# Finite-Time Boundedness Control for Nonlinear Networked Systems with Randomly Occurring Multi-Distributed Delays and Missing Measurements

Authors: Ling Hou; Dongyan Chen
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper investigates the stochastic finite-timeH∞ boundedness problem for nonlinear discrete time networked systems with randomly occurring multi-distributed delays and missing measurements. The randomly occurring multi-distributed delays and missing measurements are described as Bernoulli distributed white noise sequence. The goal of this paper is to design a full-order output-feedback controller to guarantee that the corresponding closed-loop system is stochastic finite-time H∞ bounded and with desired H∞ performance. By constructing a new Lyapunov-Krasovskii functional, sufficient conditions for the existence of output-feedback are established. The desired full-order output-feedback controller is designed in terms of the solution to linear matrix inequalities (LMIs). Finally, a numerical example is provided to show the validity of the designed method.


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## Body

## 1. Introduction

Networked control system (NCS) is a new research field originated from control engineering practice, which is composed of controller, actuator, and sensor. Due to low cost, low power consumption, and practical resource sharing, the dynamic behavior of NCSs has been widely concerned in recent years [1–5]. However, network is not a reliable communication medium. Due to the physical limitations of network bandwidth and service capability, time delay and missing measurement are inevitable problems in network transmission. These problems are an important reason for the deterioration of system performance and the instability of NCSs. Therefore, in the past few decades, a large number of new methods have been used to deal with NCSs with various time delays. At present, the following three models are mainly adopted: constant delay model [6], the upper and lower bounds with time-varying delay model [7], and random time delay with a certain probability distribution delay model (such as finite state markov process, Bernoulli distribution, etc. [8, 9]). In [8], the H∞ filtering problem of the Markov delay network system with a partially unknown distribution is studied. By using Lyapunov theory, the authors analyze the robust hybrid controller, controller design, and synchronous dissipation state estimation of the network control system with Bernoulli distribution in [10–13], respectively. On the other hand, missing measurement is also a major cause of NCSs performance degradation and system instability. Study of missing measurement: one of the most common ways to describe the missing measurement is probability method; this method assumes the missing measurement to satisfy a certain probability distribution, such as Bernoulli distribution. Many control problems are considered based on the Bernoulli distribution, such as [14–17]. In [14], recursive state estimation for time-varying complex networks with probability distribution missing measurements is investigated by using the mean square constraint method. The rapid state estimation of discrete nonlinear systems with random missing measurement is studied in [15]. In [16], the author extends the above research to the network control system with multiple missing measurement and studies the extended Kalman filter problem of the NCSs with random nonlinearity and multiple packet loss. In [17–19], the authors further extended the above results to the study of robust H∞ control, quantised recursive filtering, and event-based filtering of the NCSs with Bernoulli distribution and concluded that the use of Bernoulli distribution to describe missing measurement is more general.It should be pointed out that the above-mentioned literature on system stability considers Lyapunov’s stability at an infinite interval. In practice, however, we are concerned not only with the stability of the system on an infinite interval, but with the stability on a finite interval. The finite time stability (FTS) is different from the stability in the general sense. It studies the state behavior in the finite time; that is, the system should meet certain transient performance requirements (for example, meeting the requirements of the system orbit for a certain degree of deviation from the equilibrium point). In order to study the transient performance of the system, Dorato proposed the concept of FTS in 1961 and obtained extensive research [20]. In recent years, some scholars have extended FTS to studies on finite time boundedness (FTB) and finite time H∞ control. In [21–25], the authors studied the finite time H∞ problem of discrete time system based on linear matrix inequality. In [26–29], the authors extended the FTB and finite time H∞ control problem to the NCSs. Further, in [30], finite time H∞ control problem is extened to fuzzy discrete system. In [31, 32], the authors discussed FTB and finite time H∞ state estimation of neural networks with missing measurement based on the semini-research approach and gave the design method of the state estimator. Recently, the issues of nonfragile finite time state estimators and event-driven finite time state estimators for neural networks have been investigated [33–35].In addition, it is worth noting that, due to the influence of internal and external environmental factors, nonlinearity always exists in the actual system. At present, the control problem of the network system with nonlinearity is widely concerned [36]. Since the nonlinear perturbation phenomena may vary arbitrarily, Bernoulli distribution is more appropriate to describe the randomly occurring nonlinearity. In recent years, the problem of randomly occurring nonlinearities has been widely discussed. For example, in [37], the authors studied the H∞ control problem for randomly occurring nonlinear systems with saturation and channel fading. In [38, 39], the authors studied the reliable finite time H∞ filtering and quantized fault detection filters problem for discrete time systems, respectively.To the best of the authors’ knowledge, the SFTB control problem for a discrete-time NCS with randomly occurring multi-distributed delays and missing measurements has not been fully investigated, which motivates the main purpose of our study. The main contributions of the paper are summarized as follows:(1) The discrete-time NCS under consideration is more general networked control systems that include randomly occurring nonlinearities, multi-distributed delays, and missing measurements, where the randomly occurring nonlinearities, multi-distributed delays, and missing measurements are governed by a set of Bernoulli distributed white noise sequences; (2) definitions of SFTB and SFTH∞B are extended to discrete-time NCSs; (3) sufficient conditions are provided to ensure that the corresponding closed-loop system is stochastic finite-time bounded and H∞ performance constraint is met by using the Lyapunov-Krasovskii functional method and the LMIs technique; (4) full-order output-feedback controller, a more general controller than the commonly used state feedback controller, is designed.


