tk) = Jk(tk, χ(t−k )), k ∈ N, (37b)

where the augmented state may also include the con-troller state and some of its sampled components (state,output, etc.). Generally, for an impulsive system, (37a)is called the system’s flow dynamics while (37b) is thejump dynamics.

4.2.2 Lyapunov methods for impulsive systems

The stability of equilibria for the impulsive systems ofthe form (37) can be ensured by the existence of candi-date Lyapunov functions that depend both on the sys-tem state and on time, and evolve in a discontinuousmanner at impulse instants [17,99,187].

Theorem 10 [187] Consider system (37) and denoteτ(t) = t − tk, ∀t ∈ [tk, tk+1). Assume that Fk and Jkare locally Lipschitz functions from R+ × R

nχ to Rnχ

such that Fk(t, 0) = 0, Jk(t, 0) = 0, for all t ≥ 0. Letthere exist positive scalars c1, c2, c3, b and a Lyapunovfunction V : Rnχ × R → R, such that

c1‖χ‖b ≤ V (χ, τ) ≤ c2‖χ‖

b, (38)

for all χ ∈ Rnχ , τ ∈ [0, h]. Suppose that for any impulse

sequence σ = tkk∈N such that h ≤ tk+1 − tk ≤ h,k ∈ N, the corresponding solution χ(·) to (37) satisfies:

dV (χ(t), τ(t))

dt≤ −c3V (χ(t), τ(t)) , ∀t 6= tk, ∀k ∈ N,

and V (χ(tk), 0) ≤ limt→t−k

V (χ(t), τ(t)) , ∀k ∈ N.

Then, the equilibrium point χ = 0 of system (37) isGlobally Uniformly Exponentially Stable over the classof sampling impulse instants, i.e. there exist c, λ > 0such that for any sequence σ = tkk∈N that satisfies

h ≤ tk+1 − tk ≤ h, k ∈ N,

‖χ(t)‖ ≤ c‖χ(t0)‖e−λ(t−t0), ∀t ≥ t0.

11

The previous stability theorem requires in (38) the can-didate Lyapunov function to be positive at all times. Forthe case of system (37) with globally Lipschitz Fk, k ∈ N,the condition can be relaxed by requiring the Lyapunovfunction to be positive only at impulse times [187], i.e.c1‖χ(tk)‖

b ≤ V (χ(tk), 0) ≤ c2‖χ(tk)‖b, ∀k ∈ N, instead

of (38).

In the case of impulsive systems (35), with linear flowand jump dynamics, candidate Lyapunov functions ofthe form V (χ, τ) = χTP (τ)χ, with P : [0, h] → R

nχ×nχ

a differentiable matrix function, have been used[257,247,27,187]. Sufficient stability conditions can beobtained from Theorem 10 in terms of existence of adifferentiable matrix function P : [0, h] → R

nχ×nχ ,c1I ≺ P (τ) ≺ c2I, satisfying the parametric set of LMIs

FTP (θ1) + P (θ1)F + c3P (θ1) +∂P

∂τ(θ1) ≺ 0,

∀ θ1 ∈ [0, h], (39a)

JTP (0)J − P (θ2) ≺ 0, ∀ θ2 ∈ [h, h], (39b)

with positive scalars c1, c2, c3. This formulation is remi-niscent of the Riccati equation approach used for robustsampled-data control in [256,247]. Alternatively, stabil-ity can also be checked by analysing the behaviour atimpulse times [270,169,111,31,30,27]. The following re-sult for impulsive systems with linear flow and jump dy-namics is stated from [111].

Theorem 11 [111] Consider system (35) with tk+1 −tk ∈ [h, h], for all k ∈ N.The equilibrium point χ = 0 ofsystem (35) is Globally Uniformly Exponentially Stableif and only if there exists a positive definite function Vd :R

nχ → R+ strictly convex,

Vd(χ) = χTP[χ]χ,

homogeneous (of the second order), P[·] : Rnχ →

Rnχ×nχ , P[χ] = PT

[χ] = P[aχ] ≻ 0, ∀χ 6= 0, a ∈ R, a 6= 0,

V (0) = 0, such that

Vd(χ(tk)) > Vd(χ(tk+1)),

for all χ(tk) 6= 0, k ∈ N.

The result has been obtained using a linear differ-ence inclusion χ(tk+1) ∈ F (χ(tk)) with F (χ) =JeFθχ, θ ∈ [h, h]

representing the exact system in-

tegration between two impulse instants together withconverse Lyapunov theorems for linear difference inclu-sions [178,114]. More general results, concerning the im-pulsive systems with nonlinear dynamics, can be foundin [270,169]. Particularizing the result of Theorem 11to the case of quadratic Lyapunov candidate functionsVd(χ) = χTLχ, sufficient stability conditions are ob-tained by checking the existence of a positive definite

matrix L ≻ 0 such that

(JeFθ

)TL(JeFθ

)− L ≺ 0, ∀ θ ∈ [h, h], (40)

which is also a parametric LMI, similar to (39). Lessconservative conditions can be obtained using compos-ite quadratic Lyapunov functions of the form Vd(χ) =maxi∈1,...,M χ

TPiχ, where Pi, i = 1, . . . ,M , form a fi-nite set of symmetric positive definite matrices [111].

A relation between the existence of a continuous-timeLyapunov function, V (χ, τ) = χTP (τ)χ, and the exis-tence of a quadratic Lyapunov function, Vd (χ(tk)) =χT (tk)Lχ(tk), decreasing at impulse times, is providedin the following theorem.

Theorem 12 [27] The following statements are equiva-lent:

(a) There exists L ≻ 0 such that

(eFθJ

)TL(eFθJ

)− L ≺ 0, ∀ θ ∈ [h, h]. (41)

(b) There exists a differentiable matrix function P :[0, h] → R

nχ×nχ , P (τ) = PT (τ), P (0) ≻ 0 and a posi-tive scalar ǫ such that

FTP (θ1) + P (θ1)F +∂P

∂τ(θ1) 0, ∀ θ1 ∈ [0, h],

(42a)

JTP (0)J − P (θ2) + ǫI 0, ∀ θ2 ∈ [h, h]. (42b)

Moreover, when one of the above holds, the equilibriumχ = 0 of the impulsive system (35) is AsymptoticallyStable for any of the possible impulse sequences tkk∈N

satisfying tk+1 − tk ∈ [h, h] for all k ∈ N.

Condition (b) in the previous theorem is a sufficient con-dition for the existence of a candidate Lyapunov func-tion V (χ, τ) = χTP (τ)χ such that

dV (χ(t), τ(t))

dt≤ 0, ∀t 6= tk, ∀k ∈ N, (43)

and

V (χ(tk), 0) < limt→t−

k

V (χ(t), τ(t)) , ∀k ∈ N, (44)

which is slightly different from the conditions (39) ob-tained based on Theorem 10. For the case described inTheorem 12, (b), V (χ, τ) may be constant between im-pulse instants provided that it decays at impulse times.According to the previous theorem, the existence of sucha candidate Lyapunov function is equivalent to the exis-tence of a quadratic Lyapunov function Vd(χ) = χTLχ

12

which is strictly decreasing at impulse times. Note thatfrom Theorem 11, the latter is only a sufficient condi-tion for stability. The main point is that the existenceof a function of the form V (χ, τ) = χTP (τ)χ satisfying(43),(44) is not a necessary condition for the stabilityof the impulsive system (35). Other forms of Lyapunovfunctions need to be considered to improve the existingconditions.

4.2.3 Numerically tractable stability criteria

In practice, the difficulty of checking the existence ofcandidate Lyapunov functions using LMI formulationssuch as (39) or (41) comes from the fact that the setof LMIs are parametrized by elements in [0, h] or [h, h],which leads to an infinite number of LMIs. As followswe will discuss the derivation of a finite number of LMIsfrom (39). Numerical tools [111,31,30] for the resolutionof LMIs similar to (41), involving uncertain matrix ex-ponential terms, are discussed in Section 4.3 in the con-text of the discrete-time approach.

Concerning the parametric set of LMIs (39), a finitenumber of LMI conditions can be derived by consider-ing particular forms for the matrix function P (τ). Forexample, consider a matrix P (τ) linear with respect to τ

P (τ) = P1 + (P2 − P1)τ

h, (45)

for some positive definite matrices P1, P2, as in [120,4].There, such a Lyapunov matrix has been used forsampled-data systems with multi-rate sampling andswitched linear systems. For a candidate Lyapunov func-tion V (χ, τ) = χTP (τ)χ, with P (τ) as defined in (45), afinite set of LMIs that are sufficient for stability can beobtained from (39) using simple convexity arguments:

FTP1 + P1F + c3P1 +P2 − P1

h≺ 0, (46a)

FTP2 + P2F + c3P2 +P2 − P1

h≺ 0, (46b)

JTP1J ≺ P2, (46c)

JTP1J ≺ P1 + (P2 − P1) h/h. (46d)

For the particular case of LTI sampled-data systems rep-resented by (35),(36), Lyapunov functions of the formV (χ, τ) = χTP (τ)χ are proposed in the literature bysumming various terms such as:

V1(χ, τ) = xTP0x, (47)

V2(χ, τ) = (x− x)T Q (x− x) (h− τ), (48)

V3(χ, τ) = (x− x)TR (x− x) e−λτ , (49)

V4(χ, τ) = e−λτχT(

eFT (h−τ)SeF

T (h−τ))

χ, (50)

V5(χ, τ) = χT(∫ 0

−τ

(s+ h)(FeFs)T U(FeFs)ds)

χ, (51)

where U :=

[

U 0

0 0

]

, λ > 0 and P0, Q, R, S, U

are symmetric positive definite matrices. Using suchparticular forms of Lyapunov functions, LMI sta-bility conditions have been derived in the literature[120,187,195,205,92,65]. We point in particular to theterm (51) used in [187] which provided a significantimprovement in what concerns the conservatism re-duction. This term is inspired by Lyapunov-Krasovskiifunctionals from the input-delay approach, like the onein [72]. Note that the term (51) can also be written as∫ t

t−τ (s + h − t)xT (s)Ux(s)ds. It has been motivated

by the term∫ 0

−h

∫ t

t+θxT (s)Ux(s)dsdθ used in the time-

delay approach (see [72]). Vice versa, the hybrid systemapproach has also inspired the use of discontinuousLyapunov functionals in the time-delay approach (seefor example the functional (29)). Note that the term

(hk−τ)∫ t

t−τxT (s)Rx(s)ds in the functional (29) can be

re-written as (hk − τ)χT( ∫ 0

−τ (FeFs)T R(FeFs)ds)

χ,

with

R =

[

R 0

0 0

]

and R ≻ 0. Then, for the impulsive system (35), (36),the functional (29) can be interpreted as a Lyapunovfunction of the form V (χ, τ, hk) = χTP (τ, hk)χ. Simi-larly to the time delay approach, the LMI formulationscan be adapted to cope with uncertainties in the systemmatrices.

