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Partial State Estimation of LTI Systems with Multiple
Constant Time-Delays
Reza Mohajerpoora,∗, Hamid Abdia, Lakshmanan Shanmugama, Saeid
Nahavandia
aCentre for Intelligent Systems Research (CISR), Deakin University, Waurn Ponds,
Australia
Abstract
Functional observer design for Multi-Input Multi-Output (MIMO) Linear Time-
Invariant (LTI) systems with multiple mixed time delays in the states of the
system is addressed. Two structures for the design of a minimum-order ob-
server are considered: 1- delay-dependent, and 2- internal-delay independent.
The parameters of the delay-dependent observer are designed using the Lya-
punov Krasovskii approach. The delay-dependent exponential stability of the
observer for a specified convergence rate and delay values, is guaranteed upon
the feasibility of a set of Linear Matrix Inequalities (LMIs), together with a
rank condition. Using the descriptor transformation, a modified Jensen’s in-
equality, and improved Park’s inequality, the results can be less conservative
than the available functional observer design methods that address LTI systems
with single state delay. Furthermore, the necessary and sufficient conditions of
the asymptotic stability of the internal-delay independent observer are obtained,
which are shown to be independent of delay. Two illustrative numerical exam-
ples and simulation studies confirm the validity and highlight the performance
of the proposed theoretical achievements.
Keywords: Functional observer; Time-delay systems; Linear matrix in-
equality; Lyapunov Krasovskii functional
∗Corresponding author
Email address: reza.mohajerpoor@deakin.edu.au (Reza Mohajerpoor )
Preprint submitted to Elsevier December 22, 2015
1. Introduction
Functional observers (FOs) are a class of observers that deal with estimating
one or multiple functions of the states of a system [1, 2, 3]. This type of ob-
servers have a wide range of applications in fault diagnosis, system monitoring,
and observer based control of dynamic systems, specifically large scale systems
[4, 5, 6, 7, 8]. This is because, in these applications, it is usually not necessary
to observe the whole set of the states of the system, and the computational
costs can also be significantly reduced by decreasing the order of the observer.
Furthermore, observability requirement is replaced by a less conservative as-
sumption, called functional observability [9, 10, 11].
In addition, time-delay systems have been the focus of research for many
years [12, 13, 14, 15, 16]. This is due to their wide range of applications, and
also the theoretical difficulties that are invoked by the presence of time-delays.
Moreover, the problem of observer design for time-delay systems has recently re-
ceived a significant amount of attention (see e.g. [17, 18, 19, 20, 21]). Advanced
problems in this topic has been covered, including considering multiple stochas-
tic state delays in discrete time systems [17, 18], nonlinear systems [19, 22],
and robust fault diagnosis of sampled-data control systems [21]. Furthermore,
observer-based controller design for time-delay systems has been considered by
many researchers as in [23, 24], due to their practical importance.
On the other hand, delay-dependent functional observer design for heredi-
tary LTI systems, has been fairly overlooked, and there are a few number of
contributions in this regard (see e.g. [25, 26, 27, 28]). Due to the theoreti-
cal complexities that are invoked from both functional observer design and the
presence of time-delay, this topic still has a lot of open concerns. This in-
cludes unsolved challenges and also the improvement of the existing solutions.
The main focus in this field has been devoted to LTI systems with single state
delay [25, 27, 29, 30]. Darouach [25], considered delay-dependent and delay-
independent minimum functional observer design for LTI systems, with single
state delay, for the first time. Later on, he solved the same problem for the case
2
of unknown-input functional observers (UIFOs)[26]. There are other contribu-
tions in this regard, each one addressing a specific unsolved problem [27, 28, 30].
However, to the best of our knowledge none of the contributions that con-
cern FO design for time-delay systems, considers multiple mixed delays in the
states of the system. Moreover, most of the papers in this area consider restric-
tive assumptions in choosing and dealing with Lyapunov Krasovskii functionals
(LKFs), and the cross terms that appear in the analysis of these terms. To be
more specific, Newton-Leibniz formula, as well as Cauchy-Schwarz and Young’s
inequalities are some conservative assumptions, which are employed in the ma-
jority of the previous works on the this topic (see e.g. [25, 27, 28, 30]). This
problem can be intensified by considering multiple state delays in the system.
Furthermore, only a few of the available works consider the regulation of the
convergence rate of the observers designed through the delay-dependent stability
analysis [28, 30].
The above motivated us to propose a new methodology to design minimum-
order delay-dependent (DD) globally exponentially stable FOs for LTI systems
with multiple mixed state delays. The necessary and sufficient conditions for
the exponential stability of the delay-dependent observer structure are proposed.
In addition, for design purposes, delay-dependent sufficient conditions for the
stability of the observer for a given exponential convergence rate and the delays
values are established using the Lyapunov Krasovskii methodology, and are
expressed in the LMI format. Using the descriptor transformation [31, 32], along
with the improved Park’s inequality [33], and a modified Jensen’s inequality [34],
can result in less conservative stability conditions, compared with the recent
papers on this topic that also only consider systems with a single-state-delay.
Additionally, an internal-delay independent (IDI) functional observer is designed
for the system. The obtained necessary and sufficient conditions for the global
asymptotic stability of the IDI observer, shows a simpler design scheme with
regard to the DD observer structure. Moreover, an instructive criteria to select
the appropriate observer structure is proposed.
The paper is structured as follows. The illustration of the problem and some
3
preliminaries are given in Section 2. The design of the delay-dependent observer
scheme is demonstrated in Section 3.1. The internal-delay independent observer
design algorithm is illustrated in Section 3.2. Next, two numerical examples and
simulation results proposed in Section 4 explain the efficacy and the performance
of the proposed methodologies. Finally, the conclusions and future studies are
given.