## 2. Problem Formulation

Consider a nonlinear discrete-time NCS with randomly multi-distributed delays as shown in Figure1.Figure 1
Structure of the considered NCSs.The state-space equation is given as follows:(1)xk+1=Axk+Ad∑i=1qbikxk-dik+Buk+αkfxk+D1ωky~k=Cxk+D2ωkzk=Lxkxk=φk,k=-dM,-dM+1,…,-1,0where x(k)∈Rn is the state vector, u(k)∈Rp is the control input, y~(k)∈Rm is the measurement output, z(k)∈Rt is the controlled output, f(x(k)) is a nonlinear function, ω(k) are external disturbances, φ(k) denotes the vector-value initial condition, A, Ad, B, C, D1, D2, and L are known real constant matrices, ∑i=1qbi(k)x(k-di(k)) denotes the infinite distributed delays, di(k)(i=1,…,q) are time-varying delays satisfying (2)dm≤dik≤dM,where dm and dM are nonnegative scalars. bi(k)(i=1,…,q) are uncorrelated random variables and satisfy the following probability distribution: (3)Probbik=1=b-i,Probbik=0=1-b-i,where b-i∈[0,1] is a known constant and the variance b~i2=b-i(1-b-i).The stochastic variableα(k) is a Bernoulli distributed white noise sequence with (4)Probαk=1=α-,Probαk=0=1-α-,where α-∈[0,1] is a known constant, and the variance α~2=α-(1-α-). Throughout the paper, we assume that α(k) and bi(k)(i=1,…,q) are mutually independent.For the aforementioned system, we make the following assumptions.Assumption 1.
f ( · ) is a nonlinear function, and we assume that f(·) satisfies f(0)=0 and the sector-bounded condition (5)fx-fy-U1x-yTfx-fy-U2x-y≤0,∀x,y∈Rn,where U1,U2∈Rn×n are known real matrices and U1-U2 is a known positive definite matrix.Assumption 2.
For any given positive numberδ, external disturbances input ω(k) is time varying and satisfies (6)∑k=0NωTkωk≤δ,δ≥0.In this paper, the actual system output is expressed as(7)yk=βky~k,where the stochastic variable β(k) takes values on 0 and 1 with the following probabilities: (8)Probβk=1=β-,Probβk=0=1-β-,where β-∈[0,1] is a known constant and the variance β~2=β-(1-β-).Equation (7) can be written in the light of system (1) (9)yk=βkCxk+D2ωk.In this paper, the full-order dynamic output feedback controller for system (1) is described as (10)x^k+1=AKx^k+BKykuk=CKx^k.where x^(k)∈Rn is the state estimation of system (1) and AK, BK, and CK are the controller parameters to be determined.By combining the controller (10) with (1), we can obtain the following closed-loop system: (11)ηk+1=A-ηk+βk-β-A~ηk+C-fxk+αk-α-C~fxk+D-ωk+βk-β-D~ωk+∑i=1qA-diHηk-dik+∑i=1qbik-b-iA~diHηk-dikzk=L-ηk,where(12)ηk=xkx^k,A-=ABCKβ-BKCAK,A~=00BKC0,A-di=b-iAd0,A~di=Ad0,C-=α-I0,C~=I0,D-=D1β-BKD2,D~=0BKD2,H=I0T,L-=L0T,In this paper, our goal is to design a full-order output feedback controller to ensure that the corresponding closed-loop dynamic system is SFTB and with a desiredH∞ perfoamance. The definitions of SFTB and SFTH∞B and a lemma are introduced before sequel.Definition 3 (SFTB).
Given positive numbersc1,c2(c1<c2),δ,N, and a symmetric positive definite matrix R, the closed-loop system (11) is stochastic finite-time boundedness w.r.t. (c1,c2,δ,R,N) if (13)supi∈-dM,-dM+1,⋯-1,0EηiTRηi≤c1⇒EηkTRηk≤c2,k∈1,…,N.Definition 4 (SFTH∞B).
Given positive numbersγ,c1,c2(c1<c2),δ,N, and a symmetric positive definite matrix R, the closed-loop system (11) is stochastic finite-time H∞ boundedness w.r.t. (c1,c2,δ,γ,R,N) if the closed-loop system (11) is SFTB w.r.t. (c1,c2,δ,R,N) and under the zero-initial condition the output z(k) satisfies(14)E∑s=0NzTszs≤γ2∑s=0NωTsωsfor all ω(k) satisfying Assumption 2.Lemma 5 ([23] (Schur complement)).
Given constant matricesS1, S2, and S3, then S1+S3TS2-1S3<0 if and only if (15)S1S3T∗-S2<0or-S2S3∗S1,<0,where S1=S1T and S2=S2T>0.


## 3. Main Result

In this section, we first give sufficient conditions for the SFTB of closed-loop system (11) and then discuss the design of dynamic output feedback controller (10).