Last, note that alternatively to the LMI formulation,numerical stability criteria based on polynomial matrixfunctions P (τ) and Sum-of-Squares programming [222]have been proposed in [27].

4.2.4 More general hybrid models

A large variety of hybrid dynamical systems, includ-ing sampled-data and impulsive models, can be re-formulated in the unifying theoretical framework pro-posed by Goebel, Sanfelice and Teel [91,92]. Severalfundamental properties have been investigated in thisframework, providing a solid theory for hybrid dynami-cal systems. The main advantage of this generic hybridformulation [91,92] is that the associated theoretic prop-erties can be directly transferred to sampled-data sys-tems with aperiodic sampling. The general formulationproposed in [91,92] considers models of the form

z = Fz(z), z ∈ C, (52a)

z+ = Jz(z), z ∈ D, (52b)

with state z ∈ Rnz . The system state evolves according

to an ordinary differential equation (52a) when the stateis in some subset C of Rnz and according to a first order

13

recurrence equation (52b) when the state is in the subsetD of Rnz . z+ denotes the next value of state given as afunction of the current state z via the map Jz(·). C iscalled the flow set andD is called the jump set. Here, weassume that Fz and Jz are continuous functions from Cto Rnz andD to Rnz , respectively. C andD are assumedto be closed sets in R

nz .

Note that in the impulsive system formulation ofsampled-data systems, the system jumps are time-triggered. However, the dynamics of the triggeringmechanism is in some sense hidden. In the frameworkproposed by [35,46,195,91,92], the mechanism triggeringthe system jumps is modelled explicitly by augmentingthe system state with the clock variable τ(t) = t − tk,∀t ∈ [tk, tk+1), ∀k ∈ N. Consider the LTI sampled-data systems (12) with the notations x(t) = x(tk),τ(t) = t − tk for all t ∈ [tk, tk+1), k ∈ N. The systemcan be represented by the following hybrid model

x = Ax+BKx

˙x = 0

τ = 1

τ ∈ [0, h],

x+ = x

x+ = x

τ+ = 0

τ ∈ [h, h].

(53)

Then, system (12) with hk ∈ [h, h] (or equivalently(35),(36)) can be re-modelled in the form (52) with

zT =[

xT xT τ]

=[

χT τ]

,

C =z ∈ R

nz : τ ∈ [0, h],

D =z ∈ R

nz : τ ∈ [h, h],

Fz(z) =

Ax+BKx

0

1

, Jz(z) =

x

x

0

. (54)

Solutions φ of the general hybrid system (52) areparametrized by both the continuous time t and thediscrete time k: φ(t, k) represents the state of the hybridsystem after t time units and k jumps. Such solutionsare defined on a hybrid time domain, which for the caseof sampled-data systems is given as the union of theintervals [tk, tk+1]× k. A solution φ(·, ·) is a functiondefined on a hybrid time domain such that φ(·, k) iscontinuous on [tk, tk+1], continuously differentiable on(tk, tk+1) for each k in the domain, and such that

φ(t, k) = Fz (φ(t, k)) ,

if φ(t, k) ∈ C, t ∈ (tk, tk+1), k ∈ N, and

φ(tk+1, k + 1) = Jz (φ(tk+1, k)) ,

if φ(tk+1, k) ∈ D, k ∈ N. For sampled-data systemsas (53) such solutions may be roughly interpreted as ageneralization of the state lifting approach proposed in[269] for systems with periodic sampling.

A particularity of the model (53) in the context of sta-bility analysis is the fact that although the matrix K isdesigned such that x (and consequently x) converges tozero, the clock variable τ does not converge. For eachsampling interval [tk, tk+1), the timer τ visits succes-

sively the points of the interval [0, h] up to hk = tk+1 −tk. The main consequence is that the hybrid system(53) does not have an asymptotically stable equilibriumpoint. For such systems the stability of the compact setA = 0 × 0 × [0, h] is usually investigated instead.Studying this property allows to conclude on the con-vergence of x. One of the main results allowing to statethe asymptotic stability of a set for hybrid systems isgiven below. This results is expressed in terms of thepre-asymptotic stability of a setA (see [91] for a detaileddefinition). The prefix ”-pre” is used since the complete-ness of all system solutions 2 is not required. Only com-plete solutions need to converge to A. The concept ofpre-asymptotic stability used in the following theoremis equivalent to standard asymptotic stability of the setA when all system solutions are complete, which is thecase for sampled-data systems.

Theorem 13 [91] Consider the hybrid system (52) andthe compact set A ⊂ R

nz such that Jz (A∩D) ⊂ A. Ifthere exists a candidate Lyapunov function 3 V such that

∂V

∂zFz(z) < 0 for all z ∈ C \ A, (55a)

V (Jz(z))− V (z) < 0 for all z ∈ D \ A, (55b)

then the set A is pre-asymptotically stable.

Various relaxations of the above result are provided inChapter 3 in [92]. A converse Lyapunov theorem is givenbelow.

Theorem 14 [91] For the hybrid system (52), if thecompact set A is globally pre-asymptotically stable, thenthere exist a C∞ function V : Rnz → R+ and α1, α2 ∈K∞ such that α1 (|z|A) ≤ V (z) ≤ α2 (|z|A), ∀z ∈ R

nz ,where | · |A denotes the distance from the set A, and

∂V

∂zFz(z) ≤ −V (z), ∀z ∈ C, (56a)

V (Jz(z)) ≤ V (z)/2, ∀z ∈ D. (56b)

2 A solution φ(t, k) is called complete if dom φ is unbounded.3 V is continuous and non-negative on (C ∪D)\A ⊂ domV ,it is continuously differentiable on an open set satisfyingC \ A ⊂ domV , and limz→A,z∈domV ∩(C∪D) V (z) = 0. Fur-thermore, for global pre-asymptotic stability, the sublevelsets of V (.) are required to be compact.

14

Note that with respect to the case of sampled-data sys-tems such as (53) (or equivalently (35), (36)) where so-lutions are complete, the previous theorem shows thatasymptotic stability implies the existence of a C∞ Lya-punov function of the form V (z) = V (χ, τ), to be relatedwith the sufficient conditions for stability in Theorem 10.

4.2.5 An estimation of the MSI for nonlinear systems

For nonlinear sampled-data systems the stability prop-erties have been studied in the more general context ofNetworked Control Systems with scheduling protocols[193,35]. This approach has been particularized to thesampled-data case in [195]. Consider the plant:

x = F (x, u) ,

y = H (x, u) ,(57)

where x is the plant state, u is the control input, y is themeasured output. Suppose that asymptotic stability isguaranteed by the continuous-time output feedback:

xc = F c (xc, y) ,

u = Hc (xc, y) ,(58)

where xc is the controller state. Under an exact sampled-data implementation of the controller and a perfectknowledge of the sampling sequence σ = tkk∈N, thesampled-data implementation of the closed-loop systemcan be written in the following impulsive form:

x = F (x, u), t ∈ [tk, tk+1),

y = H(x), t ∈ R+,

xc = F c(xc, y), t ∈ [tk, tk+1),

u = Hc(xc), t ∈ R+,

˙y = 0, t ∈ [tk, tk+1),

˙u = 0, t ∈ [tk, tk+1),

y(tk) = y(t−k ),

u(tk) = u(t−k ),

(59)

where u represents the control being implemented at theplant and y the most recent plant output measurementsthat are available at the controller. In order to expressthe system in the general framework of [92], consider theaugmented state vector η(t) ∈ R

nη and the sampling-induced error e(t) ∈ R

ne :

η(t) :=

[

x(t)

xc(t)

]

, e(t) =

[

ey(t)

eu(t)

]

:=

[

y(t)− y(t)

u(t)− u(t)

]

.

The dynamics in (59) with hk ∈ [h, h] can be modelledby the following hybrid system:

η = f(η, e)

e = g(η, e)

τ = 1

τ ∈ [0, h],

η+ = η

e+ = 0

τ+ = 0

τ ∈ [h,∞),

(60)

with η ∈ Rnη , e ∈ R

ne , τ ∈ R+. The functions f andg are obtained by direct calculations from the sampled-data system (59) (see [195]):

f(η, e) =

[

F (x,Hc(xc) + eu)

F c(xc, H(x) + ey)

]

,

g(η, e) =

[

−∂H∂x F (x,Hc(xc) + eu)

−∂Hc

∂xc Fc(xc, H(x) + ey)

]

.

It should be noted that η = f(η, 0) is the closed loopsystem without the sampled-data implementation. Thefollowing theorem provides a quantitative method to es-timate the MSI, using model (60).

