# Observer-BasedH∞ Control Design for Nonlinear Networked Control Systems with Limited Information

Authors: Yilin Wang; Zhengrong Xiang; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

This paper is concerned with the problem of designing a robust observer-basedH∞ controller for discrete-time networked systems with limited information. An improved networked control system model is proposed and the effects of random packet dropout, time-varying delay, and quantization are considered simultaneously. Based on the obtained model, a stability criterion is developed by constructing an appropriate Lyapunov-Krasovskii functional and sufficient conditions for the existence of a dynamic quantized output feedback controller which are given in terms of linear matrix inequalities (LMIs) such that the augmented error system is stochastically stable with an H∞ performance level. An example is presented to illustrate the effectiveness of the proposed method.


---

## Body

## 1. Introduction

Networked control systems (NCSs) are distributed systems in which communication between sensors, actuators and controllers is supported by a shared real-time network. Compared with conventional point-to-point system connection, this new network-based control scheme reduces system wiring and has low cost, high reliability, information sharing, and remote control [1, 2]. However, NCSs also introduce many new challenges in control system design such as packet dropout, networked-induced delay and signal quantization variable transmission intervals, network security and other communication constraints [3–9]. The authors in [10] proposed different levels of network-induced imperfections and presented the main survey of the methodologies in the analysis and design of NCSs.The time-delay and packet dropouts occur in various physical, industrial, and engineering systems, which is a source of poor performance and instability of systems. Therefore, the issues of network-induced time-delays and packet dropouts have been considered by many researchers. Markov chain has been used to model the randomness of the network-induced delays (packet dropouts) [11–15]. The Markov chain takes values, which correspond to network-induced delays (packet dropouts), in a finite set based on known probabilities. In addition, the transition probabilities of Markovian jump systems (MJSs) are partly unknown due to the complexity of network [16, 17]. The authors in [16] introduced a nonlinear delayed MJSs model with partially unknown transition probabilities established by multiple channels data transmission framework. An iterative method was proposed in [17] to model NCSs with bounded packet dropout as Markovian jump linear systems with partly unknown transition probabilities. On the other hand, much attention has been devoted to the time-delayed jump linear systems with Markovian jumping parameters [18–20], which applied a less-conservative delay-range-dependent method. The stability analysis and stabilization problems for a class of discrete-time Markov jump linear systems with partially known transition probabilities, and time-varying delays are investigated in [18].Quantization process cannot be neglected in NCSs. Real communication networks only send finite precision data due to limited bandwidth. Quantizers with coarser quantization densities help in reducing the network congestion. Consequently, network-induced delays can be reduced because less information is transmitted. In [21, 22], the quantizing effects were transformed into sector-bounded uncertainties and the quantization density was given in the control synthesis. For a logarithmic quantizer, the quantization error was modeled as a norm-bounded uncertainty [23]. However, the quantization density in most studies is just neglected or given as sector bounded uncertainties, and the relationship between the quantization density and the network load has not been fully investigated in the aforementioned papers.Motivated by the above discussion, in this paper, we are aiming at investigating the study of observer-based controller design for nonlinear discrete-time networked systems subject to sensor-controller packet losses, time-varying delays, and output quantization. In this work, the quantization density is designed to be a function of the network load condition which is modeled by a Markov chain [24]. Furthermore, by applying the Lyapunov-Krasovskii functional method, a mode-dependent observer is constructed and the bounded real lemma (BRL) for the resulting error system is derived in terms of LMIs. Also, an improved version of the BRL is given by introducing additional slack matrix variables to eliminate the cross-coupling between system matrices and Lyapunov matrices among different operation modes, which can reduce the computational complexity as far as possible.The paper is organized as follows. In Section2, the NCS description, quantization error modeling, packet dropout and time-varying delay formulation are presented. Main results for stability criterion and observer-based controller synthesis are given in Section 3. A numerical example is provided in Section 4. Finally, Section 5 concludes this paper.Notation. Throughout the paper, the superscripts “-1” and “T” stand for the inverse and transpose of a matrix, respectively; Rn denotes an n-dimensional Euclidean space and the notation P>0 means that P is a real symmetric positive definite matrix. E{x} is the expectation of a stochastic variable x. I and 0 represent an identity matrix and a zero matrix with appropriate dimensions in different place, respectively. In symmetric block matrices or complex matrix expressions, we use an asterisk * to represent a term that is induced by symmetry and diag{⋯} stands for a block diagonal matrix. ∥·∥ refers to the Euclidean norm for vectors and induced 2-norm for matrices. l2[k0,∞) stands for the space of square summable infinite sequence on [k0,∞).