1.1. Notations
The following notations are employed throughout the paper: [X;Y] repre-
sents
X
Y
, where Xand Ycan be scalars, vectors, or matrices of appropriate
dimensions. Moreover, diag(a, b, · · · ) stands for a square diagonal matrix with
diagonal elements equal to a, b, · · · ;X†is the pseudo-inverse or the generalised
inverse of the matrix X; and X⊥is the right orthogonal of Xin a way that
XX⊥=0. Further, ρ(X) is the rank of the matrix X, and sym(X) = X+XT.
Additionally, In,0m×n, and 0are n×nidentity matrix, m×nzero matrix, and
a zero matrix of appropriate dimension, respectively. In addition, R,C,C+
sequentially are the set of real numbers, the set of complex numbers, and com-
plex numbers with positive real parts; Rm×nis the space of m×nreal matrices;
Snis the space of n×nreal and symmetric matrices; and Cn(Ω) is the space
of continuous functions, mapping from Ω to Rnwith the topology of uniform
convergence. Furthermore, |x|is the norm of the vector x, and ∗as the (i, j)′th
element of a symmetric matrix stands for the transpose of its (j, i)′th element.
Finally, X≻0(≺0), X⪰0(⪯0), and X≻0∈Rn×nindicate that the ma-
trix Xis positive-definite (negative definite), positive semi-definite (negative
semi-definite), and n×nsymmetric positive-definite, respectively.
2. Problem Statement and Preliminaries
LTI systems with multiple constant state delays with the below dynamics
are considered,
4
˙x(t) = A0x(t) +
2
i=1
Aix(t−hi) + Bu(t)
y(t) = Cx(t)
z(t) = Lx(t)
x(t) = ϕ(t)∀t∈[−hu,0],
(1)
where x(·)∈Rn, u(·)∈Rm, y(·)∈Rpare the vectors of the states, inputs, and
outputs of the system. Moreover, z(·)∈Rlis the functional to be estimated.
Furthermore, Ai∈Rn×n, i ={0,1,2},B∈Rn×m,C∈Rp×n, and L∈Rl×nare
known and constant distribution matrices. In addition, hi>0, i ={1,2}are
the known and constant state delays, and hu=max{hi}, i ={1,2}. Finally,
ϕ(·)∈ Cn([−hu,0]) is the initial function of the states of the system.
Remark 1. To keep the simplicity of the notations, only two time-delays are
considered. However, the extension of the proposed theories to systems with
more than two delays is straightforward.
The following assumptions are made throughout the paper:
i. The matrices Cand Lare full row rank,
ii. The matrix Cis in the canonical form [Ip,0],
iii. The time-delays are non-zero, known, and constant.
Assumptions iand ii clearly do not fail the generality of the paper. For
example, if Assumption ii is not initially satisfied, a similarity transformation
using ¯
C:= [C†, C⊥] as explained in [35] can hold the assumption. However,
including time-varying and/or unknown delays are outside the scope of this
manuscript, and require further study.
Our goal for the delay-dependent observer design problem is to obtain ob-
server parameters that satisfy delay-dependent sufficient conditions, in a way
that the estimated functional ˆz(t)exponentially tracks its true values with a
convergence rate equal to a pre-selected constant α > 0.
5
Definition 1. A minimum-order functional observer for the system (1) is α−exponentially
stable, if for any initial condition ϕ(·)∈ Cn([−hu,0]), the estimation error
e(·) := ˆz(·)−z(·) satisfies
|e(t, ϕ(θ))|c≤e−αt|e(0, ϕ(θ))|,∀t≥0,∀θ∈[−hu,0]
where |e(t)|c=sup (|e(t+θ)|,∀θ∈[−hu,0]).
In addition, in the design of the internal-delay independent FO, we aim to
obtain the necessary and sufficient conditions that the designed minimum-order
observer is asymptotically stable, i.e. limt→∞ e(t) = 0.
Lemma 2.1 (Park’s inequality [33]).For any vectors a∈R2n,b∈Rn, and
matrices Z∈S2n,R∈Sn,Y∈R2n×n, and H ∈ Rn×2n, one has
−2bTHa⪯
a
b
T
Z Y − HT
YT− H R
a
b
(2)
if and only if
Z Y
∗R
⪰0.(3)
Lemma 2.2 ([34]).Given a symmetric positive-definite matrix M, and scalars
b>a, and α > 0, the following inequality is satisfied for any function f∈
Cn([a, b]),
b
a
eα(s−b)fT(s)Mf (s)ds ≥α
eα(b−a)−1b
a
fT(s)dsMb
a
f(s)ds.(4)
6
3. Main Results
3.1. Delay-Dependent Observer Design
3.1.1. Observer Structure and Stability Analysis
A minimum-order (l′th order) observer with the below structure is employed:
˙ω(t) = F0ω(t) +
2
i=1
Fiω(t−hi) + Gu(t) + H0y(t) +
2
i=1
Hiy(t−hi)
ˆz(t) = ω(t) + V y(t)
ω(t) = 0,∀t∈[−hu,0]
(5)
where ω(·)∈Rlis the observer’s state, Fi, Hii={0,1,2}, G, and Vare con-
stant matrices of appropriate dimensions. Let us define the auxiliary error signal
ϵ(·) := ω(·)−T x(·), where T∈Rl×nis a constant matrix. The necessary and
sufficient conditions of the exponential stability of the observer, are summarized
in the following theorem.
