Robust stability of polytopic time-delay systems with
delays defined in intervals
Frederic Gouaisbaut and Dimitri Peaucelle
Abstract
Stability analysis of uncertain time-delay systems is considered in a quadratic separation framework.
Results are formulated in terms of linear matrix inequalities (LMI) which prove to be conservative but
this conservatism is significantly reduced compared to previous results by combining several tools. First
of all, the delay is splited into thinner delays to recover as much as possible some knowledge on the
functional defining the current state of the system. Secondly, a Taylor series approximation of the delay
operator is performed and the Taylor remainder is explicitly taken into account with a new uncertainty-
type approximation. Thanks to LMI formulation, and with the use of slack variables, robustness is also
guaranteed for all uncertainties and all delays included in some bounded interval which may not include
zero.
Index Terms
Linear time-delay systems, Stability, LMIs, Robustness, quadratic separation, Taylor series.
I. INTRODUCTION
Functional differential equations of retarded type with constant delays (or slowly varying)
are often used for modeling practical issues in the engineering field as well as for biology,
economics or ecology (see [21] for examples). More precisely, delays model information or
matter transportation phenomena which do not travel instantaneously and are subjected to some
delays (see for example the huge number of works dedicated to the modelisation and control
Both authors are with LAAS-CNRS, Universite de Toulouse, 7 avenue du Colonel Roche, 31077 Toulouse, FRANCE
{gouaisbaut,peaucelle}@laas.fr
of information networks [5]). They may as well serve as approximations of high dimensional
systems or hyperbolic partial differentials equations [31]. Depending on the process their value
can be assumed close to zero and analysis is needed for proving properties for the largest range
of delay values. In other cases such as in combustion [1], chatter stability in machining [18] and
others, the delay has a stabilizing effect. In that case, stabilizing delay intervals, or ”pockets”,
are searched for.
Results of this paper are intended for stability analysis with respect to uncertain delay bounded
in intervals possibly not containing zero. The main restriction is to focus on linear systems. One
single delay is assumed for sake of clarity and systems are not in neutral form. Yet, all results
may be extended to multiple-delay and neutral case without much theoretical complications (see
[16] [29] for example). Restriction to linear systems being naturally conservative, the goal is
to provide robust analysis conditions that classically allows to handle approximation made by
linearization. Constant parametric uncertainty of polytopic type is considered. Because of these
robustness issues, all results are formulated in terms of Linear Matrix Inequalities (LMIs). Indeed,
compared to analytical results such as those investigating the zeros of quasi-polynomials (see
[25], [26]) or eigenvalues problem (see the monograph [24]), LMI-based results have intrinsic
robustness characteristics keeping the analysis problem a convex polynomial-time optimization
[4].
LMI-based contributions for time-delay systems are in the literature most often related to
the production of new Lyapunov-Krasovskii functionals, [31], [10], [35], [9], relying either on
an LMI-based optimisation scheme or on a Sum-of-Square techniques [27], [30]. But in all
cases, results are conservative. Each new contribution is a step towards conservatism reduction,
usually done at the expense of increasing the numerical burden. With high numerical burden,
the Sum-of-Square technique applied to a general form for the Lyapunov-Krasovskii functional
from [16] produce results with low conservatism. But there is still a need to better understand
the construction of these functionals while keeping as low as possible the numerical burden.
For this end, results have been produced both in the IQC framework [23], [38], [19] (mainly
a frequency-domain tool) or in the Quadratic Separation framework [17] (similar technique in
the time-domain). In these two approaches Lyapunov-Krasovskii functionals are build implicitly
and are based on the Integral Quadratic Constraints which have been constructed for suitable
delay operators. More precisely, in IQC framework the operators are linear finite-dimensional
frequency domain operators issued from Pade upper and lower approximations of the infinite
dimensional delay operator, [20], [3]. Here in the paper, quadratic separation framework is
adopted by combining two differents techniques. One is Taylor series expansion of the delay
operator (e−s! = 1 ! s! + 12(s!)2 + . . . ). The second is delay fractioning (h = q h
q ). Both
these techniques are developed combining frequency-domain characteristics and time-domain
characteristics. In time-domain the mathematical formulas are reinterpreted as constraints on
augmented model signals. In frequency-domain the Taylor remainder is upper-bounded as if a
complex-valued quadratically bounded operator. The upper-bounding is studied in details and
suggested to be chosen such as to be minimal for low frequencies.
Applying a Quadratic Separation result for implicit systems [29] plus a Slack Variables result
[6], [28], [7] for dealing with robustness issues, LMI conditions are provided. All these techniques
have been exposed separately earlier in conference papers [11], [12], [14], [15]. The present paper
gathers these all and goes in further details. In particular it provides formal proof of conservatism
reduction as the Taylor series expansion and the delay fractioning grow. Numerical examples are
also exposed to illustrate the advantages of the results: for proving robust stability with respect
to uncertainties both on model parameters and on the delay; for proving stability for ”pockets”
of delays. These two features are we believe the major achievements which have no comparable
result in the literature.
The paper is organized as follows. Next section is devoted to some preliminaries about
quadratic separation. Then section III formulates precisely the Taylor series and fractioning
schemes. The following section gives the central results of the paper and discusses conservatism
reduction. Section V extends the results to robust stability with interval delays and polytopic
parametric uncertainty. Section VI illustrates the results on numerical examples.
II. PRELIMINARIES
A. Notations
For two symmetric matrices, A and B, A > (")B means that A!B is (semi-) positive definite.
