# An LMI-BasedH∞ Control Approach for Networked Control Systems with Deadband Scheduling Scheme

Authors: Hui-ying Chen; Zu-xin Li; Yan-feng Wang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Due to the bandwidth constraints in the networked control systems (NCSs), a deadband scheduling strategy is proposed to reduce the data transmission rate of network nodes. A discrete-time model of NCSs is established with both deadband scheduling and network-induced time-delay. By employing the Lyapunov functional and LMI approach, a state feedbackH∞ controller is designed to ensure the closed-loop system asymptotically to be stable with H∞ performance index. Simulation results show that the introduced deadband scheduling strategy can ensure the control performance of the system and effectively reduce the node's data transmission rate.


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## Body

## 1. Introduction

Networked control systems (NCSs) are control systems in which the control loop is closed over a wired or wireless communication network. They have received a great deal of attention in the recent years owing to their successful applications in a wide range of areas such as industrial automation, aerospace, and nuclear power station [1, 2]. Compared with the traditional point-to-point communication, NCSs have advantages of low cost, easy installation and maintenance, great reliability, and so forth. However, due to the introduction of the communication networks, they also incur some new issues such as network-induced delays, packet dropouts, and limited bandwidth resources, which make the analysis and design of NCSs become more complex [3, 4].Many results for NCSs have been reported to handle network-induced delays, packet dropouts and communication constraints in the literature; see [5–12] and the references therein. It should be pointed out that most of the available results make use of time-driven sampling and communication scheme since it is relatively easy to implement, and there is a well-established system theory for periodic signals. However, time-driven communication scheme is not desirable in many control applications. For example, in the NCSs with limited bandwidth resources, frequent data transmission will increase the network collision probability when there are many nodes on the network, thereby increasing the communication delay and data dropouts and leading to the poor performance and instability of the systems [3]. On the other hand, as is well-known, in the wireless networked control systems (WNCSs), a main constraint of wireless devices is the limited battery life, and wireless transmission consumes significantly more energy than internal computation [13]; thus reducing the data transmission rate is particularly important in the WNCSs. For the above two cases, time-driven communication scheme is not suitable since its transmission rate is generally high which results in inefficient utilization of the limited resources, such as network bandwidth and energy. Therefore, how to design a reasonable scheduling strategy to reduce the use of constrained resources and ensure the performance of NCSs becomes one of the research hotspots.Not only deadband scheduling techniques (i.e., by setting transmission deadband for the network node, the node will not transmit a new message if the node signal or signal change is within the deadband), which can effectively reduce the use of network bandwidth and energy consumption, but also the algorithms which are easy to realize have received an increasing attention in the recent years [14–18]. Besides, numerous other concepts have been proposed in the literature, such as send-on-delta sampling [19–21], event-based sampling [22–24], and event-triggered sampling [25, 26]. Despite the existence of many names, the basic principle is the same. In [14], a deadband method was introduced into the NCSs for the first time, in which the transmission deadbands were set in the sensor and controller to reduce the data transmission rate, and the deadband threshold optimization problem was also discussed. In [15], the relationship between the deadband threshold and the control performance was analyzed by simulation. The paper [23] used the deviation of two adjacent states beyond the deadband threshold to drive the nodes’ data transmission and a dynamically selected deadband threshold value in accordance with the round-trip delay. In [16, 17], the stability of the system with deadband scheduling was investigated, but the network delay was not considered in [16] the nodes should be synchronized and the network delay should be measurable in [17]. The paper [18] proposed a signal difference-based transmission deadband scheduling strategy, a continuous-time model of WNCSs was established with both the probability distribution of delay and parametric uncertainties, and the H∞ controller was designed. In [26], for a class of uncertain continuous-time NCSs with quantizations, the codesign for controller and event-triggering scheme was proposed by using a delay system approach.Until now, although some important pieces of work have been reported on deadband scheduling schemes in NCSs, which have a great significance on both theoretical development and practical applications in NCSs, it is worth noting that the obtained results on deadband scheduling in NCSs mostly focus on the system simulation and performance analysis; few papers have solved the problems of controller design and synthesis, which are more useful and challenging than the issue of performance analysis. In addition, to the best of our knowledge, few related results have been established for discrete-time NCSs with deadband scheduling, which motivates the work of this paper.In this paper, we propose a deadband scheduling scheme to save the limited bandwidth resources while guaranteeing the desiredH∞ control performance. Considering the influence of uncertain short time-delay, the NCSs with both deadband scheduling and time-delay is modeled as a discrete-time system with parameter uncertainties. By the Lyapunov functional and LMI approach, the H∞ control problem is investigated. Finally, a numerical example is given to show the usefulness of the derived results.The rest of this paper is organized as follows. Section2 gives a discrete-time model of the closed-loop system. In Section 3, a state feedback H∞ controller is designed to ensure the closed-loop system asymptotically to be stable with H∞ performance index. Section 4 demonstrates the validness of the proposed method through a numerical example. Conclusions are given in Section 5.Notation. The notations used throughout this paper are fairly standard. ℝn denotes the n-dimensional Euclidean space. ∥·∥2 refers to the Euclidean vector norm. l2[0,∞) is the space of square summable infinite sequence. I and 0 represent the identity matrix and zero matrix with appropriate dimensions, respectively. diag{⋯} stands for a diagonal matrix. The superscripts “T” and “−1” represent matrix transpose and inverse, and “*” denotes the term that is induced by symmetry.


