# Lyapunov Methods for Time-Invariant Delay Difference Inclusions.

**ABSTRACT** Motivated by the fact that Delay Difference Inclusions (DDIs) form a rich modeling class that includes, for example, uncertain time-delay systems and certain types of networked control systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. First, the Lyapunov-Krasovskii approach, which is an extension of the classical Lyapunov theory to time-delay systems, is considered. It is shown that a DDI is Kℒ-stable if and only if it admits a Lyapunov-Krasovskii Function (LKF). Second, the Lyapunov-Razumikhin method, which is a type of small-gain approach for time-delay systems, is studied. It is proved that a DDI is Kℒ-stable if it admits a Lyapunov-Razumikhin Function (LRF). Moreover, an example of a linear delay difference equation which is globally exponentially stable but does not admit an LRF is provided. Thus, it is established that the existence of an LRF is not a necessary condition for Kℒ-stability of a DDI. Then, it is shown that the existence of an LRF is a sufficient condition for the existence of an LKF and that only under certain additional assumptions is the converse true. Furthermore, it is shown that an LRF induces a family of sets with certain contraction properties that are particular to time-delay systems. On the other hand, an LKF is shown to induce a type of contractive set similar to those induced by a classical Lyapunov function. The class of quadratic candidate functions is used to illustrate the results derived in this paper in terms of both LKFs and LRFs, respectively. Both stability analysis and stabilizing controller synthesis methods for linear DDIs are proposed.

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**ABSTRACT:**Input-to-state stability (ISS) of interconnected systems with each subsystem described by a difference equation subject to an external disturbance is considered. Furthermore, special attention is given to time delay, which gives rise to two relevant problems: (i) ISS of interconnected systems with interconnection delays, which arise in the paths connecting the subsystems, and (ii) ISS of interconnected systems with local delays, which arise in the dynamics of the subsystems. The fact that a difference equation with delay is equivalent to an interconnected system without delay is the crux of the proposed framework. Based on this fact and small-gain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds. Furthermore, also using small-gain arguments, ISS for interconnected systems with local delays is established via the Razumikhin method as well as the Krasovskii approach. A combination of the results for interconnected systems with interconnection delays and local delays, respectively, provides a framework for ISS analysis of general interconnected systems with delay. Thus, a scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay.Mathematics of Control Signals and Systems 04/2012; 24. · 0.42 Impact Factor - SourceAvailable from: tue.nl[Show abstract] [Hide abstract]

**ABSTRACT:**This paper proposes a novel approach to stability analysis of discrete-time nonlinear periodi-cally time-varying systems. The contributions are as follows. Firstly, a relaxation of standard Lyapunov conditions is derived. This leads to a less conservative Lyapunov function that is required to decrease at every period rather than at each time instant. Secondly, for linear periodic systems with constraints, we show that compared to standard Lyapunov theory, the novel stability concept yields a larger estimate of the region of attraction. An example illustrates the effectiveness of the developed theory.Automatica 10/2012; · 2.92 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**For the stability analysis of time-delay systems, the Razumikhin approach provides (at the cost of some conservatism) a set of conditions that are relatively easy to verify when compared to the Krasovskii approach. Unfortunately, currently, for linear delay difference inclusions (DDIs) verification of these conditions is only possible by solving a bilinear matrix inequality (BMI). To obtain a tractable stability analysis method for DDIs, an alternative set of Razumikhin-type conditions is proposed in this paper, which are based on a technique that was developed for interconnected systems in Willems (1972). In particular, via the proper selection of storage and supply functions, these conditions can be used to establish input-to-state stability (ISS) for general DDIs. When linear DDIs and quadratic functions are considered, ℓ2ℓ2-disturbance attenuation can be established by solving a single linear matrix inequality (LMI). Moreover, this LMI is shown to be less conservative than the BMI corresponding to the existing Razumikhin-type conditions for linear DDIs.Automatica 02/2013; 49(2):619–625. · 2.92 Impact Factor

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LYAPUNOV METHODS FOR TIME-INVARIANT

DELAY DIFFERENCE INCLUSIONS∗

R.H. GIELEN†, M. LAZAR†, AND I.V. KOLMANOVSKY‡

Key words. Stability theory, Lyapunov functions and stability, Time-delay, Invariant sets

AMS subject classifications. 39A30, 37B25, 37L25

Abstract. Motivated by the fact that delay difference inclusions (DDIs) form a rich modeling

class that includes, for example, uncertain time-delay systems and certain types of networked control

systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. Firstly, the

Lyapunov-Krasovskii approach, which is an extension of the classical Lyapunov theory to time-delay

systems, is considered. It is shown that a DDI is KL-stable if and only if it admits a Lyapunov-

Krasovskii function (LKF). Secondly, the Lyapunov-Razumikhin method, which is a type of small-

gain approach for time-delay systems, is studied. It is proven that a DDI is KL-stable if it admits a

Lyapunov-Razumikhin function (LRF). Moreover, an example of a linear delay difference equation

which is globally exponentially stable but does not admit a LRF is provided. Thus, it is established

that the existence of a LRF is not a necessary condition for KL-stability of a DDI. Then, it is shown

that the existence of a LRF is a sufficient condition for the existence of a LKF and that only under

certain additional assumptions the converse is true. Furthermore, it is shown that a LRF induces

a family of sets with certain contraction properties that are particular to time-delay systems. On

the other hand, a LKF is shown to induce a type of contractive set similar to those induced by

a classical Lyapunov function. The class of quadratic candidate functions is used to illustrate the

results derived in this paper in terms of both LKFs and LRFs, respectively. Both stability analysis

and stabilizing controller synthesis methods for linear DDIs are proposed.

1. Introduction. Systems affected by time-delay can be found within many

applications in the control field, see, e.g., [25] for an extensive list of examples. Delay

difference inclusions (DDIs) form a rich modeling class that includes, for example,

uncertain systems, time-delay systems and certain types of networked control systems

[14, 45]. However, while stability analysis of delay-free systems is often based on the

existence of a Lyapunov function (LF), see, e.g., [2], for systems affected by delays the

classical Lyapunov theory does not apply straightforwardly. This is due to the fact

that the influence of the delayed states can cause a violation of the monotonic decrease

condition that a standard LF obeys.To solve this issue, two types of functions

were proposed: the Lyapunov-Krasovskii function (LKF) [27], which is an extension

of the classical LF to time-delay systems, and the Lyapunov-Razumikhin function

(LRF) see, e.g., [16], which is a function that is constructed based [40] on a type

of small-gain condition for time-delay systems. The main focus of this paper is on

discrete-time systems. Therefore, for continuous-time systems, we only give a brief

account of some Lyapunov theorems and refer to [15, 16, 25, 27, 36] and the references

therein for further reading. Theorem 4.1.3 in [25] establishes that a time-delay system

is globally exponentially stable (GES) if and only if it admits a LKF. Furthermore,

Theorem 5.19 in [15] establishes that any linear delay differential equation that is GES

admits a quadratic LKF. This result was partially extended to linear delay differential

inclusions in [24]. However, for LRFs such converse results are missing. Moreover,

LRFs can be considered [25] as particular cases of LKFs. Also, it is known [23] that

∗This paper was partially presented at the 2010 American Control Conference, Baltimore, MA.

†Department of Electrical Engineering, Eindhoven University of Technology, The Netherlands,

r.h.gielen@tue.nl, m.lazar@tue.nl.

‡Department of Aerospace Engineering, University of Michigan, Ann Arbor, Michigan, United

States of America, ilya@umich.edu.

1

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R.H. Gielen, M. Lazar and I.V. Kolmanovsky

any quadratic LRF yields a particular quadratic LKF.