AT denotes the transpose of A. 1n and 0m,n denote the respectively the identity matrix of size
+
+w
w
z
z
Figure 1. Feedback system
n and null matrix of size m # n. If the context allows it the dimensions of these matrices are
often omitted. For a given matrix B $ Rm×n such that rank(B) = r, we define B⊥ $ Rn×(n−r)
the right orthogonal complement of B by BB⊥ = 0 and B⊥B⊥T > 0. The set of complex
numbers of the right-half of the complex plane is denoted C+ = {s $ C : s + s∗ " 0}. %
stands for the Kronecker product (matrix direct product) which is such that 1k % A is a block
diagonal matrix where A is repeated k times on the diagonal. For block-diagonal matrices and
concatenated vectors the following notations are adopted
diag(
A B)
= diag
A
B
=
A 0
0 B
, vec(
x(t) y(t))
=
x(t)
y(t)
B. Review of quadratic separation
Well-posedness of feedback systems provides a fertile framework for stability analysis of
non-linear and uncertain systems. Major results for robust stability analysis have been given in
[17] and references therein. The purpose of this section is to briefly recall some new tools on
quadratic separation developed for robustness issues of descriptor systems [29], which is needed
for the main theorem of this paper.
Consider two possibly non-square matrices E and A and an uncertain constant, complex
valued, matrix & with appropriate dimensions that belongs to some set ∇. For simplicity of
notations we assume in the present paper that E is full column rank. We make no assumption
on the uncertainty set ∇.
Theorem 1: [29] The uncertain feedback system of Figure 1 is well-posed if and only if there
exists a Hermitian matrix ! = !∗ satisfying both conditions[E !A
]⊥∗!
[E !A
]⊥0 (1)
1
&
∗
!
1
&
' 0 , (& $ ∇ . (2)
If E and A are real matrices, the equivalence still holds with ! restricted to be a real matrix.
A simple corollary of that result is for LTI system stability analysis. The system x(t) =
Ax(t), where x(t) $ Rn, is stable if A has all its eigenvalues in the left half plane, which is
equivalent to the well-posedness of the feedback system of Figure 1 with E = 1n, A = A and
∇ = { s−11n , s−1 $ C+ }. Another corollary is for delay-independent stability analysis of
time-delay systems described by the equation x(t) = Ax(t) + Adx(t ! h) . Stability (for all
delays h " 0) of such system can be recast as the well-posedness of the feedback system of
Figure 1 with ∇ ={
diag(
s−11n e−sh1n
), s−1 $ C+ , h $ R+
}and appropriate E and
A matrices. Such LMI results can be found in [16] (and references therein) and are known to be
conservative due to the lossly choice of separators !. This result can also be established by using
the well-known scaled small gain theorem where the uncertainty is defined as ∇(j") = e−j"h1n
or by the use of a classical Lyapunov-Krasovskii functional V = xT (t)Px(t)+t∫
t−h
xT sQx(s)ds.
The goal of the current paper is to extend these results for reducing conservatism and proving
robust stability of time-delay systems with unknown delays in an interval.
III. APPROXIMATING THE TIME-DELAY OPERATOR
A. Taylor series and Lagrange remainder
Let s $ C and consider the function ! ) e−s! . Its Taylor series about !0 = 0 writes as
e−s! =k−1∑
i=0
1
i!(!s!)i + Rk(s, !) (3)
where Rk(s, !) is the Lagrange remainder. For any continuous time signal v(t) recall that the
e−s! and si operators are such that e−s! [v(t)] = v(t ! !) and si[v(t)] = v(i)(t). Define as well
the operator #i(s, !) = i!(!s!)−iRi(s, !) and the ri(t) signal such that #i(s, !)[v(i)(t)] = ri(t).
The Taylor series expansion imply the following formulas relating v(t! !) and the v(i)(t), ri(t)
for all i " 0 :
v(t) = v(t! !) + !r1(t) , v(i)(t) = ri(t) + !i+1ri+1(t) (4)
These formulas indicate that in order to take advantage of the Taylor series expansion of the
delay operator one needs both:
• To artificially augment the system model with higher derivatives of the state. The notations
involved in this model augmentation are given in the following section.
• To have some knowledge on the remainder representative #i(s, !). The information to be
used in the following is of norm-bounded type. But at the difference of [14] a thinner
bounded approximation is considered.
For i = 0 one has #0(s, !) = e−s! which sweeps the whole unit circle for s $ C+ but for i " 1
the domain in which the #i(s, !) operators lies is reduced and not centered at zero. For i = 1, 2
and 3 these regions are plotted in Figure 2. As the functions s *) #i(s, !) are holomorphic in
C+, the regions are delimited by the curve #i(j", !). Plots are made with a fine grid over " $ R.
Figure 2. Regions where lie the δi(s, τ) operators for all s ∈ C+ and τ ≥ 0
In order to define properly the uncertainty set which covers #i(s, !), introduce the followings
sequences.
Definition 1: Two sequences c = {ci $ R}i≥0 and $ = {$i $ R}i≥0 are said to be Taylor-
remainder valid if |#i(s, !)! ci| ' $i for all s $ C+, ! " 0 and i " 0.
This definition means simply that the complex scalars #i(s, !) belong to the covering disk centered
at ci with radius $i. As shown in Figure 2 the actual sets are not discs. Except for the trivial
case (#0(s, !) = e−s! , c0 = 0, $0 = 1), we have not found any methodology in the literature to
prove formally that a sequence is Taylor-remainder valid. But this may be done graphically. For
example, in [38] it is proposed to take $1 = 0.751, c1 = 0.249. Yet, to contribute to the issue,
two results are now provided. The first one proves by induction that the discs may be chosen
smaller or equal as i grows.
