# Study and Application of State Estimator for Time-Delay Systems with State Saturation Nonlinearities

Authors: Yeong-Jeu Sun
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The design and application of state estimator for a class of time-delay systems with state saturation nonlinearities are firstly explored. Based on the Lyapunov-Razumikhin functions, delay-dependent sufficient conditions are established to guarantee the asymptotic stability of the error between the state estimate and the true state. Besides, an upper bound of arbitrary time-varying delays is derived to assure that the resulting error can converge asymptotically to zero. Finally, applications to the secure communication and simulation results are given to demonstrate the feasibility and effectiveness of the main results.


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## Body

## 1. Introduction

It is well known that time delays are frequently encountered in various physical dynamic systems and may lead to instability or oscillation. Recently, time-delay systems have been extensively studied; see, for instance, [1–8], and they are often encountered in various areas, such as chemical engineering systems, electrical system, mechanical system, biological system, transportation system, the nuclear reactor, AIDS epidemic, and systems with lossless transmission lines. However, time-delay systems are, in general, infinite dimensional systems, which make the analysis and synthesis complicated. As is well known, there have been two key approaches to deal with the problems of time-delay systems, namely, Lyapunov-Razumikhin function approach [1, 3, 6] and Lyapunov-Krasovskii functional approach [4, 5, 8]. In general, the Lyapunov-Krasovskii approach controller design and stability analysis are more complex than the Lyapunov-Razumikhin approach. Besides, the Lyapunov-Krasovskii approach frequently requires the time-varying delay hi(t) to meet some conservational conditions such that hi(t) is bounded and h˙i(t)≤1 but the Lyapunov-Razumikhin approach only requires the time-varying delay to be bounded. To the author’s knowledge, there have not been any reported results on the use of the Lyapunov-Razumikhin method for state estimator of time-delay systems with state saturation nonlinearities. Therefore it is important to develop a novel state estimate for a class of time-delay systems with state saturation nonlinearities and time-varying delays via the Lyapunov-Razumikhin technique.On the other hand, saturation nonlinearities frequently appear in most physical systems and have drawn the attention of several researchers; reader is referred to [2, 7, 9, 10] and the references therein; for example, the states are constrained to stay within a bounded set due to physical limitations of the devices or by protection equipment. For nonlinear systems with state saturation nonlinearities, it is of great difficulty and challenge to design state estimator, in that such nonlinearities will generate a complex nonlinear system.Since not all of internal states can be measured directly, some observing structure should be used to reconstruct the state-variables. Nevertheless, the state estimator design of dynamic systems with saturation nonlinearities is in general not as easy as that without saturation nonlinearities. Due to above-mentioned reasons, the estimator design of time-delay systems employing saturation nonlinearities is indeed an important and challenging task.The main contributions of this paper are summarized as follows: (I) searching a suitable state estimator for a class of time-delay systems with state saturation nonlinearities to achieve the asymptotic stability of the error between the state estimate and the true state; (II) establishing delay-dependent sufficient conditions to guarantee the existence of state estimator for a class of time-delay systems with state saturation nonlinearities and time-varying delays; (III) calculating an upper bound of arbitrary time-varying delays without destroying the state estimator; (IV) constructing an observer-based secure communication architecture to demonstrate the practicability and effectiveness of the main results.