It is not immediately clear how the Lyapunov-Razumikhin method and Lyapunov-

Krasovskii approach are to be used for stability analysis of discrete-time systems. One

of the most commonly used approaches [1] to stability analysis of DDIs is to augment

the state vector with all delayed states/inputs that affect the current state, which

yields a standard difference inclusion of higher dimension. Thus, stability analysis

methods for difference inclusions based on Lyapunov theory, see, e.g., [2, 21], become

applicable. Recently, in [17] it was pointed out that such a LF for the augmented

state system provides a LKF for the original system affected by delay. Moreover, in

[17] it was also shown that all existing methods based on the Lyapunov-Krasovskii

approach provide a particular type of LF for the augmented state system. As such, an

equivalent notion of LKFs for discrete-time systems was obtained. Examples of con-

troller synthesis methods based on this approach can be found in, among many others,

[7, 8, 12, 26, 43]. However, converse results for the Lyapunov-Krasovskii approach,

such as the ones mentioned above for continuous-time systems, are missing. For LRFs

the situation is more complicated. The exact translation of this approach to discrete-

time systems yields a non-causal constraint [11, 44]. An alternative, Razumikhin-like

condition for discrete-time systems was proposed in [33], where the LRF was required

to be less than the maximum over its past values for the delayed states. Stability

analysis and controller synthesis methods based on the existence of a LRF can be

found in, e.g., [13, 32, 34]. For discrete-time systems, a result on the connection be-

tween LKFs and LRFs is missing. Moreover, for both continuous and discrete-time

systems, it remains an open question whether there exist systems that are KL-stable

or even globally exponentially stable (GES) but do not admit a LRF.

Given that DDIs form a rich and relevant modeling class (that was recently shown

to include networked control systems) while an overview of the corresponding counter-

part of the Lyapunov methods for delay differential inclusions is missing, the purpose

of this paper is to provide a comprehensive collection of Lyapunov methods for DDIs.

To this end, firstly, using the augmented state system, a converse Lyapunov theorem

for the Lyapunov-Krasovskii approach is established. Secondly, for the Lyapunov-

Razumikhin method, the results of [11] and [33] are extended to delay difference

inclusions. Thirdly, via an example of a linear delay difference equation that is GES

but does not admit a LRF, it is shown that the existence of a LRF is a sufficient con-

dition but not a necessary condition for KL-stability of DDIs. Then, it is established

that the existence of a LRF is a sufficient condition for the existence of a LKF and

that only under certain additional assumptions the converse is true. Furthermore, it

is shown that a LRF induces a family of sets with certain contraction properties that

are particular to time-delay systems. On the other hand, a LKF is shown to induce

a standard contractive set for the augmented state system, similar to the contractive

set induced by a classical LF. The class of quadratic candidate functions is used to

illustrate the application of the results derived in this paper to both stability analysis

and stabilizing controller synthesis for linear polytopic DDIs in terms of LKFs as well

as LRFs.

The remainder of the paper is organized as follows. Section 2 contains some use-

ful preliminaries. Section 3.1 and Section 3.2 present stability conditions in terms of

LKFs and LRFs, respectively. In Section 4 relations between LKFs and LRFs are

discussed. Next, Section 5 deals with contractive sets for DDIs. In Section 6 synthe-

sis techniques are provided for quadratic LKFs and LRFs, respectively. Moreover, in

Section 6.1 these techniques are illustrated via an example. Conclusions are drawn in

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Lyapunov methods for time-invariant delay difference inclusions

3

Section 7 and the two appendices contain the proof of a technical lemma and some

numerical data, respectively.

2. Preliminaries.

2.1. Notation and basic definitions. Let R, R+, Z and Z+denote the field

of real numbers, the set of non-negative reals, the set of integers and the set of non-

negative integers, respectively. For every c ∈ R and Π ⊆ R, define Π≥c:= {k ∈ Π |

k ≥ c} and similarly Π≤c. Furthermore, RΠ:= Π and ZΠ:= Z ∩ Π. For a vector x ∈

Rn, let [x]i, i ∈ Z[1,n]denote the i-th component of x and let ?x?p:= (?n

the infinity-norm. Let x := {x(l)}l∈Z+with x(l) ∈ Rnfor all l ∈ Z+ denote an

arbitrary sequence and define ?x? := sup{?x(l)? | l ∈ Z+}. Furthermore, x[c1,c2]:=

{x(l)}l∈Z[c1,c2], with c1,c2∈ Z, denotes a sequence that is ordered monotonically with

respect to the index l ∈ Z[c1,c2]. Similarly, col({x(l)}l∈Z[c1,c2]) := [ x(c2)?... x(c1)?]?

is also ordered monotonically (albeit in a decreasing fashion from top to bottom)

with respect to the index l. For a symetric matrix Z ∈ Rn×nlet Z ? 0 (Z ≺ 0)

denote that Z is positive (negative) definite and let λmax(Z) (λmin(Z)) denote the

largest (smallest) eigenvalue of Z. Moreover, ∗ is used to denote the symmetric part

of a matrix, i.e.,

b c

0n×m∈ Rn×mdenote a matrix with all elements equal to zero. Let Sh:= S×...×S for

any h ∈ Z≥1denote the h-times cross-product of an arbitrary set S ⊆ Rn. Moreover,

let int(S) denote the interior of S, let ∂S denote the boundary of S and let cl(S) denote

the closure of S. For a λ ∈ R define λS := {λx | x ∈ S}. Let co(·) denote the convex

hull. A continuous function ϕ : R[0,a)→ R+, for some a ∈ R>0, is said to belong to

class K if it is strictly increasing and ϕ(0) = 0. Moreover, ϕ ∈ K∞if ϕ : R+→ R+,

ϕ ∈ K and limr→∞ϕ(r) = ∞. A continuous function β : R[0,a)×R+→ R+, for some

a ∈ R>0, is said to belong to class KL if for each fixed s ∈ R+, β(r,s) ∈ K with

respect to r and for each fixed r ∈ R[0,a), β(r,s) is decreasing with respect to s and

lims→∞β(r,s) = 0.

2.2. Delay difference inclusions. Consider the DDI

i=1|[x]i|p)

1

p,

p ∈ Z>0, denote an arbitrary p-norm. Moreover, let ?x?∞:= maxi∈Z[1,n]|[x]i| denote

?a b?

?= [a ∗

b c]. Let In∈ Rn×ndenote the identity matrix and let

x(k + 1) ∈ F(x[k−h,k]),k ∈ Z+,

(2.1)

where x[k−h,k]∈ (Rn)h+1, h ∈ Z≥1 is the maximal delay and F : (Rn)h+1⇒ Rn

is a set-valued map with the origin as equilibrium point, i.e., F(0[k−h,k]) = {0}.

Next, consider the following standing assumption which is a common assumption for

difference inclusions without delay as well, see, e.g., [21].

Assumption 1.

The set F(x[−h,0]) ⊂ Rnis compact and nonempty for all

x[−h,0]∈ (Rn)h+1.

Note that while the DDI (2.1) is time-invariant, uncertain time-varying delays

can be incorporated, similarly as in, e.g., [17]. It is worth to point out that the

aforementioned technique does not introduce any conservatism since the map F is

not required to be convex.

Let S(x[−h,0]) denote the set of all trajectories of (2.1) that correspond to initial

condition x[−h,0]∈ (Rn)h+1. Furthermore, let Φ(x[−h,0]) := {φ(k,x[−h,0])}k∈Z≥−h∈

S(x[−h,0]) denote a trajectory of (2.1) such that φ(k,x[−h,0]) = x(k) for all k ∈ Z[−h,0]

Page 4

4

R.H. Gielen, M. Lazar and I.V. Kolmanovsky

and φ(k + 1,x[−h,0]) ∈ F(φ[k−h,k](x[−h,0])) for all k ∈ Z+. Above, the notation

φ[k−h,k](x[−h,0]) := {φ(l,x[−h,0])}l∈Z[k−h,k]was used.