Theorem 3.1. The functional observer (5) is globally α-exponentially stable if
and only if
(a) The error dynamics
˙ϵ(t) = F0ϵ(t) +
2
i=1
Fiϵ(t−hi)
ϵ(θ) = −T ϕ(θ),∀θ∈[−hu,0]
(6)
is α-exponentially stable.
(b) There exists a matrix T, such that the observer parameters obey the fol-
lowing set of matrix equations,
T+V C −L=0(7a)
FiT−T Ai+HiC=0, i ={0,1,2}(7b)
G=T B (7c)
7
Proof.Differentiating ϵ(t) along the solutions of (5) and (1) gives,
˙ϵ(t) = F0ϵ(t) +
2
i=1
Fiϵ(t−hi)+(F0T−TA0+H0C)x(t)
+
2
i=1
(FiT−T Ai+HiC)x(t−hi)+(G−TB )u(t)
(8)
Hence, if there exists a matrix T, such that conditions (7b) to (7c), as well
as Condition (a) are satisfied, then ϵ(t) is globally α-exponentially stable. Next,
the calculation of the error signal e(t) gives,
e(t) = ω(t) + V Cx(t)−Lx(t)
=ϵ(t)+(T+V C −L)x(t)
(9)
Consequently, if condition (7a) is achieved, then the estimated functional,
ˆz(t), globally exponentially tracks its actual value with a convergence rate equal
to α. This is due to the α-exponential stability of the error signal ϵ(t).
The proof of the necessity part is straightforward, and can be followed as
explained in [25]. The theorem is thus verified.
Now, a delay-dependent criteria is established for the exponential stability
of the error dynamics (6) in the following theorem.
Theorem 3.2. For given positive scalars h1, h2, and α, the system (6) is glob-
ally α-exponentially stable if there exist matrices P1≻0∈Rl×l,P2∈Rl×l,
P3∈Rl×l,Yi∈R2l×l,Si≻0∈Rl×l,Ri≻0∈Rl×l,¯
Ri∈Sland Zi∈S2l,
i={1,2}, that satisfy the following matrix inequalities,
Γ≺0,(10)
ZiYi
∗¯
Ri
⪰0, i ={1,2},(11)
8
where
Γ :=
Γ10 0 Γ2Γ3
∗ −e−2αh1S10 0 0
∗ ∗ −e−2αh2S20 0
∗ ∗ ∗ ¯
R1−2α
e2αh1−1R10
∗ ∗ ∗ ∗ ¯
R2−2α
e2αh2−1R2
,
Γ1:= PT
0I
2
i=0
Fi−I
+
0
2
i=0
FT
i
I−I
P+
2
i=1
Zi+
2
i=1
Si0
0hiRi
+2αEP,
Γ2:= Y1−PT
0
F1
, and Γ3:= Y2−PT
0
F2
.
Proof.The theorem is proved in Appendix A.
Remark 2. The restrictive assumptions that is adopted in deriving the matrix
inequalities of Theorem 3.2 come from the bounding techniques that are used re-
garding to the cross terms in (A.4) and (A.6). However, employing the improved
Park’s inequality and the descriptor transformation, the results can be less con-
servative than the DD analysis of other papers in this area such as [25, 27, 30].
This is because they generally use Young’s inequality, and/or Cauchy-Schwarz
inequality in conjunction with the conventional Leibniz-Newton transformation
formula. The extra restrictive nature of the latter methods, is shown and dis-
cussed in different papers, such as [36]. This issue is magnified for the problem
of this note, which is addressing multiple number of delays.
More specifically, the descriptor transformation is an effective alternative
to the Leibniz-Newton transformation that is applied to the LKF (A.1). This
transformation provides two extra slack matrices P2and P3, and also add to the
dimension of the matrix inequality (10) by two block rows and columns. As a
result, additional diagonal and off-diagonal terms in Γ is created in an effective
9
way, which increase the degrees of freedom of the matrix inequality (10), and
can eventually result in larger stability regions.
To find the observer parameters, it is sufficient to find a solution for the
interconnected equations (7), such that for given h1, h2, and α, there exist pa-
rameters Pj, Yi, Si, Ri,¯
Ri, and Zi, j ={0,1,2}, i ={1,2}, such that the matrix
inequalities (10) and (11) are satisfied. Nevertheless, due to the presence of
the terms
2
i=0
P2Fi, and
2
i=0
P3Fiin the definition of Γ, the inequality (10) is
nonlinear. This problem, is resolved in the next section.