### 3.1. Stochastic Finite-Time Boundedness Analysis

The following theorem provides sufficient conditions for the SFTB of closed-loop system (11) by employing Lyapunov-Krasovskii functional approach.Theorem 6.
Given positive constantsc1,c2(c1<c2),δ,N, a symmetric positive-definite matrix R, closed-loop system (11) is SFTB w.r.t.(c1,c2,δ,R,N) if, for a scalar μ≥1, there exist symmetric positive definite matrices P, Qi(i=1,…,q) and positive scalars λj(j=1,2,3) such that the following inequalities hold: (16)Σ11Σ12Σ13∗Σ22Σ23∗∗Σ33<0,(17)rδ-c2μλ1c1μN+1λ2ρλ3∗-c1μN+1λ20∗∗-ρλ3<0,where(18)Σ11=diag-μP,-μdmQ1,…,-μdmQq,Σ22=diag-I,-γμNI,Σ33=diag-P,-P,-Q^,Σ12=-HTR~2000⋮⋮00,Σ13=A-TPβ~A~TPHTQ^A-d1TPb~1A~d1TP0⋮⋮⋮A-dqTPb~qA~dqTP0,Σ23=C-TPα~C~TP0D-TPβ~D~TP0,Q^=dM-dm+1∑i=1qQi-R~1,ρ≔c1qμN+dM-1dMμ+dMdM-1-dmdm-12,P-=R-1/2PR-1/2,λ1=λminP-,λ2=λmaxP-,Q-i=R-1/2HTQiHR-1/2,λ3=maxi∈1,…,qλmaxQ-i,R~1=U1TU2+U2TU12,R~2=-U1T+U2T2.Proof.
Choose the following Lyapunov-Krasovskii functional candidate:(19)Vk=V1k+V2k+V3k,where(20)V1k=ηTkPηk,V2k=∑j=1q∑s=k-djkk-1μk-1-sηTsHTQjHηs,V3k=∑j=1q∑s=-dM+2-dm+1∑t=k+s-1k-1μk-1-tηTtHTQjHηt.
Let us assume that(21)EηkTRηk≤c1,∀k=-dM,-dM+1,…,0.The following goal is to show that if conditions (16) and (17) hold, then (22)EηkTRηk≤c2,∀k=1,…,N.
First, calculating the difference variation ofVi(k)(i=1,2,3), and taking the mathematical expectation, one has (23)EV1k+1-μV1k=EξTkΦTPΦξk+ξTkΠTPΠξk-μηTkPηk,where(24)ξTk=ηkxk-d1k⋯xk-dqkfxkωk,Φ=A-A-d1⋯A-dqC-D-,Π=β~A~b~1A~d1⋯b~qA~dqα~C~β~D~.
Similarly, it can be obtained that(25)EV2k+1-μV2k=E∑j=1q∑s=k+1-djk+1kμk-sηTsHTQjHηs-∑j=1q∑s=k-djkk-1μk-sηTsHTQjHηs=∑j=1qEηTkHTQjHηk+∑s=k+1-dmk-1μk-sηTsQjηs+∑s=k+1-djk+1k-dmμk-sηTsQjηs-∑s=k+1-djkk-1μk-sηTsQjηs-μdmxTk-djkQjxk-djk.Considering dm≤dj(k)≤dM, (26)E∑s=k+1-dmk-1μk-sηTsQjηs-∑s=k-djk+1k-1μk-sηTsQjηs≤0.It follows from (25) that (27)EV2k+1-μV2k≤E∑j=1qηTkHTQjHηk+∑s=k+1-djk+1k-dmμk-sηTsQjηs-μdmxTk-djkQjxk-djk.Similarly, we have (28)EV3k+1-μV3k≤E∑j=1qdM-dmηTkHTQjHηk-∑s=k-dM+1k-dmμk-sηTsHTQjHηs.It is obtained from (23), (27), and (28) that (29)
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,where Δ=diag{-μP+(dM-dm+1)∑i=1qHTQiH, -μdmQ1,…,-μdmQq,0,-γ/μNI}.
Notice that (5) implies the following inequality: (30)ηkfxkTHTR~1HR~2TH∗Iηkfxk≤0.Combining (29) with (30), we have (31)EVk+1-μVk≤EξTkΦTPΦ+ΠTPΠ+Θξk+γμNωTkωk,where (32)Θ=-μP+dM-dm+1∑i=1qHTQiH-HTR~1H0⋯0-HTR~20∗-μdmQ1⋯000⋮⋮⋱⋮⋮⋮∗∗⋯-μdmQq00∗∗⋯∗-I0∗∗⋯∗∗-γμN.By applying Lemma 5 to (16), we deduce that ΦTPΦ+ΠTPΠ+Θ<0. Thus, we obtain from (31) that (33)EVk+1-μVk≤γμNωTkωk,∀k∈Z+.That is, (34)EVk+1≤μEVk+γμNωTkωk,∀k∈Z+.It follows from (34) that (35)EVk≤μEVk-1+γμNωTk-1ωk-1,∀k∈Z+.
By iteration and taking Assumption2 into account, it can be obtained that (36)EVk+1-μVk≤μkEV0+γμN∑s=0k-1μk-1-sωTsωs≤μNEV0+γμNμN-1∑s=0N-1ωTsωs≤μNEV0+γμδ.Noting definition of λi(i=2,3), we have (37)EV0=EηT0Pη0+∑j=1q∑s=-dj0-1μ-1-sηTsHTQjHηs+∑j=1q∑s=-dM+2-dm+1∑t=s-1-1μ-1-tηTtHTQjHηt<λ2EηT0Rη0+qλ3μdM-1∑s=-dM-1EηTsRηs+qλ3μdM+2∑s=-dM-2-dm+1∑t=-1+s-1EηTtRηt<λ2+qλ3dMμdM-1+qλ3dMdM-1-dmdm-12μdM-2c1.By taking (36) and (37) into account, we have (38)EVk+1-μVk≤μNσ+γμδ,∀k∈Z+,where σ≔λ2+qλ3dMμdM-1+qλ3dM(dM-1)-dm(dm-1)/2μdM-2c1.
On the other hand, also from the definition ofλi(i=1,2,3), we have (39)EVk+1-μVk≥λ1EηTkRηk,∀k∈Z+.Consequently, we get from (38) and (39) that (40)EηTkRηk<1λ1μμN+1σ+γδ.
Note that(41)μN+1σ+γδ-c2μλ1<0⇔γδ-c2μλ1+c1μN+1λ2+c1qμN+dM-1dMμ+dMdM-1-dmdm-12λ3<0⇔γδ-c2μλ1+c1μN+1λ2+ρλ3<0⇔γδ-c2μλ1+c1μN+1λ2ρλ3c1μN+1λ2-100ρλ3-1c1μN+1λ2ρλ3<0⇔γδ-c2μλ1+c1μN+1λ2ρλ3c1μN+1λ200ρλ3-1c1μN+1λ2ρλ3<0.By Lemma 5, (41) is equivalent to LMI (23); it can be verified that (42)EηTkRηk<1λ1μμN+1σ+γδ<c2.Therefore, according to Definition 3, closed-loop system (11) is SFTB. The above discussion completes the proof of Theorem 6.Remark 7.
If nonlinearities and perturbation are not considered in system (11), that is, f(k)=0 and ω(k)=0, the SFTB problem of system (11) is deduced by analyzing the SFTB of the following system: (43)ηk+1=A-ηk+βk-β-A~ηk+∑i=1qA-diHηk-dik+∑i=1qA~diHηk-dikzk=L-ηk,where system parameters are the same as before.The following criterion for system (43) is given based on Theorem 6.Corollary 8.
Given positive constantsc1,c2,(c1<c2),δ,N, a symmetric positive-definite matrix R, closed-loop system (43) is SFTB w.r.t.(c1,c2,δ,R,N) if, for a scalar μ≥1, there exist symmetric positive definite matrices P, Qi and positive scalars λi(i=1,2,3) such that the following inequalities hold: (44)Θ=-μP000A-TPβ~A~TPHTQ^∗-μdmQ100A-d1TPb~1A~d1TP0∗∗⋱0⋮⋮0∗∗∗-μdmQqA-dqTPb~qA~dqTP0∗∗∗∗-P00∗∗∗∗∗-P0∗∗∗∗∗∗-Q^<0,-c2μλ1c1μN+1λ2ρλ3∗-c1μN+1λ20∗∗-ρλ3<0.