Definition 2.1.

(i) System (2.1) is called a linear delay difference equation

(DDE) if F(x[k−h,k]) := {?0

cl(Mθ)} with Mθ⊂ Rn×nand Mθbounded for all θ ∈ Z[−h,0].

Definition 2.2. System (2.1) is called D-homogeneous of order t, t ∈ Z+, if for

any s ∈ R it holds that F(sx[−h,0]) = stF(x[−h,0]) for all x[−h,0]∈ (Rn)h+1.

Definition 2.3. Let λ ∈ R[0,1). A convex and compact set X ⊂ Rnwith 0 ∈

int(X) is called λ-D-contractive for the DDI (2.1) if F(x[−h,0]) ⊆ λX for all x[−h,0]∈

Xh+1.

Moreover, consider the following notions of stability.

Definition 2.4. (i) The origin of the DDI (2.1) is called globally attractive if

limk→∞?φ(k,x[−h,0])? = 0 for all x[−h,0]∈ (Rn)h+1and all Φ(x[−h,0]) ∈ S(x[−h,0]);

(ii) the origin of (2.1) is called Lyapunov stable (LS) if for every ε ∈ R>0there exists

a δ(ε) ∈ R>0such that if ?x[−h,0]? ≤ δ, then ?φ(k,x[−h,0])? ≤ ε for all Φ(x[−h,0]) ∈

S(x[−h,0]) and all k ∈ Z+; (iii) System (2.1) is called globally asymptotically stable

(GAS) if its origin is both globally attractive and LS.

Definition 2.5. (i) System (2.1) is called KL-stable if there exists a function

β : R+× R+ → R+, β ∈ KL, such that ?φ(k,x[−h,0])? ≤ β(?x[−h,0]?,k) for all

x[−h,0]∈ (Rn)h+1, Φ(x[−h,0]) ∈ S(x[−h,0]) and all k ∈ Z+. (ii) System (2.1) is called

globally exponentially stable (GES) if it is KL-stable with β(r,s) := crµs, for some

c ∈ R≥1and µ ∈ R[0,1).

Note that the above definitions define global and strong properties, i.e., properties

that hold for all x[−h,0]∈ (Rn)h+1and all Φ(x[−h,0]) ∈ S(x[−h,0]). The following

lemma relates DDIs that are GAS to DDIs that are KL-stable.

Lemma 2.6. The following two statements are equivalent:

(i) The DDI (2.1) is GAS and δ(ε) in Definition 2.4 can be chosen to satisfy

limε→∞δ(ε) = ∞;

(ii) The DDI (2.1) is KL-stable.

The proof of Lemma 2.6 can be obtained mutatis mutandis from the proof of

Lemma 4.5 in [22], a result for continuous-time systems without delay. The relevance

of the result of Lemma 2.6 comes from the fact that KL-stability, as opposed to

mere GAS, is a standard assumption in converse Lyapunov theorems, see, e.g., [2, 21,

38]. Note that, if the DDI (2.1) is upper semicontinuous [20], then it can be shown,

similarly to Proposition 6 in [20], that GAS is equivalent to KL-stability.

With the above equivalence established, in the next section various conditions

under which a DDI is KL-stable are established.

3. Stability of delay difference inclusions.

3.1. The Lyapunov-Krasovskii approach. As pointed out in the introduc-

tion, a standard approach for studying stability of delay discrete-time systems is to

augment the state vector and then to obtain a LF for the resulting augmented state

system. Hence, let ξ(k) := col({x(l)}l∈Z[k−h,k]) and consider the difference inclusion

ξ(k + 1) ∈¯F(ξ(k)),

where the map¯F : R(h+1)n⇒ R(h+1)nis obtained from the map F in (2.1), i.e.,

¯F(ξ) = col({x(l)}l∈Z[−h+1,0],F(x(−h),...,x(0))), with ξ = col({x(l)}l∈Z[−h,0]). There-

fore,¯F(ξ) is compact and non-empty for all ξ ∈ R(h+1)nand¯F(0) = {0}. We use

θ=−hAθx(k + θ)} where Aθ∈ Rn×nfor all θ ∈ Z[−h,0];

(ii) System (2.1) is called a linear DDI if F(x[k−h,k]) := {?0

θ=−hAθx(k + θ) | Aθ∈

k ∈ Z+,

(3.1)

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Lyapunov methods for time-invariant delay difference inclusions

5

¯S(ξ) to denote the set of all trajectories of (3.1) from initial condition ξ ∈ R(h+1)n.

Let¯Φ(ξ) := {¯φ(k,ξ)}k∈Z+∈¯S(ξ) denote a trajectory of (3.1) such that¯φ(0,ξ) = ξ

and¯φ(k + 1,ξ) ∈¯F(¯φ(k,ξ)) for all k ∈ Z+.

Definition 3.1. A function g : Rl⇒ Rp, possibly set-valued, is called homoge-

neous (positively homogeneous) of order t, t ∈ Z+, if g(sx) = stg(x) (g(sx) = |s|tg(x))

for all x ∈ Rland all s ∈ R.

Definition 3.2. Let λ ∈ R[0,1). A convex and compact set¯X ⊂ R(h+1)nwith

0 ∈ int(¯X) is called λ-contractive for system (3.1) if¯F(ξ) ⊆ λ¯X for all ξ ∈¯X.

Remark 1.

Throughout this paper, uniformly strict Lyapunov conditions are

sought for, as opposed to classical Lyapunov conditions. Such conditions yield uni-

formly strict LFs, which in turn induce contractive sets, as opposed to merely invariant

sets. The reader interested in more details on uniformly strict LFs is referred to [31].

The following lemma relates stability of the DDI (2.1) to stability of the difference

inclusion (3.1). Thus, stability of the set-valued map F : (Rn)h+1⇒ Rnis related to

stability of the set-valued map¯F : R(h+1)n⇒ R(h+1)n.

Lemma 3.3. The following claims are true:

(i) The DDI (2.1) is GAS if and only if the difference inclusion (3.1) is GAS;

(ii) The DDI (2.1) is KL-stable if and only if the difference inclusion (3.1) is

KL-stable;

(iii) The DDI (2.1) is GES if and only if the difference inclusion (3.1) is GES.

The proof of Lemma 3.3 can be found in Appendix A. In the standard approach,

e.g., [7, 8, 12, 14, 17, 26, 43], a LF for the difference inclusion (3.1) is obtained. This

LF is then used to conclude that the DDI (2.1) is KL-stable. Lemma 3.3 enables a

formal characterization of this conjecture. Moreover, the converse is also obtained.

Theorem 3.4. Let ¯ α1, ¯ α2∈ K∞. The following statements are equivalent:

(i) There exists a function¯V : R(h+1)n→ R+and a constant ¯ ρ ∈ R[0,1)such that

¯ α1(?ξ?) ≤¯V (ξ) ≤ ¯ α2(?ξ?),

¯V (ξ+) ≤ ¯ ρ¯V (ξ),

for all ξ ∈ R(h+1)nand all ξ+∈¯F(ξ).

(ii) The difference inclusion (3.1) is KL-stable.

(iii) The DDI (2.1) is KL-stable.