## 2. Problem Description and Modeling

The networked control system with deadband scheduling in this paper is shown in Figure1, where deadband schedulers (DS1, DS2) are set in the sensor and controller, respectively, τksc is sensor-to-controller delay, and τkca is controller-to-actuator delay.Figure 1
Model of networked control system with deadband scheduling.Consider the following continuous plant model:(1)x˙(t)=Ax(t)+B1v(t)+B2w(t),y(t)=Cx(t),
where x(t)∈ℝn is the state vector of plant, v(t)∈ℝp is the input vector, y(t)∈ℝq is the output vector, and w(t)∈ℝp is the disturbance input. A, B1, B2, and C are known real constant matrices with appropriate dimensions.Make the following assumptions for the system.(1)
In the smart sensor, the sampler is time-driven, with a sampling periodh; both the controller and actuator are event-driven.(2)
The total network-induced time-delayτk=τksc+τkca is time varying and nondeterministic, which satisfies 0≤τk≤h.Thus, the discretized equation of plant can be described as [27]
(2)x(k+1)=Gx(k)+H0(τk)v(k)+H1(τk)v(k-1)+Hww(k),y(k)=Cx(k),
where G=eAh, H0(τk)=∫0h-τkeAtdtB1, H1(τk)=∫h-τkheAtdtB1, and Hw=∫0heAtdtB2. By mathematical transformation, H0(τk), H1(τk) can be expressed as
(3)H0(τk)=H0+DF(τk′)E,H1(τk)=H1-DF(τk′)E,
where
(4)H0=∫0h/2eAsds·B,H1=∫h/2heAsds·B,δ=maxτk′∥∫0-τk′eAsds∥2=∥∫0h/2eAsds∥2,D=δeA(h/2),E=B,τk′∈[-h2,h2],FT(τk′)F(τk′)≤I.Remark 1.
Limited by space, the detailed discretization process for system (1) with uncertain short time-delay is omitted in this paper and can be found in [27]. From (2) and (3), we know that the continuous plant with uncertain short time-delay in NCSs can be modeled as a discrete linear system with parameter uncertainties.


### 2.1. Description of Deadband Schedulers

The signal will be transmitted only when the difference between the current signal and the previous transmission signal is greater than the error threshold. According to this, the deadband schedulers designed in this paper are the error threshold-based deterministic schedulers. The working mechanism of the deadband scheduler 1(DS1) in the sensor can be described as(5)x-i(k)={xi(k),|x-i(k-1)-xi(k)|>δi|xi(k)|,x-i(k-1),|x-i(k-1)-xi(k)|≤δi|xi(k)|,
where i=1,2,…,n; δi∈[0,1], x-i(k), and xi(k) are given error threshold values, output signals, and input signals of DS1, respectively.SetΛ1=diag{δ1,δ2,…,δn},  F1(k)=diag{f11(k),f12(k),…,f1n(k)},  f1i(k)∈[-1,1], i=1,2,…,n; then from (5), the input-output relationship of DS1 can be converted to
(6)x-(k)=x(k)+Λ1F1(k)x(k),
where F1T(k)F1(k)≤I.Remark 2.
In this paper, the effect of active packet dropouts under the deadband scheduling scheme is modeled as a bounded uncertain item of the transmission signal. The main advantages of this modeling method are as follows:(1) a non-linear relationship between the input and output of the deadband scheduler is converted to a linear relationship with uncertain parameters; (2) due to the bounded ranges of uncertain parameters related to the deadband threshold values, it is easier to merge scheduling policy parameters into the system model.
Similarly, the working mechanism of deadband scheduler 2(DS2) in the controller can be described as:(7)vj(k)={uj(k),|vj(k-1)-uj(k)|>σj|uj(k)|,vj(k-1),|vj(k-1)-uj(k)|≤σj|uj(k)|,
where j=1,2,…,p; σj∈[0,1], vj(k) and uj(k) are given error threshold values, output signals and input signals of DS2, respectively.
SetΛ2=diag{σ1,σ2,…σp},  F2(k)=diag{f21(k),f22(k),…,f2p(k)}, f2j(k)∈[-1,1], j=1,2,…,p, then from (7) the input-output relationship of DS2 can be converted to:
(8)v(k)=u(k)+Λ2F2(k)u(k),
where F2T(k)F2(k)≤I.
From the above, we know that after the introduction of deadband schedulers into NCSs, the signals do not need to be transmitted at each sampling period so as to achieve the purpose of reducing data transmission rate and the effect of bandwidth constraints on the system. In addition, the principles of the considered schedulers are simple, which do not require a lot of computing and data storage.