Theorem 15 [195] Assume that f and g in (60) are con-

tinuous. Suppose there exist ∆η, ∆e > 0, a locally Lips-chitz function W : Rne → R+, a locally Lipschitz, posi-tive definite, radially unbounded function V : Rnη → R+,a continuous function Θ : R

nη → R+, real numbersL > 0, γ > 0, functions αW , αW ∈ K∞ and a continu-ous, positive definite function such that, for all e ∈ R

ne :

αW (‖e‖) ≤ W (e) ≤ αW (‖e‖),

and for almost all ‖η‖ ≤ ∆η and ‖e‖ ≤ ∆e:

∂W

∂eg(η, e) ≤ LW (e) + Θ(η),

∂V

∂ηf(η, e) ≤ −(‖η‖)− (W (e))−Θ2(η) + γ2W 2(e).

Finally, consider that 0 < h ≤ h < T (γ,L), with

T (γ, L) :=

1Lrarctan(r), γ > L,

1L , γ = L,

1Lrarctanh(r), γ < L,

and r =√∣∣ γ

2

L2 − 1∣∣. Then, for all sampling intervals less

than h the set A = (η, e, τ) : η = 0, e = 0, τ ∈ [0, h] isUniformly Asymptotically Stable for system (60).

15

Theorem 15 provides an explicit formulation of the MSIfor nonlinear sampled-data systems. It is applicable forboth constant and variable sampling intervals. For aconstructive application to the case of bilinear systemssee [205]. A numerical formulation using Sum-of-Squares[222] has also been provided in [22].

4.2.6 Further reading

In the impulsive system framework, control design condi-tions have been proposed in [27]. For observer design con-ditions we point to the works in [5,49,3,186,220,59].Someextensions of the hybrid systems approach for sampled-data systemswith delay can be found in [73,188] and [22].When constructing a hybrid model for linear sampled-data system, it is also possible to consider as state vari-able a decreasing counter θ(t) = hk − τ(t) with θ = −1.Aside from the methodology presented here, the sta-bility of impulsive systems can also be analysed usingdiscrete-time approaches [111] based on convex embed-ding methods and looped Lyapunov functionals [31,30].These methods are presented for the particular case ofsampled-data systems in Section 4.3.

4.3 Discrete-time approach and convex-embeddings

In this sub-section we present several approaches whichuse either the system integration over the sampling in-terval or convex embeddings of the transition matrix be-tween sampling times in order to derive stability condi-tions. We will start with extensions of the discrete-timeapproach for the case of linear systems with aperiodicsampling; next a technique based on Delta operators willbe given, followed by a continuous-time approach and adiscrete-time approach based on functionals, similar tothe ones used in the time-delay approach. At last, an ex-tension of the discrete-time approach to nonlinear sys-tems with aperiodic sampling will be presented.

4.3.1 Theoretical results for LTI systems using thediscrete-time approach

Let us consider the LTI systemwith sampled linear staticstate feedback (12) where hk = tk+1 − tk takes values inthe set T = [h, h]. Recall the notations xk = x(tk),

Λ(θ) = eAθ +

∫ θ

0

eAsdsBK, (61)

for θ ∈ R. One can verify that the closed-loop system(12) satisfies

xk+1 = Λ(hk)xk (62)

with hk ∈ T = [h, h]. Model (62) belongs to the class ofdiscrete-time Linear Parameter Varying (LPV) systems[227,127,178]. It captures the behaviour of system (12)

at sampling times, without consideration of the inter-sample behavior. However, in [81], the following propo-sition has shown that for LTI sampled-data system, theasymptotic stability in continuous-time and in discrete-time are equivalent.

Proposition 16 [81] Consider the sampled-data system(12) with hk = tk+1 − tk ∈ [h, h]. For a given x(t0), thefollowing conditions are equivalent:

(1) limt→∞ x(t) = 0(2) limk→∞ x(tk) = 0.

Various methods are available for studying the sta-bility of discrete-time LPV systems. Stability criteriahave been proposed by analysing the joint spectralradius [24,1] or by checking the existence of quasi-quadratic [178,121], parameter dependent [45], path-dependent [144], non-monotonic [167,141,2] and com-posite quadratic [121] Lyapunov functions.

The following theorem from [114] addresses specificallythe case of model (62).

Theorem 17 [114] Consider the continuous-time sys-tem (12) and the discrete-time model (62) with T =

[h, h]. The following statements are equivalent:

1) The equilibrium point x = 0 of (62) is Globally Uni-formly Exponentially Stable.

2) There exist a P ≻ 0 and N > 0 such that

(N∏

i=1

Λ(θi)

)T

P

(N∏

i=1

Λ(θi)

)

− P ≺ 0, (63)

for any N -length sequence θiNi=1 with values in T , i.e.

the function V (x) = xTPx satisfies V (xk+N ) < V (xk)for all xk 6= 0, k ∈ N.

3) There exists a positive definite function V : Rn →R

+ strictly convex, homogeneous (of the second order),V (x) = xTP[x]x, with P[·] : R

n → Rn×n, P[x] = PT

[x] =

P[ax], ∀x 6= 0, a ∈ R, a 6= 0 such that :

V (x)−maxθ∈T

V (Λ(θ)x) > 0, ∀x 6= 0. (64)

Condition 2) in Theorem 17 corresponds to the existenceof a non-monotonic Lyapunov function V (x) = xTPx,[167,141,2] which is decreasing every N samples. If thesystem is stable, then necessarily there exists a finite Nand a matrix P such that (63) holds. However, checkingthe existence of a matrix P satisfying (63) for a givenN represents a set of LMIs which are sufficient only forstability. Condition 3) corresponds to the existence of

16

a quasi-quadratic Lyapunov function [178,121] V (x) =xTP[x]x. Theorem 17 shows the equivalence betweenquasi-quadratic Lyapunov functions and non-monotonicLyapunov functions and provides necessary and suffi-cient conditions for the exponential stability of system(62). A simple stability criterion which is sufficient forstability can be obtained using classical quadratic Lya-punov functions, which are decreasing at each sample.

Theorem 18 [275] The origin of system (62) is Glob-ally Uniformly Exponentially Stable for all sampling se-quences σ = tkk∈N with hk = tk+1− tk ∈ [h, h], k ∈ N,if there exists P ≻ 0 such that

ΛT (θ)PΛ(θ) − P ≺ 0, ∀θ ∈ T = [h, h]. (65)

The LMI (65) is a particular case of condition 2) in The-orem 17 withN = 1. In a similar way, it is also a particu-lar case of condition 3) with P[x] = P, for all x ∈ R

n.Thecondition ensures that the candidate Lyapunov functionV (x) = xTPx satisfies the relation

∆V (k) = V (xk+1)− V (xk) < 0, ∀xk 6= 0. (66)

Finally, let us note that, similarly to conditions (39) or(41) used for the hybrid system approach, the stabilityconditions (63) and (65) represent sets of LMIs that are

parametrized by θ ∈ T = [h, h]. They are not computa-tionally tractable problems by themselves. Approximatesolutions, based on evaluation of the condition for a finiteset of values of θ have been presented in [275,228,241].A finite set of sufficient tractable numerical conditionscan be obtained using normed-bounded and/or poly-topic convex embeddings of the transition matrix Λ(θ).

4.3.2 Tractable criteria

In what follows, we try to give an idea about the mannerto solve parametric LMIs involving matrix exponentialssuch as the one in (65). First, we present briefly the ap-proach proposed by Fujioka in [79]. Consider a nominal

sampling interval T0 ∈ [h, h]. For a scalar δ, the transi-tion matrix Λ(·) satisfies the relation

Λ(T0 + δ) = Λ(T0) + ∆(δ)Ψ(T0), (67)

where ∆(δ) :=∫ δ

0 eAsds, Ψ(T0) = AΛ(T0) +BK. Usingclassical properties of the matrix exponential [155], theinduced Euclidean norm of ∆(δ) can be over-bounded

‖∆(δ)‖2 ≤

∫ δ

0

eµ(A)sds, (68)

where µ(A) is the maximum eigenvalue of A+AT

2 . System(62) can be expressed as a nominal discrete-time LTI

system with a norm-bounded uncertainty

xk+1 = Λ(T0)xk +∆(δk)Ψ(T0)xk (69)

where δk = hk − T0, for which classical H∞ criteria [86]can be used. A simplified version of the main result in[79] is given as follows.

Theorem 19 [79] Let T0 > 0 be given. If there existX ≻ 0 and γ > 0 satisfying

M(T0, X, γ) :=[

Λ(T0) I

Ψ(T0) 0

][

X 0

0 I

][

Λ(T0) I

Ψ(T0) 0

]T

−

[

X 0

0 γ2I

]

≺ 0, (70)

then (65) is satisfied with P = X−1 for all θ ∈T (T0, γ) :=

[h(T0, γ), h(T0, γ)

]∩ (0,∞), with

h(T0, γ) =

T0 − γ−1, if µ(−A) = 0,

−∞, if µ(−A) ≤ −γ,

T0 −log(1+γ−1µ(−A))

µ(−A) , otherwise,

(71)

h(T0, γ) =

T0 + γ−1, if µ(A) = 0,

∞, if µ(A) ≤ −γ,

T0 +log(1+γ−1µ(A))

µ(A) , otherwise.

(72)

Condition (70) is sufficient for the asymptotic stability ofsystem (62) under time-varying sampling intervals hk ∈

[h, h] with h and h given in (71) and (72), respectively.Other norm-bounded approximations of the transitionmatrix Λ(·) exist in the literature [18,245,129,85,277].For example, stability conditions have been provided us-ing the Schur decomposition in [245] while [277] usesthe Jordan normal form. In [85] the transition matrixΛ(T0 + δ) is decomposed as

Λ(T0 + δ) = Λ(T0) + δL(T0) + ∆2(δ)AL(T0)

with L(T0) = eAT0(A + BK), ∆2(δ) :=∫ δ

0

∫ ρ

0eAsdsdρ,

and stability conditions are provided by computing theinduced Euclidean norm of ∆2(δ). See also [129] wherestability conditions have been derived using IntegralQuadratic Constraints (IQC), by studying the positiverealness of ∆(δ). More general Lyapunov functions havebeen used in [84].