Lemma 1: If for all s $ C+ and all % $ [ 0 ! ] the complex number #i(s, %) belongs to the
disc centered at ci with radius $i, then the same property holds for #j(s, %) with j " i.
Proof : Assume #i(s, %) is in the disc centered at ci with radius $i: $−1i (#i(s, %)! ci)2 ' $i .
Apply a Schur complement argument to get
$i #i(s, %)! ci
#i(s, %)! ci $i
" 0 which also
writes as
"i(s, %) =
$i%i/i! (!s)−iRi(s, %)! ci%i/i!
(!s)−iRi(s, %)! ci%i/i! $i%i/i!
" 0 .
Note that∫ #
0 Ri(s, %)d% =∫ #
0 e−s# !∑k−1
i=01i!(!s%)id% = (!s)−1Ri+1(s, %), hence taking the
integral of "i(s, %) gives for any % $ [ 0 ! ]: "i+1(s, %) =∫ #
0 "i(s, %)d% " 0. Applying a
converse Schur complement argument implies that #i+1(s, %) belongs to the disc centered at ci
with radius $i. By induction the property holds for all j " i. !
The conclusion of Lemma 1 is that the sequences {ci = c0 = 0}k≥0 and {$i = $0 = 1}k≥0
are Taylor-remainder valid. The second result, given below, gives the formulas for the disk that
approximates the best the Taylor-remainder at low frequencies.
Lemma 2: For all i " 0, the osculating circle of #i(j", !) at point " = 0 is centered at
cosci = i
2(i+1) with radius $osci = i+2
2(i+1) .
Proof : Due to (3) one has
Ri(s, !)!Ri+3(s, !) =(!s!)i
i!+
(!s!)i+1
(i + 1)!+
(!s!)i+2
(i + 2)!
which implies due to definition of #i(s, !):
#i(s, !) = 1! !s!
i + 1+
(!s!)2
(i + 1)(i + 2)+
(!s!)3
(i + 1)(i + 2)(i + 3)#i+3(s, !) .
Lemma 1 indicates that |#i+3(s, !)| ' 1 for all s $ C+ and therefore the last terms is o(s3).
Hence the second order expansion of #i(j", !) is
#i(j", !) = 1 +j"!
i + 1! ("!)2
(i + 1)(i + 2)+ o("3)
which means that the real and imaginary parts of the function and its two first derivatives are
at point " = 0 exactly equal to:
Re(#i(0, !)) = 1, Re(#i(0, !)) = 0, Re(#i(0, !)) = 2 −!2
(i+1)(i+2) ,
Im(#i(0, !)) = 0, Im(#i(0, !)) = !i+1 , Im(#i(0, !)) = 0 .
The radius and center of the osculating circle of a parametric equation (f("), g(")) are respec-
tively defined by
$osc =(f 2 + g2)3/2
gf ! f g, cosc =
(f ! (f 2 + g2)g
gf ! f g, g +
(f 2 + g2)f
gf ! f g
)
which taking f = Re(#k(0, !)) and g = Im(#k(0, !)) proves the lemma. !
Note that the values $1 = 0.751, c1 = 0.249 proposed in [38] are close to the ones suggested
by Lemma 2: cosc1 = 0.25, $osc
1 = 0.75. Note as well (see figure 2) that graphically sequences
cosc = {cosci }, $osc = {$osc
i } are Taylor-remainder valid. Although we cannot prove it formally,
we conjecture it is indeed the case.
B. Fractioning the delay
The quality of the truncated Taylor series at order k depends on the approximation of the
#k(s, !) representative of the remainder which is exposed upper. Yet another property of Taylor
series is that the truncation is less and less conservative as ! tends to zero. For this purpose,
assuming a delay h, the methodology is applied to a fraction of the delay ! = h/q. As in [12],
based on the property that
v(t! h) = e−sh/q[v(t! (q−1)hq )] = e−s2h/q[v(t! (q−2)h
q )]= · · · = e−sh[v(t)],
the fractionning implies to augment the system model with all v(t! lhq ) signals were l = 0 . . . q.
In [11] it is demonstrated that these augmented signals allow implicitly to prove stability with
Lyapunov functions involving the q samples v(t! lhq ) of the signal v(%) for % $ [t! h t]. The
whole state of a time-delay system defined as x(%), % $ [t!h t] can hence be taken into account
asymptotically as q goes to infinity.
Note that the operators #i(s, h/q) = #i(s/q, h) are drawn towards zero frequencies as q grows.
This indicates that the approximations of these operators by osculating circles chosen at point
" = 0 are even more relevant as the fractioning q is increased. The next section exposes the
complete description of the interconnected model of system (5) with the proposed methodologies,
Taylor series expansion and delay fractionning.