Proof. The equivalence of (i) and (ii) was proven in [21], Theorem 2.7, under the

additional assumptions that the map¯F is upper semicontinuous and the function V

is smooth. However, these assumptions were only used to prove certain robustness

properties and can therefore be omitted. Alternatively, this equivalence can be shown

following mutatis mutandis the reasoning used in the proof of Lemma 4 in [38], which

is a result for difference equations. Furthermore, the equivalence of (ii) and (iii) follows

from Lemma 3.3.

A function¯V that satisfies the hypothesis of Theorem 3.4 is called a LKF for the

DDI (2.1). From Theorem 3.4 the following two corollaries are obtained.

Corollary 3.5. Let c1∈ R>0and let c2∈ R≥c1. Suppose that the DDI (2.1) is

a linear DDE and hence also that the corresponding system (3.1) is a linear difference

equation. Then the following statements are equivalent:

(i) There exist a quadratic function¯V (ξ) := ξ?¯Pξ, for some symmetric matrix

¯P ∈ R(h+1)n×(h+1)n, and a constant ¯ ρ ∈ R[0,1)such that

c1?ξ?2

¯V (ξ+) ≤ ¯ ρ¯V (ξ),

(3.2a)

(3.2b)

2≤¯V (ξ) ≤ c2?ξ?2

2,

(3.3a)

(3.3b)

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R.H. Gielen, M. Lazar and I.V. Kolmanovsky

for all ξ ∈ R(h+1)nand all ξ+∈¯F(ξ).

(ii) The linear difference equation (3.1) is GES.

(iii) The linear DDE (2.1) is GES.

Corollary 3.6. Let c1∈ R>0, c2∈ R≥c1and let p ∈ Z≥(h+1)n. Suppose that

the DDI (2.1) is a linear DDI and hence also that the corresponding system (3.1) is

a linear difference inclusion. Then the following statements are equivalent:

(i) There exist a polyhedral function¯V (ξ) := ?¯Pξ?∞, for some¯P ∈ Rp×(h+1)n,

and a constant ¯ ρ ∈ R[0,1)such that

c1?ξ?∞≤¯V (ξ) ≤ c2?ξ?∞,

¯V (ξ+) ≤ ¯ ρ¯V (ξ),

for all ξ ∈ R(h+1)nand all ξ+∈¯F(ξ).

(ii) The linear difference inclusion (3.1) is GES.

(iii) The linear DDI (2.1) is GES.

The proof of Corollary 3.5 follows from Corollary 3.1* in [19] and Lemma 3.3.

Furthermore, the proof of Corollary 3.6 follows from the Corollary in [3], Part III,

and Lemma 3.3. Note that, the set cl(Mθ) is closed and bounded by assumption

but not necessarily convex, which is exactly what is required for the Corollary in

[3], Part III. A function¯V (ξ) = ξ?¯Pξ that satisfies the hypothesis of Corollary 3.5

is called a quadratic Lyapunov-Krasovskii function (qLKF). Moreover, a function

¯V (ξ) = ?¯Pξ?∞ that satisfies the hypothesis of Corollary 3.6 is called a polyhedral

Lyapunov-Krasovskii function (pLKF). The following example illustrates the results

derived above.

Example 1. Consider the linear DDE

(3.4a)

(3.4b)

x(k + 1) = ax(k) + bx(k − 1),k ∈ Z+,

(3.5)

where x[k−1,k]∈ R × R and a,b ∈ R. Let ξ(k) := [x(k),x(k − 1)]?, which yields

ξ(k + 1) =¯Aξ(k),

where¯A = [a b

the spectral radius of¯A is strictly less than one and hence (3.6) is GES, see, e.g., [19].

Therefore, it follows from Corollary 3.5 that, if a,b ∈ R with |b| < 1 and |a| < 1 − b,

then there exist a ¯ ρ ∈ R[0,1)and a symmetric¯P ∈ R2×2such that

¯A?¯P¯A − ¯ ρ¯P ≺ 0,

Moreover, it also follows from Corollary 3.5 that if a,b ∈ R with |b| < 1 and |a| < 1−b,

then (3.5) is GES and admits a qLKF. For example, let a = 1 and b = −0.5. As

¯ ρ = 0.95 and¯P =

−0.5 0.7

b = −0.5, is GES. Hence, the linear DDE (3.5), with a = 1 and b = −0.5, is GES.

Moreover, the function¯V (ξ) = ξ?¯Pξ is a quadratic LF for (3.6) and the function

¯V (ξ) =¯V (x[−1,0]) = 1.3x(0)2− x(0)x(−1) + 0.7x(−1)2is a qLKF for (3.5).

Unfortunately, the sublevel sets of a LKF do not provide a contractive set in the

original state space, i.e., Rn, but rather a contractive set in the higher dimensional

state space corresponding to the augmented state system, i.e., R(h+1)nor equivalently

(Rn)h+1. Moreover, as the LKF is a function of the current state and all delayed states,

it becomes increasingly complex when the size of the delay, i.e., h ∈ Z≥1, increases.

Therefore, it would be desirable to construct a function satisfying particular Lyapunov

conditions that involve the non-augmented system, rather than the augmented one.

k ∈ Z+,

(3.6)

1 0]. Note that for all b ∈ R with |b| < 1 and all a ∈ R with |a| < 1 − b,

¯P ? 0.

(3.7)

?1.3 −0.5

?

is a solution to (3.7), system (3.6), with a = 1 and

Page 7

Lyapunov methods for time-invariant delay difference inclusions

7

3.2. The Lyapunov-Razumikhin approach. The Razumikhin approach is a

Lyapunov technique for time-delay systems that satisfies Lyapunov conditions that

directly involve the DDI (2.1), as opposed to the augmented state system (3.1).

Theorem 3.7.

Let α1,α2 ∈ K∞ and let π : R+ → R+ be a function such

that π(s) > s for all s ∈ R>0 and π(0) = 0. Suppose that there exists a function

V : Rn→ R+and a constant ρ ∈ R[0,1)such that

α1(?x?) ≤ V (x) ≤ α2(?x?),

and, for all x[−h,0]∈ (Rn)h+1, if π(V (x+)) ≥ maxθ∈Z[−h,0]V (x(θ)), then

V (x+) ≤ ρV (x(0)),

Then, the DDI (2.1) is KL-stable.

The proof of the above theorem, which is omitted here for brevity, is similar in

nature to the proof of Theorem 6 in [11] by replacing mutatis mutandis the difference

equation with the difference inclusion as in (2.1). It is obvious that the LRF defined

in Theorem 3.7 is non-causal, i.e., (3.8b) imposes a condition on V (x+) if V (x+)

satisfies some other condition. Note that the corresponding Lyapunov-Razumikhin

theorem for continuous-time systems, e.g., Theorem 4.1 in [16], is causal, because it

imposes a condition on the derivative of V (x) if V (x) satisfies a certain condition.

Next, an extension of Theorem 3.2 in [33], which provides a causal sufficient condition

for stability of the DDI (2.1), will be presented.

Theorem 3.8. Let α1,α2∈ K∞. If there exists a function V : Rn→ R+and a

constant ρ ∈ R[0,1)such that

α1(?x?) ≤ V (x) ≤ α2(?x?),

V (x+) ≤ ρ

∀x ∈ Rn,

(3.8a)

∀x+∈ F(x[−h,0]).

(3.8b)

∀x ∈ Rn,

(3.9a)

(3.9b) max

θ∈Z[−h,0]V (x(θ)),

for all x[−h,0]∈ (Rn)h+1and all x+∈ F(x[−h,0]), then the DDI (2.1) is KL-stable.