Alternatively to the use of norm bounded approxima-tions, tractable numerical conditions can also be ob-tained using polytopic embeddings of the transition ma-trix Λ(·) in system (62). The set

W[h,h] := Λ(θ), θ ∈ [h, h]

17

is embedded in a larger convex polytope with a finitenumber of vertices Λi, i ∈ I := 1, · · · , Nv,

W :=

Nv∑

i=1

αiΛi |αi ≥ 0, i ∈ I,

Nv∑

i=1

αi = 1

, (73)

in such a way that W[h,h] ⊆ W . Using a polytopic em-

bedding, system (62) can be expressed as a

xk+1 =

Nv∑

i=1

αi(hk)Λixk, (74)

where∑Nv

i=1 αi(hk) = 1, αi(hk) ≥ 0, i ∈ I. This is aclassical discrete-time systemwith polytopic uncertainty

[45]. Here α(hk) =[

α1(hk) α2(hk) . . . αNv(hk)

]T

rep-

resent the barycentric coordinates of Λ(hk) in the poly-tope W. The properties of the over-approximating poly-topic set W make it possible to derive a finite number ofsufficient stability conditions from (65), by writing sim-ple LMIs over the polytope vertices:

P ≻ 0, ΛTi PΛi − P ≺ 0, ∀i ∈ I. (75)

The same procedure can also be applied for the condi-tion (63) - see [114] for details. However, for the caseof conditions (63), the numerical complexity is increas-ing in an exponential manner with respect to the chosenparameter N. One of the advantages of the polytopicembedding is the fact that it allows the use of parame-ter dependent Lyapunov functions [45,108,43] V (x, α) =

xTP (α)x, P (α) =∑Nv

i=1 αiPi, which lead to refined sta-bility conditions under a reasonable numerical complex-ity:

∃ Pi = PTi ≻ 0, ΛT

i PjΛi−Pi ≺ 0, ∀ (i, j) ∈ I×I. (76)

With respect to Theorem 17, the set of conditions (76)represents also a sufficient criterion for the existenceof a quasi-quadratic Lyapunov function [111], V (x) =xTP[x]x = maxi∈1,...,Nv x

TPix.

Themain difficulty in constructing the polytope W is theexponential dependence of the transition matrix Λ(θ) =

eAθ+∫ θ

0 eAsdsBK in the parameter θ over the the inter-

val [h, h]. Several approaches exist for the computationof a convex polytope embedding an uncertain matrix ex-ponential. See for example [201,200,42,43,157] for tech-niques based on the real Jordan form, [90] for a construc-tion that uses the Cayley-Hamilton theorem and [41] foran approach studying intervalmatrices. Onemay remarkthat the transition matrix Λ(·) can be re-expressed as

Λ(θ) = I +∆(θ) (A+BK) (77)

which involves only one uncertain matrix term ∆(θ) =∫ θ

0 eAsds. Then the stability problem can be addressedby constructing a polytopic approximation of ∆(θ) forθ ∈ [h, h]. To give an idea about the manner such aconvex polytope can be constructed, let us consider asimple case where the matrix A has n real eigenvaluesλi 6= 0, i ∈ 1, . . . , n with multiplicity equal to one, i.e.

A = T−1

λ1 0 0 . . . 0

0 λ2 0 . . . 0

.... . .

...

0 . . . . . . 0 λn

T (78)

for some invertible matrix T ∈ Rn×n. Then the uncer-

tain matrix ∆(θ) takes the form:

∆(θ) = T−1

ρ1(θ) 0 0 . . . 0

0 ρ2(θ) 0 . . . 0

.... . .

...

0 . . . . . . 0 ρn(θ)

T (79)

where ρi(θ) = 1λi

(eλiθ − 1

), i = 1, . . . , n. By comput-

ing ρmini and ρmax

i the minimum and maximum values

of ρi(θ) over [h, h], the uncertain matrix ∆(θ) is embed-ded in a convex polytope with Nv = 2n vertices

∆(θ) ∈ conv D1, D2, . . . , DNv

:= convT−1diag(ρ1, . . . , ρn)T : ρi ∈ ρmin

i , ρmaxi ,

i = 1, . . . , n .

Using (77), the polytopic set (73) can be constructedwith Λi = I +Di(A+BK), i ∈ I. A similar embeddingprocedure can be applied in the general case (when theeigenvalues of A have multiplicity different than one orwhen they are complex) - see [43].

As the numerical complexity of the obtained LMI con-ditions depends significantly on the number of verticesNv of the polytopic approximation, one of the challengesis to provide accurate convex polytopes while reducingthe number of vertices. For the Jordan decompositionprocedure, the number of vertices Nv increases expo-nentially with the order of the system. A method forreducing the number of vertices has been provided in[201,157,156]. However, the method provides a largerpolytopic embedding and may result in a conservativestability condition. Methods that are independent of theorder of the systems have been proposed by combiningpolytopic embeddings with norm bounded approxima-tions [108,109,53,52]. We present briefly an adaptationof the approach based on Taylor series approximation in[108,109], originally used for sampled-data systems with

18

input delay. Note that the transition matrix Λ(hk) withhk ∈ [h, h] can be rewritten as

Λ(hk) = Λ(h) + ∆(ρk)Ψ(h)

where ρk = hk − h ∈ [0, h − h], ∆(ρ) =∫ ρ

0 eAsds andΨ(h) = AΛ(h) +BK. Using a Taylor series approxima-tion of the matrix exponential, ∆(ρ) can be expressed as

∆(ρ) = TM (ρ) +RM (ρ)

where TM (ρ) =∑M

i=1Ai−1ρi

i! is the M th order Taylorseries approximation and RM (ρ) is the remainder. Theprocedure proposed in [108,109] allows to embed TM (ρ)in a convex polytope with Nv = M + 1 vertices

TM (ρ) ∈ conv Ui, i = 1, . . . ,M + 1 , ∀ ρ ∈ [0, h− h],

where U0 = 0, Ui+1 =(h−h)iAi−1

i! + Ui, i = 1, . . . ,M .Furthermore, an upper bound on the induced Eu-clidean norm of RM (ρ) can be computed using themethod proposed in [149]. To obtain an embedding with‖RM (ρ)‖2 < γR for all ρ ∈ [0, h− h] the approximationorder M must be chosen such that

‖A‖2 (h− h)

M + 2< 1,

and

∥∥AM

∥∥2(h− h)M+1

(M + 1)!

M + 2

M + 2− ‖A‖2 (h− h)≤ γR.

For this approach the number of vertices is linear in theorder M of the Taylor approximation. Stability crite-ria are obtained in a direct manner by combining LMImethods for polytopic systems with the ones for systemswith norm-bounded uncertainty.

Note that for both norm-bounded and polytopic em-beddings approaches, the accuracy of the approximationmay be significantly increased by dividing [h, h] into sev-eral subintervals and applying the embedding procedurelocally [79,200,111,52]. For example, in the case of thenorm-bounded embedding used in Theorem 19, the ideais to consider a grid of r ”nominal” sampling intervalsT1 < T2 < · · · < Tr and to verify the existence of asymmetric positive definite matrix X and of r parame-ters γi, i = 1, . . . , r, such that M(Ti, X, γi) ≺ 0 for alli = 1, . . . , r.When this condition is satisfied, system (62)is stable for any time-varying sampling interval hk ∈∪ri=1T (Ti, γi) where T (Ti, γi) =

[h(Ti, γi), h(Ti, γi)

]are

defined using (71), (72). Furthermore, it has been shownin [79] that using this approach one can approximatethe condition (65) as accurately as desired, in the sense

that if the condition (65) holds for θ ∈ [h, h], then nec-essarily there exists a matrix X = P−1, a sufficiently

tight grid of parameters Ti, i = 1, . . . , r and positivescalars γi, i = 1, . . . , r, such thatM(Ti, X, γi) ≺ 0 for all

i = 1, . . . , r, and [h, h] ⊂ ∪ri=1T (Ti, γi). Such an asymp-

totic exactness property has also been discussed for otherembedding approaches [52,200,241]. The main issue isthat, using convex embeddings, the conservatism withrespect to the quadratic stability condition (65) can bereduced to any degree at the cost of increased computa-tional complexity. However, the analysis of the asymp-totic exactness property does not take into account allnumerical implementation aspects. Most of the meth-ods are based on the computation of the matrix expo-nential for nominal sampling intervals, on the use of theeigenvalues/eigenvectors of the state matrix A or of itscharacteristic polynomial, etc. Computing any of theseelements introduces approximations [179] which mightinfluence the numerical implementation of the embed-ding. The effect of these approximations on the accuracyof the stability analysis needs to be further analysed.

4.3.3 Analysis based on Delta-operators

One of the drawbacks of the discrete-time analysis as theone proposed in (65) is the fact that the matrix Λ(θ) isclose to identity when θ is small. For small values of thelower bound of the sampling interval h, the inequalitymay be difficult to handle numerically. To avoid this nu-merical drawback, a condition which encompasses (65)has been proposed in [200].