IV. MAIN RESULT
A. Augmented model
Let the following time-delay system with one delay h assumed constant:
x(t) = Ax(t) + Adx(t! h), (t " 0
x(t) = &(t), (t $ [!h, 0](5)
where x(t) $ Rn is the instantaneous state and & is the initial condition. To reduce numerical
complexity of the results in case Ad is not full rank, let any factorization Ad = BC. The
effectively delayed part of the state is denoted v(t! h) = Cx(t! h) with v $ Rp with p ' n:
x(t) = Ax(t) + Bv(t! h)
v(t) = Cx(t). (6)
The augmented model with Taylor series stopped at degree k with delay fractioning q which
includes all possible relations described above is now defined. First, let the vectors x and v of
all derivatives up to order k ! 1:
x(t) = vec(
x(k−1)(t) · · · x(t) x(t))
, v(t) = vec(
v(k−1)(t) · · · v(t) v(t))
and let the vector v of delayed signals and their derivatives for all delays of the fractioning:
v(t) = vec(
v(t! q−1q h) · · · v(t! 1
qh) v(t))
.
Second, let the vectors of remainder signals with their derivatives and the vectors of signals on
which apply the operators #i:
ri(t) = vec(
r(k−i)i (t) · · · ri(t) ri(t)
), vi(t) = vec
(v(k)(t) · · · v(i+1)(t) v(i)(t)
).
The relationships between these vectors can all be formulated in terms of a feedback intercon-
nected system of Figure 1. Choosing the vectors:
z(t) = vec(
˙x(t) v(t) v1(t) · · · vk(t))
w(t) = vec(
x(t) v(t! hq ) r1(t) · · · rk(t)
) (7)
the ”uncertainty” that gathers all involved operators is defined as
& = diag(
s−11kn #0(s,hq )1kqp #1(s,
hq )1kp · · · #k(s,
hq )1p
)(8)
and the matrices E and A are constructed by the following equations:
• Augmented system equations (6)
˙x(t) = (1k % A)x(t) + (1k %B)[
1kp 0kp,kp(q−1)
]v(t! h
q ) (9)[
0kp,kp(q−1) 1kp
]v(t) = (1k % C)x(t) (10)
(1k % C) ˙x(t)! v1(t) = 0 (11)
• Internal relationships between the signals:[
0(k−1)n,n 1(k−1)n
]˙x(t) =
[1(k−1)n 0(k−1)n,n
]x(t) (12)
[1kp(q−1) 0kp(q−1),kp
]v(t) =
[0kp(q−1),kp 1kp(q−1)
]v(t! h
q ) (13)[
1p(k−i) 0p(k−i),kp
]vi(t)! vi+1(t) = 0 (14)
• Equations obtained from the Taylor series formula (4)[
0kp,kp(q−1) 1kp
]v(t) =
[0kp,kp(q−1) 1kp
]v(t! h
q ) + hq r1(t) (15)
[0p(k−i),kp(q−1) 1p(k−i) 0p(k−i),pi
]v(t) =
[0p(k−i),p 1p(k−i)
]ri(t) + h
(i+1)q ri+1(t) (16)
If one concatenates the equalities in the following order (13), (10), (9), (12), (11), the equations
(14) with increasing index i, (15) and equations (16) with increasing index i, one obtains the
expression Ez = Aw with
E = A =
0 1kp(q−1) 0 0
0 0 1kp 0
1kn 0 0 0[0 1(k−1)n
]0 0 0
1k % C
0
0 0 L1
0 0 L2 0
,
0 0 1kp(q−1) 0
1k % C 0 0 0
1k % A 1k %B 0 0[1(k−1)n 0
]0 0 0
0 0 0 0
0 0
0 1kp
0 0
L3(h)
(17)
where
L1 =
!1kp 0[1(k−1)p 0
]!1(k−1)p
. . . . . .
0[
1p 0]!1p
, L2 =
1kp[1(k−1)p 0
]
...[1p 0
]
,
L3(h) =
hq 1kp 0[
0 1(k−1)p
]h2q1(k−1)p
. . . . . .
0[
0 1p
]hkq1p
.
Note that L3(h) is affine with respect to h and that the matrix E is full rank.
B. LMI conditions for stability analysis
Based on the modeling described above, Theorem 1 can be applied.
Theorem 2: Given Taylor series maximal degree k, delay fractioning q, and a Taylor-remainder
valid sequences c, $, let L(k, q, c,$) be the LMI problem composed of equation (1) with
! =
!11 !12
!∗12 !22
,
!11 = diag(
0 !Q0 (c21 ! $2
1)Q1 · · · (c2k ! $2
k)Qk
)
!12 = diag(!P 0 !c1Q1 · · · !ckQk
)
!22 = diag(
0 Q0 Q1 · · · Qk
)(18)
where the matrices P , Qi are all symmetric, P is positive definite, the Qi are positive semi-
definite and E ,A are described by (17). The time-delay system (5) is stable if L(k, q, c,$) is
feasible.
Proof : Take ! as (18) with & defined in (8):
1
&
∗
!
1
&
= diag
!(s + s∗)P
(e−2sh/q ! 1)Q0
(|#1(s,hq )! c1|2 ! $2
1)Q1
...
(|#k(s,hq )! ck|2 ! $2
k)Qk
' 0
which proves (2) for s $ C+. !
Stability proved by Theorem 2 corresponds implicitly to some Lyapunov function V (x) such
that V (x) ' !(
z∗ w∗)
!(
z∗ w∗)∗
for all trajectories of the system Ez = Aw. The
Lyapunov function depends both on the k first derivatives of the state and on the delayed state
for the q fractions of the interval [ t! h t ].
For the most simple instance of Taylor-remainder valid circles (ci = 0, $i = 1) the Lyapunov-
Krasovskii functional expression is formally identified as given in the following lemma. The
constructed Lyapunov-Krasovskii functionals encompass several Lyapunov-Krasovskii proposed
in the litterature. The descriptor form Lyapunov-Krasovskii functional proposed by Fridman et
al, [8], [34] correspond to q = 1, k = 1, c1 = 0, $1 = 1. Others such as in [37],[36] are also
sub-cases (see [13] for a complete proof).