### 2.2. The Model of Closed-Loop System

Employ a memoryless state feedback controller(9)u(k)=Kx-(k),
where x-(k)∈ℝn, u(k)∈ℝp, and K is the state feedback gain with appropriate dimensions.Selecting the augmented vectorz(k)=[xT(k)vT(k-1)]T and synthesizing (2), (3), (6), (8), and (9), the closed-loop system can be described as
(10)z(k+1)=Φkz(k)+H-ww(k),y(k)=C-z(k),
where(11)Φk=[G+(H0+DF(τk′)E)(I+Λ2F2(k))K(I+Λ1F1(k))H1-DF(τk′)E(I+Λ2F2(k))K(I+Λ1F1(k))0]=G-+H-K1I-+D-F(τk′)EK1I-+D-F(τk′)E-,where(12)G-=[GH100],C-=[C0],H-=[H0I],D-=[D0],I-=[I0],E-=[0-E],H-w=[Hw0],K1=(I+Λ2F2(k))K(I+Λ1F1(k)).


## 2.1. Description of Deadband Schedulers

The signal will be transmitted only when the difference between the current signal and the previous transmission signal is greater than the error threshold. According to this, the deadband schedulers designed in this paper are the error threshold-based deterministic schedulers. The working mechanism of the deadband scheduler 1(DS1) in the sensor can be described as(5)x-i(k)={xi(k),|x-i(k-1)-xi(k)|>δi|xi(k)|,x-i(k-1),|x-i(k-1)-xi(k)|≤δi|xi(k)|,
where i=1,2,…,n; δi∈[0,1], x-i(k), and xi(k) are given error threshold values, output signals, and input signals of DS1, respectively.SetΛ1=diag{δ1,δ2,…,δn},  F1(k)=diag{f11(k),f12(k),…,f1n(k)},  f1i(k)∈[-1,1], i=1,2,…,n; then from (5), the input-output relationship of DS1 can be converted to
(6)x-(k)=x(k)+Λ1F1(k)x(k),
where F1T(k)F1(k)≤I.Remark 2.
In this paper, the effect of active packet dropouts under the deadband scheduling scheme is modeled as a bounded uncertain item of the transmission signal. The main advantages of this modeling method are as follows:(1) a non-linear relationship between the input and output of the deadband scheduler is converted to a linear relationship with uncertain parameters; (2) due to the bounded ranges of uncertain parameters related to the deadband threshold values, it is easier to merge scheduling policy parameters into the system model.
Similarly, the working mechanism of deadband scheduler 2(DS2) in the controller can be described as:(7)vj(k)={uj(k),|vj(k-1)-uj(k)|>σj|uj(k)|,vj(k-1),|vj(k-1)-uj(k)|≤σj|uj(k)|,
where j=1,2,…,p; σj∈[0,1], vj(k) and uj(k) are given error threshold values, output signals and input signals of DS2, respectively.
SetΛ2=diag{σ1,σ2,…σp},  F2(k)=diag{f21(k),f22(k),…,f2p(k)}, f2j(k)∈[-1,1], j=1,2,…,p, then from (7) the input-output relationship of DS2 can be converted to:
(8)v(k)=u(k)+Λ2F2(k)u(k),
where F2T(k)F2(k)≤I.
From the above, we know that after the introduction of deadband schedulers into NCSs, the signals do not need to be transmitted at each sampling period so as to achieve the purpose of reducing data transmission rate and the effect of bandwidth constraints on the system. In addition, the principles of the considered schedulers are simple, which do not require a lot of computing and data storage.


## 2.2. The Model of Closed-Loop System

Employ a memoryless state feedback controller(9)u(k)=Kx-(k),
where x-(k)∈ℝn, u(k)∈ℝp, and K is the state feedback gain with appropriate dimensions.Selecting the augmented vectorz(k)=[xT(k)vT(k-1)]T and synthesizing (2), (3), (6), (8), and (9), the closed-loop system can be described as
(10)z(k+1)=Φkz(k)+H-ww(k),y(k)=C-z(k),
where(11)Φk=[G+(H0+DF(τk′)E)(I+Λ2F2(k))K(I+Λ1F1(k))H1-DF(τk′)E(I+Λ2F2(k))K(I+Λ1F1(k))0]=G-+H-K1I-+D-F(τk′)EK1I-+D-F(τk′)E-,where(12)G-=[GH100],C-=[C0],H-=[H0I],D-=[D0],I-=[I0],E-=[0-E],H-w=[Hw0],K1=(I+Λ2F2(k))K(I+Λ1F1(k)).


## 3. Design ofH∞ Controller

In this section, we will investigateH∞ control problem for the closed-loop system (10). Throughout this paper, we will use the following lemmas.