3.1.2. Observer Design
To solve interconnected matrix equations (7), a transformation-based ap-
proach using the transformation matrix ¯
Cdefined in Section 2 is chosen. First,
the following parameters are introduced,
T1T2:= T¯
C(12)
L1L2:= L¯
C(13)
Ai
11 Ai
12
Ai
21 Ai
22
:= ¯
C−1Ai¯
C, i ={0,1,2}(14)
where T1∈Rl×p, T2∈Rl×(n−p), L1∈Rl×p, L2∈Rl×(n−p),Ai
11 ∈Rp×p, Ai
12 ∈
Rp×(n−p), and Ai
22 ∈R(n−p)×(n−p),i={0,1,2}. Next, post-multiplying (7a)
and (7b) by ¯
C, results in the below set of equations,
T2=L2(15)
T1+V−L1=0(16)
FiT1−T1Ai
11 −T2Ai
21 +Hi=0, i ={0,1,2}(17)
FiT2−T1Ai
12 −T2Ai
22 =0, i ={0,1,2}(18)
Now, considering (15) and (18), we have,
10
F0F1F2−T1Ω = Φ (19)
where
Ω :=
L20 0
0L20
0 0 L2
A0
12 A1
12 A2
12
and Φ := L2A0
22 L2A1
22 L2A2
22 . It can be shown that (19) has a solution
if and only if the below condition is fulfilled [37],
Condition I
ρ
L2A0
22 L2A1
22 L2A2
22
A0
12 A1
12 A2
12
L20 0
0L20
0 0 L2
=ρ
A0
12 A1
12 A2
12
L20 0
0L20
0 0 L2
(20)
If Condition I is achieved, then it is concluded from (19) that,
F0F1F2−T1=U1+¯
ZU2(21)
where U1:= ΦΩ†,U2:= I3l+p−ΩΩ†, and ¯
Z∈Rl×(3l+p)is an arbitrary param-
eter. Hence, the below can be written from (21),
Fi=U1i+¯
ZU2i, i ={0,1,2}(22)
and
−T1=U13 +¯
ZU23 (23)
where U1iand U2i,i={0,1,2,3}are the partitions of U1and U2with appro-
priate dimensions. Substituting Fis, i={0,1,2}into Γ, it is observed that the
matrix inequality (10) has nonlinear terms. The main achievement of the paper
11
resolves this problem, and is outlined in the following theorem.
Theorem 3.3. Upon the satisfaction of Condition I, the functional observer
(5) is globally α-exponentially stable if for given positive scalars h1, h2, and α,
(˜a) there exist parameters Q1≻0∈Rl×l,Q2∈Rl×l,Q3∈Rl×l,¯
Zi
11 ∈Sl,
¯
Zi
22 ∈Sl,¯
Zi
12 ∈Rl×l,¯
Yi∈R2l×l,˜
Ri∈Sl,i={1,2},¯
S≻0∈Rl×l,
Ki∈Rl×l,i={0,1,2}, and positive scalars αr, αs, and αq, such that the
following LMIs are satisfied,
˜
Γ≺0,(24)
and
¯
Zi¯
Yi
∗˜
Ri
⪰0, i ={1,2},(25)
where
¯
Zi:=
¯
Zi
11 ¯
Zi
12
∗¯
Zi
22
,
˜
Γ :=
˜
Γ10 0 ˜
Γ2˜
Γ3
∗ −e−2αh1¯
S0 0 0
∗ ∗ −αse−2αh2¯
S0 0
∗ ∗ ∗ ˜
R1−2ααq
e2αh1−1Q10
∗ ∗ ∗ ∗ ˜
R2−αr
2ααq
e2αh2−1Q1
,
(26)
˜
Γ1:=
˜
Ξ1˜
Ξ2Q1QT
2
∗ −sym(Q3) +
2
i=1
¯
Zi
22 0QT
3
∗ ∗ − 1
1+αs
¯
S0
∗ ∗ ∗ − αq
h1+αrh2Q1
,(27)
12
˜
Γ2:=
¯
Y1−
0
αqU11Q1+αqK1
0
,˜
Γ3:=
¯
Y2−
0
αqU12Q1+αqK2
0
,
˜
Ξ1:= sym(Q2)+
2
i=1
¯
Zi
11 +2αQ1, and ˜
Ξ2:= Q3+
2
i=0
Q1UT
1i−QT
2+
2
i=1
¯
Zi
12 +
2
i=0
KT
i.
(˜
b) the below rank condition is fulfilled,
Condition II
ρ
U20Q1U21 Q1U22Q1
K0K1K2
=ρ U20Q1U21 Q1U22Q1
(28)
In addition, the observer design parameter ¯
Zcan be computed from the below
equation
¯
Z=¯
K؆(29)
where ¯
K:= K0K1K2, and Ψ := U20Q1U21 Q1U22Q1.
Proof.The theorem is proved in Appendix B.
Remark 3. Although the satisfaction of Condition II is necessary for solving
(B.1), if it is not achieved, a matrix ¯
Zcan still be found from (29) such that the
constrained equations (7) are satisfied. As a result, a different set of matrices
Ki,i={0,1,2}are found, which still might satisfy the LMI (24). However,
finding the conditions under which this scenario works, is beyond the scope of
this paper, and needs further analysis.
Remark 4. The parameters αr, αs, and αqare tuning parameters that cannot
be directly calculated via solving LMIs (24) and (25). Otherwise the inequal-
ity (24) becomes nonlinear. However, optimization techniques such as genetic
algorithm can still be utilized in parallel, in order to adequately tune these
parameters.
3.2. Internal-Delay Independent Observer Design
The following observer structure is considered for this part,
13
˙ω(t) = F0ω(t) + Gu(t) + H0y(t) +
2
i=1
Hiy(t−hi)
ˆz(t) = ω(t) + V y(t)
ω(t) = 0,∀t∈[−hu,0]
(30)
where F0,Hi, i ={0,1,2},G,Vare as defined for the observer structure (5).
Corollary 3.4. The observer (30) is a globally asymptotically stable functional
observer for the system (1), if and only if
(ˆa) the matrix F0is strictly stable,
(ˆ
b) there exists a matrix T, such that the below equations are satisfied,
T+V C −L=0(31a)
F0T−T A0+H0C=0(31b)
−T Ai+HiC=0, i ={1,2}(31c)
G=T B (31d)
Proof.The proof is similar to the proof of Theorem 3.1. Thus it is omitted.
Following the same procedure as explained in Section 3.1.2, the following
equation is obtained for the observer parameters,
F0−T1Ω1= Φ1(32)
where Ω1:=
L20 0
A0
12 A1
12 A2
12
, and Φ1:= L2A0
22 L2A1
22 L2A2
22 .