Theorem 20 System (12) is Exponentially Stable for

any arbitrary sampling sequence with tk+1 − tk ∈ [h, h]if there exists X ≻ 0 such that

Ω(θ)X +XΩT (θ) + θΩT (θ)XΩ(θ) ≺ 0, (80)

for all θ ∈ [h, h] where

Ω(θ) =1

θ(Λ(θ)− I) =

1

θ

∫ θ

0

eAsds (A+BK) . (81)

This condition is obtained from an extension of theDelta-operator approach [171]. System (12) withhk ∈ [h, h] satisfies the relation

δkx = Ω(hk)x(tk), ∀ k ∈ N, (82)

where δkx := x(tk+1)−x(tk)hk

, ∀k ∈ N, and Ω(·) as de-

fined in (81). Note that when h tends to zero, the ma-trix Ω(h) converges to A + BK. The model (82) pro-vides a smooth transition from the continuous-time con-trol loop x = (A +BK)x (obtained when the samplinginterval tends to zero) and the discrete-time represen-tation (62). Condition (80) implies that the candidate

19

Lyapunov function V (x) = xTX−1x satisfies

δkV :=V (xk+1)− V (xk)

hk< 0 (83)

for all xk 6= 0, hk ∈ [h, h], k ∈ N. When θ tends to zero,(80) converges to the classical stability conditions for thecontinuous-time control loop:

(A+BK)X +X(A+BK)T ≺ 0. (84)

Similarly to the discrete-time analysis in condition (65),the parametric conditions (80) can be replaced by a finitenumber of LMIs using a convex embedding of the matrixΩ(θ). See [200], where a polytopic embedding procedurebased on the Jordan normal form and the mean valuetheorem has been proposed. See also [29] for an approachusing more general Lyapunov functions.

4.3.4 Continuous-time analysis using embeddings

In practice it is important to provide an estimate of thesystem’s performance in between sampling instants. Acontinuous-time approach based on convexification ar-guments has been proposed in [114,61] for LTI systems.The approach takes into account the relation

x(t) = Λ(t− tk)x(tk), ∀ t ∈ [tk, tk+1), k ∈ N, (85)

still referring to the definition of the transition matrix

Λ(t − tk) = I +∫ t−tk0 eAsds(A + BK) of system (12).

The classical condition V (x(t)) < −λV (x(t)) ensuringthe exponential decay of a candidate Lyapunov functionV (x) = xTPx for some positive λ can be expressed asthe following parametric LMI

Λ(θ)

I

T

ATP + PA+ λP PBK

KTBTP 0

Λ(θ)

I

≺ 0,

for all θ ∈ [0, h]. A finite number of LMIs can be obtainedsimilarly to the discrete-time case, by embedding thematrix Λ(θ) in a convex polytope

W := convΛ1, Λ2, · · · , ΛNv, (86)

that is Λ(θ) ∈ W , ∀ θ ∈ [0, h]. Sufficient stability condi-tions [114] are given by the existence of a matrix P ≻ 0and of matrices G1, G2 ∈ R

n×n solution to

ATP + PA+ λP +G1 +GT1 PBK −G1Λi +GT

2

KTBTP − ΛTi G

T1 +G2 −G2Λi − ΛT

i GT2

≺ 0,

for all i = 1, . . . , Nv. For a less conservative approachusing the Lyapunov-Razumikhin method, see [61]. A ro-bust analysis faced to perturbations has been proposedin [63].

4.3.5 Looped functionals

While convex embeddings allow for approximating thestability conditions (65) as accurately as desired, includ-ing parametric uncertainties in the analysis is quite dif-ficult. Recently, a discrete-time approach based on theso-called looped-functionals has been considered in [232]to deal with this issue. The key idea is to apply the func-tional framework, as in the time-delay approach, whilepreserving the accuracy of discrete-time stability con-ditions. The approach relies on an unusual formulationof sampled-data systems inspired from the lifting mod-elling [269]. For all k ∈ N, consider a lifted state functionχk(·) satisfying

χk+1(0) = χk(hk), ∀ k ∈ N, hk ∈ [h, h],

ddτ χk(τ) = Aχk(τ) +BKχk(0), ∀τ ∈ [0, hk].

(87)

The function χk(τ), τ ∈ [0, hk] represents the trajectoryof the sampled-data system over the interval [tk, tk+1].Define the set K =

⋃

h∈[h,h] C0 ([0, h],Rn) so to rep-

resent such a class of functions. The following theoremestablishes a relation between a discrete-time stabilityanalysis (as in (66)) and an analysis using functionals(as in the time-delay approach).

Theorem 21 [232] Consider system (12) and the liftedmodel (87). Let V : Rn → R+ be a differentiable functionfor which there exist positive scalars µ1 < µ2 such thatfor all x ∈ R

n, µ1 ‖x‖2≤ V (x) ≤ µ2 ‖x‖

2. The two

following statements are equivalent.

(i) The increment of the candidate Lyapunov functionV (·) is strictly negative at sampling instants, i.e.

∆V (k) = V (χk(hk)) − V (χk(0)) < 0,

for all k ∈ N and hk ∈ [h, h];(ii) There exists a continuous and differentiable func-

tional V : [0, h] × K → R which satisfies for all h ∈[h, h], z ∈ C0 ([0, h],Rn) ⊂ K

V(h, z(·)) = V(0, z(·)), (88)

and such that, for all (k, hk, τ) ∈ N× [h, h]× [0, hk],

W(τ, χk) =d

dτ[V (χk(τ)) + V(τ, χk)] < 0. (89)

Moreover, if one of these two statements is satisfied, thenthe null solution of system (12) is Asymptotically Stablefor any sampling sequence with tk+1 − tk ∈ [h, h].

The main difference compared with the Lyapunov-Krasovkii approach used in the time-delay frameworkremains in the design of the functional. The positive

20

definiteness condition of the LKF is exchanged with alooping-condition, a two-point algebraic equality (88)that the functional should verify; see e.g. [232,30–32,237]. There are two main methods for building loopfunctionals. The first manner follows the constructionof discontinuous Lyapunov-functionals, as for instance(48), (50) or the ones provided in [67,187,230,154]. Theprevious theorem first states that if a term of the func-tional meets the looping condition (88), then the posi-tive definiteness of this term can be relaxed. The secondmethod enters into the framework of polynomials func-tions [31,30,32,237] and of sum of squares tools [222].For example, in [237], a polynomial looped-functionalwas introduced and it has the following form

V(τ, χk) =

[

χk(τ)

χk(0)

]T

M(τ, hk)

[

χk(τ)

χk(0)

]

where M is a polynomial matrix function from [0, h] ×[h, h] to the set of 2n × 2n symmetric matrices. Thelooping condition (88) is ensured by adding the followingconstraints on M :

∀h ∈ [h, h],

[

I

I

]T

M(0, h)

[

I

I

]

= 0, and M(h, h) = 0.

This method can easily deal with polytopic uncertaintiesin the system matrices.

4.3.6 A discrete-time approach for nonlinear systems

Results on discrete-time approaches for the control ofnonlinear systems with time-varying sampling intervalsare quite rare.We present as follows an adaptation of theresult from [258] which extends earlier stability criteriafrom [191,196,190]. Consider the nonlinear system

x(t) = F (x(t), u(t)) (90)

with F (x, u) globally Lipschitz, i.e. there exists βf > 0such that

‖F (xa, ua)− F (xb, ub)‖ ≤ βf (‖xa − xb‖+ ‖ua − ub‖)

for all xa, xb ∈ Rn and ua, ub ∈ R

m. The control takesthe form u(t) = uk for all t ∈ [tk, tk+1) and the samplinginterval is bounded hk = tk+1−tk ∈ T = [h, h], ∀ k ∈ N.The exact discrete-time model of the system over thesampling interval is given by

xk+1 = F ehk(xk, uk) := xk +

∫ tk+hk

tk

F (x(s), uk) ds

(91)where xk = x(tk). Note however that (91) is not knownin general since it is rare to obtain an analytic solution

to a nonlinear initial value problem. In practical prob-lems, approximations are usually used [244,191]. A sim-ple example is given by the Euler model of (90):

xk+1 = xk + hkF (xk, uk) .

Other approximations can be found in standard books[244] and tutorials [182,183]. The approach in [258] con-siders an approximate model

xk+1 = F ah∗ (xk, uk) , (92)

obtained for some nominal sampling interval h∗ ∈ [h, h].Model (92) is assumed to be one-step consistent[244] with the exact discrete-time plant, i.e. there ex-ists ρ ∈ K∞ such that ‖F a

h∗ (x, u) − F eh∗ (x, u) ‖ ≤

h∗ρ(h∗) (‖x‖+ ‖u‖) , for all x ∈ Rn, u ∈ R

m. It is con-sidered that the approximate model (92) has been usedto design a controller

uk = Kh∗ (xk) (93)

parametrized by the nominal sampling interval h∗, andthat the closed-loop system (92),(93) is asymptoticallystable. More formally, it is assumed that there existsa candidate Lyapunov function for the approximateclosed-loop system (92),(93), i.e. a function Vh∗(x) andαi > 0, i = 1, 2, 3 such that the involved conditionsholds for some r > 1 : α1 ‖x‖

r≤ Vh∗(x) ≤ α2 ‖x‖

rand

Vh∗ (F ah∗ (x,Kh∗(xk)))− Vh∗(x)

h≤ −α3‖x‖

r (94)

for all x ∈ Rn. Furthermore, the control law Kh∗(·) is

considered to be linearly bounded, i.e. there exists βu >0 such that ‖Kh∗(x)‖ ≤ βu‖x‖ for all x ∈ R

n. Thefollowing theorem provides generic results for the robuststability of the exact closed-loop system

xk+1 = F ehk

(xk,Kh∗(xk)) , (95)

using the fact that the control law uk = Kh∗(xk) is astabilizer for the approximate model (92).

Theorem 22 [258] Consider system (95) with hk ∈

[h, h] for all k ∈ N. Consider the following notation

βa =(

2 + βu + (1 +max(1, βu))(eβfh − 1)

)

+h∗ρ(h∗)(1 + βu). (96)

Assume that the Lyapunov candidate function Vh∗(x) islocally Lipschitz and there exists βv > 0 such that

supz∈∂Vh∗ (x)

‖z‖ ≤ βv‖x‖r,

21

for all x ∈ Rn, where ∂Vh∗(x) denotes the generalized

differential of Clarke. If there exists β ∈ (0, 1) such that

βvβr−1a

h∗

(

h∗ρ(h∗)(1 + βu) + ρh(h∗,Mh)

)

≤ (1− β)α3

(97)is satisfied where

ρh(h∗,Mh) = eβfh

∗(

(1 + βu)(eβfMh − 1

) )

with Mh = maxh∈[h,h] |h− h∗|, then there exist c, λ > 0

such that ‖xk‖ ≤ c‖x0‖e−λkh. In other words, system

(95) is Globally Exponentially Stable, Uniformly for allhk ∈ [h, h] and all k ∈ N.