### 3.1. Related Lemmas

Lemma 3 (see [28]).
For the given matricesA, Q=QT, and P=PT>0, ATPA+Q<0 hold if and only if
(13)[QATA-P-1]<0,[-P-1AATQ]<0.Lemma 4 (see [28]).
LetW, M, N, F(k) be real matrices of appropriate dimensions with FT(k)F(k)≤I and W=WT; then
(14)W+MF(k)N+NTFT(k)MT<0
holds if and only if there exists a real scalar ε>0, satisfying
(15)W+εMMT+ε-1NTN<0.
More especially, when F(k) is a diagonal matrix, there also exists the following lemma.Lemma 5 (see [29]).
LetW, M, N be real matrices of appropriate dimensions, let F(k) be a diagonal matrix with FT(k)F(k)≤I and W=WT; then the following two conditions are equivalent:(1)
W+MF(k)N+NTFT(k)MT<0(2)
there exists a matrixS=ST>0, satisfying W+MSMT+NTS-1N<0.Remark 6.
Due to the introduction of a symmetric positive-definite matrix instead of a scalar in Lemma5, problem solving is expected to have a less conservatism compared with Lemma 4.
We are now in a position to formulateH∞ control problem for NCSs with both deadband scheduling and uncertain short time-delay. More specifically, given a disturbance attenuation level γ, we design a state feedback controller of the form (9) such that the closed-loop system (10) with w(k)=0 is asymptotically stable and under zero initial condition; the output y(k) satisfies ∥y∥2≤γ∥w∥2 for all nonzero w(k)∈l2[0,∞).