## 2. Problem Formulation and Main Results

Nomenclature
Rm×n≔ the set of all real m by n matrices,
AT≔ the transpose of the matrix A,
I≔ the unit matrix,
A≔ the induced Euclidean norm of the matrix A,
λmax⁡(Q)(resp.,λmin⁡Q)≔ the maximum (resp., minimum) eigenvalue of the symmetric matrix Q,
Q>0≔ the symmetric matrix Q is positive and definite,
p_≔1,2,…,p,
p-≔0,1,2,…,p.Nonlinear time-delay systems with state saturation nonlinearities considered in this paper are described by the following state-space representation:(1a)x˙t=B0xt+∑i=1pBixt-hit+∑i=1qEifλ1,λ2,…,λnxt-τit+Dut,t≥0,(1b)yt=C0xt+∑i=1pCixt-hit+∑i=1qHifλ1,λ2,…,λnxt-τit,t≥0,(1c)xt=θt,t∈-H,0,where x∈Rn×1 is the state vector, y∈Rr×1 is the output vector, u∈Rd×1 is the chaotic input vector, hit′s,for  alli∈p_, and τit′s,for  alli∈q_, are arbitrary delay arguments with 0≤hit≤H and 0≤τit≤H for some constant H, Bi, Ci, Ei, Hi, and D are matrices of appropriate dimensions, and θt is a given continuous vector-valued initial function. As we know, saturation nonlinearities may result in the instability or complexity of the designed system. Saturation overflow arithmetic is still one of the well-known nonlinear phenomena observed in the real world [9]. In this paper, the saturation nonlinearity fλ1,λ2,…,λnx:Rn×1→Rn×1, with λi>0,for  alli∈n_, is defined as follows [2, 7, 10]:(2)fλ1,λ2,…,λnx≔sat⁡λ1x1sat⁡λ2x2⋮sat⁡λnxn,with x≔x1x2⋯xnTand(3)sat⁡λz≔λ,z≥λz,-λ<z<λ-λ,z≤-λ.Throughout this paper, the following assumption is made on systems (1a), (1b), and (1c).(A1)
There exists a matrixK∈Rd×r such that(4)A≔∑i=0pBi+DK∑i=0pCiis  a  Hurwitz  matrix;λmin⁡Pλmax⁡3P>∑i=1phit·Bi+DKCi·∑i=0pBi+DKCi+∑i=1qEi+DKHi+∑i=1qEi+DKHi,∀t≥0,where P>0 is the unique solution to the following Lyapunov equation:(5)ATP+PA=-2I.Here we propose a full state estimator of the form(6a)x^̇t=B0x^t+∑i=1pBix^t-hit+∑i=1qEifλ1,λ2,…,λnx^t-τit+Dut-DKyt-y^t,t≥0,(6b)y^t=C0x^t+∑i=1pCix^t-hit+∑i=1qHifλ1,λ2,…,λnx^t-τit,t≥0,where x^(t) expresses the reconstructed state of the system and the constant matrix K∈Rd×r is defined in (A1).Remark 1.
It has been shown in [1, 3–6, 8] that Lyapunov-Razumikhin function approach and Lyapunov-Krasovskii functional approach are the key tools in the analysis of time-delayed dynamical systems. In [4], some nonnegative inequalities and Lyapunov-Krasovskii functional approach have been used to design the switching control for a class of time-delay systems. In [5], based on a new Lyapunov-Krasovskii functional, linear matrix inequality technique, and free weighting matrix approach, a control law has been presented to make the neutral time-delayed system robustly asymptotically stable. Besides, Razumikhin-type uniform stability has been developed in [1] to guarantee the stability of functional fractional-order time-delay systems. Meanwhile, a necessary and sufficient stability condition joining the Razumikhin and the Lyapunov-Krasovskii approaches has been provided in [6] for linear systems with several delays. The condition allows estimating the stability domains in the parameter space. On the basis of Lyapunov-Krasovskii functional approach, less conservative delay-dependent stability criteria have been proposed in [8] to guarantee the stability of the neutral time-delay system.Remark 2.