## 2. Problem Formulation

A simple-networked control system is shown in Figure1. A class of discrete-time nonlinear systems under consideration is described by the following equations:
(1)x(k+1)=Ax(k)+Bu(k)+f(k,x(k))+Hw(k),y(k)=Cx(k),z(k)=Gx(k),
where x(k)∈Rn, u(k)∈Rm, z(k)∈Rp, and y(k)∈Rs are the state vector, control input vector, controlled output vector, and measured output, respectively. w(k)∈Rd is the exogenous disturbance signal belonging to l2[0,∞). A, B, H, C, and G are known real matrices with appropriate dimensions. The nonlinear functionf(·) satisfies f(·,0)=0 and the following sector-bounded condition [25]:
(2)[f(k,x(k))-S1x(k)]T[f(k,x(k))-S2x(k)]≤0,
where S1, S2∈Rn×n are known real constant matrices, and S=S1-S2 is a symmetric positive definite matrix.Figure 1
Configuration of the networked control systems with quantizer.The network load condition is related to the network-induced delay. In [26], a Markov chain has been used to model the network load condition. Varying network load can be modeled by making the Markov chain do a transition and transfer every time over the communication network. Modes 1, 2, and 3 are, respectively, corresponded to light load, medium load, and heavy load.Then, a discrete homogeneous Markov chain{σ(k),k} with a finite number of states ℜ={1,2,3} is used to model a network load. The transition probability from mode i at k to mode j at time k+1 is governed by
(3)πij=prob{σ(k+1)=j∣σ(k)=i},i,j∈ℜ,
where πij≥0(i,j∈ℜ) is the transition probability element and satisfies ∑j=13πij=1 (for all i∈ℜ).In addition, the transition probability considered in this paper is partly accessed, and some elements are unknown. For notation clarity, we denoteℜ=ℜKi+ℜUKi (for all i∈ℜ) with ℜKi≜{j:πijisknown} and ℜUKi≜{j:πijisunknown}.Also, we can obtain(4)∑j∈ℜπijPj=PKi+∑j∈ℜUKiπijPj,
where PKi=∑j∈ℜKiπijPj.In this paper, the quantization density is designed to be a function of the network load condition. Hence, the quantization density functionδ=δ(σ(k)) is described as a finite state Markov chain. On the basis of this technique, a networked load dependent on the logarithmic quantizer [23, 24] is proposed as follows:
(5)q(υ,i)={ρh(i)if11+δ(i)ρh(i)<υ≤11-δ(i)ρh(i),υ>0,h=0,±1,±2,…0ifυ=0-q(-υ,i)ifυ<0,
where 0<ρ(i)<1 is the quantization density of q(·,·), and δ(i) is related to ρ(i) by
(6)δ(i)=1-ρ(i)1+ρ(i).This means that the smallerρ(i) results in the smaller δ(i). For this reason, we will call δ(i) the quantization density in the rest of this paper.The associated quantized setU is given by
(7)U={±ρh(i),h=0,±1,±2,…}⋃{0}.Now define the quantization error as(8)ey(k,i)=q(y(k),i)-y(k)=Δq(k,i)y(k),
where y(k) is the signal to be quantized and q(y(k),i) is the quantized signal. The uncertainty matrix Δq(k,i)≜diag(Δq1(k,i),…,Δqs(k,i)) satisfies Δqι(k,i)∈[-δ(i),δ(i)], ι=1,2,…,s.The measured outputy(k) which is used as feedback information for controller may not be available all the time due to packet dropouts. To describe these packet dropouts, a stochastic variable following Bernoulli sequence is used. In this paper, the measured output received in the controller side is involved with the effects of quantization, packet losses, and time-varying delay, and it is described by