Lemma 3: If L(k, q, ci = 0, $i = 1) is feasible then stability of system (5) is proved by the
following Lyapunov-Krasovskii functional
V (x) = xT (t)Px(t) + V0(x) +k∑
i=1
Vi(x)
where V0(x) =∫ t
t−h/q vT (%1)Q0v0(%1)d%1 and for i " 1
Vi(x) = i!( q
h
)it∫
t−h/q
t∫
#1
. . .
t∫
#i
vTi (%i+1)Qivi(%i+1)d%i+1 . . . d%2d%1 .
To prove this lemma one needs first the next technical result:
Lemma 4: For all i " 1 on hast∫
t−!
t∫
#2
. . .
t∫
#i
v(i)(%i+1)d%i+1 . . . d%2 =! i
i!ri(t) .
Proof : For i = 1 the result is trivial due to the left-hand side equality of (4):t∫
t−!
v(%1)d%1 = v(t)! v(t! !) = !r1(t).
Now assume the property holds for some i! 1 " 1 and we shall prove it holds for i.∫ t
t−!
∫ t
#2. . .
∫ t
#iv(i)(%i+1)d%i+1 . . . d%2
=∫ t
t−!
∫ t
#2. . .
∫ t
#i!1
(v(i−1)(t)! v(i−1)(%i)
)d%i . . . d%2
= ! i!1
(i−1)!v(i−1)(t)! ! i!1
(i−1)!ri−1(t)
which, thanks to right-hand side equality of (4), is exactly ! i
i! ri(t). The property is proved by
induction for all i " 1. !
Proof of Lemma 3: First, not that the first positivity condition (see [16]) Vi(x) " '+x(t)+2
is trivially fulfilled since P is positive definite. Then, consider the time-derivatives of Vi(x):
Vi(x) = !i!(
qh
)i ∫ t
t−h/q
∫ t
#2. . .
∫ t
#ivT
i (%i+1)Qivi(%i+1)d%i+1 . . . d%2
+i!(
qh
)i ∫ t
t−h/q
∫ t
#1. . .
∫ t
#i!1vT
i (t)Qivi(t)d%i . . . d%1
= vTi (t)Qivi(t)! i!
(qh
)i ∫ t
t−h/q
∫ t
#2. . .
∫ t
#ivT
i (%i+1)Qivi(%i+1)d%i+1 . . . d%2
Since Qi " 0, a simple Schur complement argument indicates that the following matrix is
positive semi-definite
vTi (%i+1)Qivi(%i+1) vT
i (%i+1)Qi
Qivi(%i+1) Qi
" 0
thus one gets that
i!( q
h
)it∫
t−h/q
t∫
#2
. . .
t∫
#i
vTi (%i+1)Qivi(%i+1) vT
i (%i+1)Qi
Qivi(%i+1) Qi
d%i+1 . . . d%2 " 0
which due to definition of vi and to lemma 4 is exactly
vTi (t)Qivi(t)! Vi(x) rT
i (t)Qi
Qiri(t) Qi
" 0.
Thus by a reverse Schur complement argument, the following inequality holds
Vi(x) ' vTi (t)Qivi(t)! rT
i (t)Qiri(t).
This method for deriving this upper bound corresponds to Jensen’s inequality. The upper bounds
being used on time-derivatives of all Vi(x) functions, one obtains the global upper bound
V (x) ' !(
z∗ w∗)
!(
z∗ w∗)∗
.
Applying Finsler lemma implies that the derivative of the Lyapunov function is negative definite
if the matrices P , Qi are solutions of L(k, q, ci = 0, $i = 1). !
Numerically, the LMI problem L(k, q, c,$) has the following characteristics:
• The constraint (1) has n+kqp rows and columns. The additional constraints P > 0, Q0 " 0
and Qi " 0 have respectively nk, kqp and (k! i + 1)p rows and columns. The overall size
of the LMIs is (k + 1)n + (2q + (k + 1)/2)kp .
• The decision variables are the symmetric matrices constituting !. The number of scalar
decision variables is kn(kn+1)/2 for P , kqp(kqp+1)/2 for Q0 and (k! i+1)(k! i+2)/2
for the Qi matrices. The overall number of decision variables is kn2 (kn + 1) + kpq
2 (kpq +
1) + kp6 (2k2p + 3k(p + 1) + p + 3) .
These comments on the dimensions of the LMI problem L(k, q, c,$) clearly indicate that even if
one can prove some asymptotic property as both k and q grow to infinity, it would be numerically
intractable. Still some properties with respect to conservatism reduction can be produced.
C. Conservatism reduction
Proposition 1: Let two couple of vectors (c, $) and (c, $) such that, for all i, the disc centered
at ci with radius $i is included in the disc centered at ci with radius $i. In such a case, if
L(k, q, c,$) is feasible, L(k, q, c, $) is feasible as well.
Proof : #i lies in the circle centered at ci with radius $i if and only if the following quadratic
constraints holds(
1 #i
)
c2i ! $2
i !ci
!ci 1
1
#i
' 0 .
S-procedure indicates that the conditions assumed on (c, $) and (c, $) imply that there exists a
positive scalar 'i such that
c2i ! $2
i !ci
!ci 1
" 'i
c2i ! $2
i !ci
!ci 1
which suffices to conclude taking !(c, $) defined by P = P , Q0 = Q0 and Qi = 'iQi. !