### 3.2. Stochastic Finite-TimeH∞ Boundedness Analysis

In this subsection, we will provide sufficient condition to ensure SFTH∞B of closed-loop system (11) based on Theorem 6.Theorem 9.
Given positive constantsc1, c2(c1<c2), δ, γ, N, a symmetric positive-definite matrix R, closed-loop system (11) is SFTH∞B w.r.t. (c1,c2,δ,γ,R,N) if, for a scalar μ≥1, there exist symmetric positive definite matrices P and Qi(i=1,…,q), and positive scalars λj(j=1,2,3) such that the following inequalities hold: (45)Ω11Ω12Ω13∗Ω22Ω23∗∗Ω33<0,(46)rδ-c2μλ1c1μN+1λ2ρλ3∗-c1μN+1λ20∗∗-ρλ3<0,where(47)Ω11=diag-μP,-μdmQ1,…,-μdmQq,Ω22=diag-I,-γμNI,Ω33=diag-P,-P,-Q^,-IΩ12=-HTR~2000⋮⋮00,Ω13=A-TPβ~A~TPHTQ^LTA-d1TPb~1A~d1TP00⋮⋮⋮⋮A-dqTPb~qA~dqTP00,Ω23=C-TPα~C~TP00D-TPβ~D~TP00.Other parameters are shown in Theorem 6.Proof.
Obviously LMI (45) implies LMI (16); according to Theorem 6, we can easily conclude that the closed-loop system (11) is SFTB w.r.t. (c1,c2,δ,R,N).
In the following, we shall prove that inequality (14) holds. By introducing the same Lyapunov-Krasovskii functional, we can get from (33) (48)EVk+1≤μEVk+γμNωTkωk,∀k∈Z+.And by iteration, we obtain that (49)EVk≤μk-1EV0+∑s=0k-1μk-1-sγμNωTsωs-EzTszs.Under the zero initial condition E{V(0)}=0, and noting that E{V(k)}≥0,∀k∈{1,2,…,N}, we get (50)∑s=0k-1μk-1-sEzTszs≤∑s=0k-1μk-1-sγμNωTsωs.Letting k=N+1, we then have (51)∑s=0NμN-sEzTszs≤∑s=0NμN-sγμNωTsωs.Notice that 1≤μN-s≤μN,∀s∈0,1,…,N; (51) immediately yields(52)E∑s=0NzTszs≤γ∑s=0NωTsωs.which implies that system (11) is SFTH∞B. This completes proof of Theorem 9.


### 3.3. Stochastic Finite-TimeH∞ Output Feedback Control

In this subsection, we will deal with the problem of the design of output-feedback controller for system (1).Theorem 10.
Consider the discrete-time NCS (1) with randomly occuring multi-distrbuted delays and missing measurement. For given positive constants c1, c2(c1<c2), δ and N, and a symmetric positive-definite matrix R=diagR1,R2, closed-loop system (11) is SFTH∞B w.r.t. (c1,c2,δ,γ,R,N) if, for a scalar μ≥1, there exist symmetric positive definite matrices U, Y, Qi(i=1,…,q), matrices Ωj(j=1,2,3), and positive scalars λt′(t=1,2,3,), such that the following inequalities hold: (53)J10H1TH2TH5T∗J20H3T0∗∗J3H4T0∗∗∗J40∗∗∗∗J5<0,(54)rδ-c2μλ1′c1μN+1λ2′ρλ3′∗-c1μN+1λ2′0∗∗-ρλ3′<0,(55)λ1′R1<Y<λ2′R1,0<Qi<λ3′R1,i=1,…,q,where(56)J1=-μU-μI-μI-μY,J2=diag-μdmQ1,…,-μdmQq,J3=diag-I,-γμNI,J4=-U-I00-I-Y0000-U-I00-I-Y,J5=diag-I,-Q^,H1T=UR~2T0R~2T0,H2T=UAT+Ω3BTΩ10β~UCTΩ2TATATR+β-CTΩ20CTΩ2T,H3T=b-1AdTb-1AdTYb~1AdTb~1AdTY⋮⋮⋮⋮b-qAdTb-qAdTYb~qAdTb~qAdTY,H4T=α-Iα-Yα~Iα~YD1TD1TW+β-D2TΩ20D2TΩ2T,H5T=ULTUQ^LTI.
R 1 and R2 are symmetric positive matrices.
In this case, the parameters of the output feedback controller can be designed as(57)AK=X12-1Ω1-YAU-β-Ω2CU-YBΩ3Y12-T,BK=X12-1Ω2,CK=Y12-TΩ3,where X12 and Y12 are any nonsingular matrices satisfying X12Y12T=I-UY.Proof.
Evidently, from Theorem9, system (11) is SFTH∞B if, for a scalar μ≥1, LMIs (45) and (46) hold. To do this, let P and P-1 be as follows: (58)P=YX12X12TX22,P-1=S-1Y12Y12TY22,where the partitioning of P and P-1 is compatible with that of A-, A~, A-di, A~di and X22 and Y22 are symmetric positive matrices.
Define(59)T1=S-1IY12T0,T2=IY0X12T,and this implies that (60)PT1=T2,T1TPT1=T1TT2.Thus (61)P=T2T1-1,X22=Y12-1S-1Y-SS-1Y12-T>0.
It is clear that, by Lemma5, if LMI (53) is feasible, then we have J1<0; thus Y-S<0 and therefore I-YS-1 is nonsingular, which implies that P>0.
First, it can be verified that the LMI (53) can be rewritten as (62)Σ=Σ110Σ130Σ15Σ16Σ17∗Σ2200Σ25Σ260∗∗Σ330Σ35Σ360∗∗∗Σ44Σ45Σ460∗∗∗∗Σ5500∗∗∗∗∗Σ660∗∗∗∗∗∗Σ77<0,where AK, BK, CK are defined in (57),(63)Σ11=-μS-1-μI∗-μY,Σ22=diag-μdmQ1,…,-μdmQq,Σ33=-I,Σ44=-γμN,Σ55=Σ66=-S-1-I∗-Y,Σ77=-I0∗-Q^,Σ13=S-1R~2R~2,Σ15=S-1AT+Ω2TBTΩ1ATATY+β-CTΩ2T,Σ16=0β~S-1CTΩ2T0CTΩ2T,Σ17=S-1LTS-1Q^LTI,Σ25=b-1AdTb-1AdTY⋮⋮b-qAdTb-qAdTY,Σ26=b~1AdTb~1AdTY⋮⋮b~qAdTb~qAdTY,Σ35=α-Iα-Y,Σ36=α~Iα~Y,Σ45=D1TD1TY+β-D2TΩ2T,Σ46=0α~D2TΩ2T.That is, (64)Σ′=Σ11′Σ12′Σ13′Σ14′∗Σ22′Σ23′0∗∗Σ33′0∗∗∗Σ44′<0,where(65)Σ11′=diag-μT1TPT1,-μdmQ1,…,-μdmQq,Σ22′=diag-I,-γμN,Σ33′=diag-T1TP-1T1,-T1TP-1T1,Σ44′=diag-I,-Q^,Σ12′=-R~2HT10⋯000⋯0T,Σ13′=T1TA-TT2β~T1TA~TT2A-d1TT2A~d1TT2⋮⋮A-dqTT2A~dqTT2,Σ14′=T1TLTT1THTQ^00⋮⋮00,Σ23′=C-TT2C~TYD-TT2D~TY.Now letting U=S-1 and applying the congruence transformations diag{T1-T,I,…,I,I,T2-T,T2-T,I,I} to (64), we can obtain LMI (45).
In addition, sinceY and Qi are fringed matrices of P and Q^i=HTQiH, by the property of fringed matrices, we have (66)λmaxY≥λmaxP,λminY≤λminP,λmaxQi≥λmaxQ^i,λminQi≤λminQ^i.Thus, (46) follows from inequalities (54) and (55). This completes the proofRemark 11.
Being different from literature [25], on the one hand, we consider NCSs with the stochastic and nonlinearity. On the other hand, state feedback controller is designed in [25]. However when not all the state variables are available, the results in [25] cannot be applicable. In this paper, output feedback controllers are designed in the stochastic context.Remark 12.
In order to design a controller independent of the matrixX12, we can make X12=Y; then AK, BK, and CK become(67)AK=Y-1Ω1-YAU-β-Ω2CU-YBΩ3I-UY-1Y,BK=Y-1Ω2,CK=I-UY-1YΩ3.Remark 13.
The feasibility of Theorem10 can be represented by the following basic LMIs with a parameter μ: (68)minc2+γs.t.53,54and55.Remark 14.
So far, we have discussed problem of SFTH∞B analysis for a class of nonlinear discrete networked systems with randomly occurring multi-distributed delays and missing measurements. Also, some sufficient conditions are given to reflect the impacts from the involved phenomena. When the error analysis of the proposed theoretical result becomes a concern, additional efforts will be made with some criterion which will appear in the near future.