(9)y-(k)=θkq(y(k-τk)),
where τ_=minτk and τ-=maxτk are the lower and upper bounds of the delay, respectively. Stochastic variable θk∈R is a Bernoulli distributed white sequence with a probability distribution as follows:
(10)Prob{θk=1}=E{θk}=θ,Prob{θk=0}=1-E{θk}=1-θ,
where 0≤θ≤1 is a known positive constant which is used to denote the probability that the packet will be transmitted successfully from sensor to controller, and we have
(11)E{θk-θ}=0,E{(θk-θ)2}=θ(1-θ)=θ-2.Remark 1.
For NCSs, network-induced delay, packet dropout, and quantization error often occur simultaneously due to limited channel capacity and noise. Actually, pattern (9) describes a class of NCSs with data loss, time-varying delay and the quantization error simultaneously, so it is more comprehensive than separate consideration of delay pattern [27] or data loss pattern [28].In this work, the following dynamic observer-based control scheme is employed for system (1):
(12)x^(k+1)=Ax^(k)+Bu(k)+Lσ(k)(y-(k)-y^(k))y^(k)=θCx^(k)u(k)=Kσ(k)x^(k),
where x^(k)∈Rn and y^(k)∈Rs represent the state estimation and output estimation vectors of system (1), respectively. Lσ(k)∈Rn×s and Kσ(k)∈Rm×n are the observer gain and controller feedback gain to be designed, respectively.We define the following error variable as(13)e(k)=x(k)-x^(k).Combining (12) with (1) and considering (9) and (11), the error system is derived as follows:
(14)e(k+1)=θLσ(k)Cx(k)+(A-θLσ(k)C)e(k)-θLσ(k)(I+Δq(k,σ(k)))Cx(k-τk)-(θk-θ)Lσ(k)(I+Δq(k,σ(k)))Cx(k-τk)+f(k,x(k))+Hw(k).On the other hand, system (1) can be rewritten as
(15)x(k+1)=(A+BKσ(k))x(k)-BKσ(k)e(k)+f(k,x(k))+Hw(k).By settingη(k)=[xT(k)eT(k)]T, the augmented system can be obtained as follows:
(16)η(k+1)=Φ1σ(k)η(k)+θΦ2σ(k)η(k-τk)+(θk-θ)Φ2σ(k)η(k-τk)+DF(k,η(k))+H-w(k),z(k)=G-η(k)η(k)=ϕ(k);k=-τ-,-τ-+1,…,0,
where
(17)Φ1σ(k)=A-+θI1Lσ(k)C-1+B-Kσ(k)I2,Φ2σ(k)=-I1Lσ(k)(I+Δq(k,σ(k)))C-2,G-=[G0],H-=[HH],D=[II],F(k,η(k))=f(k,Eη(k)),
with
(18)A-=[A00A],I1=[0I],C-1=[C-C],B-=[B0],I2=[I-I],C-2=[C0],E=[I0].Lemma 2 (see [29]).
LetD, E, and F be matrices with appropriate dimensions. Suppose FTF≤I, and then for any scalar ε>0, we have
(19)DFE+ETFTDT≤εDDT+ε-1ETE.Definition 3 (see [30]).
System (16) with w(k)≡0 is said to be stochastically stable if, for every finite η0=η(0), the following inequality holds:
(20)E{∑k=0∞∥η(k)∥2∣η0}<∞.Definition 4 ([31]).
For a given scalarγ>0, system (16) is stochastically stable and has an H∞ noise attenuation performance index γ if the following conditions are satisfied. (a)
System (16) with ω(k)≡0 is stochastically stable.(b)
Under the zero-initial condition, it holds that(21)∑k=0∞E{zT(k)z(k)}<γ2∑k=0∞E{wT(k)w(k)},∑k=0∞E{zT(k)z(k)}<γ2mii∀w(k)∈l2[0,∞).In this paper, the main objective is to design controller (12) for system (1) in the presence of output quantization, sensor-controller packet losses, and time-varying delays such that system (16) is stochastically stable with a prescribed H∞ performance index γ.


