Lyapunov Methods for TimeInvariant Delay Difference Inclusions.
ABSTRACT Motivated by the fact that Delay Difference Inclusions (DDIs) form a rich modeling class that includes, for example, uncertain timedelay systems and certain types of networked control systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. First, the LyapunovKrasovskii approach, which is an extension of the classical Lyapunov theory to timedelay systems, is considered. It is shown that a DDI is Kℒstable if and only if it admits a LyapunovKrasovskii Function (LKF). Second, the LyapunovRazumikhin method, which is a type of smallgain approach for timedelay systems, is studied. It is proved that a DDI is Kℒstable if it admits a LyapunovRazumikhin 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 Kℒstability 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 timedelay 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.

Article: Necessary and Sufficient RazumikhinType Conditions for Stability of Delay Difference Equations
[Show abstract] [Hide abstract]
ABSTRACT: This technical note considers stability analysis of timedelay systems described by delay difference equations (DDEs). All existing analysis methods for DDEs that rely on the Razumikhin approach provide sufficient, but not necessary conditions for asymptotic stability. Nevertheless, LyapunovRazumikhin functions are of interest because they induce invariant sets in the underlying state space of the dynamics. Therefore, we propose a relaxation of the Razumikhin conditions and prove that the relaxed conditions are necessary and sufficient for asymptotic stability of DDEs. For linear DDEs, it is shown that the developed conditions can be verified by solving a linear matrix inequality. Moreover, it is indicated that the proposed relaxation of LyapunovRazumikhin functions has an important implication for the construction of invariant sets for linear DDEs.IEEE Transactions on Automatic Control 10/2013; 58(10):26372642. · 3.17 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The stabilisation of linear delay difference inclusions is often complicated by computational issues and the presence of constraints. In this study, to solve this problem, a receding horizon control scheme is proposed based on the Razumikhin approach and timevarying control Lyapunov functions. By allowing the control Lyapunov function to be time varying, the computational advantages of the Razumikhin approach can be exploited and at the same time the conservatism associated with this approach is avoided. Thus, a control scheme is obtained which takes constraints into account and requires solving online a lowdimensional semidefinite programming problem. The effectiveness of the proposed results is illustrated via an example that also shows the computational limitations of existing control strategies.IET Control Theory and Applications 08/2012; 6(12):19581964. · 1.84 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This paper proposes a new stability analysis method called the HybrideventTime Lyapunov Function (HTLF) approach for Discretetime Delayed Switched Systems (DDSS). Two types of stability notions (GUAS and eventGUAS) are proposed to reflect the effects on stability from events. A basic stability result is derived: a DDSS has eventGUAS if and only if there exists an HTLF which is strictly decreasing and converges to zero. Moreover, some sufficient conditions expressed by HTLFRazumikhintype stability theorems are established. The issue of impulsive stabilization for DDSS is studied. It is proved that an unstable DDSS can be stabilized by impulsive control. And the impulsive stabilization to inputtostate stability of DDSS is also achieved.SIAM Journal on Control and Optimization 04/2014; 52(2). · 1.39 Impact Factor
Page 1
LYAPUNOV METHODS FOR TIMEINVARIANT
DELAY DIFFERENCE INCLUSIONS∗
R.H. GIELEN†, M. LAZAR†, AND I.V. KOLMANOVSKY‡
Key words. Stability theory, Lyapunov functions and stability, Timedelay, 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 timedelay systems and certain types of networked control
systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. Firstly, the
LyapunovKrasovskii approach, which is an extension of the classical Lyapunov theory to timedelay
systems, is considered. It is shown that a DDI is KLstable if and only if it admits a Lyapunov
Krasovskii function (LKF). Secondly, the LyapunovRazumikhin method, which is a type of small
gain approach for timedelay systems, is studied. It is proven that a DDI is KLstable if it admits a
LyapunovRazumikhin 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 KLstability 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 timedelay 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 timedelay 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, timedelay systems and certain types of networked control systems
[14, 45]. However, while stability analysis of delayfree 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 LyapunovKrasovskii function (LKF) [27], which is an extension
of the classical LF to timedelay systems, and the LyapunovRazumikhin function
(LRF) see, e.g., [16], which is a function that is constructed based [40] on a type
of smallgain condition for timedelay systems. The main focus of this paper is on
discretetime systems. Therefore, for continuoustime 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 timedelay 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 LyapunovRazumikhin method and Lyapunov
Krasovskii approach are to be used for stability analysis of discretetime 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 LyapunovKrasovskii
approach provide a particular type of LF for the augmented state system. As such, an
equivalent notion of LKFs for discretetime 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 LyapunovKrasovskii approach,
such as the ones mentioned above for continuoustime systems, are missing. For LRFs
the situation is more complicated. The exact translation of this approach to discrete
time systems yields a noncausal constraint [11, 44]. An alternative, Razumikhinlike
condition for discretetime 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 discretetime systems, a result on the connection be
tween LKFs and LRFs is missing. Moreover, for both continuous and discretetime
systems, it remains an open question whether there exist systems that are KLstable
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 LyapunovKrasovskii 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 KLstability 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 timedelay 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 timeinvariant 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 nonnegative 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 ith component of x and let ?x?p:= (?n
the infinitynorm. 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 htimes crossproduct 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]ip)
1
p,
p ∈ Z>0, denote an arbitrary pnorm. 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 setvalued 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 timeinvariant, uncertain timevarying 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 Dhomogeneous 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 λDcontractive 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 KLstable 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 KLstable 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 KLstable.