## 3.1. Stochastic Finite-Time Boundedness Analysis

The following theorem provides sufficient conditions for the SFTB of closed-loop system (11) by employing Lyapunov-Krasovskii functional approach.Theorem 6.
Given positive constantsc1,c2(c1<c2),δ,N, a symmetric positive-definite matrix R, closed-loop system (11) is SFTB w.r.t.(c1,c2,δ,R,N) if, for a scalar μ≥1, there exist symmetric positive definite matrices P, Qi(i=1,…,q) and positive scalars λj(j=1,2,3) such that the following inequalities hold: (16)Σ11Σ12Σ13∗Σ22Σ23∗∗Σ33<0,(17)rδ-c2μλ1c1μN+1λ2ρλ3∗-c1μN+1λ20∗∗-ρλ3<0,where(18)Σ11=diag-μP,-μdmQ1,…,-μdmQq,Σ22=diag-I,-γμNI,Σ33=diag-P,-P,-Q^,Σ12=-HTR~2000⋮⋮00,Σ13=A-TPβ~A~TPHTQ^A-d1TPb~1A~d1TP0⋮⋮⋮A-dqTPb~qA~dqTP0,Σ23=C-TPα~C~TP0D-TPβ~D~TP0,Q^=dM-dm+1∑i=1qQi-R~1,ρ≔c1qμN+dM-1dMμ+dMdM-1-dmdm-12,P-=R-1/2PR-1/2,λ1=λminP-,λ2=λmaxP-,Q-i=R-1/2HTQiHR-1/2,λ3=maxi∈1,…,qλmaxQ-i,R~1=U1TU2+U2TU12,R~2=-U1T+U2T2.Proof.
Choose the following Lyapunov-Krasovskii functional candidate:(19)Vk=V1k+V2k+V3k,where(20)V1k=ηTkPηk,V2k=∑j=1q∑s=k-djkk-1μk-1-sηTsHTQjHηs,V3k=∑j=1q∑s=-dM+2-dm+1∑t=k+s-1k-1μk-1-tηTtHTQjHηt.
Let us assume that(21)EηkTRηk≤c1,∀k=-dM,-dM+1,…,0.The following goal is to show that if conditions (16) and (17) hold, then (22)EηkTRηk≤c2,∀k=1,…,N.
First, calculating the difference variation ofVi(k)(i=1,2,3), and taking the mathematical expectation, one has (23)EV1k+1-μV1k=EξTkΦTPΦξk+ξTkΠTPΠξk-μηTkPηk,where(24)ξTk=ηkxk-d1k⋯xk-dqkfxkωk,Φ=A-A-d1⋯A-dqC-D-,Π=β~A~b~1A~d1⋯b~qA~dqα~C~β~D~.
Similarly, it can be obtained that(25)EV2k+1-μV2k=E∑j=1q∑s=k+1-djk+1kμk-sηTsHTQjHηs-∑j=1q∑s=k-djkk-1μk-sηTsHTQjHηs=∑j=1qEηTkHTQjHηk+∑s=k+1-dmk-1μk-sηTsQjηs+∑s=k+1-djk+1k-dmμk-sηTsQjηs-∑s=k+1-djkk-1μk-sηTsQjηs-μdmxTk-djkQjxk-djk.Considering dm≤dj(k)≤dM, (26)E∑s=k+1-dmk-1μk-sηTsQjηs-∑s=k-djk+1k-1μk-sηTsQjηs≤0.It follows from (25) that (27)EV2k+1-μV2k≤E∑j=1qηTkHTQjHηk+∑s=k+1-djk+1k-dmμk-sηTsQjηs-μdmxTk-djkQjxk-djk.Similarly, we have (28)EV3k+1-μV3k≤E∑j=1qdM-dmηTkHTQjHηk-∑s=k-dM+1k-dmμk-sηTsHTQjHηs.It is obtained from (23), (27), and (28) that (29)
E
V
k
+
1
-
μ
V
k
≤
E
ξ
T
k
Φ
T
P
Φ
ξ
k
+
ξ
T
k
Π
T
P
Π
ξ
k
-
μ
η
T
k
P
η
k
+
d
M
-
d
m
+
1
η
T
k
∑
j
=
1
q
Q
j
η
k
-
μ
d
m
∑
j
=
1
q
x
T
k
-
d
j
k
Q
j
x
k
-
d
j
k
+
z
T
k
z
k
-
γ
μ
N
ω
T
k
ω
k
-
z
T
k
z
k
+
γ
μ
N
ω
T
k
ω
k
≤
E
ξ
T
k
Φ
T
P
Φ
+
Π
T
P
Π
+
Δ
ξ
k
,where Δ=diag{-μP+(dM-dm+1)∑i=1qHTQiH, -μdmQ1,…,-μdmQq,0,-γ/μNI}.