Proposition 1 indicates that by improving the approximation of the #i operators, the LMI
conditions become less conservative. The results with osculating circles are therefore less con-
servative than those of [14] in which all the #i operators were restricted to lie in the unit circle
centered at zero.
Proposition 2: If L(k, q, c,$) is feasible then for all larger degrees of the Taylor series k " k,
L(k, q, c,$) is feasible as well.
This result is demonstrated in [14] for q = 1. Here it is extended to any fractioning q.
Proof : Consider the separator defined by (18). First, decompose Q0 in blocks according
to the decomposition of v(t): Q0 = (Q0,i,j)(i,j)∈{1...q}. Second, pre and post-multiply equation
(1) by(
zT (t) wT (t))
and its transpose respectively to obtain the equivalent condition to
L(k, q, c,$): for all non identically zero vectors defined by (7) the following holds
2 ˙xT (t)Px(t) +∑q
i,j=1 vT (t! i−1q h)Q0,i,j v(t! j−1
q h)! vT (t! iqh)Q0,i,j v(t! j
qh)
+∑k
i=1($2i ! c2
i )vTi (t)Qivi(t) + 2civT
i (t)Qiri(t)! rTi (t)Qiri(t) < 0 .
The inequality being strict, there exists a small perturbation ' such that the following holds as
well
2 ˙xT (t)Px(t) + 2'x(k+1)T (t)x(k)(t)
+∑q
i,j=1 vT (t! i−1q h)Q0,i,j v(t! j−1
q h)! vT (t! iqh)Q0,i,j v(t! j
qh)
+'(∑q
i=1 +v(k)(t! i−1q h)+2 ! +v(k)(t! i
qh)+2)
+∑k
i=1($2i ! c2
i )vTi (t)Qivi(t) + 2civT
i (t)Qiri(t)! rTi (t)Qiri(t)
+'(∑k
i=1($2i ! c2
i )+v(k+1)(t)+2 + 2civ(k+1)T (t)r(k−i+1)i (t)! +r(k−i+1)
i +2)
< 0 .
Conversely it proves that L(k + 1, q, c,$) is feasible with a separator ! constituted of matrices
P =
'1 0
0 P
, Q0,i,i =
'1 0
0 Q0,i,i
, Q0,i,j '=i =
0 0
0 Q0,i,j
, Qi =
'1 0
0 Qi
.
The result extends to any k " k by induction. !
Proposition 3: If L(k, q, c,$) is feasible then for any thinner fractioning q " q, L(k, q, c, $)
is feasible as well.
This result is demonstrated in [12] for sequences {ci = 0} and {$i = 1} and for k = 0,
zero order Taylor series expansion. Here it is demonstrated in another way and extended to any
Taylor-admissible sequences and any k " 0.
Proof : Let M be the[E !A
]matrix for a fractioning q and M be the same matrix for
fractioning q = q + 1. Let ! be a solution to the LMI problem L(k, q, c,$). Due to inverse
elimination lemma [32] applied to (1), there exists a matrix Y such that ! > Y M + MT Y T . In
case q > 1 (otherwise the formulas are slightly different but the reasoning is the same), M can
be partitioned as
M =
0 1kp(q−1) 0 !1kp(q−1) 0
N1 0 N2 N3 N4
and according to this partitioning define
Y T =
Y11 Y12 Y13 Y14 Y15
Y21 Y22 Y23 Y24 Y25
.
Let as well the partitioning
Q0 =
Q01 Q02
QT02 Q03
where Q01 $ Rkp×kp and Q03 $ Rkp(q−1)×kp(q−1). For the fractioning q = q + 1, M is exactly
M =
0 1kp 0 0 !1kp 0 0
0 0 1kp(q−1) 0 0 !1kp(q−1) 0
N1 0 0 N2 0 N3 N4
.
Let the matrices
Y T =
0 !12Q01 !1
2Q02 0 !12Q01 !1
2Q02 0
Y11 !12Q
T02 Y12 Y13 !1
2QT02 Y14 Y15
Y21 0 Y22 Y23 0 Y24 Y25
, Q0 =
Q01 0 Q02
0 Q01 Q02
QT02 QT
02 Q03
and let ! be constituted as (18) with matrices P = P , Q0 and Qi = Qi. The matrices defined
in this way happen to satisfy ! " Y M + MT Y T . Compared with the inequality for fractioning
q it simply corresponds to insertion of zero rows and columns. Inverse elimination lemma and
a small perturbation argument gives (1) for q = q + 1. The result extends to any q " q by
induction. !
D. Some important sub-cases
One major sub-case of time-delay systems is when stability is independent of the value of the
delay (IOD-stability). The apparently delay-dependent LMI formulations given above do allow
to handle this case.
Proposition 4: If L(k, q, c,$) is feasible with a ! matrix restricted to have Qi = 0 for all
i = 1 . . . k, then the system is stable whatever the value of the delay h.
This result is based on the fact that the LMI conditions become delay-independent when
Qi∈{1...k} = 0. A particularity is that the fractioning does not reduce conservatism in such a case:
one can restrict to q = 1. Aside from that, introducing higher order derivatives by increasing
k does reduce conservatism. The problem L(k, 1, c, $) with Qi = 0 for all i = 1 . . . k happens
to be exactly that of [2]. Note as well that L(0, q, c,$) may be feasible only for IOD-stable
systems. In all other cases one has to take k " 1.