### 3.2. Main Results

Based on Lyapunov functional method andH∞ theory [28], the following conclusions can be obtained.Theorem 7.
For a given scalarγ>0, under the given deadband scheduling schemes (5) and (7), the closed-loop system (10) is asymptotically stable with H∞ performance γ if there exist symmetric positive-definite matrices P, W1, W2, feedback gain matrix K, and scalar ε1>0 such that
(16)[Π11G-+H-KI-H-wΠ14H-KΛ10*Π220(EKI-+E-)T0(KI-)T**-γ2I000***Π44EKΛ10****-W1(KΛ1)T*****-W2]<0,
where
(17)Π11=-P-1+ε1D-D-T+(H-Λ2)W2(H-Λ2)T,Π14=(H-Λ2)W2(EΛ2)T,Π22=-P+C-TC-+I-TW1I-,Π44=-ε1I+(EΛ2)W2(EΛ2)T.Proof .
(i) We first show that system (10) with w(k)=0 is asymptotically stable. To the end, defining a Lyapunov functional as V(k)=zT(k)Pz(k), we have that
(18)ΔV(k)=V(k+1)-V(k)=zT(k+1)Pz(k+1)-zT(k)Pz(k)=zT(k)ΦkTPΦkz(k)-zT(k)Pz(k)=zT(k)(ΦkTPΦk-P)z(k).
Obviously, if ΦkTPΦk-P<0, then ΔV(k)<0, and the closed-loop system (10) is asymptotically stable.
(ii)  Next, we prove that system (10) has H∞ performance γ. Define
(19)J=∑k=0∞[yT(k)y(k)-γ2wT(k)w(k)]=∑k=0∞[yT(k)y(k)-γ2wT(k)w(k)+ΔV(k)]-V(∞)+V(0).
Under zero initial conditions, we have that V(0)=0, but V(∞)≥0; therefore,
(20)J≤∑k=0∞[yT(k)y(k)-γ2wT(k)w(k)+ΔV(k)]=∑k=0∞[zT(k)wT(k)]Π[z(k)w(k)],
where
(21)Π=[ΦkTPΦk+C-TC--PΦkTPH-w*H-wTPH-w-γ2I].
IfΠ<0, then ΦkTPΦk+C-TC--P<0; therefore, ΦkTPΦk-P<-C-TC-<0; the condition in (i) holds, and system (10) with w(k)=0 is asymptotically stable. In addition, If Π<0, then J<0; therefore, ∥y∥22=∑k=0∞yT(k)y(k)<γ2∑k=0∞wT(k)w(k)=γ2∥w∥22.
According toΠ<0, substituting Φk=G-+H-K1I-+D-F(τk′)EK1I-+D-F(τk′)E- into (21) and applying Lemma 3, we have that
(22)[-P-1G-+H-K1I-H-w*-P+C-TC-0**-γ2I]+[D-00]F(τk′)[0EK1I-+E-0]+[0EK1I-+E-0]TFT(τk′)[D-00]T<0.
Due to FT(τk′)F(τk′)≤I and by the use of Lemma 4, we can get that
(23)[-P-1G-+H-K1I-H-w*-P+C-TC-0**-γ2I]+ε1[D-00][D-00]T+ε1-1[0EK1I-+E-0]T[0EK1I-+E-0]<0.
According to Lemma 3, (23) is equivalent to
(24)[-P-1+ε1D-D-TG-+H-K1I-H-w0*-P+C-TC-0(EK1I-+E-)T**-γ2I0***-ε1I]<0.
Substituting K1=(I+Λ2F2(k))K(I+Λ1F1(k)) into (24), we have that(25)[-P-1+ε1D-D-TG-+H-(I+Λ2F2(k))KI-H-w0*-P+C-TC-0(E(I+Λ2F2(k))KI-+E-)T**-γ2I0***-ε1I]+[H-(I+Λ2F2(k))KΛ100E(I+Λ2F2(k))KΛ1]F1(k)[0I-00]+[0I-00]TF1T(k)[H-(I+Λ2F2(k))KΛ100E(I+Λ2F2(k))KΛ1]T<0.
Considering thatF1(k) is a diagonal matrix with F1T(k)F1(k)≤I, and so employing Lemma 5, we can get that(26)[-P-1+ε1D-D-TG-+H-(I+Λ2F2(k))KI-H-w0*-P+C-TC-0(E(I+Λ2F2(k))KI-+E-)T**-γ2I0***-ε1I]+[H-(I+Λ2F2(k))KΛ100E(I+Λ2F2(k))KΛ1]W1-1[H-(I+Λ2F2(k))KΛ100E(I+Λ2F2(k))KΛ1]T+[0I-00]TW1[0I-00]<0.
According to Lemma3, (26) is equivalent to(27)[-P-1+ε1D-D-TG-+H-(I+Λ2F2(k))KI-H-w0H-(I+Λ2F2(k))KΛ1*-P+C-TC-+I-TW1I-10(E(I+Λ2F2(k))KI-+E-)T0**-γ2I00***-ε1IE(I+Λ2F2(k))KΛ1****-W1]<0.
Similarly, we can eliminateF2(k). By use of Lemma 5, we have that(28)[-P-1+ε1D-D-TG-+H-KI-H-w0H-KΛ1*-P+C-TC-+I-TW1I-10(EKI-+E-)T0**-γ2I00***-ε1IEKΛ1****-W1]+[H-Λ200EΛ20]W2[H-Λ200EΛ20]T+[0KI-00KΛ1]TW2-1[0KI-00KΛ1]<0.
Furthermore, applying Lemma3, (28) can be converted to (16).
The proof is completed.Remark 8.
Notice that the matrix inequality (16) in Theorem 7 is a bilinear matrix inequality due to the existence of P-1. Generally, it can be solved by the linear approach [30] or the cone complementarity linearization (CCL) method [31]. By contrast, the CCL result is less conservative [32] and so is employed in this paper.Corollary 9.
The bilinear matrix inequality (16) can be transformed to the following objective function minimization problems by the CCL method.
Find(29)P>0,X>0,W1>0,W2>0,ε1>0,KminTrace(PX)s.t.[Π^11G-+H-KI-H-wΠ14H-KΛ10*Π220(EKI-+E-)T0(KI-)T**-γ2I000***Π44EKΛ10****-W1(KΛ1)T*****-W2]<0,[PIIX]≥0,
whereΠ^11=-X+ε1D-D-T+(H-Λ2)W2(H-Λ2)T.Since Corollary9 has turned the nonconvex feasibility problem of Theorem 7 into a minimization problem of nonlinear objective function with linear matrix inequalities constraints, it can be solved by the following iterative algorithm.Algorithm 10.
Step  1. Find a set of feasible solutions Ξ0={P0,X0,W10,W20,ε10,K0}, which satisfyies (29), and set the iterative number l=0.
Step  2. Use LMI toolbox of mincx solver to solve the following linear objective function minimization problem:
(30)minTrace(PlX+PXl)s.t.(29).
The solutions are set Ξ*={P*,X*,W1*,W2*,ε1*,K*}.
Step  3. Substituting the solutions Ξ* into the matrix inequality (16) in Theorem 7 to check if (16) is satisfied, then K* becomes the state feedback gain matrix and the iteration terminates. Otherwise, enter into Step 4.
Step  4. If the iterative number satisfies l≤L(L is a predetermined iterative number upper bound), set Ξl+1=Ξ*, l=l+1, and return to Step 2 for the next iteration. Otherwise, enter into Step 1 and reselect a set of feasible solutions Ξ0 to calculate.Thus, a state feedbackH∞ controller can be obtained for NCSs with both deadband scheduling and uncertain short time-delay. More especially, if there are no deadband schedulers in the NCSs shown in Figure 1, the closed-loop system in (10) reads
(31)z(k+1)=Φkz(k)+H-ww(k),y(k)=C-z(k),
where
(32)Φk=[G+(H0+DF(τk′)E)KH1-DF(τk′)EK0]=G-+H-KI-+D-F(τk′)EKI-+D-F(τk′)E-.Then, we have the following corollary, which can be proved along similar lines as in the proof of Theorem7.Corollary 11.
Consider the NCSs in Figure1, but without the deadband schedulers. For a given scalar γ>0, the closed-loop system (31) is asymptotically stable with H∞ performance γ if there exist symmetric positive-definite matrix P, feedback gain matrix K, and scalar ε1>0 such that
(33)[-P-1+ε1D-D-TG-+H-KI-H-w0*-P+C-TC-0(EKI-+E-)T**-γ2I0***-ε1I]<0.
Similarly, the bilinear matrix inequality (33) can be solved by the above CCL method and the iterative algorithm in Corollary 9 and is thus omitted.