Notice that (5) implies the following inequality: (30)ηkfxkTHTR~1HR~2TH∗Iηkfxk≤0.Combining (29) with (30), we have (31)EVk+1-μVk≤EξTkΦTPΦ+ΠTPΠ+Θξk+γμNωTkωk,where (32)Θ=-μP+dM-dm+1∑i=1qHTQiH-HTR~1H0⋯0-HTR~20∗-μdmQ1⋯000⋮⋮⋱⋮⋮⋮∗∗⋯-μdmQq00∗∗⋯∗-I0∗∗⋯∗∗-γμN.By applying Lemma 5 to (16), we deduce that ΦTPΦ+ΠTPΠ+Θ<0. Thus, we obtain from (31) that (33)EVk+1-μVk≤γμNωTkωk,∀k∈Z+.That is, (34)EVk+1≤μEVk+γμNωTkωk,∀k∈Z+.It follows from (34) that (35)EVk≤μEVk-1+γμNωTk-1ωk-1,∀k∈Z+.
By iteration and taking Assumption2 into account, it can be obtained that (36)EVk+1-μVk≤μkEV0+γμN∑s=0k-1μk-1-sωTsωs≤μNEV0+γμNμN-1∑s=0N-1ωTsωs≤μNEV0+γμδ.Noting definition of λi(i=2,3), we have (37)EV0=EηT0Pη0+∑j=1q∑s=-dj0-1μ-1-sηTsHTQjHηs+∑j=1q∑s=-dM+2-dm+1∑t=s-1-1μ-1-tηTtHTQjHηt<λ2EηT0Rη0+qλ3μdM-1∑s=-dM-1EηTsRηs+qλ3μdM+2∑s=-dM-2-dm+1∑t=-1+s-1EηTtRηt<λ2+qλ3dMμdM-1+qλ3dMdM-1-dmdm-12μdM-2c1.By taking (36) and (37) into account, we have (38)EVk+1-μVk≤μNσ+γμδ,∀k∈Z+,where σ≔λ2+qλ3dMμdM-1+qλ3dM(dM-1)-dm(dm-1)/2μdM-2c1.
On the other hand, also from the definition ofλi(i=1,2,3), we have (39)EVk+1-μVk≥λ1EηTkRηk,∀k∈Z+.Consequently, we get from (38) and (39) that (40)EηTkRηk<1λ1μμN+1σ+γδ.
Note that(41)μN+1σ+γδ-c2μλ1<0⇔γδ-c2μλ1+c1μN+1λ2+c1qμN+dM-1dMμ+dMdM-1-dmdm-12λ3<0⇔γδ-c2μλ1+c1μN+1λ2+ρλ3<0⇔γδ-c2μλ1+c1μN+1λ2ρλ3c1μN+1λ2-100ρλ3-1c1μN+1λ2ρλ3<0⇔γδ-c2μλ1+c1μN+1λ2ρλ3c1μN+1λ200ρλ3-1c1μN+1λ2ρλ3<0.By Lemma 5, (41) is equivalent to LMI (23); it can be verified that (42)EηTkRηk<1λ1μμN+1σ+γδ<c2.Therefore, according to Definition 3, closed-loop system (11) is SFTB. The above discussion completes the proof of Theorem 6.Remark 7.
If nonlinearities and perturbation are not considered in system (11), that is, f(k)=0 and ω(k)=0, the SFTB problem of system (11) is deduced by analyzing the SFTB of the following system: (43)ηk+1=A-ηk+βk-β-A~ηk+∑i=1qA-diHηk-dik+∑i=1qA~diHηk-dikzk=L-ηk,where system parameters are the same as before.The following criterion for system (43) is given based on Theorem 6.Corollary 8.
Given positive constantsc1,c2,(c1<c2),δ,N, a symmetric positive-definite matrix R, closed-loop system (43) is SFTB w.r.t.(c1,c2,δ,R,N) if, for a scalar μ≥1, there exist symmetric positive definite matrices P, Qi and positive scalars λi(i=1,2,3) such that the following inequalities hold: (44)Θ=-μP000A-TPβ~A~TPHTQ^∗-μdmQ100A-d1TPb~1A~d1TP0∗∗⋱0⋮⋮0∗∗∗-μdmQqA-dqTPb~qA~dqTP0∗∗∗∗-P00∗∗∗∗∗-P0∗∗∗∗∗∗-Q^<0,-c2μλ1c1μN+1λ2ρλ3∗-c1μN+1λ20∗∗-ρλ3<0.