Proof. Suppose that ρ ?= 0. Let ˆ ρ := ρ

θopt(k,φ[k−h,k](x[−h,0])) := arg

U(k,φ[k−h,k](x[−h,0])) :=

1

h+1∈ R(0,1)and let

max

θ∈Z[−h,0]ˆ ρ−(k+θ)V (φ(k + θ,x[−h,0])),

max

θ∈Z[−h,0]ˆ ρ−(k+θ)V (φ(k + θ,x[−h,0])),

(3.10)

where x[−h,0]∈ (Rn)h+1, Φ(x[−h,0]) ∈ S(x[−h,0]) and k ∈ Z+. Next, it will be proven

that

U(k + 1,φ[k−h+1,k+1](x[−h,0])) ≤ U(k,φ[k−h,k](x[−h,0])),

for all x[−h,0]∈ (Rn)h+1, Φ(x[−h,0]) ∈ S(x[−h,0]) and all k ∈ Z+. Therefore, suppose

that θopt(k + 1,φ[k−h+1,k+1](x[−h,0])) = 0 for some x[−h,0]∈ (Rn)h+1, Φ(x[−h,0]) ∈

S(x[−h,0]) and k ∈ Z+. Then, (3.9b) yields

U(k + 1,φ[k−h+1,k+1](x[−h,0])) = ˆ ρ−(k+1)V (φ(k + 1,x[−h,0]))

≤ ˆ ρ−(k+1)

max

(3.11)

max

θ∈Z[−h,0]ˆ ρ(h+1)V (φ(k + θ,x[−h,0]))

θ∈Z[−h,0]ˆ ρ−(k+θ)V (φ(k + θ,x[−h,0])) = U(k,φ[k−h,k](x[−h,0])).

≤

(3.12)

Page 8

8

R.H. Gielen, M. Lazar and I.V. Kolmanovsky

Furthermore, if θopt(k + 1,φ[k−h+1,k+1](x[−h,0])) ∈ Z[−h,−1]it holds that

U(k + 1,φ[k−h+1,k+1](x[−h,0])) =max

θ∈Z[−h,−1]ˆ ρ−(k+θ+1)V (φ(k + θ + 1,x[−h,0]))

max=

θ∈Z[−h+1,0]ˆ ρ−(k+θ)V (φ(k + θ,x[−h,0]))

≤ U(k,φ[k−h,k](x[−h,0])).

(3.13)

Therefore, from (3.12) and (3.13) it follows that (3.11) holds. Applying (3.11) recur-

sively, yields

U(k,φ[k−h,k](x[−h,0])) ≤ U(0,φ[−h,0](x[−h,0])) ≤

max

θ∈Z[−h,0]V (x(θ)).

(3.14)

Next, combining (3.10) and (3.14) yields

V (φ(k,x[−h,0])) ≤ ˆ ρkU(k,φ[k−h,k](x[−h,0])) ≤ ˆ ρk

max

θ∈Z[−h,0]V (x(θ)),

for all x[−h,0]∈ (Rn)h+1, Φ(x[−h,0]) ∈ S(x[−h,0]) and all k ∈ Z+. Observing that

maxθ∈Z[−h,0]α2(?x(θ)?) = α2(?x[−h,0]?) and applying (3.9a) yields

?φ(k,x[−h,0])? ≤ α−1

1(ˆ ρkα2(?x[−h,0]?)),

(3.15)

for all x[−h,0] ∈ (Rn)h+1, Φ(x[−h,0]) ∈ S(x[−h,0]) and all k ∈ Z+. As β(r,s) :=

α−1

Suppose that ρ = 0. Then, it follows from (3.9b) and (3.9a) that ?x+? = 0 for

all x[−h,0]∈ (Rn)h+1and all x+∈ F(x[−h,0]). Hence, ?φ(k,x[−h,0])? ≤ ?x[−h,0]?1

for all x[−h,0]∈ (Rn)h+1, all Φ(x[−h,0]) ∈ S(x[−h,0]) and all k ∈ Z+. Observing that

β(r,s) := r1

2

A function that satisfies the hypothesis of Theorem 3.7 is called a non-causal LRF

and one that satisfies the hypothesis of Theorem 3.8 is called a LRF. The following

corollary follows directly from (3.15).

Corollary 3.9. Let c1∈ R>0, c2∈ R≥c1and λ ∈ Z>0. If there exist a function

V : Rn→ R+ and a constant ρ ∈ R[0,1)that satisfy the hypothesis of Theorem 3.8

with α1(s) = c1sλand α2(s) = c2sλ, then the DDI (2.1) is GES.

Next, Example 1 is used to show that the converse of Theorem 3.7 and Theo-

rem 3.8 is not true in general.

Proposition 3.10. Consider the linear DDE (3.5) and suppose that b ∈ R(−1,0)

and a = 1. Then, the following statements are true:

(i) The linear DDE (3.5) is GES;

(ii) The linear DDE (3.5) does not admit a non-causal LRF;

(iii) The linear DDE (3.5) does not admit a LRF.

Proof. It was shown in Example 1 that the DDE (3.5) with a,b ∈ R and such that

|b| < 1 and |a| < 1 − b admits a qLKF. Hence, the DDE (3.5) with b ∈ R(−1,0)and

a = 1 admits a qLKF. Therefore, it follows from Corollary 3.5 that the DDE (3.5)

with b ∈ R(−1,0)and a = 1 is GES. The proof of claim (ii) and claim (iii) proceeds by

contradiction.

To prove claim (ii), suppose that there exists a non-causal LRF V : R → R+

for the DDE (3.5) with b ∈ R(−1,0)and a = 1. Let x(0) = 1, x(−1) = 0 and let

1(α2(r)ˆ ρs) ∈ KL, it follows that (2.1) is KL-stable.

2

k

s∈ KL completes the proof.

Page 9

Lyapunov methods for time-invariant delay difference inclusions

9

π : R+ → R+ be any function such that π(s) > s for all s ∈ R>0 and π(0) = 0.

Hence, (3.5) yields that x(1) = 1. As

π(V (x(1))) = π(V (1)) ≥

max

θ∈Z[−1,0]V (x(θ)) = V (1),

it follows from (3.8b) that

V (x(1)) = V (1) ≤ ρV (x(0)) = ρV (1).

Obviously, as ρ ∈ R[0,1)a contradiction has been reached and hence, V is not a

non-causal LRF for the DDE (3.5). As the functions V and π and the constant ρ

were chosen, with the restriction that π(s) > s for all s ∈ R>0and that ρ ∈ R[0,1),

arbitrarily, it follows that the second claim has been established.

The same initial conditions as the ones used in the proof of claim (ii) can be used

to establish, by contradiction, claim (iii).

While it can be verified using the conditions in Theorem 3.7 if a function is a non-

causal LRF, these conditions can not be reformulated into an optimization problem

which can be used to obtain a non-causal LRF. The conditions in Theorem 3.8, on

the other hand, can be reformulated as a semi-definite programming problem whose

solution yields a LRF, as it will be shown in Section 6. Therefore, in what follows, we

will focus on LRFs and disregard non-causal LRFs. The interested reader is referred to

[33] for a detailed discussion on LRFs, non-causal LRFs and their differences. Therein,

it is indicated why LRFs form a less conservative test for stability when compared to

non-causal LRFs, which provides another reason for disregarding non-causal LRFs.

In the next section, it will be shown that the existence of a LRF implies the existence

of a LKF and that only under certain additional assumptions the converse is true.