## 3. Main Results

Theorem 5.
For a given scalarγ>0, quantization densities ρ(i) and a packet loss rate 0<θ<1, system (16) with partly unknown transition probabilities is stochastically stable with a prescribed H∞ performance index γ, if there exist matrices Pi>0(i∈ℜ) and Q>0 such that
(22)[πKiΘ-1iΓ-1iPKiΓ-2iPKi*-PKi0**-PKi]<0,[Θ-1iΓ-1iPjΓ-2iPj*-Pj0**-Pj]<0,∀j∈ℜUKi,
where
(23)Θ-1i=[(τ--τ_+1)Q-Pi-S-1+G-TG-0-S-20*-Q00**-I0***-γ2I],Γ-1i=[Φ1iθΦ2iDH-]T,Γ-2i=[0θ-Φ2i00]T,πKi=∑j∈ℜKiπij,PKi=∑j∈ℜKiπijPj.Proof.
Choose a Lyapunov-Krasovskii functional of the form:(24)V(η(k),k)=V1(η(k),k)+V2(η(k),k)+V3(η(k),k),
with
(25)V1(η(k),k)=ηT(k)Pσ(k)η(k)V2(η(k),k)=∑l=k-τkk-1ηT(l)Qη(l)V3(η(k),k)=∑h=-τ-+1-τ_∑l=k+hk-1ηT(l)Qη(l),
where Pσ(k)>0(σ(k)∈ℜ) and Q>0 are symmetric positive definite matrices.
Let the mode at timesk and k+1 be i and j, respectively. That is, σ(k)=i and σ(k+1)=j for any i,j∈ℜ. Taking the difference of above functional along the solution of system (16), we have
(26)E[ΔV(η(k),k)]=E[V(η(k+1),k+1)∣η(k)]-V(η(k),k)=ΔV1(η(k),k)+ΔV2(η(k),k)+ΔV3(η(k),k),
with
(27)ΔV1(η(k),k)=ηT(k+1)∑j∈ℜπijPjη(k+1)-ηT(k)Piη(k)ΔV2(η(k),k)=∑l=k+1-τk+1kηT(l)Qη(l)-∑l=k-τkk-1ηT(l)Qη(l)=ηT(k)Qη(k)-ηT(k-τk)Qη(k-τk)+∑l=k+1-τk+1k-1xT(l)Qx(l)-∑l=k+1-τkk-1xT(l)Qx(l)≤ηT(k)Qη(k)-ηT(k-τk)Qη(k-τk)+∑l=k+1-τ-k-τ_ηT(l)Qη(l),ΔV3(η(k),k)=∑h=-τ-+1-τ_∑l=k+1+hkηT(l)Qη(l)-∑h=-τ-+1-τ_∑l=k+hk-1ηT(l)Qη(l)=(τ--τ_)ηT(k)Qη(k)-∑l=k+1-τ-k-τ_ηT(l)Qη(l).
Therefore,(28)E[ΔV(η(k),k)]≤ηT(k+1)∑j∈ℜπijPjη(k+1)-ηT(k)Piη(k)+(τ--τ_+1)ηT(k)Qη(k)-ηT(k-τk)Qη(k-τk).
From inequality (2), we know
(29)[η(k)F(k,η(k))]T[S-1S-2*I][η(k)F(k,η(k))]≤0,
where
(30)S-1=ETS1TS2+S2TS12E,S-2=-ETS1T+S2T2.
Denotingζ(k)=[ηT(k),ηT(k-τk),FT(k,η(k))]T and combining (28) and (29) lead to
(31)E[ΔV(η(k),k)]≤ηT(k+1)∑j∈ℜπijPjη(k+1)+ζT(k)[∑j∈ℜKiπij+∑j∈ℜUKiπij]Θ1iζ(k)=ηT(k+1)[PKi+∑j∈ℜUKiπijPj]η(k+1)+ζT(k)[πKi+∑j∈ℜUKiπij]Θ1iζ(k)=ζT(k){[πKiΘ1i+Θ2i]+∑j∈ℜUKiπij(Θ1i+Θ3i)}ζ(k),
where
(32)Θ1i=[(τ--τ_+1)Q-Pi-S-10-S-2*-Q0**-I],Θ2i=[Φ1iTPKiΦ1iθΦ1iTPKiΦ2iΦ1iTPKiD*(θ2+θ-2)Φ2iTPKiΦ2iθΦ2iTPKiD**DTPKiD],Θ3i=[Φ1iTPjΦ1iθΦ1iTPjΦ2iΦ1iTPjD*(θ2+θ-2)Φ2iTPjΦ2iθΦ2iTPjD**DTPjD].
By Schur complement, (22) implies πKiΘ1i+Θ2i<0 and Θ1i+Θ3i<0, and then we obtain from (31) that
(33)E[ΔV(ζ(k),k)]≤-λmin[-πKiΘ1i-Θ2i]ζT(k)ζ(k)-∑j∈ℜUKiπijminj{λmin[-(Θ1i+Θ3i)]}ζT(k)ζ(k)≤-(β1+β2)ζT(k)ζ(k),