In practice, the saturation nonlinearities can be regarded as uncertainties with Lipschitz property. Besides, Lyapunov-Razumikhin function method is a useful and general approach for studying the stability of time-delay systems with uncertainties. Therefore, motivated by Lyapunov-Razumikhin function approach in [3], an asymptotic state estimator for time-delay systems with saturation nonlinearities of (1a), (1b), and (1c) will be presented. By the way, a sufficient condition for the asymptotic convergence to zero of the state observation error will be derived.Now we present the first main result for the state reconstruction of systems (1a), (1b), and (1c).Theorem 3.
If condition (A1) is satisfied, systems (6a) and (6b) are an asymptotic state estimator for systems (1a), (1b), and (1c); that is, limt→∞⁡x(t)-x^(t)=0.Proof.
Define(7)et=xt-x^t,Ai=Bi+DKCi,∀i∈p¯,(8)ki=Ei+DKHi,∀i∈q_,(9)Δft=∑i=1qEi+DKHifλ1,λ2,…,λnxt-τit-fλ1,λ2,…,λnx^t-τit.Then, from (1a), (1b), and (1c)–(9), it is easy to see that the error dynamic system is given by(10)e˙t=B0+DKC0et+∑i=1pBi+DKCiet-hit+∑i=1qEi+DKHi·fλ1,λ2,…,λnxt-τit-fλ1,λ2,…,λnx^t-τit=A0et+∑i=1pAiet-hit+Δft,with(11)Δft=∑i=1qEi+DKHifλ1,λ2,…,λnxt-τit-fλ1,λ2,…,λnx^t-τit≤∑i=1qEi+DKHi·xt-τit-x^t-τit=∑i=1qEi+DKHi·et-τit,in view of the global Lipschitz of sat⁡λz. From (10), we have(12a)e˙t=A0et+∑i=1pAiet-∑i=1pAi∫t-hitte˙zdz+Δft=∑i=0pAiet-∑i=1pAi·∫t-hitt∑i=0pAiez-hiz+Δfzdz+Δft,∀t≥0,(12b)xt=θt,t∈-H,0.Define the dynamic system(13a)y˙t=A0yt+∑i=1pAiyt-∑i=1pAi∫t-hitty˙zdz+Δft=∑i=0pAiyt-∑i=1pAi·∫t-hitt∑i=0pAiyz-hiz+Δfzdz+Δft,∀t≥0,(13b)yt=et,t∈-H,0,yt=e-H,t∈-2H,-H.For brevity, let us define yts=yt+s, for all s∈-2H,0, and yts≔sup-2H≤r≤0⁡y(t+r). By comparing (13a) and (13b) with (12a) and (12b), it is easy to see that yt=et, for all t≥0. Thus it can be deduced that(14)-∑i=1pAi·∫t-hitt∑i=0pAiyz-hiz+Δfzdz≤∑i=1pAi·hit·∑i=0pAi·yts+∑i=1qki·yts=∑i=1pAi·hit·∑i=0pAi+∑i=1qki·yts,∀t≥0.Let(15)Vyt=yTtPyt.The time derivative of Vy(t) along the trajectories of systems (13a) and (13b) is given by(16)V˙yt=yTtATP+PAyt-2yTtP∑i=1pAi·∫t-hitt∑i=0pAiyz-hiz+Δfzdz+2yTtPΔft=-2yt2-2yTtP∑i=1pAi·∫t-hitt∑i=0pAiyz-hiz+Δfzdz+2yTtPΔft,∀t≥0.Applying (14) to (16) yields(17)V˙yt≤-2yt2+2yt·λmax⁡P·∑i=1pAi·hit·∑i=0pAi+∑i=1qki·yts+2ytλmax⁡P·∑i=1qki·yts=-2yt2+2yt·λmax⁡P·∑i=1pAi·hit·∑i=0pAi+∑i=1qki+∑i=1qki·yts,∀t≥0.By (A1), (7), and (8), there exists a sufficiently small constant ε1>0 such that(18)ε2≔1-1+ε1λmax⁡3Pλmin⁡P·∑i=1phit·Ai·∑i=0pAi+∑i=1qki+∑i=1qki>0,∀t≥0.In the spirit of Theorem 4.2 in [3], with p(s)=1+ε12s, we suppose that(19)yTt+rPyt+r<1+ε12yTtPyt,∀-2H≤r≤0;this implies that(20)λmin⁡Pyt+r2<1+ε12λmax⁡Pyt2,∀-2H≤r≤0.This shows that(21)yt+r<1+ε1λmax⁡Pλmin⁡Pyt,∀-2H≤r≤0.Substituting (21) into (17), it can be shown that(22)V˙yt≤-2yt2+2yt·λmax⁡P·∑i=1pAi·hit∑i=0pAi+∑i=1qki+∑i=1qki·1+ε1.λmax⁡Pλmax⁡P·yt=-2ε2yt2,∀t≥0,in view of (18). Thus, by Theorem 4.2 in [3] with (15) and (22), we conclude that systems (13a) and (13b) and systems (12a) and (12b) are both globally asymptotically stable with(23)limt→∞⁡et=limt→∞⁡xt-x^t=0.This completes our proof.Remark 4.
By Theorem3, an upper bound of arbitrary time-varying delays without destroying the state estimator is given by H<H-, where(24)H-=α4-α3α1α2+α3,ifα1α2+α3≠0,α4>α3,∞,ifα1α2+α3=0,α4>α3,α1=∑i=1pBi+DKCi,α2=∑i=0pBi+DKCi,α3=∑i=1qEi+DKHi,α4=λmin⁡Pλmax⁡3P.