Thereby, (32) has a solution if and only if the below condition is met,
14
Condition III
ρ
L2A0
22 L2A1
22 L2A2
22
A0
12 A1
12 A2
12
L20 0
=ρ
A0
12 A1
12 A2
12
L20 0
(33)
If Condition III is satisfied, then the unknown parameters of (32) can be
obtained from,
F0−T1= Φ1Ω†
1+˜
Z(Ip+l−Ω1Ω†
1) (34)
where ˜
Z∈Rl×(l+p)is an arbitrary matrix. Analogously, defining ˜
U1:= Φ1Ω†
1
and ˜
U2:= Ip+l−Ω1Ω†
1,
F0=˜
U10 +˜
Z˜
U20 (35)
and
−T1=˜
U11 +˜
Z˜
U21 (36)
where ˜
U1=: ˜
U10 ˜
U11 and ˜
U2=: ˜
U20 ˜
U21 . Now, Condition (ˆa) of
Corollary 3.4 is satisfied, if and only if the pair ( ˜
U10,˜
U20) is detectable, or
equivalently the below is achieved,
Condition IV
ρ
sIp−˜
U10
˜
U20
=l, ∀s∈C+(37)
Accordingly, after finding a suitable matrix ˜
Z, such that the matrix F0is
Hurwitz, possibly with some desired eigenvalues, the other observer parameters
can be readily found in a way that Condition (ˆ
b) of Corollary 3.4 is satisfied.
The following theorem summarizes the above design algorithm.
Theorem 3.5. There exists a globally asymptotically stable internal-delay in-
15
dependent functional observer (30) for the system (1) if and only if Conditions
III and IV are satisfied. Moreover, if Condition IV is met for every point s
in the complex plane, then arbitrary convergence rate can be regulated for the
observer.
Remark 5. The important advantage that is brought by the observer structure
(30) is that the dynamics of the error signal ϵ(·) is an ODE, ˙ϵ(t) = F0ϵ(t), t ≥0.
Thereby, the stability analysis of this system is much easier, and also results
in the necessary and sufficient stability conditions. However, the conditions are
considerably more conservative than those obtained from the delay-dependent
observer structure. This has been emphasized in Section 4 through a numerical
example. Hence, in practice it can be advised to use the IDI structure if Con-
ditions III and IV are fulfilled. Otherwise, upon the satisfaction of conditions
of Theorem 3.3, the delay-dependent observer structure (5) can be employed.
4. Numerical Examples
4.1. Example 1
An LTI system with the following parameters was studied,
A0=
0 0 −1 0
0−1 0 0.1
2 3 −1 0
2−1 0 −1
, A1=
03×4
−3 0.120
, A2=
0 0 0 0
0 0.8 0 0
0 0 0 0
0 0.200.4
,
C=I202,B= [0; 1; 0; 0], and L=−3 0.1 2 0 .
4.1.1. Delay-dependent observer design
Firstly, it can be observed that Condition I is satisfied. Now, consider the
case α= 0.001. Considering h1= 5sand h2= 6s, and by applying Theorem
3.3, the system of LMIs (24) and (25) were successfully solved, when αr=
αs=αq= 1. As a result, the genuine observer parameters were obtained
16
as, F0=−0.0883, F1=F2=G= 0, H0=3.8390 6 ,H1=H2=
0 0 , and V=−1.1766 0.1. In addition, the LMI parameters were
achieved as Q1= 4.8157, Q2=−0.4253, Q3= 0.4237, ¯
S= 152.6392, K0=
3.4272, K1=−3.8293e−14, K2= 1.7534e−14, ˜
R1= 0.4792, ˜
R2= 0.3989,
¯
Y1= [5.2696e−17; −1.9146e−14], ¯
Y2= [1.7629e−17; 8.7672e−15], ¯
Z1=
[0.0485,−7.2807e−16; −7.2807e−16,0.1418], and ¯
Z2= [0.0485,−4.5996e−
16; −4.5996e−16,0.1418]. Moreover, it can be seen that these parameters satisfy
Condition II.
As the second scenario, a faster convergence rate was demanded by setting
α= 2. Assuming the same values of delays as h1= 5sand h2= 6s, and
the tuning parameters αr, αs, and αqequal to 1, it was seen that the LMIs
(24) and (25) are not feasible. However, setting αr=αs= 1, and αq= 50,
results in the feasibility of the LMIs. Accordingly, the observer parameters
were attained as, F0=−4.4364, F1=F2=G= 0, H0=34.4912 6 ,H1=
H2=0 0 , and V=−9.8729 0.1. Moreover, the LMI parameters
were obtained as Q1= 266.5977, Q2=−1158.4, Q3= 563.1807, ¯
S= 2053,
K0=−969.4684, K1=−2.0821e−20, K2=−2.3782e−21, ˜
R1= 5.4950e−
05, ˜
R2= 1.0064e−06, ¯
Y1= [7.2625e−22; −5.1186e−19], ¯
Y2= [4.2278e−
22; −5.8190e−20], ¯
Z1= [24.7248,−5.7710e−12; −5.7710e−12,288.9564], and
¯
Z2= [24.7248,−6.8490e−12; −6.8490e−12,288.9564]. The satisfaction of
Condition II can also be observed for this case.
4.1.2. Internal delay-independent observer design
Examining Conditions III and IV, it is observed that the system parame-
ters satisfy both conditions. Hence, according to Theorem 3.5, there exists an
asymptotic FO with structure (30) for the system. The observer equations (31)
were solved by setting the desired eigenvalue of the observer at the point −1.