where β1=infi∈ℜ{-λmin[-πKiΘ1i-Θ2i]} and β2=infi∈ℜ{-(1-πKi)minj{λmin[-(Θ1i+Θ3i)]}}. From (33), if we do mathematical expectations in both sides, we obtain that for any N≥0(34)E[V(ζ(N+1),N+1)-V(ζ(0),0)]≤-(β1+β2)∑k=0NE[ζT(k)ζ(k)],
which yields
(35)∑k=0∞E[ζT(k)ζ(k)]≤1β1+β2E[V(ζ(0),0)]<∞.
Thus, from Definition3, the closed-loop system (16) with w(k)=0 is stochastically stable.
Now, to establish anH∞ performance, consider the following performance index:
(36)JN≜E{∑k=0N[zT(k)z(k)-γ2wT(k)w(k)]}.
Under zero-initial condition, we have
(37)JN=E∑k=0N[zT(k)z(k)-γ2wT(k)w(k)+ΔV(η(k),k)]-E[V(N+1)]≤E∑k=0∞[zT(k)z(k)-γ2wT(k)w(k)+ΔV(η(k),k)]=∑k=0∞ξT(k)[[πKiΘ-1i+Θ-2i]+∑j∈ℜUKiπij(Θ-1i+Θ-3i)]ξ(k),
where
(38)ξ(k)=[ζT(k),wT(k)]T,Θ-2i=[Φ1iTPKiΦ1iθΦ1iTPKiΦ2iΦ1iTPKiDΦ1iTPKiH-*(θ2+θ-2)Φ2iTPKiΦ2iθΦ2iTPKiDθΦ2iTPKiH-**DTPKiDDTPKiH-***H-TPKiH-],Θ-3i=[Φ1iTPjΦ1iθΦ1iTPjΦ2iΦ1iTPjDΦ1iTPjH-*(θ2+θ-2)Φ2iTPjΦ2iθΦ2iTPjDθΦ2iTPjH-**DTPjDDTPjH-***H-TPjH-].
Similarly, by Schur complement, it can be obtained from (22) that [πKiΘ-1i+Θ-2i]+∑j∈ℜUKiπij(Θ-1i+Θ-3i)<0, which means that JN<0.
Then,(39)∑k=0∞E{∥z(k)∥2}≤γ2∑k=0∞∥w(k)∥2.
This completes the proof.Remark 6.
Note thatβ1+β2 will be reduced to only β1 (resp., β2) if all the transition probabilities are known (resp., unknown).Remark 7.
The considered systems are more general than these systems with completely known or completely unknown transition probabilities, which can be viewed as two special cases of the ones tackled here; that is, when all the transition probabilities are known or unknown, the underlying systems are the traditional MJSs (ℜKi=ℜ, ℜUKi=⌀) or the switched systems under arbitrary switching (ℜKi=⌀, ℜUKi=ℜ), respectively. Therefore, observer-based controller with partly unknown transition probabilities of this paper is a more natural assumption to the Markovian jump systems and hence covers the existing ones.Corollary 8.
For a given scalarγ>0, quantization densities ρ(i) and a packet losses rate 0<θ<1, the closed-loop system (16) with completely known transition probabilities is stochastically stable with an H∞ attenuation level if there exist matrices Pi>0(i∈ℜ) and Q>0 such that
(40)[πKiΘ-1iΓ-1iPKiΓ-2iPKi*-PKi0**-PKi]<0,
where PKi=∑j∈ℜKiπijPj with ℜKi=ℜ and ℜUKi=⌀.Proof.