## 3. Application to Secure Communication System

Motivated by the time-delay systems (1a), (1b), and (1c) with their state estimator of (6a) and (6b), let us consider the following secure communication system and the architecture of such a system is shown in Figure 1.Figure 1
Secure communication system (m1(t) is the message vector and m2(t) is the recovered vector).Transmitter:
(25a)
x
˙
t
=
B
0
x
t
+
∑
i
=
1
p
B
i
x
t
-
h
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+
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=
1
q
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f
λ
1
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λ
2
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λ
n
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t
-
τ
i
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D
u
t
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t
≥
0
,
(25b)
y
t
=
C
0
x
t
+
∑
i
=
1
p
C
i
x
t
-
h
i
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H
i
f
λ
1
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λ
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x
t
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i
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≥
0
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(25c)
ϕ
m
t
=
T
x
t
+
m
1
t
,where m1(t)∈Rv×1 is the message vector, T∈Rv×n, and systems (25a) and (25b) are the same as systems (1a) and (1b), respectively.Receiver:
(26a)
x
^
̇
t
=
B
0
x
^
t
+
∑
i
=
1
p
B
i
x
^
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1
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λ
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λ
n
x
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t
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u
t
-
D
K
y
t
-
y
^
t
,
t
≥
0
,
(26b)
y
^
t
=
C
0
x
^
t
+
∑
i
=
1
p
C
i
x
^
t
-
h
i
t
+
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i
=
1
q
H
i
f
λ
1
,
λ
2
,
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λ
n
x
^
t
-
τ
i
t
,
t
≥
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,
(26c)
m
2
t
=
ϕ
m
t
-
T
x
^
t
,where dynamics of (26a) and (26b) are the same as those of (6a) and (6b), respectively. In addition, m2(t)∈Rv×1 is the signal recovered from m1(t). It is worthwhile to note that our proposed secure communication is motivated by the time-delay systems (1a), (1b), and (1c) with their state estimator of (6a) and (6b). For any information vector m1(t) in the transmitter system, the objective of secure communication system is to recover the message m2(t) in the receiver system.Theorem 5.
The global synchronization of signalsm1(t) and m2(t) can be realized by the transmitters (25a), (25b), and (25c) with the receiver of (26a), (26b), and (26c), provided that (A1) is satisfied.Proof.
By Theorem3 with (23)–(26c), one can see that(27)limt→∞⁡m1t-m2t=limt→∞⁡ϕmt-Txt-ϕmt+Tx^t≤limt→∞⁡T·et=0.This implies that one can recover the information m1(t) in the receiver system. This completes our proof.


## 4. Illustrative Example with Numerical Simulations

Example.
Consider the transmitter system of (25a), (25b), and (25c) with parameters as follows:(28)p=q=2,r=1,H=10,m1t=3sin⁡0.7t+3cos⁡0.5t+e-0.01t,B0=-0.1-20.10,B1=-2-200,B2=02-0.1-0.1,E1=2000.1,E2=-220.10.1,D=-10,C0=01,C1=11,C2=0-1,H1=10,H2=-11,T=1-1.Furthermore, the input u(t) is a Lorenz chaotic signal, shown in Figure 2. Condition (A1) is evidently satisfied if we let K=-2. Consequently, by Theorem 3, we conclude that the state estimator of systems (25a)-(25b) is of existence and a suitable state estimator is given by (26a)-(26b). Additionally, the real information m1(t), the recovered information m2(t), and the error signal are depicted in Figures 3–5, respectively. From the foregoing simulations results, it is seen that the global synchronization of signals m1(t) and m2(t) can be achieved by the transmitters (25a), (25b), and (25c) with the receiver of (26a), (26b), and (26c). By the way, from Remark 4, an upper bound of arbitrary time-varying delays without destroying the state estimator is given by H<H-=10.2.Figure 2
Signal ofu(t).Figure 3
Real information ofm1(t) described in the transmitter of (25a), (25b), and (25c).Figure 4
Recovered information ofm2(t) described in the receiver of (26a), (26b), and (26c).Figure 5
Error signal ofm1t-m2(t).Remark 6.
It should be emphasized that there exist an infinite number of secure communication systems of (25a), (25b), and (25c) with (26a), (26b), and (26c) since there exist infinitely many systems (1a), (1b), and (1c) satisfying the condition of (A1).


## 5. Conclusion

In this paper, the design and application of state estimator for a class of time-delay systems with state saturation nonlinearities have been firstly investigated. Based on the Lyapunov-Razumikhin functions, sufficient conditions have been established to guarantee the asymptotic stability of the error between the state estimate and the true state. Besides, an upper bound of arbitrary time-varying delays has been derived to guarantee that the resulting error can converge asymptotically to zero. Applications of the proposed state estimator technique to secure communication as well as some numerical simulations have also been presented to demonstrate the practicability and effectiveness of the main results.



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