A second major sub-case is known as ”delay-dependent” stability (DD-stability). It corresponds
to systems stable for zero delay, the problem being to find the maximal delay h such that the
system is stable for all 0 ' h ' h.
Proposition 5: If L(k = 1, q, c,$) is feasible for a delay h, then the system is stable whatever
h $ [ 0 h ].
Note that this result is demonstrated in [12] for ($1, c1) = (1, 0). Here it is extended to any
Taylor-admissible couple ($1, c1).
Proof : When k = 1 and for all h ,= 0 the matrix[E !A
]⊥can be easily written as:
[E !A
]⊥=
AT 0 CT AT CT 1n 0 0 qhCT
BT 0 0 BT CT 0 1p 0 0
0 1p(q−1) 0 0 0 0 1p(q−1)qh
0
1p
T
.
Define Q1 = 1hQ1 and let
Q =
A B 0
1n 0 0
T
0 P
P 0
A B 0
1n 0 0
+
0 0 1p(q−1)
C 0 0
T
Q0
0 0 1p(q−1)
C 0 0
![
0 1q
]T
Q0
[0 1q
]
+
CA CB 0
C 0[
0 1p
]
T
0 c1qQ1
c1qQ1 0
CA CB 0
C 0[
0 1p
]
inequality (1) can be rewritten as
Q+ ($21! c2
1)h[
CA CB 0]T
Q1
[CA CB 0
]<
q2
h
[C 0 1p
]T
Q1
[C 0 1p
].
(19)
The region where lies #1(s, !) plotted on Figure 2 includes the origin which makes that if ($1, c1)
is Taylor-remainder valid then $21 ! c2
1 " 0. Therefore if (19) holds for a value h, it also holds
for any smaller value h ' h. L(k = 1, q, c,$) is therefore feasible for the delay h with the
separator build with matrices P = P , Q0 = Q0 and Q1 = hhQ1. !
Proposition 5 also indicates that if the system is not stable for h = 0, then L(k, q, c,$) cannot
be feasible unless k " 2. LMI conditions may prove interval-delay stability only for k " 2.
V. ROBUSTNESS ISSUES
As for most analysis problems formulated in terms of LMIs, the results of the previous section
may be extended including robustness issues. In the current paper, affine polytopic uncertain
models with uncertain time-delays are considered:
x(t) = A(()x(t) + B(()v(t! h(())
v(t) = C(()x(t)(20)
where the uncertainty is defined by the vector ( =(
(1 . . . (N
)that belongs to the set
# ={
(i " 0 ,∑N
i=1 (i = 1}
and the model parameters dependency is such that
A(() B(()
C(() 0
=N∑
i=1
(i
A[i] B[i]
C [i] 0
, h(() =N∑
i=1
(ih[i] .
The data with subscripts [i] are the vertices of the polytope and the uncertain domain is the
convex hull of these vertices. Note that the E and A matrices are linear with respect to both the
model parameters A, B and C and with respect to the delay h. More precisely, denote M(()
the parameter dependent[E !A
]matrix. It can decomposed as
M(() = M0 + M1(() = M0 +N∑
i=1
(iM[i] .
Theorem 3: If there exist N separators (![i]) structured as (18) and a matrix Y such that the
following LMIs hold for all i = 1 . . . N
![i] > Y (M0 + M [i]) + (M0 + M [i])T Y T (21)
then the time delay system (20) is robustly stable.
Proof : Define the parameter-dependent separator !(() =∑N
i=1 (i![i] . The inequalities
(21) are all linear with respect to ![i] and M [i]. Therefore, taking linear combinations of these
inequalities implies that
!(() > Y (M0 + M1(()) + (M0 + M1(())T Y T = Y M(() + MT (()Y T
holds for all ( $ #. Post and pre multiply the last inequality by M⊥(() and its transpose
respectively to get (1) which concludes the proof. !
Result of Theorem 3 produces robustness properties by means of additional ”slack” variables
Y as in [6], [28]. The conditions are conservative if compared to the original infinite dimensional
problem that would be to test conditions of Theorem 2 for the infinite number of values of the
parameters ( $ #. This conservatism is due to the choice of parameter-independent ”slack”
variables Y and a special choice of !((). Nonetheless, the obtained LMI result have easy-to-
code properties and shows good behavior on examples without increasing too rapidly the size
of the LMI problem.
VI. NUMERICAL EXAMPLES
The latest version of SeDuMi [33] is used to solve the LMIs along with the parser YALMIP
[22]. Computation is done on a Linux PC computer with i686 processor and 2GB memory.
The system of [38] is tested. The data is such that
A =
0 0 1 0
0 0 0 1
!(10 + K) 10 0 0
5 !15 0 !0.25
, B =
0
0
K
0
, C =
1
0
0
0
T
.
First tests are lead for K = 1. The system is known to be DD-stable. To check this fact h is
assumed uncertain. It is chosen to lie in the interval [ h[1] h[2] ] with h[1] = 0 fixed. The goal
is by dichotomy line search to find the maximal upper bound of the interval, h[2]. Results for
various values of k and q are given in Table I. The LMIs fail for k < 1 as expected (see section
IV-D). Tests are limited to problems with less than 160 decision variables. For this example,
increasing both the Taylor series degree and the fractioning has a major contribution for getting
close to the actual maximal bound 1.4247.