## 3.1. Related Lemmas

Lemma 3 (see [28]).
For the given matricesA, Q=QT, and P=PT>0, ATPA+Q<0 hold if and only if
(13)[QATA-P-1]<0,[-P-1AATQ]<0.Lemma 4 (see [28]).
LetW, M, N, F(k) be real matrices of appropriate dimensions with FT(k)F(k)≤I and W=WT; then
(14)W+MF(k)N+NTFT(k)MT<0
holds if and only if there exists a real scalar ε>0, satisfying
(15)W+εMMT+ε-1NTN<0.
More especially, when F(k) is a diagonal matrix, there also exists the following lemma.Lemma 5 (see [29]).
LetW, M, N be real matrices of appropriate dimensions, let F(k) be a diagonal matrix with FT(k)F(k)≤I and W=WT; then the following two conditions are equivalent:(1)
W+MF(k)N+NTFT(k)MT<0(2)
there exists a matrixS=ST>0, satisfying W+MSMT+NTS-1N<0.Remark 6.
Due to the introduction of a symmetric positive-definite matrix instead of a scalar in Lemma5, problem solving is expected to have a less conservatism compared with Lemma 4.
We are now in a position to formulateH∞ control problem for NCSs with both deadband scheduling and uncertain short time-delay. More specifically, given a disturbance attenuation level γ, we design a state feedback controller of the form (9) such that the closed-loop system (10) with w(k)=0 is asymptotically stable and under zero initial condition; the output y(k) satisfies ∥y∥2≤γ∥w∥2 for all nonzero w(k)∈l2[0,∞).