Accordingly, the following parameters were obtained, F0=−1, H0=4 6 ,
H1=H2=0 0 ,G= 0, and V=−3 0.1.
Simulations were performed with an arbitrary input signal equal to:
17
u(t) = 2 + 10e−0.4tcos(2t), t ≥0 (38)
Fig. 1 demonstrates the results obtained from applying different observers
illustrated above. According to the figure, the designed IDI observer performs
better than the DD observer with α= 0.001, but it is slower than the the other
one with α= 2.
However, it should be emphasized that the internal delay-independent ob-
server significantly outperforms the delay-dependent observer in this example.
There are three main reasons for this claim:
1. The IDI observer has a simpler structure, and needs much less computa-
tional burden (since it does not need solving a semi-definite programming
optimization problem).
2. The IDI observer can easily provide a much better performance by reg-
ulating the location of the eigenvalue of the observer at a larger negative
point. Roughly speaking F0=−1 a performance close to the second DD
observer scenario with α= 2.
3. More importantly, the IDI observer’s stability conditions are completely
independent of the values of the time-delays.
Hence, using the IDI observer is always preferable to the DD one, in case
conditions of Theorem 3.5 are satisfied. This point has also been emphasized in
Remark 5.
4.2. Example 2
Consider the time-delay system (1) with the below parameters,
A0=
−10 1 0 0
−48.6−1.26 48.6 0
0 0 −22 1
1.95 0 −19.5−6
, A1=
2−1 0 0
−1 3 0 0
1−1 0 0
−2 1 0 0
,
18
0 10 20 30 40 50 60
−2
−1.5
−1
−0.5
0
0.5
time (sec)
ˆz(t)−z(t)
internal delay−independent
α=0.001, h1=5s, h2=6s
α=2, h1=5s, h2=6s
0 5 10 15 20 25 30
0
10
20
30
40
time (sec)
ˆz(t)
true values
α=0.001, h1=5s, h2=6s
internal delay−independent
0 1 2 3 4 5 6
0
10
20
30
40
time (sec)
ˆz(t)
true values
α=2, h1=5s, h2=6s
internal delay−independent
Figure 1: The comparison between the convergence rates of the DD and IDI observers illus-
trated in Example 1
A2=
0.1 0 0 0
0 0 −5 0
0 0 0 0
0 0 0 −2
, B = [1; 0; 2; 0.5], C=I202, and L=
02I2. Testifying Condition III, it can be readily seen that this condi-
tion does not hold, as the right-hand-side of (33) is equal to 3, which is not
equal to its left-hand-side that is 4. Hence, according to Theorem 3.5, an inter-
nal delay-independent functional observer, cannot be designed for the system.
However, Condition I is satisfied. Therefore, the proposed design scheme were
applied to design a delay-dependent observer for the system. Similar to Example
1, two scenarios were studied: 1) α= 0.001, and 2) α= 2.
In the first scenario, the LMIs (24) and (25) were feasible for delays equal
19
to h1=h2= 9s, when αr=αs= 0.001, and αq= 50. In addition,
the observer parameters for the mentioned values of α, h1, and h2were ob-
tained as F0=
−4.1848 1
−0.2042 −6
,F1=02,F2=
−1.8328 0
−1.9852 −2
,H0=
−17.8152 0.6751
−17.3458 1.9568
,H1=
0.6334 0.0997
−2.3970 2.1911
,H2=
0 0.6719
0 1.5218
,
V=
0−0.3666
0−0.3970
and G=
2
0.5
. Interestingly, Condition II, de-
scribed in Theorem 28 is not satisfied for the initially obtained parameters
Q1, Ki, i ={0,1,2}. As a consequence, (B.1) did not have a unique solution.
However, considering Remark 3, the values of Ki, i ={0,1,2}were updated
from the value of ¯
Zobtained from (29). Accordingly, the LMIs (24) and (25) still
hold. Therefore, Theorem 3.3 is still satisfied, and the exponential convergence
rate is guaranteed.
In the second case, i.e. α= 2, the delays were again considered as h1=h2=
9s, and the tuning parameters were set at αr=αs= 0.01, and αq= 100. In
consequence, the LMIs (24) and (25) were feasible, and resulted in the observer
parameters F0=
−6.8538 1
−0.1991 −6
,F1=0,F2=
−1.5583 0
−1.9857 −2
,H0=
−15.1462 1.3462
−17.3509 1.9445
,H1=
0.6883 −0.0650
−2.3971 2.1914
,H2=
0 0.4856
0 1.4131
,
V=
0−0.3117
0−0.3971
, and G=
2
0.5
. Nevertheless, the same situation
was observed regarding to Condition II as explained in the first scenario. In
summary, this example shows that the delay-dependent observer is less conser-
vative than the internal delay-independent functional observer. In addition, the
example is a good illustration of Remark 3.
Simulation results in the MATLAB/Simulink environment, are shown in
Fig. 2. The results were obtained via applying the input signal (38), and
similar initial conditions for both of the scenarios. Clearly, in both cases the
designed observers successfully estimate the desired functional, which is the last
20
two states of the system. However, the observer associated to the larger αis
considerably faster, and converges close to t= 50s, whereas the other observer
accomplishes this aim after 80 seconds of the simulation time.