4. Relations between LKFs and LRFs. For delay differential equations, i.e.,

delay continuous-time systems, it was shown in [25], Section 4.8, that LRFs form a

particular case of LKFs, when only Lyapunov stability (see Definition 2.4) rather than

KL-stability is of concern. A similar reasoning as the one used in [25] can be applied

to DDIs as well. Suppose that the function V satisfies the hypothesis of Theorem 3.8

with ρ = 1. Then, it can be easily verified that

¯V (x[−h,0]) =max

θ∈Z[−h,0]V (x(θ)),

satisfies the hypothesis of Theorem 3.4 with ¯ ρ = 1. Thus, it follows from (3.2b) that

¯V (φ[k−h,k](x[−h,0])) ≤¯V (x[−h,0]),

for all k ∈ Z+. From this observation one can show, using (3.2a), that (2.1) is LS.

However, the same candidate LKF does not satisfy the assumptions of Theorem 3.4

for ¯ ρ ∈ R[0,1), i.e., when KL-stability is imposed. Furthermore, in [23] an example was

provided where the above result was generalized to ¯ ρ ∈ R[0,1)for quadratic candidate

functions and continuous-time systems. Next, it will be shown how the continuous-

time result of [25] can be extended for DDIs to allow for ¯ ρ ∈ R[0,1), via a more complex

candidate LKF.

Theorem 4.1. Suppose that V : Rn→ R+ satisfies the hypothesis of Theo-

rem 3.8. Then,

∀x[−h,0]∈ (Rn)h+1,

∀Φ(x[−h,0]) ∈ S(x[−h,0]),

¯V (x[−h,0]) :=max

θ∈Z[−h,0]ρh+1+θV (x(θ)),

(4.1)

Page 10

10

R.H. Gielen, M. Lazar and I.V. Kolmanovsky

where ρi:=ρ+i

Proof. First, it is established that

i+1, i ∈ Z[1,h], and ρh+1:= 1, satisfies the hypothesis of Theorem 3.4.

ρ < ρ1< ... < ρh< ρh+1= 1.

(4.2)

As ρ < 1 it holds that ρ < 1 = (i + 1)2− (i + 2)i, which is equivalent to

(i + 2)(ρ + i) < (i + 1)(ρ + (i + 1)).

Therfore, it follows that ρi< ρi+1, for all i ∈ Z[1,h]. Obviously, ρi<1+i

establishes that (4.2) holds. Next, let πi:=

as ρi−1< ρiit follows that πi<ρi

Next, consider any x[−h,0]∈ (Rn)h+1. Then,

¯V ({x[−h+1,0],x+}) = max{ρh+1V (x+),

≤ max{ max

= max{ρV (x(−h)),

=max

i+1= 1, which

ρi−1

ρi, i ∈ Z[1,h+1], and let ρ0:= ρ. Then,

ρi= 1. Letting π := maxi∈Z[1,h+1]πi, yields π < 1.

max

θ∈Z[−h+1,0]ρh+θV (x(θ))}

max

θ∈Z[−h,0]ρV (x(θ)),

θ∈Z[−h+1,0]ρh+θV (x(θ))}

max

θ∈Z[−h+1,0]ρh+θV (x(θ))}

θ∈Z[−h,0]πh+θ+1ρh+θ+1V (x(θ)) ≤ π¯V (x[−h,0]),

for all x+∈ F(x[−h,0]). Let ¯ ρ := π, ¯ α1(s) := ρ1α1(s) and ¯ α2(s) := α2(s). As

¯ α1, ¯ α2∈ K∞and ¯ ρ < 1, it follows that¯V satisfies the hypothesis of Theorem 3.4.

Next, it is established under what conditions the existence of a LKF implies the

existence of a LRF.

Proposition 4.2. Suppose that¯V : R(h+1)n→ R+ satisfies the hypothesis of

Theorem 3.4. Moreover, let α3,α4∈ K∞ be such that α3(s) ≤ α4(s) and α3(ρs) ≥

¯ ρα4(s) for some ρ ∈ R[0,1)and all s ∈ R>0. If there exists a function V : Rn→ R+

satisfying (3.9a) and

0

?

θ=−h

α3(V (x(θ))) ≤¯V (x[−h,0]) ≤

0

?

θ=−h

α4(V (x(θ))),

(4.3)

then V satisfies the hypothesis of Theorem 3.8.

Proof. Applying (4.3) in (3.2b) yields

α3(V (x+)) − ¯ ρα4(V (x(−h))) +

0

?

θ=−h+1

α3(V (x(θ))) − ¯ ρα4(V (x(θ))) ≤ 0,

(4.4)

for all x+∈ F(x[−h,0]). Note that α3(s) > ¯ ρα4(s) for all s ∈ R+and hence

0

?

θ=−h+1

α3(V (x(θ))) − ¯ ρα4(V (x(θ))) > 0.

(4.5)

The inequality (4.5) in combination with V (x(−h)) ≤ maxθ∈Z[−h,0]V (x(θ)) yields that

(4.4) is a sufficient condition for

α3(V (x+)) − ¯ ρα4( max

θ∈Z[−h,0]V (x(θ))) ≤ 0,

(4.6)

Page 11

Lyapunov methods for time-invariant delay difference inclusions

11

for all x+∈ F(x[−h,0]). Then, using that there exists a ρ ∈ R[0,1)such that ρs ≥

α−1

3(¯ ρα4(s)) yields

V (x+) − ρ

max

θ∈Z[−h,0]V (x(θ)) ≤ 0,

for all x+∈ F(x[−h,0]). Hence, the hypothesis of Theorem 3.8 is satisfied and the

proof is complete.

The following corollary is a slight modification of Proposition 4.2.

Corollary 4.3. Suppose that the hypothesis of Proposition 4.2 holds with (4.3)

replaced by

max

θ∈Z[−h,0]α3(V (x(θ))) ≤¯V (x[−h,0]) ≤

Then V satisfies the hypothesis of Theorem 3.8.

Proof. Using the bounds (4.7) in (3.2b) yields

max

θ∈Z[−h,0]α4(V (x(θ))).

(4.7)

max{

max

θ∈Z[−h+1,0]α3(V (x(θ))),α3(V (x+))} − ¯ ρ

for all x+∈ F(x[−h,0]). As max{s1,s2} ≥ s2for any s1,s2∈ R+, (4.8) is sufficient for

α3(V (x+)) − ¯ ρα4( max

max

θ∈Z[−h,0]α4(V (x(θ))) ≤ 0,

(4.8)

θ∈Z[−h,0]V (x(θ))) ≤ 0,

for all x+∈ F(x[−h,0]). Hence, (4.6) is recovered, which completes the proof.

The hypothesis and conclusion of Theorem 4.1, Proposition 4.2 and Corollary 4.3

might not seem very intuitive. However, when quadratic or polyhedral candidate

functions are considered, these results do provide valuable insights. For example,

suppose that V (x) = ?Px?∞is a polyhedral Lyapunov-Razumikhin function (pLRF).

Then, it follows from Theorem 4.1 that

¯V (x[−h,0]) =max

θ∈Z[−h,0]ρh+1+θ?Px(θ)?∞=

?????

?ρh+1P0

...

0ρ1P

?

ξ

?????

∞

,

(4.9)

is a pLKF. Conversely, suppose that the function (4.9) is a pLKF for some ¯ ρ ∈ R[0,1)

such that ¯ ρ < ρ1. Then, it follows from Corollary 4.3, i.e., by taking α3(s) = ρ1s and

α4(s) = s, that V (x) = ?Px?∞is a pLRF.

In contrast, given a quadratic Lyapunov-Razumikhin function (qLRF), Theo-

rem 4.1 does not yield a qLKF but rather a more complex LKF, i.e., the maximum

over a set of quadratic functions. On the other hand, Proposition 4.2 can provide a

qLRF constructed from a qLKF. Indeed, consider the qLKF

¯V (x[−h,0]) =

0

?

θ=−h

x(θ)?Px(θ) = ξ?

?P0

...

0P

?