## 3.2. Main Results

Based on Lyapunov functional method andH∞ theory [28], the following conclusions can be obtained.Theorem 7.
For a given scalarγ>0, under the given deadband scheduling schemes (5) and (7), the closed-loop system (10) is asymptotically stable with H∞ performance γ if there exist symmetric positive-definite matrices P, W1, W2, feedback gain matrix K, and scalar ε1>0 such that
(16)[Π11G-+H-KI-H-wΠ14H-KΛ10*Π220(EKI-+E-)T0(KI-)T**-γ2I000***Π44EKΛ10****-W1(KΛ1)T*****-W2]<0,
where
(17)Π11=-P-1+ε1D-D-T+(H-Λ2)W2(H-Λ2)T,Π14=(H-Λ2)W2(EΛ2)T,Π22=-P+C-TC-+I-TW1I-,Π44=-ε1I+(EΛ2)W2(EΛ2)T.Proof .
(i) We first show that system (10) with w(k)=0 is asymptotically stable. To the end, defining a Lyapunov functional as V(k)=zT(k)Pz(k), we have that
(18)ΔV(k)=V(k+1)-V(k)=zT(k+1)Pz(k+1)-zT(k)Pz(k)=zT(k)ΦkTPΦkz(k)-zT(k)Pz(k)=zT(k)(ΦkTPΦk-P)z(k).
Obviously, if ΦkTPΦk-P<0, then ΔV(k)<0, and the closed-loop system (10) is asymptotically stable.
(ii)  Next, we prove that system (10) has H∞ performance γ. Define
(19)J=∑k=0∞[yT(k)y(k)-γ2wT(k)w(k)]=∑k=0∞[yT(k)y(k)-γ2wT(k)w(k)+ΔV(k)]-V(∞)+V(0).
Under zero initial conditions, we have that V(0)=0, but V(∞)≥0; therefore,
(20)J≤∑k=0∞[yT(k)y(k)-γ2wT(k)w(k)+ΔV(k)]=∑k=0∞[zT(k)wT(k)]Π[z(k)w(k)],
where
(21)Π=[ΦkTPΦk+C-TC--PΦkTPH-w*H-wTPH-w-γ2I].
IfΠ<0, then ΦkTPΦk+C-TC--P<0; therefore, ΦkTPΦk-P<-C-TC-<0; the condition in (i) holds, and system (10) with w(k)=0 is asymptotically stable. In addition, If Π<0, then J<0; therefore, ∥y∥22=∑k=0∞yT(k)y(k)<γ2∑k=0∞wT(k)w(k)=γ2∥w∥22.
According toΠ<0, substituting Φk=G-+H-K1I-+D-F(τk′)EK1I-+D-F(τk′)E- into (21) and applying Lemma 3, we have that
(22)[-P-1G-+H-K1I-H-w*-P+C-TC-0**-γ2I]+[D-00]F(τk′)[0EK1I-+E-0]+[0EK1I-+E-0]TFT(τk′)[D-00]T<0.
Due to FT(τk′)F(τk′)≤I and by the use of Lemma 4, we can get that
(23)[-P-1G-+H-K1I-H-w*-P+C-TC-0**-γ2I]+ε1[D-00][D-00]T+ε1-1[0EK1I-+E-0]T[0EK1I-+E-0]<0.
According to Lemma 3, (23) is equivalent to
(24)[-P-1+ε1D-D-TG-+H-K1I-H-w0*-P+C-TC-0(EK1I-+E-)T**-γ2I0***-ε1I]<0.
Substituting K1=(I+Λ2F2(k))K(I+Λ1F1(k)) into (24), we have that(25)[-P-1+ε1D-D-TG-+H-(I+Λ2F2(k))KI-H-w0*-P+C-TC-0(E(I+Λ2F2(k))KI-+E-)T**-γ2I0***-ε1I]+[H-(I+Λ2F2(k))KΛ100E(I+Λ2F2(k))KΛ1]F1(k)[0I-00]+[0I-00]TF1T(k)[H-(I+Λ2F2(k))KΛ100E(I+Λ2F2(k))KΛ1]T<0.
Considering thatF1(k) is a diagonal matrix with F1T(k)F1(k)≤I, and so employing Lemma 5, we can get that(26)[-P-1+ε1D-D-TG-+H-(I+Λ2F2(k))KI-H-w0*-P+C-TC-0(E(I+Λ2F2(k))KI-+E-)T**-γ2I0***-ε1I]+[H-(I+Λ2F2(k))KΛ100E(I+Λ2F2(k))KΛ1]W1-1[H-(I+Λ2F2(k))KΛ100E(I+Λ2F2(k))KΛ1]T+[0I-00]TW1[0I-00]<0.
According to Lemma3, (26) is equivalent to(27)[-P-1+ε1D-D-TG-+H-(I+Λ2F2(k))KI-H-w0H-(I+Λ2F2(k))KΛ1*-P+C-TC-+I-TW1I-10(E(I+Λ2F2(k))KI-+E-)T0**-γ2I00***-ε1IE(I+Λ2F2(k))KΛ1****-W1]<0.
Similarly, we can eliminateF2(k). By use of Lemma 5, we have that(28)[-P-1+ε1D-D-TG-+H-KI-H-w0H-KΛ1*-P+C-TC-+I-TW1I-10(EKI-+E-)T0**-γ2I00***-ε1IEKΛ1****-W1]+[H-Λ200EΛ20]W2[H-Λ200EΛ20]T+[0KI-00KΛ1]TW2-1[0KI-00KΛ1]<0.
Furthermore, applying Lemma3, (28) can be converted to (16).
The proof is completed.Remark 8.
Notice that the matrix inequality (16) in Theorem 7 is a bilinear matrix inequality due to the existence of P-1. Generally, it can be solved by the linear approach [30] or the cone complementarity linearization (CCL) method [31]. By contrast, the CCL result is less conservative [32] and so is employed in this paper.Corollary 9.
The bilinear matrix inequality (16) can be transformed to the following objective function minimization problems by the CCL method.
Find(29)P>0,X>0,W1>0,W2>0,ε1>0,KminTrace(PX)s.t.[Π^11G-+H-KI-H-wΠ14H-KΛ10*Π220(EKI-+E-)T0(KI-)T**-γ2I000***Π44EKΛ10****-W1(KΛ1)T*****-W2]<0,[PIIX]≥0,
whereΠ^11=-X+ε1D-D-T+(H-Λ2)W2(H-Λ2)T.Since Corollary9 has turned the nonconvex feasibility problem of Theorem 7 into a minimization problem of nonlinear objective function with linear matrix inequalities constraints, it can be solved by the following iterative algorithm.Algorithm 10.
Step  1. Find a set of feasible solutions Ξ0={P0,X0,W10,W20,ε10,K0}, which satisfyies (29), and set the iterative number l=0.
Step  2. Use LMI toolbox of mincx solver to solve the following linear objective function minimization problem:
(30)minTrace(PlX+PXl)s.t.(29).
The solutions are set Ξ*={P*,X*,W1*,W2*,ε1*,K*}.
Step  3. Substituting the solutions Ξ* into the matrix inequality (16) in Theorem 7 to check if (16) is satisfied, then K* becomes the state feedback gain matrix and the iteration terminates. Otherwise, enter into Step 4.
Step  4. If the iterative number satisfies l≤L(L is a predetermined iterative number upper bound), set Ξl+1=Ξ*, l=l+1, and return to Step 2 for the next iteration. Otherwise, enter into Step 1 and reselect a set of feasible solutions Ξ0 to calculate.Thus, a state feedbackH∞ controller can be obtained for NCSs with both deadband scheduling and uncertain short time-delay. More especially, if there are no deadband schedulers in the NCSs shown in Figure 1, the closed-loop system in (10) reads
(31)z(k+1)=Φkz(k)+H-ww(k),y(k)=C-z(k),
where
(32)Φk=[G+(H0+DF(τk′)E)KH1-DF(τk′)EK0]=G-+H-KI-+D-F(τk′)EKI-+D-F(τk′)E-.Then, we have the following corollary, which can be proved along similar lines as in the proof of Theorem7.Corollary 11.
Consider the NCSs in Figure1, but without the deadband schedulers. For a given scalar γ>0, the closed-loop system (31) is asymptotically stable with H∞ performance γ if there exist symmetric positive-definite matrix P, feedback gain matrix K, and scalar ε1>0 such that
(33)[-P-1+ε1D-D-TG-+H-KI-H-w0*-P+C-TC-0(EKI-+E-)T**-γ2I0***-ε1I]<0.
Similarly, the bilinear matrix inequality (33) can be solved by the above CCL method and the iterative algorithm in Corollary 9 and is thus omitted.