## 3.2. Stochastic Finite-TimeH∞ Boundedness Analysis

In this subsection, we will provide sufficient condition to ensure SFTH∞B of closed-loop system (11) based on Theorem 6.Theorem 9.
Given positive constantsc1, c2(c1<c2), δ, γ, N, a symmetric positive-definite matrix R, closed-loop system (11) is SFTH∞B w.r.t. (c1,c2,δ,γ,R,N) if, for a scalar μ≥1, there exist symmetric positive definite matrices P and Qi(i=1,…,q), and positive scalars λj(j=1,2,3) such that the following inequalities hold: (45)Ω11Ω12Ω13∗Ω22Ω23∗∗Ω33<0,(46)rδ-c2μλ1c1μN+1λ2ρλ3∗-c1μN+1λ20∗∗-ρλ3<0,where(47)Ω11=diag-μP,-μdmQ1,…,-μdmQq,Ω22=diag-I,-γμNI,Ω33=diag-P,-P,-Q^,-IΩ12=-HTR~2000⋮⋮00,Ω13=A-TPβ~A~TPHTQ^LTA-d1TPb~1A~d1TP00⋮⋮⋮⋮A-dqTPb~qA~dqTP00,Ω23=C-TPα~C~TP00D-TPβ~D~TP00.Other parameters are shown in Theorem 6.Proof.
Obviously LMI (45) implies LMI (16); according to Theorem 6, we can easily conclude that the closed-loop system (11) is SFTB w.r.t. (c1,c2,δ,R,N).
In the following, we shall prove that inequality (14) holds. By introducing the same Lyapunov-Krasovskii functional, we can get from (33) (48)EVk+1≤μEVk+γμNωTkωk,∀k∈Z+.And by iteration, we obtain that (49)EVk≤μk-1EV0+∑s=0k-1μk-1-sγμNωTsωs-EzTszs.Under the zero initial condition E{V(0)}=0, and noting that E{V(k)}≥0,∀k∈{1,2,…,N}, we get (50)∑s=0k-1μk-1-sEzTszs≤∑s=0k-1μk-1-sγμNωTsωs.Letting k=N+1, we then have (51)∑s=0NμN-sEzTszs≤∑s=0NμN-sγμNωTsωs.Notice that 1≤μN-s≤μN,∀s∈0,1,…,N; (51) immediately yields(52)E∑s=0NzTszs≤γ∑s=0NωTsωs.which implies that system (11) is SFTH∞B. This completes proof of Theorem 9.


## 3.3. Stochastic Finite-TimeH∞ Output Feedback Control

In this subsection, we will deal with the problem of the design of output-feedback controller for system (1).Theorem 10.
Consider the discrete-time NCS (1) with randomly occuring multi-distrbuted delays and missing measurement. For given positive constants c1, c2(c1<c2), δ and N, and a symmetric positive-definite matrix R=diagR1,R2, closed-loop system (11) is SFTH∞B w.r.t. (c1,c2,δ,γ,R,N) if, for a scalar μ≥1, there exist symmetric positive definite matrices U, Y, Qi(i=1,…,q), matrices Ωj(j=1,2,3), and positive scalars λt′(t=1,2,3,), such that the following inequalities hold: (53)J10H1TH2TH5T∗J20H3T0∗∗J3H4T0∗∗∗J40∗∗∗∗J5<0,(54)rδ-c2μλ1′c1μN+1λ2′ρλ3′∗-c1μN+1λ2′0∗∗-ρλ3′<0,(55)λ1′R1<Y<λ2′R1,0<Qi<λ3′R1,i=1,…,q,where(56)J1=-μU-μI-μI-μY,J2=diag-μdmQ1,…,-μdmQq,J3=diag-I,-γμNI,J4=-U-I00-I-Y0000-U-I00-I-Y,J5=diag-I,-Q^,H1T=UR~2T0R~2T0,H2T=UAT+Ω3BTΩ10β~UCTΩ2TATATR+β-CTΩ20CTΩ2T,H3T=b-1AdTb-1AdTYb~1AdTb~1AdTY⋮⋮⋮⋮b-qAdTb-qAdTYb~qAdTb~qAdTY,H4T=α-Iα-Yα~Iα~YD1TD1TW+β-D2TΩ20D2TΩ2T,H5T=ULTUQ^LTI.
R 1 and R2 are symmetric positive matrices.
In this case, the parameters of the output feedback controller can be designed as(57)AK=X12-1Ω1-YAU-β-Ω2CU-YBΩ3Y12-T,BK=X12-1Ω2,CK=Y12-TΩ3,where X12 and Y12 are any nonsingular matrices satisfying X12Y12T=I-UY.Proof.
Evidently, from Theorem9, system (11) is SFTH∞B if, for a scalar μ≥1, LMIs (45) and (46) hold. To do this, let P and P-1 be as follows: (58)P=YX12X12TX22,P-1=S-1Y12Y12TY22,where the partitioning of P and P-1 is compatible with that of A-, A~, A-di, A~di and X22 and Y22 are symmetric positive matrices.
Define(59)T1=S-1IY12T0,T2=IY0X12T,and this implies that (60)PT1=T2,T1TPT1=T1TT2.Thus (61)P=T2T1-1,X22=Y12-1S-1Y-SS-1Y12-T>0.
It is clear that, by Lemma5, if LMI (53) is feasible, then we have J1<0; thus Y-S<0 and therefore I-YS-1 is nonsingular, which implies that P>0.
First, it can be verified that the LMI (53) can be rewritten as (62)Σ=Σ110Σ130Σ15Σ16Σ17∗Σ2200Σ25Σ260∗∗Σ330Σ35Σ360∗∗∗Σ44Σ45Σ460∗∗∗∗Σ5500∗∗∗∗∗Σ660∗∗∗∗∗∗Σ77<0,where AK, BK, CK are defined in (57),(63)Σ11=-μS-1-μI∗-μY,Σ22=diag-μdmQ1,…,-μdmQq,Σ33=-I,Σ44=-γμN,Σ55=Σ66=-S-1-I∗-Y,Σ77=-I0∗-Q^,Σ13=S-1R~2R~2,Σ15=S-1AT+Ω2TBTΩ1ATATY+β-CTΩ2T,Σ16=0β~S-1CTΩ2T0CTΩ2T,Σ17=S-1LTS-1Q^LTI,Σ25=b-1AdTb-1AdTY⋮⋮b-qAdTb-qAdTY,Σ26=b~1AdTb~1AdTY⋮⋮b~qAdTb~qAdTY,Σ35=α-Iα-Y,Σ36=α~Iα~Y,Σ45=D1TD1TY+β-D2TΩ2T,Σ46=0α~D2TΩ2T.That is, (64)Σ′=Σ11′Σ12′Σ13′Σ14′∗Σ22′Σ23′0∗∗Σ33′0∗∗∗Σ44′<0,where(65)Σ11′=diag-μT1TPT1,-μdmQ1,…,-μdmQq,Σ22′=diag-I,-γμN,Σ33′=diag-T1TP-1T1,-T1TP-1T1,Σ44′=diag-I,-Q^,Σ12′=-R~2HT10⋯000⋯0T,Σ13′=T1TA-TT2β~T1TA~TT2A-d1TT2A~d1TT2⋮⋮A-dqTT2A~dqTT2,Σ14′=T1TLTT1THTQ^00⋮⋮00,Σ23′=C-TT2C~TYD-TT2D~TY.Now letting U=S-1 and applying the congruence transformations diag{T1-T,I,…,I,I,T2-T,T2-T,I,I} to (64), we can obtain LMI (45).
In addition, sinceY and Qi are fringed matrices of P and Q^i=HTQiH, by the property of fringed matrices, we have (66)λmaxY≥λmaxP,λminY≤λminP,λmaxQi≥λmaxQ^i,λminQi≤λminQ^i.Thus, (46) follows from inequalities (54) and (55). This completes the proofRemark 11.
Being different from literature [25], on the one hand, we consider NCSs with the stochastic and nonlinearity. On the other hand, state feedback controller is designed in [25]. However when not all the state variables are available, the results in [25] cannot be applicable. In this paper, output feedback controllers are designed in the stochastic context.Remark 12.
In order to design a controller independent of the matrixX12, we can make X12=Y; then AK, BK, and CK become(67)AK=Y-1Ω1-YAU-β-Ω2CU-YBΩ3I-UY-1Y,BK=Y-1Ω2,CK=I-UY-1YΩ3.Remark 13.
The feasibility of Theorem10 can be represented by the following basic LMIs with a parameter μ: (68)minc2+γs.t.53,54and55.Remark 14.
So far, we have discussed problem of SFTH∞B analysis for a class of nonlinear discrete networked systems with randomly occurring multi-distributed delays and missing measurements. Also, some sufficient conditions are given to reflect the impacts from the involved phenomena. When the error analysis of the proposed theoretical result becomes a concern, additional efforts will be made with some criterion which will appear in the near future.