0 20 40 60 80 100 120
−2
−1.5
−1
−0.5
0
0.5
1
1.5
time (sec)
ˆz1(t)−z1(t)
α=0.001, h1=h2=9s
α=2, h1=h2=9s
0 10 20 30 40 50 60 70 80
−1
−0.5
0
0.5
1
1.5
2
2.5
3
time (sec)
ˆz1(t) & z1(t)
true values
α=0.001, h1=h2=9s
α=2, h1=h2=9s
0 20 40 60 80 100 120
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
time (sec)
ˆz2(t)−z2(t)
α=0.001, h1=h2=9s
α=2, h1=h2=9s
0 10 20 30 40 50 60 70 80
−6
−5
−4
−3
−2
−1
0
1
2
time (sec)
ˆz2(t) & z2(t)
true values
α=0.001, h1=h2=9s
α=2, h1=h2=9s
Figure 2: The comparison of the convergence speeds of the DD observers obtained from the
two scenarios studied in Example 2
5. Conclusions and Future Works
Minimum-order functional observers for LTI systems with multiple constant
state delays have been designed. Delay-dependent and internal-delay indepen-
dent observer structures have been proposed as two alternate options. Firstly,
the delay-dependent observer has been designed using the Lyapunov Krasovskii
approach. The necessary and sufficient conditions of the stability of the ob-
server have been proposed. A new LKF has been established to design the
observer parameters in a way that the exponential convergence of the observer
21
for a given convergence rate and delay values, is guaranteed. Due to using less
restrictive techniques such as descriptor transformation and Jensen’s inequality,
the obtained criteria can be less conservative than the existing ones. Secondly,
the necessary and sufficient conditions for the existence of an asymptotically
stable functional observer without internal delays have been obtained. The IDI
observer has a simpler design procedure, and can perform better than the delay-
dependent one, and further it can be applied to systems with any delays values.
Nevertheless, due to its independent of delays stability conditions, this class of
FOs can be more restrictive than the delay-dependent structure. Two numerical
examples and simulation results have explained the above achievements in more
details.
Future work will concentrate on less conservative FO design for systems
with both input and state delays, as well as systems with output delays. More
importantly, FO and UIFO design for LTI systems with unknown time delays
is an open problem.
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Appendix A. Proof of Theorem 3.2
Consider the following LKF candidate,
V(t, ¯ϵt) = V1(t) + V2(t, ˙ϵt) + V3(t, ϵt) (A.1)
where
V1(t) = ¯ϵT(t)E P ¯ϵ(t),
V2(t, ˙ϵt) =
2
i=1 0
−hit
t+θe2α(s−t)˙ϵT(s)Ri˙ϵ(s)dsdθ,
V3(t, ϵt) =
2
i=1 t
t−hie2α(s−t)ϵT(t)Siϵ(s)ds,
ϵt(θ) = ϵ(t+θ), θ ∈[−hu,0], ¯ϵ(t) :=
ϵ(t)
˙ϵ(t)
,E:=
Il0
0 0
, and P:=
P10
P2P3
. Apparently, there exists positive constants α1and α2, such that
α1|¯ϵ(t)| ≤ V(t, ¯ϵt)≤α2|¯ϵt|,∀t≥0.
26
Differentiating (A.1) along the solution of (6) gives
d
dt V(t, ¯ϵt)|(6) =d
dt V1(t)|(6) +d
dt V2(t, ˙ϵt)|(6) +d
dt V3(t, ϵt)|(6) (A.2)
Computing the first term on the right-hand-side of (A.2), and employing the
descriptor transformation [31, 32], yields the following,
d
dt V1(t)|(6) = 2¯ϵ(t)P1˙ϵ(t)
= 2¯ϵT(t)PT
˙ϵ(t)
−˙ϵ(t) +
2
i=0
Fiϵ(t)−
2
i=1
Fit
t−hi˙ϵ(s)ds
.
(A.3)
Substituting from (6) into (A.3), results
d
dt V1(t)|(6) = ¯ϵT(t)
PT
0I
2
i=0
Fi−I
+
0
2
i=0
FT
i
I−I
P
¯ϵ(t)
−2
2
i=1
¯ϵT(t)PT
0
Fi
t
t−hi˙ϵ(s)ds.
(A.4)
In order to find an upper-bound in the squared form for the cross terms ap-
pearing in the right-hand-side of (A.4), the improved Park’s inequality (Lemma
2.1), is used. This may result in less conservative conditions than when us-
ing Young’s inequality to this aim. Accordingly, assuming a= ¯ϵ(t), Hi=
0
Fi
T
P, and bi=t
t−hi˙ϵ(s)ds,i={1,2}, upon satisfaction of LMIs (11),
we have
−2bT
iHia≤
a
bi
T
ZiYi− HT
i
YT
i− Hi¯
Ri
a
bi
, i ={1,2}.(A.5)
27
In addition, the second and the third terms on the right-hand-side of (A.2)
are calculated as,
d
dt V2(t, ˙ϵt)|(6) =−2αV2(t, ˙ϵt)+
2
i=1 0
−hi
˙ϵT(t)Ri˙ϵ(t)dθ −0
−hi
e2αθ ˙ϵT(t+θ)Ri˙ϵ(t+θ)dθ,
(A.6)
and
d
dt V3(t, ϵt)|(6) =−2αV3(t, ϵt) +
2
i=1
(ϵT(t)Siϵ(t)−e−2αhiϵT(t−hi)Siϵ(t−hi)).
(A.7)
Considering (A.4)-(A.7), Lemma 2.2, and after the handling of the algebraic
expressions, it is obtained that
d
dt V(t, ¯ϵt)|(6) ⪯ζT(t)Γζ(t)−2αV (t, ¯ϵt) (A.8)
where
ζ(t) := ¯ϵ(t); ϵ(t−h1); ϵ(t−h2); t
t−h1˙ϵ(s)ds;t
t−h2˙ϵ(s)ds .