## 4. Numerical Example

In this section, a numerical example is introduced to demonstrate the effectiveness of the proposed method. Consider a ball and beam system with [33]
(34)x˙(t)=[0100]x(t)+[01]u(t)+[01]w(t),y(t)=[10]x(t).In this example, we chooseh=0.5s, and τk∈[0,h] is time varying and nondeterministic. According to (2) and (3), we have that
(35)G=[10.501],H0=[0.03130.25],H1=[0.09380.25],D=[0.26610.066500.2661],E=[01],Hw=[0.1250.5].For a given disturbance attenuation levelγ=5, based on the LMI toolbox, and applying Corollaries 9 and 11, the feedback gain matrix K values are given with different error threshold values Λ1, Λ2 in Table 1. It is obvious from Table 1 that for a given level γ we can find the feasible feedback gain matrix K values when Λ1, Λ2 are within certain ranges.Table 1
The feedback gain matrixK values for various Λ1, Λ2.
Case
Λ
1
Λ
2
K
1
Without scheduling
Without scheduling
K
=
[
-
0.5462
-
1.0978
]
2
[
0.02
0
;
0
0.02
]
0.02
K
=
[
-
0.4942
-
1.0314
]
3
[
0.05
0
;
0
0.05
]
0.05
K
=
[
-
0.4393
-
0.9733
]
4
[
0.07
0
;
0
0.07
]
0.05
K
=
[
-
0.4185
-
0.9605
]
5
[
0
.
1
0
;
0
0.1
]
0.07
No solutionIn addition, we choose the initial valuex0=[2-0.5]T, the disturbance
(36)w(k)={0.1,50≤k≤60,0,other,
and the total runtime 100 seconds. Define MTR=nsent/ntotal and IAE=∑∥e(k)∥2·h, in which MTR denotes the mean data transmission rate (nsent and ntotal denote the number of data transmitted with and without deadband schedulers in the runtime, respectively.) and IAE denotes the control performance of the system. Under three different random time-delay sequences, the performance of MTR and IAE is shown for the system with the above four error threshold values in Figures 2 and 3, respectively. It can be easily seen that compared with the system without deadband schedulers (case 1), although the control performance of the system by using deadband scheduling scheme (case 2–case 4) is slightly worse (in Figure 3), the mean data transmission rate of the system is greatly reduced (in Figure 2).Figure 2
The performance of MTR.Figure 3
The performance of IAE.Figures4–6 show the simulation results for the system in which the error threshold values take Λ1=diag{0.07,0.07}, Λ2=0.05, the feedback gain matrix K=[-0.4185-0.9605] according to Table 1, and the time-delay takes the first series. It can be seen that the closed-loop system is asymptotically stable (in Figure 4), and only part of the sampled data and control signal are transmitted with the proposed deadband scheduling scheme (in Figures 5 and 6, here the transmission interval of x-2(k) is similar to x-1(k) in DS1 and is thus omitted).Figure 4
State response curves of the closed-loop system.Figure 5
Comparisons of the transmission interval ofx-1(k).Figure 6
Comparisons of the transmission interval ofv(k).Furthermore, under zero initial conditions, we get that∥y∥2=0.7310, ∥w∥2=0.3317, which yields γ*=2.20. It means that the practical disturbance attenuation level is smaller than the given level γ=5, which shows the effectiveness of the proposed H∞ controller design method.


## 5. Conclusions

In this paper, a discrete-time model for NCSs with both deadband scheduling and time-delay has been established and theH∞ control problem has been investigated. Based on the LMI approach, a state feedback H∞ controller has been designed to ensure the closed-loop system asymptotically to be stable with H∞ performance index. A numerical example has been provided to show the validness of the derived results. Since the principles and algorithms of deadband schedulers in this paper are very simple, the smart sensor and controller are easy to implement. In addition, simulation results show that it can effectively reduce the node’s data transmission rate, so it is very suitable for applying in the NCSs with limited bandwidth resources.



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*Source: 430630-2013-11-11.xml*