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 KLstable.
The proof of Lemma 2.6 can be obtained mutatis mutandis from the proof of
Lemma 4.5 in [22], a result for continuoustime systems without delay. The relevance
of the result of Lemma 2.6 comes from the fact that KLstability, 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 KLstability.
With the above equivalence established, in the next section various conditions
under which a DDI is KLstable are established.
3. Stability of delay difference inclusions.
3.1. The LyapunovKrasovskii approach. As pointed out in the introduc
tion, a standard approach for studying stability of delay discretetime 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 nonempty 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)
Page 5
Lyapunov methods for timeinvariant 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 setvalued, is called homoge
neous (positively homogeneous) of order t, t ∈ Z+, if g(sx) = stg(x) (g(sx) = stg(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 λcontractive 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 setvalued map F : (Rn)h+1⇒ Rnis related to
stability of the setvalued 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 KLstable if and only if the difference inclusion (3.1) is
KLstable;
(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 KLstable. 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 KLstable.
(iii) The DDI (2.1) is KLstable.
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 LyapunovKrasovskii function (qLKF). Moreover, a function
¯V (ξ) = ?¯Pξ?∞ that satisfies the hypothesis of Corollary 3.6 is called a polyhedral
LyapunovKrasovskii 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 nonaugmented 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 timeinvariant delay difference inclusions
7
3.2. The LyapunovRazumikhin approach. The Razumikhin approach is a
Lyapunov technique for timedelay 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 KLstable.
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 noncausal, i.e., (3.8b) imposes a condition on V (x+) if V (x+)
satisfies some other condition. Note that the corresponding LyapunovRazumikhin
theorem for continuoustime 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 KLstable.
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 noncausal 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 noncausal 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 noncausal 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 KLstable.
2
k
s∈ KL completes the proof.
Page 9
Lyapunov methods for timeinvariant 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
noncausal 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 noncausal LRF. The conditions in Theorem 3.8, on
the other hand, can be reformulated as a semidefinite 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 noncausal LRFs. The interested reader is referred to
[33] for a detailed discussion on LRFs, noncausal LRFs and their differences. Therein,
it is indicated why LRFs form a less conservative test for stability when compared to
noncausal LRFs, which provides another reason for disregarding noncausal 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 continuoustime 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
KLstability 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 KLstability 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 continuoustime 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 timeinvariant 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 LyapunovRazumikhin 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 LyapunovRazumikhin 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 KLstable
(2) is KLstable
(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 wellknown that the sublevel sets of a LF are
λcontractive sets. Next, it is established that the existence of a λcontractive set and
a λDcontractive 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 λcontractive 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 normbased
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 KLstable, a
LKF does not provide a contractive set in the original, nonaugmented state space.
A LRF is based on particular Lyapunov conditions that involve the nonaugmented
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 nonaugmented system. The
above discussion indicates, apart from a lower complexity, another advantage of the
LyapunovRazumikhin method over the LyapunovKrasovskii approach.
Proposition 5.2. Suppose that the DDI (2.1) is Dhomogeneous2of order 1.
The following two statements are equivalent:
1For example, linear difference inclusions are homogeneous of order 1.
2For example, linear DDIs are Dhomogeneous of order 1.
Page 13
Lyapunov methods for timeinvariant 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 λDcontractive 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 λDcontractive set with λ := ρ
Next, the relation (ii)⇒(i) is proven. Let V denote a λDcontractive 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 Dhomogeneous 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 λDcontractive set then νV is a λDcontractive set as well. As the set νV is
λDcontractive 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 Dhomogeneous and homoge
neous of order 1, respectively. Moreover, suppose that the DDI (2.1) admits a set
V ⊂ Rnwhich is λDcontractive. 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 Dhomogeneous 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 λDcontractive.
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 semidefinite 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 nth 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 ∈
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Lyapunov methods for timeinvariant 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 closedloop 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 closedloop 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,