Similar to the proof line of Theorem5, the result can be obtained. The detailed proof is omitted here.Theorem5 provides Markov-chain-element-dependent sufficient conditions which guarantee the H∞ performance as well as the stochastic stability of the augmented error system (16). The following theorem gives sufficient conditions on the existence of a mode-dependent observer-based controller.Theorem 9.
For a given scalarγ>0, quantization densities ρ(i) and a packet loss rate 0<θ<1, if there exist matrices B~i, L~i, Ri, Pi>0(i∈ℜ), Q>0 and scalar εi>0 such that
(41)[Θ--1iΣ1iΣ2i0*Υj-Ri-RiT0-θL~i**Υj-Ri-RiTθ-L~i***-εiI]<0,
where
(42)Θ--1i=[(τ--τ_+1)Q-Pi-S-1+G-TG-0-S-20*-Q+εiδ2(i)C-2TC-200**-I0***-γ2I],Σ1i=[RiA-+B~iI2+θL~iC-1-θL~iC-2RiDRiH-]T,Σ2i=[0θ-L~iC-200]T,
then, there exists a mode-dependent observer-based controller as (12) such that the augmented system (16) is stochastically stable with a prescribed H∞ performance index γ. Moreover, if LMIs (41) have a feasible solution, the desired controller gains and observer gains are given by
(43)B~i=RiB-Ki,L~i=RiI1Li,i∈ℜ.Proof.
From Theorem9, inequalities (22) can be rewritten as
(44)[Θ-1iΓ-1iΥjΓ-2iΥj*-Υj0**-Υj]<0,
where
(45)Υj≜{1πKiPKi,∀j∈ℜKiPj,∀j∈ℜUKi.
Performing a congruence transformation to (44) by diag{I,…,I︸4,Υj-1RiT,Υj-1RiT}, one has
(46)[Θ-1iΓ-1iRiTΓ-2iRiT*-RiΥj-1RiT0**-RiΥj-1RiT]<0.
For an arbitrary matrix Ri, for all i∈ℜ, we have the following fact:
(47)(1πKiPKi-Ri)(1πKiPKi)-1(1πKiPKi-Ri)-1≥0,(Pj-Ri)Pj-1(Pj-Ri)-1≥0.
Then, we obtain Υj-RiT-Ri≥-RiΥj-1RiT, so (46) holds if the following condition is satisfied:
(48)[Θ-1iΓ-1iRiTΓ-2iRiT*Υj-Ri-RiT0**Υj-Ri-RiT]<0.
Then, using the condition (8), we have
(49)Δq(k,i)=δ(i)Δi,
where Δi=diag{Δq1(k,i)/δ(i),…,Δqs(k,i)/δ(i)} and satisfies ΔiTΔi≤I.
Substituting (49) into (48) and using Lemma 2, we get that (48) is equivalent to
(50)Λ+Π1iΔiTΠ2i+Π2iTΔiΠ1iT≤Λ+εiΠ1iΠ1iT+εi-1Π2iTΠ2i<0,
where
(51)Λ=[(τ--τ_+1)Q-Pi-S-10-S-20(RiΦ1i)T0*-Q00-(θRiI1LiC-2)T(θ-RiI1LiC-2)T**-I0(RiD)T0***-γ2I(RiH-)T0****Υj-Ri-RiT0*****Υj-Ri-RiT],Π1i=[0δ(i)C-20000]T,Π2i=[0000-(θRiI1Li)T(θ-RiI1Li)T].
Applying Schur complement and substitutingB~i=RiB-Ki and L~i=RiI1Li into (50), we have that (50) is equivalent to (41). Therefore, we obtain that system (16) is stochastically stable with a prescribed H∞ performance index γ. The proof is completed.