The above theorem is a natural extension of the result in[190,191] for sampled-data systems with constant sam-pling intervals. The main condition (97) involves twoterms. The first term βvβ

r−1a ρ(h∗)(1 + βu) reflects the

effect of approximatively discretizing the nominal sys-tem using a nominal sampling interval h∗; the second

one,βvβ

r−1a

h∗ ρh(h∗,Mh) reflects the effect of uncertainty

in the sampling interval.

4.3.7 Further reading

Control design methodologies based on convex em-beddings have been presented in [108,43,109,83,185].See also [226] for an LPV design of controllers thatare adapted in real time to the value of the samplinginterval and [110] for the case of systems with delayscheduled controllers. Extensions of the discrete-timeapproach for networked control systems with schedul-ing protocols can be found in [53,146,52,147,36]. Formodel predictive control of networked control systemssee also [201,89,157]. Lie algebraic criteria for the anal-ysis of systems with time varying sampling have beenproposed in [58]. A mixed continuous-discrete approachhas also been proposed in [148]. The relation betweenclock dependent and looped Lyapunov functionals hasbeen investigated in [28].

4.4 Input/Output stability approach

In this subsection we present several methods thatstudy sampled-data systems from a robust control pointof view. The main idea of the Input/Output stabilityapproach is to consider the sampling error as a per-turbation with respect to a nominal continuous-timecontrol-loop. Classical robust control tools are usedin order to assess the stability of the sampled-datasystems [272,278,168,217]. Some of the presented meth-ods are reminiscent from the Input/Output stabilityapproach used for the analysis of time delay systems[122,124,197,96,130,77,131], and have been further de-veloped independently of the time delay approach.

∆sh

y

G

e+

+

f

Fig. 6. Equivalent representation of the sampled-data sys-tem, from a robust control theory point of view.

4.4.1 Basic idea

Note that the LTI sampled-data system (12) can be re-expressed in the form [174]

x(t) =(A+BK︸ ︷︷ ︸

:=Acl

)x(t) + BK

︸︷︷︸

:=Bcl

(x(tk)− x(t)︸ ︷︷ ︸

:=e(t)

), (98)

whereAcl corresponds to the state matrix of the nominalcontinuous-time control loop while e(t) represents theerror induced by sampling. An essential fact in this ap-proach is that the sampling induced error e(t) = x(tk)−x(t) can be equivalently re-expressed as

e(t) = −

∫ t

tk

x(θ)dθ, ∀t ∈ [tk, tk+1). (99)

Considering y(t) = x(t) as an auxiliary output for sys-tem (98), the sampled-data system (12) can be repre-sented equivalently by the feedback interconnection ofthe operator ∆sh : Ln

2e[0,∞) → Ln2e[0,∞), ∆sh : y → e,

defined by:

e(t) = (∆sh y)(t) = −

∫ t

tk

y(θ)dθ, ∀t ∈ [tk, tk+1),

(100)with the system

x(t) = Aclx(t) +Bcle(t), x(0) = x0 ∈ R

n,

y(t) = Cclx(t) +Dcle(t) = x(t),(101)

where Ccl = Acl = A+BK and Dcl = Bcl = BK. Notethat the nominal system (101) is LTI. It represents thedynamics of the continuous-time systemwith an additiveinput perturbation e. The operator ∆sh captures boththe effects of sampling and its variations. An alternativemodel can also be derived by considering the actuationerror eu(t) = K(x(tk) − x(t)) (see [81]). The stabilityof the sampled-data system (12) can then be studied byanalysing the interconnection (100),(101).

4.4.2 Small gain conditions

To provide constructive stability conditions, the SmallGain Theorem [272,278,122,96] constitutes a simple and

22

powerful tool in the robust control framework. Let G :Ln2 [0,∞) → Ln

2 [0,∞) be the linear operator describedby the transfer function

G(s) = s(sI −Acl)−1Bcl (102)

associated to system (101). The operator G cap-tures the behaviour of (101) for null initial condi-tions. Considering the free response of system (101),f(t) = Acle

Acltx0, ∀ t ≥ 0, the interconnection(100),(101) can be re-expressed as

y = Ge+ f

e = ∆shy(103)

(see Figure 6). A direct consequence of the Small GainTheorem is the fact that if

‖G‖2,2‖∆sh‖2,2 < 1, (104)

then the interconnection (103) is L2 stable, i.e. thereexist a positive scalar C such that

∫ t

0

(

‖y(θ)‖2+ ‖e(θ)‖

2)

dθ ≤ C

∫ t

0

‖f(θ)‖2dθ (105)

for any t > 0. Here ‖G‖2,2, ‖∆sh‖2,2 denote the inducedL2 norms ofG and ∆sh, respectively

4 . Inequality (104)is known as the small gain condition. Due to the linearityofG, its induced L2 norm can be readily computed [278]using the H∞ norm of its transfer function:

‖G‖2,2 = ‖G‖∞ := supω∈R

σ(

G(jω))

.

Furthermore, for the case of LTI sampled-data systems,L2 stability of the interconnection (103) implies asymp-totic stability 5 of the sampled-data control loop (12):

Theorem 23 [81] Suppose that Acl is Hurwitz. System(12) is Uniformly Asymptotically Stable if the feedbackinterconnection (103) is L2 stable.

Therefore, providing tractable stability conditions forsystem (12) leads to providing an estimate for the in-duced L2 norm of the operator ∆sh. An upper boundof this norm has been computed in [130] using a moregeneral uncertain delay operator:

∆d : y(t) → e(t) = (∆dy)(t) := −

∫ t

t−τ(t)

y(θ)dθ,

(106)

4 Given an operator G : Ln2 [0,∞) → Ln

2 [0,∞), its induced

L2 norm is defined as ‖G‖2,2 := supu 6=0

‖Gu‖L2

‖u‖L2

.5 For relations with exponential stability see also [68].

where τ(t) ∈ [0, h]. The operator ∆sh is a particular caseof ∆d with τ(t) = t− tk, ∀t ≥ 0, k ∈ N.

Lemma 24 [130] The L2-induced norm of the operator∆d in (106) is bounded by h.

Using this property, and the fact that the operator ∆d

satisfies M∆d = ∆dM for all M ∈ Rn×n, Mirkin [174]

provided the following L2 stability conditions

∃M ∈ Rn×n, M ≻ 0 such that ‖MG(s)M−1‖∞ <

1

h,

(107)which is a consequence of the Scaled Small Gain The-orem [242]. Interestingly, it is also shown that (107) isrelated to the condition in [72] which is obtained usingthe input-delay approach and the Lyapunov-Krasovskiifunctional (23). The same LMI can be used to checkboth conditions. Mirkin then showed that the bound onthe L2 induced norm can be enhanced by exploiting theproperties of ∆sh.

Lemma 25 [174]The L2-induced norm of the operator

∆sh is bounded by δ0 = 2πh, and thus

∫ +∞

0

‖(∆shy)(θ)‖2dθ ≤

∫ +∞

0

δ20‖y(θ)‖2dθ, (108)

for all y ∈ Ln2 [0,∞).

This bound on the induced L2 norm of ∆sh is actuallyexact and it is attained when there exists an index k ∈ N

such that tk+1 − tk = h. This leads to the followingsufficient L2 stability condition, improving (107):

∃M ∈ Rn×n, M ≻ 0 such that ‖MG(s)M−1‖∞ <

π

2h.

(109)Note that the upper bound on induced L2 norm of ∆shcan also be related to the Wirtinger’s inequalities [154]used in the time delay approach. In practice, condition(109) is readily verifiable via standard LMI for the esti-mation of the H∞ norm of LTI systems [174,242,96]

XAcl + ATclX

2πhXBK AT

clY

∗ −Y 2πhKTBTY

∗ ∗ −Y

≺ 0 (110)

to be solved for X,Y ≻ 0 (obtained with Y = M2).

Recently, an extension of the Input-Output stability ap-proach has been proposed [40] for nonlinear systems

x = f(x) + g(x)u,

y = H(x),(111)

23

with a sampled-data output feedback

u(t) = K (y(tk)) , ∀t ∈ [tk, tk+1). (112)

Here f(0) = 0 and all the functions are supposed tobe continuously differentiable. The closed-loop system(111), (112) can be re-expressed as

x(t) =(f(x) + g(x)K (H(x))︸ ︷︷ ︸

:=fcl(x)

)

−g(x)(K (y(t))−K (y(tk))︸ ︷︷ ︸

:=eu(t)

), (113)

where fcl(x) represents the dynamics of a nominalclosed-loop system and eu the error of sampling at con-troller level. This model can be related with (60) usedin the impulsive approach by [193]. The sampling errorsatisfies

eu(t) =

∫ t

tk

z(θ)dθ, ∀t ∈ [tk, tk+1), (114)

where z(t) = dK (y(t)) /dt. Stability conditions can bederived based on Input-to-State stability and small gainanalysis.

Theorem 26 [40] Consider system (111) under the con-trol law (112). Assume that tk+1 − tk ∈ [0, h]. Supposethat the following auxiliary system

x(t) = fcl (x(t)) − g (x(t)) eu(t)

z(t) = dK(H(x(t)))dt

(115)

with input eu and output z has the following Input-to-State and Input-to-Output Stability properties

‖x(t)‖ ≤maxβx (‖x(τ0)‖, t− τ0) , γx(‖eu[τ0,t]‖

),(116)

‖z(t)‖≤maxβ (‖x(τ0)‖, t− τ0) , γ(‖eu[τ0,t]‖

), (117)

∀t ≥ τ0 ≥ 0, where ‖eu[τ0,t]‖ := supt∈[τ0,t] ‖eu(t)‖, β, βx

are class KL functions and γ, γx are class K functions.If γ(s) < s/h for all s > 0, then the equilibrium pointx = 0 of the closed-loop system (111), (112) is GloballyAsymptotically Stable for any initial condition in R

n.