As a result, since (10) holds, it is deduced from (A.8) that
d
dt V(t, ¯ϵt)|(6) + 2αV (t, ¯ϵt)<0 (A.9)
Now, let us define v(t) := e2αtV(t, ¯ϵt), and differentiate it along the solution of
(6). It is obtained that ˙v(t)<0. Integrating the latter inequality from 0 to t,
and substituting from the definition of v(t) results
V(t, ¯ϵt)< e−2αt V(0,¯
ϕ) (A.10)
where ¯
ϕ(θ) := ¯ϵ0(θ),∀θ∈[−hu,0]. Therefore, the Lyapunov Krasovskii theo-
rem helps us to conclude that ϵ(t) and ˙ϵ(t) exponentially converge to zero with
28
the rate of α. More specifically,
|¯ϵ(t)|c<α2
α1
e−αt|¯
ϕ(0)|.
Appendix B. Proof of Theorem 3.3
To convert the matrix inequality (10) to an equivalent LMI, it is pre and post
multiplied by a matrix ∆1:= diag(Q, Il, Il, Il, Il), where Q:= P−1. Analogous
to the definition of P,Qcan be defined as,
Q=
Q10
Q2Q3
where Q1≻0∈Rl×l,Q2∈Rl×l, and Q3∈Rl×l. Hence, denoting ¯
Γ :=
∆T
1Γ∆1, (10) is equivalent to ¯
Γ≺0, where
¯
Γ :=
¯
Γ10 0 QTY1−
0
F1
QTY2−
0
F2
∗ −e−2αh1S10 0 0
∗ ∗ −e−2αh2S20 0
∗ ∗ ∗ ¯
R1−2α
e2αh1−1R10
∗ ∗ ∗ ∗ ¯
R2−2α
e2αh2−1R2
,
and
¯
Γ1:= QTΓ1Q
=
0I
2
i=0
U1i−I
Q+QT
0
2
i=0
UT
1i
I−I
+
2
i=1
QT
Si0
0hiRi
Q
+
2
i=1
QTZiQ+ 2αQTE+
0 0
2
i=0
¯
ZU2iQ10
+
0
2
i=0
Q1UT
2i¯
ZT
0 0
29
In order to linearize the quadratic terms in ¯
Γ1,Schur complement [38] is
useful. However, due to the summation of different variables of the form QTχiQ,
this technique is not directly applicable. To comply with this limitation, some
simplifications are considered. In this line, it is assumed that S:= S1,S2=αsS,
R:= R1,R2=αrRand ¯
Zi:= QTZiQ, i ={1,2}, where αsand αrare
arbitrary positive scalars that are tuning parameters. Now, let us define Ki:=
¯
ZU2iQ1, i ={0,1,2},¯
R:= R−1,¯
S:= S−1, and ˜
Ri:= ¯
R¯
Ri¯
R,i={1,2}. Then,
it is obtained after pre and post multiplying ¯
Γ by ∆2:= diag(Il,¯
S, ¯
S, ¯
R, ¯
R),
using Schur complement, and the definition of ¯
Zithat ¯
Γ≺0 is analogous to
ˆ
Γ≺0, where
ˆ
Γ :=
ˆ
Γ10 0 ˆ
Γ2ˆ
Γ3
∗ −e−2αh1¯
S0 0 0
∗ ∗ −αse−2αh2¯
S0 0
∗ ∗ ∗ ˜
R1−2α
e2αh1−1¯
R0
∗ ∗ ∗ ∗ ˜
R2−2ααr
e2αh2−1¯
R
,
ˆ
Γ1:=
ˆ
Ξ1ˆ
Ξ2Q1QT
2
∗ −sym(Q3) +
2
i=1
¯
Zi
22 0QT
3
∗ ∗ − 1
1+αs
¯
S0
∗ ∗ ∗ − 1
h1+αrh2
¯
R
,
ˆ
Γ2:=
QTY1¯
R−
0
U11 +¯
ZU21
¯
R
0
,
ˆ
Γ3:=
QTY2¯
R−
0
U12 +¯
ZU22
¯
R
0
,
ˆ
Ξ1:= Q2+QT
2+
2
i=1
¯
Zi
11 +2αQ1, and ˆ
Ξ2:= Q3+
2
i=0
Q1UT
1i−QT
2+
2
i=1
¯
Zi
12 +
2
i=0
KT
i.
30
Nevertheless, ˆ
Γ still has nonlinear terms QTYi¯
R, and ¯
ZU2i¯
R, i ={1,2}. To
resolve this problem, further simplification is applied via assuming ¯
R=αqQ1,
where αq>0 is an arbitrary scalar. Defining ¯
Yi:= QTYi¯
R, i ={1,2},ˆ
Γ and ˆ
Γ1
can be rewritten as ˜
Γ and ˜
Γ1defined in (26) and (27), respectively. Moreover,
employing the above similarity transformations, it can be shown that LMIs (11)
are equivalent to the LMIs (25). Hence, upon the satisfaction of Condition
(˜a), Theorem 3.2, and in consequence Condition (a) of Theorem 3.1 is satisfied.
Finally, according to the definitions of Ki(i={0,1,2}), it can be shown that
¯
ZΨ = ¯
K(B.1)
The parameter ¯
Zhas a unique solution as (29), if and only if Condition II is
satisfied. As a result, the observer parameters Fi,T1,Hi, and V(i={0,1,2})
can be respectively computed from (22), (23), (17), and (A.10), which implies
that Condition (b) of Theorem 3.1 is also satisfied. This completes the proof of
the theorem.
31