## 4. Numerical Example

Consider the following state-space representation of a nonlinear networked system in (1) with the following parameters:
(52)A=[-0.35-0.10.10.2],B=[10],H=[-0.40.2],C=[0.10],G=[0.2-0.3],f(k,x(k))=[f1T(k,x(k))f2T(k,x(k))]T,withf1(k,x(k))=-tanh(x1)+0.5x1+0.1x2,f2(k,x(k))=0.1x1-tanh(x2)+0.5x2.
It is easy to verify that
(53)S1=[-0.50.10.1-0.1],S2=[0.50.10.10.5].For the given system with data loss and time-varying delay, delay bounds are given asτ-=3 and τ_=1, and the packet losses parameters are selected as θ=0.8. The network load is modeled by a Markov chain which takes values from a finite set {1,2,3} and the transition probability matrix is
(54)Ω=[0.10.60.3?0.4?0.5??],
where ? stands for the unknown elements.The quantization density for each mode is assumed asδ(1)=0.2, δ(2)=0.3, and δ(3)=0.4. Then the corresponding augmented system (16) can be obtained.Takingγ=1.2, then solving (41) in Theorem 9, we obtain the observer gain as
(55)L1=[0.42780.1599],L2=[0.58820.8301],L3=[-0.7726-0.5458],
and controller gains can be computed as
(56)K1=[0.7088-0.2844],K2=[0.4107-0.2327],K3=[0.7023-0.2856].
Thus according to Theorem 9, we can conclude that, under the designed controller, the corresponding augmented system (16) is stochastically stable with a given H∞ performance level γ=1.2.The initial condition is selected asx(0)=[1-1], e(0)=[00], and the noise signal w(k)=0.01exp(-0.01k)sin(0.02πk). The switching sequence and the trajectories of plant state x(k) and error e(k) are shown in Figures 2, 3, 4, and 5.Figure 2
Markovian switching signal.Figure 3
Trajectory of system statex(k).Figure 4
Trajectory of error statee1(k).Figure 5
Trajectory of error statee2(k).It is clearly observed from the simulation results that the corresponding augmented system (16) is stochastically stable. This demonstrates the effectiveness of the proposed results.


## 5. Conclusions

In this work, the problem of robust observer-basedH∞ control for discrete-time nonlinear networked systems with limited information is proposed. The quantization density is designed to be a function of the network load condition which is modeled by a Markov chain with partly unknown transition probability. Bernoulli random sequences are used to model packet dropouts in the network. An estimation technique is then developed to obtain the estimation of system state and the observer-based controller is proposed to stabilize the networked controlled systems by using the Lyapunov-Krasovskii functional approach. An example is presented to illustrate the effectiveness of the proposed method.



---

*Source: 604249-2013-09-19.xml*