ξ,

then it follows from Proposition 4.2 that V (x) = x?Px is a qLRF.

Figure 4.1 presents a schematic overview of all results derived in Section 2, Sec-

tion 3.1, Section 3.2 and Section 4. Interestingly, the existence of a qLRF implies the

existence of a qLKF under the additional assumption that the system under study is

a linear DDE only. The existence of a LRF and the existence of a pLRF, on the other

hand, do imply the existence of a LKF and pLKF, respectively, for general DDIs (as

opposed to for linear DDEs only).

In the next section results on contractive sets for DDIs will be established.

Page 12

12

R.H. Gielen, M. Lazar and I.V. Kolmanovsky

(1) is KL-stable

(2) is KL-stable

(1) is GAS(2) is GAS

(1) is GES(2) is GES

A2

A3

A4

A4

A1A1

A2

(1) admits

a LRF

(1) admits

a LKF

(1) admits

a pLKF

(1) admits

a pLRF

(1) admits

a qLRF

(1) admits

a qLKF

Fig. 4.1. A schematic overview of all relations established in this paper. B → C means that B

implies C, B ? C means that B does not necessarily imply C and B

C under the additional assumption A. The employed assumptions are as follows: (A1) – δ(ε) in

Definition 2.4 can be chosen to satisfy limε→∞δ(ε) = ∞; (A2) – the DDI (2.1) is a linear DDE;

(A3) – the DDI (2.1) is a linear DDI; (A4) – the LKF has certain structural properties.

A

− → C means that B implies

5. Contractive sets for DDIs. Contractive sets are at the basis of many con-

trol techniques, see, e.g., [5], and it is well-known that the sublevel sets of a LF are

λ-contractive sets. Next, it is established that the existence of a λ-contractive set and

a λ-D-contractive set is equivalent to the existence of a LKF and LRF, respectively.

Both results are established via the sublevel sets of a LKF and LRF, respectively.

Recall that a contractive set is by assumption a convex and compact set with the

origin in its interior, see Definition 2.3 and Definition 3.2.

Proposition 5.1. Suppose that system (3.1) is homogeneous1of order 1. The

following two statements are equivalent:

(i) The difference inclusion (3.1) admits a continuous and convex LF that is

positively homogeneous of order t, for some t ∈ Z≥1.

(ii) The difference inclusion (3.1) admits a λ-contractive set, for some λ ∈ R[0,1).

The proof of Proposition 5.1 can be obtained from the results derived in [4, 5,

37]. Note that the most common LF candidates, such as quadratic and norm-based

functions, are inherently continuous and convex. Moreover, continuity is a desirable

property as continuous LFs guarantee that the corresponding type of stability does

not have zero robustness, see, e.g., [31].

Unfortunately, it remains unclear what a contractive set¯V ⊂ R(h+1)nimplies for

the DDI (2.1) and for the trajectories Φ(x[−h,0]) ∈ S(x[−h,0]) in the original state

space Rn, in particular. The above observation indicates an important drawback of

LKFs. While the DDI (2.1) admits a LKF if and only if the system is KL-stable, a

LKF does not provide a contractive set in the original, non-augmented state space.

A LRF is based on particular Lyapunov conditions that involve the non-augmented

system, rather than the augmented one. As such, in contrast to a LKF, a LRF,

if it exists, provides a type of contractive set for the non-augmented system. The

above discussion indicates, apart from a lower complexity, another advantage of the

Lyapunov-Razumikhin method over the Lyapunov-Krasovskii approach.

Proposition 5.2. Suppose that the DDI (2.1) is D-homogeneous2of order 1.

The following two statements are equivalent:

1For example, linear difference inclusions are homogeneous of order 1.

2For example, linear DDIs are D-homogeneous of order 1.

Page 13

Lyapunov methods for time-invariant delay difference inclusions

13

(i) The DDI (2.1) admits a continuous and convex LRF that is positively homo-

geneous of order t, for some t ∈ Z≥1.

(ii) The DDI (2.1) admits a λ-D-contractive set, for some λ ∈ R[0,1).

Proof. First, the relation (i)⇒(ii) is proven. Consider a sublevel set of V , i.e.,

V := {x ∈ Rn| V (x) ≤ 1}. As V : Rn→ R+ is continuous and convex the set

V is [6] closed and convex, respectively. Moreover, boundedness follows from the

K∞upperbound on the function V . Furthermore, if maxθ∈Z[−h,0]V (x(θ)) ≤ 1 then

it follows from (3.9b) that V (x+) ≤ ρ.

V (ρ−1

Hence, V is a λ-D-contractive set with λ := ρ

Next, the relation (ii)⇒(i) is proven. Let V denote a λ-D-contractive set for the

DDI (2.1) and consider the Minkowski function, see, e.g., [35], of V, i.e.,

Hence, as V is positively homogeneous,

tV for all x[−h,0]∈ Vh+1and all x+∈ F(x[−h,0]).

1

t for the DDI (2.1).

tx+) ≤ 1, which yields x+∈ ρ

1

V (x) := inf{µ ∈ R>0| x ∈ µV}.

(5.1)

Then, it follows from claim 4 and claim 2 and 3 of Lemma 5.12.1 in [35] that

the function V is continuous and convex, respectively. Furthermore, letting a1 :=

maxx∈V?x? > 0 and a2:= minx∈∂V?x? > 0 yields

a−1

1?x? ≤ V (x) ≤ a−1

2?x?.

Next, consider any ν ∈ R>0and let x[−h,0]∈ (νV)h+1. Then, ν−1x[−h,0]∈ Vh+1and

therefore F(ν−1x[−h,0]) ⊆ λV. As the DDI (2.1) is assumed to be D-homogeneous of

order 1 it follows that F(x[−h,0]) = νF(ν−1x[−h,0]) ⊆ λ(νV). Thus, it was shown that

if V is a λ-D-contractive set then νV is a λ-D-contractive set as well. As the set νV is

λ-D-contractive for all ν ∈ R>0, it follows that if x+∈ ∂(µV), for some µ ∈ R>0and

some x+∈ F(x[−h,0]), then there exists a θ ∈ Z[−h,0]such that x(θ) ∈ ∂(µ(λ−1V)).

The above implies that

V (x+) = inf{µ ∈ R>0| x+∈ µV}

≤

max

θ∈Z[−h,0]inf{µ ∈ R>0| x(θ) ∈ µ(λ−1V)} = max

θ∈Z[−h,0]λV (x(θ)),

for all x[−h,0]∈ (Rn)h+1and all x+∈ F(x[−h,0]). Therefore, the candidate function

(5.1) satisfies the hypothesis of Theorem 3.8 with α1(s) := a−1

a−1

proof is complete.

Note that the assumptions under which the statements of Proposition 5.1 and

Proposition 5.2 were proven, i.e., regarding the properties of the contractive sets and

the homogeneity of the systems, are standard assumptions for the type of results

derived in this section, see, e.g., [4, 5, 37]. Furthermore, Proposition 5.2 recovers

Proposition 5.1 and similar results in [4, 5, 37] as a particular case, i.e., for h = 0.

Suppose that the DDI (2.1) and system (3.1) are D-homogeneous and homoge-

neous of order 1, respectively. Moreover, suppose that the DDI (2.1) admits a set

V ⊂ Rnwhich is λ-D-contractive. Then, it follows from Proposition 5.2 that the

DDI (2.1) admits a LRF. Moreover, it follows from Theorem 4.1 that the DDI (2.1)

admits a LKF which in turn, via Proposition 5.1, guarantees the existence of a λ-

contractive set for the augmented state system (3.1).