## 4. Numerical Example

In this section, a numerical example is used to verify the validity of the controller design approach proposed. Consider a discrete-time NCS described by system (1) and its parameters are selected as(69)A=0.4-0.1-0.10.035,Ad=-0.1250.10.05-0.275,B=0.40.2,C=-0.1-0.1T,D1=0.50.1,D2=0.2,L=-0.10.5T,U1=-0.3000,U2=0000.5,R1=1001.We letq=2 and the stochastic parameters are chosen as (70)Eαk=α-=0.8,Eβk=β-=0.8,Eb1k=b-1=0.6,Eb2k=b-2=0.8.Moreover, the time-varying delays satisfy 1≤di(k)≤4(i=1,2), the exogenous disturbance ω(k) is chosen as ω(k)=1/0.1+k2, and the nonlinear function f(x(k)) is selected as fxk=tanh⁡-0.3x1ktanh⁡0.5x2k.For givenN=40, δ=1, c1=1, and μ=1.002, it can be easily checked that E{xT(k)R1x(k)}≤c1 is satisfied under the initial condition x(0)=0.5-0.2T and x^(0)=00T.By solving the minimization problem involving LMIs conditions (53) and (54) and using LMI toolbox in Matlab, we can obtain the optimal value c2min=39.5855 and the optimal H∞ performance γmin=0.0779. By Theorem 10, the resulting controller gain matrices are expressed as (71)AK=1011,BK=0.016300350,CK=2.11002.1324T.Figures2 and 3 show the system state trajectories and the curve of E{xT(k)R1x(k)}. From Figures 2 and 3, we can get that all the state variables converge to zero quickly in the finite-time, which means that system (1) is stochastic finite-time stable. Moreover, for all x(0) that satisfy the initial condition E{xT(0)R1x(0)}≤c1, we can get that the system state trajectory cannot exceed the given optimal value c2min=39.5855, which means the obtained output-feedback controller can guarantee that system (11) is SFTB with prescribed properties. This proves the validity of the theoretical results.Figure 2
The trajectory of statex(k) when E{α(k)}=0.8,E{β(k)}=0.8,E{b1(k)}=0.6,E{b2(k)}=0.8.Figure 3
The curve ofE{xT(k)R1x(k)}.In the following, for further assessment of the control method, we considerEα(k)=α-=0.3, Eβ(k)=β-=0.6, Eb1(k)=b-1=0.5, Eb2(k)=b-2=0.5. Other parameters are chosen based on the values represented above.By solving the minimization problem, we can get the optimal valuesc2min=39.6743, γmin=0.1313, and the resulting controller gain matrices are (72)AK=1011,BK=-0.11200.2499,CK=1.34802.1717T.Simulation results are illustrated in Figures4 and 5. From Figures 4 and 5, it is apparent that the state responses satisfy E{xT(k)R1x(k)}≤39.6743,  k∈{1,…,N}, which means that the system is also SFTB in this case. Compared with the first case, we can get that when randomly occurring multi-distributed delays and missing measurements are more serious, the state trajectory of the system converges to zero for a long time. In addition, the maximum value of E{xT(k)R1x(k)} obtained in the second case is much greater than that in the first case. In short, from the above results, randomly occurring multi-distributed delays and missing measurements have an important effect on the stability of the system. In addition, the adequate performance of NCS (1) is achieved by using the proposed controller, which confirms that the STFH∞B of the NCS (1) is secured in this paper.Figure 4
The trajectory of statex(k) when E{α(k)}=0.3, E{β(k)}=0.6, E{b1(k)}=E{b2(k)}=0.5.Figure 5
The curve ofE{xT(k)R1x(k)}.


## 5. Conclusions

This paper has investigated the SFTH∞B control problem for a class of discrete-time NCS. In this addressed problem, the effects of randomly occurring multi-distributed delays and missing measurement have been taken into account simultaneously. Firstly, the concept of FTB has been extended to SFTB in the discrete-time NCSs. Then, sufficient conditions with less conservative have been established to ensure the existence of the output feedback controller by constructing a set of improved Lyapunov-Krasovskii functional with power function μk-1-s. And the obtained conditions are expressed based on LMI. The gain of controller has been designed by solving LMIs. Lastly, a numerical example is given to show the validity of the verified results. It is worth mentioning that the main results obtained in this paper can be extended to the stochastic finite-time passive or stochastic finite-time dissipative control problems. The results will appear in the near future. In addition, the methods proposed in this paper can be further used to deal with SFTH∞B control problems for discrete-time NCS with more complicated network-induced phenomena such as randomly occurring faults [6] and randomly occurring quantized effect [13, 22, 35].



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*Source: 5109646-2018-12-09.xml*