4.4.3 Integral Quadratic Constraints and extensions

For the case of LTI sampled-data systems (12), the prop-erties of the operator ∆sh in (100) can be further ex-ploited in the framework of Integral Quadratic Con-straints (IQC) [168]. Less conservative stability condi-tions can be obtained. While very general definitionsof IQCs are available in the literature [168], we restrictourselves here to IQCs defined by symmetric matrices

Π with real elements, that have been used for stabilityanalysis of systems with aperiodic sampling. Roughlyspeaking, the bounded operator ∆sh in (100), with in-put y and output e, is said to satisfy the IQC defined bythe symmetric matrix Π if

∫ ∞

0

[

y(θ)

e(θ)

]T

Π

[

y(θ)

e(θ)

]

dθ ≥ 0 (118)

for all y ∈ Ln2 [0,∞) and e = ∆shy.We present as follows

a simplified version of the classical IQC Theorem [168]that can be used in order to derive stability conditionsfor the interconnection (103).

Theorem 27 [168] Consider the interconnection (103)describing the LTI sampled-data system (12) and thebounded operator ∆sh in (100). Suppose that Acl = A+BK is Hurwitz and assume that there exists a matrix

Π =

[

Π11 Π12

ΠT12 Π22

]

(119)

with Π11,Π12,Π22 ∈ Rn×n, Π11 0, Π22 0, such that

the operator ∆sh satisfies the IQC defined by Π; thereexists ǫ > 0 such that

[

G(jω)

I

]⋆

Π

[

G(jω)

I

]

−ǫI, ∀ ω ∈ R. (120)

Then the interconnection (103) is L2 stable.

Using Theorem 23, the conditions of Theorem 27 alsoimply uniform asymptotic stability of the sampled-datasystem (12). Condition (120) can be converted into afrequency independent finite dimensional LMI using theKalman-Yakubovich-Popov Lemma [223]:

ATclP + PAcl PBcl

BTclP 0

+

Ccl Dcl

0 I

T

Π

Ccl Dcl

0 I

≺ 0

(121)to be solved for P ≻ 0.

As an example, a simple IQC can be obtained directlyfrom Lemma 25. Note that inequality (108) implies that∆sh satisfies the IQC defined by

Π =

(

2hπ

)2

I 0

0 −I

. (122)

For this IQC, condition (120) yields to the standardsmall gain criteria

(2h

π

)2

G⋆(jω)G(jω) ≺ I, ∀ ω ∈ R, (123)

24

which corresponds to a simple condition on theH∞ normof G: ‖G(s)‖∞ < π

2h.

Fujioka [81] showed that the operator ∆sh also satisfiesthe following passivity-like property.

Lemma 28 [81] The operator ∆sh defined in (100) sat-isfies

∫ +∞

0

yT (θ)(∆shy)(θ)dθ ≤ 0, (124)

for all y ∈ Ln2 [0,∞).

It is important to note that if ∆sh satisfies several IQCdefined by matrices Π1,Π2, . . . ,Πr, then a sufficientcondition for stability that takes into account all theproperties is given by the existence of positive scalarsα1, α2, . . . , αr such that condition (120) holds withΠ = α1Π2 +α2Π2 + . . . , αrΠr. The properties of ∆sh inLemma 25 and Lemma 28 can be generalized [81] usingscaling matrices 0 Y ∈ R

n×n, 0 ≺ X ∈ Rn×n and

grouped into the following IQC:

∫ ∞

0

[

y(θ)

e(θ)

]T [

δ20X −Y

−Y −X

][

y(θ)

e(θ)

]

dθ ≥ 0 (125)

which holds for all y ∈ Ln2 [0,∞) and e = ∆shy with

δ0 = 2hπ . Using the integral property (125) and Theo-

rem 23, Fujioka [81] has proposed the following stabilitycondition.

Theorem 29 [81] The system (12) is Globally Uni-formly Asymptotically Stable for any sampling sequencewith tk+1 − tk ≤ h if there exist 0 ≺ P ∈ R

n×n,0 ≺ X ∈ R

n×n, 0 Y ∈ Rn×n satisfying

ATclP + PAcl PBcl

BTclP 0

+

Ccl Dcl

0 I

T

δ20X −Y

−Y −X

Ccl Dcl

0 I

≺ 0. (126)

Taking into account more properties of the operator ∆sh

may lead to less conservative results. Nevertheless, sincethe analysis is of a frequency domain nature, the IQC ap-proach is only applicable to LTI systems. However, onemay note that input delays, several performance specifi-cations and classical nonlinearities (sector bounded, sat-urations, etc.) can be characterized by elementary op-erators and IQCs [168]. A more complex system can bedescribed by an interconnection of an LTI system anda single block diagonal operator representing the differ-ent perturbing elements. Once the IQCs for the differ-ent perturbing elements are available, stability of morecomplex systems is then a rather straightforwardmatter

of defining a single aggregate IQC. This point enhancesthe applicability of the IQC approach.

In the nonlinear case, an extension [206,207,203] of theIQC approach is possible using methods inspired by theDissipativity Theory [267,262]. Consider the followingnonlinear affine system:

x(t) = f(x(t)

)+g(x(t)

)K(x(tk)

), ∀t ∈ [tk, tk+1), k ∈ N.

(127)The functions f : Rn → R

n with f(0) = 0, and g : Rn →R

n×m are considered to be sufficiently smooth and thecontroller K : Rn → R

m is a continuously differentiablefunction. Considering fcl(x) = f(x)+ g(x)K(x), w(t) =K(x(tk)

)−K

(x(t)

)and an auxiliary output y = ∂K

∂x x,system (127) can be represented by

x = fcl (x) + g (x)w

y = ∂K∂x (fcl (x) + g (x)w)

w = ∆shy.

(128)

Here we consider that ∆sh : Lm2e[0,∞) → Lm

2e[0,∞),with ∆sh defined similarly to (100). The main idea in[206,207,203] is to re-interpret the properties of the op-erator ∆sh in terms of ”supply” functions S

(y, w

)such

that

∫ t

tk

S(y(θ), w(θ)

)dθ ≤ 0, ∀t ∈ [tk, tk+1). (129)

The following result provides an extension of Theorem29to the nonlinear affine case.

Theorem 30 [206] Consider the sampled-data system(127), and the equivalent representation (128), with ∆sh

as given in (100). Consider the quadratic form

S(y, w

)=

[

y

w

]T [

−δ20X Y

Y X

][

y

w

]

, (130)

with δ0 = 2πh, 0 ≺ X ∈ R

m×m, and 0 Y ∈ Rm×m.

Consider a neighbourhood D ⊂ Rn of the equilibrium

point x = 0, and suppose that there exist a differentiablepositive definite function V : D → R

+, a scalar α > 0,and class K functions β1 and β2, such that

β1(‖x‖) ≤ V (x) ≤ β2(‖x‖), ∀x ∈ D, (131)

and the following inequality is satisfied:

∂V

∂x

(fcl(x) + gcl(x)w

)+ αV

(x)≤ S

(y, w

)e−αθ, (132)

for any θ ∈ 0, h, x ∈ D, w ∈ Rm. Then, the equilibrium

x = 0 of system (127) is Locally Uniformly Asymptoti-cally Stable for any sampling sequence with tk+1−tk ≤ h.

25

For particular classes of systems, such as LPV, bilinearor more general polynomial sampled-data systems, theconditions of Theorem 30 can be re-written as tractablenumerical criteria (LMIs or Sum-Of-Squares decompo-sition) [206,207,203,202]. These conditions can be en-hanced by giving more insight into the mathematicalmodel of the sampling operator ∆sh. This would leadto new characterizations of supply functions used inthe dissipativity-based approach. However, finding newproperties for the operator∆sh or a better way to rewritethe sampled-data system as an interconnected systemhas been proven to be difficult, and research is still inprogress.

4.4.4 Further reading

Some of the elements presented in Section 4.3 concern-ing the use of norm-bounded approximations of the ma-trix exponential [79] can also be interpreted in the In-put/Output approach as the application of the SmallGain Theorem to a discrete-time model. Other IQCscan be found in [80,82]. An approach based on IQCsfor the discrete-time model has been proposed recentlyin [129]. For more general nonlinear networked systems,approaches considering sampling as a perturbations canbe found in [263,193]. The boundedness properties of thesampling operator ∆sh from Lemma 25 from [174] canbe related with the Wirtinger’s inequalities used in thetime delay approach [154,233,234].Motivated by the ap-proach presented in [67] in the input delay framework,the sampling effect has been recently described by a newoperator in [128].

5 Sampling as a control parameter - an emerg-ing area

In this section we briefly present the main research di-rections and some problems concerning the case whenthe sampling interval hk (or equivalently the sequence ofsampling σ = tkk∈N) is considered to be a control pa-rameter that can be modified in order to ensure desiredproperties in terms of stability and resource utilization.From the real-time control point of view, this formula-tion corresponds to designing a scheduling mechanismthat triggers the sampler [12,259]. The problem has at-tracted sporadically the attention of the control system’scommunity since the early ages of sampled-data control[125,55].With the spring of event- and self-triggered con-trol techniques [10,13,259] it has become a very populartopic [103,177].