Table I
h[2] OBTAINED BY THEOREM 3 (NUMBER OF DECISION VARIABLES GIVEN BETWEEN BRACKETS)
q=1 q=2 q=3 q=4 q=5
k = 1 0.2528 (37) 0.2541 (43) 0.2543 (49) 0.2544 (55) 0.2544 (61)
k = 2 1.1953 (79) 1.3229 (91) 1.3432 (103) 1.3494 (115) 1.3519 (127)
k = 3 1.3606 (127) 1.4195 (145)
The interesting feature of Theorem 3 is that stability is guaranteed for all the interval [ 0 h[2] ].
If removing this robustness specification and searching by dichotomy line search the maximal
stabilizing delay with the help of Theorem 2 gives even better bounds (higher values of k can
be tested because numerical complexity is about 10 times lower). Tests are done for various
values of k = q and for two choices of the (c, $) Taylor-remainder valid vectors. Results are
given in Table II. As expected due to Proposition 1 proper approximation of the regions where
lie the #i(s, !) operators has a non negligible effect on conservatism. This effect seems to vanish
for large values of k and q but this is at the expense of increasing the numerical burden. For
this example, combining the two techniques (Taylor series expansion and fractioning of delays)
gives results with vanishing conservatism as both k and q grow. We have tested other examples
and in all cases this vanishing conservatism was observed.
Table II
h[2] OBTAINED BY THEOREM 2 (NUMBER OF DECISION VARIABLES GIVEN BETWEEN BRACKETS)
k=q=1 (18) k=q=2 (39) k=q=3 (68) k=q=4 (105) k=q=5 (150)
(ci, αi) = (0, 1) 0.1268 1.1644 1.3844 1.4019 1.4236
(ci, αi) = (cosci , αosc
i ) 0.2528 1.3229 1.4237 1.4243 1.424626
Although only DD-stability has been tested for this example in the literature, it happens that
depending on the value of gain K, the system is either IOD-stable, DD-stable and sometimes has
”pockets” of stabilizing values of delay. The structure of the data (p = 1) makes it possible to
evaluate the stability regions of the systems in the (K, h) plane by a grid over K and computing
delay-margins with the allmargin function of the Control Toolbox of Matlab c-. In Figure
3 the filled regions of the (K, h) plane correspond to destabilizing values. With the help of
Theorem 3 inner approximations of the stabilizing regions are obtained (polytopes in Figure 3):
Figure 3. Stability regions in the (K, h) plane
• For k = 1, q = 1, with the additional constraint Q1 = 0, the LMIs of Theorem 3 are
feasible for the polytope K $ [0 0.2959], thus proving IOD-stability of the system for all
such gains (at the left-hand side of the vertical line in Figure 3).
• For k = 1, q = 3, the LMIs of Theorem 3 are feasible for the polytopes defined by the
following sets of vertices (computation time is less than 1 second)
(K [i], h[i]) $ { (0.2959, 0.70) , (0.4, 0.70) } , (K [i], h[i]) $ { (0.4, 0.43) , (0.6, 0.43) }
(K [i], h[i]) $ { (0.6, 0.25) , (1, 0.25) }
thus proving DD-stability in the regions covered by these ranges for K and these maximal
values on h (regions below the three horizontal lines in Figure 3).
• For k = q = 2, the LMIs of Theorem 3 are feasible for the polytopes defined by the
following sets of vertices (computation time is about 3 seconds)
(K [i], h[i]) $ { (0.2959, 0) , (0.2959, 1.7) , (1.2, 1.2) , (1.2, 0) }
(K [i], h[i]) $ { (1.2, 0) , (1.2, 0.7) , (10, 0.5) , (10, 0) }
(K [i], h[i]) $ { (1.2, 0.7) , (1.2, 1.2) , (1.6, 1) , (1.6, 0.67) }
thus proving stability in these regions that improve the upper bounds found for DD-stability
(see the quadrilaterals in Figure 3).
• For k = q = 3, the LMIs of Theorem 3 are feasible for the polytopes defined by the
following sets of vertices (computation time is about 200 seconds)
(K [i], h[i]) $ { (1.2, 1.2) , (1.74, 1.2) , (1.74, 0.93) }
(K [i], h[i]) $ { (0.2959, 2.6) , (0.2959, 4.2) , (1.3, 3) }
thus proving stability in the convex hull regions (see the two bottom triangles in Figure
3). The first region improves the regions found for DD-stability. The second is included
inside a ”pocket”. Note that applying Theorem 2 for a grid over K with k = q = 3 gives
exactly the DD-stability curve with less than 1% error (not shown of the figure). This result
is comparable to the best results from the literature [38].
• For k = q = 4, the LMIs of Theorem 3 are feasible for the polytopes defined by the
following sets of vertices (computation time is about 2300 seconds)
(K [i], h[i]) $ { (1.74, 1.2) , (2, 1.2) , (1.74, 1.05) }
(K [i], h[i]) $ { (0.2959, 5.5) , (0.2959, 6.5) , (0.6, 6) }
thus proving stability in these ”pocket” regions (the two triangles at the top of Figure 3).
VII. CONCLUSIONS
A novel method for stability analysis of time-delay systems is provided in the quadratic
separation framework. Inherent conservatism of LMI-based methods is reduced by the means of
three techniques. One is fractioning of the delay. The second is Taylor series representation of
the delay operator. The third is norm-bounded representation of the Taylor series remainders in
circles not centered at zero. Many properties of the main result are derived, including robustness
issues. Numerical examples show the effectiveness of the results and illustrate possible vanishing
conservatism as both fractioning index and degree of Taylor series representation are increased.
Proving this vanishing conservatism is an important issue for future work.
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