Suppose again that the DDI (2.1) is D-homogeneous of order 1 and it admits

a LKF that satisfies the hypothesis of Proposition 4.2 or Corollary 4.3. Then, from

1s ∈ K∞, α2(s) :=

2s ∈ K∞and ρ := λ ∈ R[0,1). As (5.1) satisfies V (sx) = sV (x) for all s ∈ R+, the

Page 14

14

R.H. Gielen, M. Lazar and I.V. Kolmanovsky

Proposition 4.2 or Corollary 4.3 it follows that there exists a LRF and hence a V ⊂ Rn

which is λ-D-contractive.

In the next section we proceed to the illustration of the applicability of the devel-

oped Lyapunov methods to stability analysis and stabilizing controller synthesis for

linear polytopic DDIs.

6. Synthesis of quadratic Lyapunov functions. The synthesis problem for

a quadratic LF can be solved efficiently via semi-definite programming. Therefore, in

what follows we restrict ourselves to this class of candidate functions. However, the

results derived in the preceding sections are not restricted to a particular type of LF

candidate. In fact, since the augmented state system (3.1) is a standard difference

inclusion, synthesis techniques for LF candidates such as polyhedral LFs [5, 30], com-

posite LFs [18] and polynomial LFs [39] can be applied directly to obtain a LKF of a

corresponding type. In what follows we consider the linear DDI

?

i=−h

with k ∈ Z+and where Mi:= co({(ˆAi,li,ˆBi,li)}li∈Z[0,Li]) ⊂ Rn×n× Rn×m, Li∈ Z≥1

and i ∈ Z[−h,0].

Remark 2. Linear DDIs, such as (6.1), can be found within many fields. Apart

from the obvious class of uncertain linear systems, networked control systems can be

modeled [14, 45] by linear DDIs as well.

Next, several hypotheses which include a linear matrix inequality that yields,

if feasible, a LKF for system (6.1), will be presented. Firstly, stability analysis of

system (6.1) with zero input, i.e., u(k) = 0 for all k ∈ Z≥−h, is discussed. Therefore,

let¯Al0,...,l−h:=

Ihn

In ∈ Rn×nand 0n×m ∈ Rn×mdenote the n-th dimensional identity matrix and a

rectangular matrix with all elements equal to zero, respectively.

Proposition 6.1. If there exists a symmetric matrix¯P ∈ R(h+1)n×(h+1)nsuch

that

?

then system (6.1) with zero input is GES.

Proof. Letting ξ(k) = col({x(l)}l∈Z[k−h,k]) yields

ξ(k + 1) ∈?¯Aξ(k) |¯A ∈¯

where ¯

ment to (6.2) yields¯P ? 0 and

¯A?

x(k + 1) ∈

0

?

(Aix(k + i) + Biu(k + i))

????(Ai,Bi) ∈ Mi, i ∈ Z[−h,0]

?

,

(6.1)

?ˆA0,l0

...

ˆA−h+1,l−h+1

ˆA−h,l−h

0hn×n

?

and let ¯ ρ ∈ R[0,1). Recall that

¯ ρ¯P

∗

¯P

¯P¯Al0,...,l−h

?

? 0,

∀li∈ Z[0,Li],

∀i ∈ Z[−h,0],

(6.2)

M?,k ∈ Z+,

(6.3)

M := co(?¯Al0,...,l−h

?

(l0,...,l−h)∈Z[0,L0]×...×Z[0,L−h]). Applying the Schur comple-

l0,...,l−h¯P¯Al0,...,l−h− ¯ ρ¯P ≺ 0,

¯

M are a convex combination of¯Al0,...,l−h, li ∈ Z[0,Li], i ∈ Z[−h,0], it

follows that the candidate LKF¯V (x[−h,0]) =¯V (ξ) = ξ?¯Pξ satisfies (3.2b) for system

(6.3). Moreover, this candidate LKF also satisfies (3.2a) with α1(s) := λmin(¯P)s2and

α2(s) := λmax(¯P)s2. From Corollary 3.5 it then follows that system (6.1) with zero

input, i.e., u(k) = 0, k ∈ Z≥−h, is GES.

∀li∈ Z[0,Li],

∀i ∈ Z[−h,0].

As all¯A ∈

Page 15

Lyapunov methods for time-invariant delay difference inclusions

15

When stabilizing controller synthesis is of concern, different augmentations of the

state vector lead to different controller synthesis problems. Firstly, let ¯ ρ ∈ R[0,1),

ξ(k) = col({x(l)}l∈Z[k−h,k]) and let˜Ai,li:=ˆAi,liG +ˆBi,liY .

Proposition 6.2. Suppose there exist a symmetric matrix¯P ∈ R(h+1)n×(h+1)n

a matrix G ∈ Rn×nand a matrix Y ∈ Rm×nsuch that

for all li ∈ Z[0,Li]and all i ∈ Z[−h,0]. Then, system (6.1) in closed-loop with the

controller u(k) = Kx(k), k ∈ Z+, where K = Y G−1, is GES.

Proof. Substituting Y = KG, transposing and using Theorem 1 in [10] or Theo-

rem 3 in [9] yields¯P ? 0 and

for all li∈ Z[0,Li]and all i ∈ Z[−h,0]. The remainder of the proof can then be obtained

from the proof of Proposition 6.1.

Augmenting the state vector with the delayed states and the delayed inputs, i.e.,

ξ(k) = col({u(l)}l∈Z[k−h,k−1],{x(l)}l∈Z[k−h,k]), yields

?¯Aξ(k) +¯Bu(k) | (¯A,¯B) ∈˜

where ˜

M := co({(˜Al0,...,l−h,˜Bl0)}(l0,...,l−h)∈Z[0,L0]×...×Z[0,L−h]) and

Proposition 6.3. Let ¯ ρ ∈ R[0,1). Suppose there exist a matrix Y ∈ Rm×((h+1)n+hm)

and a symmetric matrix Z ∈ R((h+1)n+hm)×((h+1)n+hm)such that

?

for all li ∈ Z[0,Li]and all i ∈ Z[−h,0]. Then, system (6.4) in closed-loop with the

controller u(k) = Kξ(k), k ∈ Z+, where K = Y Z−1, is GES.

Proof. Substituting Y = KZ, applying a congruence transformation with a matrix

that has Z−1on its diagonal and zero elsewhere, and applying the Schur complement

yields Z−1? 0 and

(˜Al0,...,l−h+˜Bl0K)?Z−1(˜Al0,...,l−h+˜Bl0K) − ¯ ρZ−1≺ 0,

¯ ρ¯P

∗

˜A0,l0

G

...

˜A−h+1,l−h+1

0

˜A−h,l−h

0

...

0

...

0

G

?

G

0

...

0

G

+

G

0

...

0

G

?

−¯P

? 0,

(ˆA0,l0+ˆB0,l0K)?

...

...

(ˆA−h,l−h+ˆB−h,l−hK)?

Ihn

0n×hn

¯P

(ˆA0,l0+ˆB0,l0K)?

...

...

(ˆA−h,l−h+ˆB−h,l−hK)?

Ihn

0n×hn

?

− ¯ ρ¯P ≺ 0,

ξ(k + 1) ∈

M

?

,k ∈ Z+,

(6.4)

˜Al0,...,l−h:=

ˆA0,l0

...

ˆA−h,l−h

0hn×n

ˆB−1,l−1

...

ˆB−h,l−h

Ihn

0hn×hm

0(h−1)m×m

I(h−1)m

0hm×(h+1)n

0m×m

0(h−1)m×m

,

˜Bl0:=

ˆB0,l0

0hn×m

Im

0(h−1)m×m

.

¯ ρZ

∗

Z

˜Al0,...,l−hZ +˜Bl0Y

?

? 0,