Let us consider the nonlinear system (2) and the con-troller (3) with a given sampling sequence σ = tkk∈N.Clearly, the asymptotic stability of system holds whenthe sampling sequence σ satisfies hk = tk+1 − tk ∈ (0, h]for all k ∈ N, where h represents the MSI for whichthe system is asymptotically stable under arbitrary sam-pling. A basic problem in designing a sampling sequence

σ = tkk∈N is to ensure the stability of the system whileoptimizing some Performance Index associated to thefrequency of sampling. Most of the time, sampling se-quences are compared in simulation based on the meansampling interval. Given σ, one possible choice of Per-formance Index to be maximized could be

J (σ) = lim infN→∞

1

N

N−1∑

k=0

(tk+1 − tk). (133)

Note that the limit inferior of the sequence is neededsince, given a nonlinear system (2),(3), it is not obviousthat over an infinite time horizon the mean value intervalconverges. Generally, the goal is to find sequences thatensure stability and have the mean sampling intervallarger then the maximum sampling interval admissiblein the periodic and arbitrary varying case. Using thePerformance Index (133), the following basic problemcan be formulated:

• Problem B (Optimal sampling sequence): Considerthe nonlinear system (2) and the controller (3). Designa sampling sequence σ maximizing the PerformanceIndex J (σ) in (133) while ensuring the stability of theclosed-loop system (1),(2),(3),(4).

Of course, the problem makes sense only for systemswhere the control action is necessary. It is not meaning-ful for open-loop stable systems, where the sampling in-terval can be made arbitrarily large. Various alternativeformalizations of Problem B can be imagined by consid-ering other performance indexes or Cost Functions (e.g.Jc(σ) =

∑∞k=0 e

−(tk+1−tk)) to be maximized or mini-mized (see for instance [117,160] for a finite horizon for-mulation). A stochastic formulation of the problem canbe found in [44,180]. It is possible to formulate morecomplex problems in which one needs to find simultane-ously the sampling sequence and system input, as in theminimum attention control formulation [33,54,162], orto optimise not only the sampling cost but also a moreclassical performance index (LQR, LQG, L2-gain, etc.)[14,104,8,93].

While the research in the case of arbitrary sampling hasreached an advanced phase of development, Problem Bis largely open. Due to the complexity of Problem B,simplified versions are under study. For example, sta-bility of sampled-data systems over periodic sequencesof sampling has been investigated in [125,146,232]. Theoptimization of sampling sequences over a finite horizonhas been considered since the early works in [117,160].For both practical and theoretical reasons, the designof state-dependent (closed-loop) sampling sequences, inwhich the sampling is triggered according to the systemstate, represents a topic of interest. Basic ideas appearedin the ’60s in the context of adaptive sampling [55,47]and the topic is currently under study in the frameworkof event-/self-triggered control [10,13,259,12,103,177].

26

5.1 Event-Triggered (ET) Control

The basic idea of event-triggered control schemes[10,13,14,103,177] is to continuously monitor the systemstate and to trigger the sampling only when neces-sary, according to the desired performance of the sys-tem. A sampling event is generated when the system’sstate crosses some frontier in the state-space. Let usre-consider the hybrid model of an LTI sampled-datasystem

x = Ax+BKx

˙x = 0

τ = 1

(

x, x, τ)

∈ C,

x+ = x

x+ = x

τ+ = 0

(

x, x, τ)

∈ D,

(134)

where x represents the sampled version of the state andτ the clock measuring the time since the last samplinginstant. In the classical time-triggered sampling context(53), the sets C and D implicitly indicating the sam-pling moments are defined only according to the clockvariable τ : when uniform sampling with period T is con-sidered, C is defined by τ ∈ [0, T ] and D by τ = T . Inevent-triggered control the idea is to define the samplingtriggering sets according to the state variable x and x.For example, it may be of interest to trigger only whenthe error x − x becomes too large with respect to thesystem state, i.e. when ‖x(t) − x(tk)‖ ≥ γ‖x(t)‖ whereγ > 0 is a design parameter (see [250]). For this examplethe sets C and D are:

C = (x, x, τ) ∈ Rn × R

n × R : ‖x− x‖ ≤ γ‖x‖ ,

D= (x, x, τ) ∈ Rn × R

n × R : ‖x− x‖ ≥ γ‖x‖ .

Various other types of triggering conditions have beenproposed in the literature: send-on-delta (Lesbeguesampling, absolute triggering) [14,209,37], send-on-energy [175], send-on-area [176], Lyapunov sampling[260,238,63,221], etc.

Note that in event-triggering control, the sampling se-quence σ = tkk∈N is implicitly defined as:

tk+1 = min t : t ≥ tk, (x, x, τ) ∈ D . (135)

The value h∗ for which tk+1 − tk ≥ h∗ for all k ∈ N

and all initial conditions is called the minimum inter-event time. In the general case the implicit definitionof the sampling sequence does not guarantee anythingabout the ”well posedness” of the closed-loop systemin terms of existence of solutions, or concerning the ex-istence of a minimum interval between two consecu-tive events. In particular cases of event-triggered con-

trol Zeno phenomena may occur, i.e. the minimum inter-event time h∗ is zero 6 [162,51,25]. This represents animportant drawback since the system is converging toa continuous-time control implementation instead of asampled-data one. To avoid it, various systematic de-sign methodologies for event-triggered control with sta-bility guarantees and no Zeno behavior have been pro-posed: see [250,266,264,159] based on the Input/Outputstability approach, [51,238,65,221] using hybrid models,[271,215,63] based on the time-delay approach. Note thatZeno phenomena can be easily avoided by including re-strictions on the clock variable when defining the jumpsetD. For example, one may add next to the constraintson x and x, a constraint that guarantees that samplingoccurs only if τ is greater than some minimum desiredinter-execution time [65,63,221]. Additionally, the trig-gering condition may be verified on a discrete sequenceof time, as in the Periodic Event-Trigger (PET) control[101,218,102], or in [57], where the event-triggered con-trol problem is formulated directly in discrete-time.

5.2 Self-Triggered (ST) Control

The term self-triggered control was initially proposed by[259] in the context of real time systems. The recent ar-ticles [264,6] have attracted the attention of the controlsystem community. Note that basic ideas related to self-triggered control appeared in the ’60s (see [55,117,47]and the references therein). We point also to the pio-neering work in [118] where elements concerning the useof Lyapunov arguments for the design of self-triggeringcontrol laws can be found.

In self-triggering, at each sampling time it is computedboth the sampled-data control value (to be sent to theactuators) and the next sampling instant. The main ideais to use the value of the state at sampling times andknowledge about the system dynamics in order to pre-dict the next time instant a control update is needed. Aself-triggering control scheme is described by a samplingfunction h : Rn → R

+ \ 0 which, at each samplingtime tk, k ∈ N, indicates the value of the current sam-pling interval according to the system state. The sam-pling sequence σ = tkk∈N is formulated explicitly as

tk+1 = tk + h (xk) , (136)

where xk = x(tk). Very often, the synthesis of a self-triggered control scheme is based on a pre-existing event-triggered control mechanism. In this context, it is aimedat designing the sampling function by pre-computing, ateach sampling instant, an estimation of the next timea sampling event has to be generated. For the exampleof the LTI system (12) with the event-triggered controlcondition ‖x(t) − x(tk)‖ ≥ γ‖x(t)‖, one may want to

6 the system requires infinitely fast sampling

27

design the sampling function:

h(xk) = sup θ > 0 : ‖(Λ(s)− I)xk‖ < γ‖Λ(s)xk‖,

∀ s ≤ θ , (137)

where Λ(θ) = eAθ +∫ θ

0eAsdsBK. An important issue

is the complexity of the algorithms used for the onlineimplementation of the sampling function h(x). Even forthe simple case (137), the algorithms may be quite com-plex since they involve solving hyperbolic inequalities. Inpractice, simple approximations of such sampling func-tion must be used.

Self-triggered control mechanisms with stability guaran-tees have been proposed in [264,265,6,64,23] using theInput/Output stability approach, in [166,161,255] usingdiscrete-time Lyapunov functions, in [61,63] using con-vex embeddings, in [219] using a hybrid formulation andin [62] using a time-delay system approach.

5.3 Relations with the arbitrary sampling problem

A basic problem in both event- and self-triggered con-trol is to design the trigger (or the sampling map h(xk))so as to enlarge the minimum inter-event time h∗ whileguaranteeing the stability of the system. Providing aquantitative estimation of the minimum inter-eventtime h∗ guarantees the existence of a sub-optimal solu-tion to Problem B with a performance index J ≥ h∗.Recently, connections between the arbitrary samplingproblem (Problem A) and Problem B have been madein [60,61,63,221,252,50]. It has been shown that, forsome Lyapunov-based triggering conditions, the mini-mum inter-event time h∗ corresponds to the MaximumSampling Interval h admissible in the arbitrary sam-pling configuration. This issue is interesting since trig-gering control schemes could be constructed by upper-bounding the derivatives of Lyapunov functions, as theones used for solving Problem A. See, for instance, theresults in [63,62] where triggering mechanisms are op-timized off-line using LMI criteria so as to enlarge theminimum inter-event time. However, the potential ofthe approaches used for the arbitrary sampling problemis far from being fully exploited. The tools presentedin Section 4 may be useful for various aspects in Prob-lem B: deriving new event-/self-triggering mechanisms,providing less conservative estimations of the minimuminter-event time h∗, etc.

6 Conclusion

This article has presented basic concepts and recent re-search directions in sampled-data systems: time-delay,hybrid, discrete-time and input-output models; Lya-punov and frequency domain methods for the stabilityof systems with arbitrary sampling intervals; converse

Lyapunov theorems and constructive numerical crite-ria. It is to be emphasized that this overview is far frombeing exhaustive. The research topic of systems withtime-varying sampling is still wide open and continu-ously growing. In particular, the control of sampling ispresently receiving a lot of attention, as it was shownin Section 5. It is worth noticing that the subject liesat the intersection of four important axes in ControlTheory (time-delay, hybrid, LPV and input-outputapproaches) and we hope this will have a stimulatingimpact in the control community. Methods and toolsare being transferred from one approach to another andthe perspectives of cross-fertilisation and generalizationare numerous.

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