𝐻_∞ Control for Multirate Networked Control Systems with Deadband Scheduling

Abstract

This paper is concerned with the problem of 𝐻_∞ control for multirate networked control systems with deadband scheduling policy and transmission delay. The transmission deadbands are set at the sensor nodes and the controller nodes to reduce the network traffic, and then the systems are modeled as discrete-time switch system with uncertain parameters. Using the Lyapunov theory, a robust 𝐻_∞ controller is designed to maintain the asymptotic stability of the closed-loop systems. Finally, a simulation example is presented to show the advantage of the deadband scheduling policy and the efficiency of the proposed theoretical results.

1. Introduction

Networked control systems (NCSs) are the complex systems in which the controlled plants, sensors, controllers, and actuators are distributed in different places and connected through a communication network [1]. Compared with the traditional control systems, NCSs have many advantages such as the flexibility of structure, high reliability, and simple installation. As the rapid development of information technology, NCSs have been applied to various applications including mobile robots, unmanned aerial vehicles, and remote surgeries [2–5]. However, the data transmission through a network brings many challenges, such as time delay, packet dropout, and data disorder, which will cause the system performance degradation and even, make the systems unstable. In recent years, the stability analysis and controller design of NCSs have been researched by many scholars [6–10].

When the bandwidth of NCSs is constrained, the packet collision becomes more severe. It is difficult to guarantee the stabilization and acceptable performance of systems only by means of controller design [11, 12]. In this case, the network scheduling becomes particularly important. The network nodes’ unnecessary data traffic could be reduced by introducing appropriate network scheduling strategy; thus, the network-induced delay and packet dropouts can be reduced remarkably [13–17]. In [13], the problem of integrated design of controller and communication sequences is addressed for a class of multiple inputs multiple outputs NCSs based on switch system theory, but not taking the network-induced delay into account, and the results are extended to NCSs with transmission delay in [14, 15]. In [16], a coordinate design scheme of the try-once-discard (TOD) dynamic scheduling strategy and robust  𝐻_∞  controller for a class of networked control systems with communication constraints and random time-delay is proposed. In [17], the wireless NCSs with communication delay and packet dropouts, which are based on the deadband scheduling policy, are modeled as continuous time-delay systems and an 𝐻_∞ controller is designed. Meanwhile, the event-triggering NCSs attract many researchers’ attention [18–20]. Compared with the widely used time-triggering sampling scheme, the event-triggering sampling scheme, whose main property is that the signal is sampled only when some functions of the system state or output measurement exceeds a threshold, can also save network bandwidth for the NCSs.

It is worth pointing that all of the aforementioned results are under the assumption that NCSs work in a single rate mode (either the actuators are event driven or the updating period of actuators is the same as the sampling period). The multirate NCSs, in which the updating rate of actuators is several times the sampling rate, have many advantages: (1) compared with the NCSs with event-driven actuators, the actuator updating moments are fixed in the multirate NCSs; thus, the analysis and synthesis of the multirate NCSs become more easier; (2) compared with the NCSs in which the actuator updating rate is the same as the sampling rate, the control input can be implemented to the plant more sooner in the multirate NCSs; therefore, better system performance can be ensured [21–23]. In [21], the multirate NCSs are models as the discrete switch systems, and an 𝐻_∞ controller is designed by using the average dwell time method. The model-predictive control problem for multirate NCSs is proposed in [22]. To the best of the authors’ knowledge, 𝐻_∞ control problem of multirate NCSs with deadband scheduling has not been fully investigated to date, which motivates the present study.

In this paper, we focus on the problem of 𝐻_∞ control for multirate NCSs with deadband scheduling and transmission delay. By analyzing the influences of transmission deadband, a discrete-time switch system model with uncertain parameters is proposed to describe the multirate NCSs. The design method of a robust 𝐻_∞ controller that stabilizes this class of networked control systems is proposed. A simulation example is presented to show the advantage of the deadband scheduling policy and the efficiency of the proposed method.

This paper is organized as follows. In Section 2, the problem is stated and some useful definitions and lemmas are given, and then the main results of this paper are given in Section 3. Section 4 provides a simulation example to illustrate the effectiveness of our results. Finally, Section 5 gives some concluding remarks.

Notation. ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, and 𝐼 is identity matrix. 𝐴^𝑇  stands for the transpose of the corresponding matrix  𝐴. The notation 𝐴≥0(𝐴>0) means that the matrix 𝐴 is a positive semidefinite (positive definite) matrix. For an arbitrary matrix 𝑌 and two symmetric matrices  𝑋  and  𝑍, 𝑋𝑌∗𝑍  denotes a symmetric matrix, where ∗ denotes a block matrix entry implied by symmetry.

2. Problem Formulation and Preliminaries

Consider NCSs whose setup is depicted in Figure 1. The controlled plant, sensor, and controller are connected with a communication network. The transmission deadbands are set at the sensor node and the controller node.

Figure 1 

Illustration of NCSs over communication network.

The plant is described by 𝑥̇(𝑡)=𝐴_𝑝𝑥(𝑡)+𝐵_𝑝𝑢(𝑡)+𝐵_𝑝𝑤𝑤(𝑡)𝑦(𝑡)=𝐶_𝑝𝑥(𝑡), (1) where 𝑥(𝑡)∈ℝ^𝑛  is the state, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝑦(𝑡)∈ℝ^𝑙 is the output, and 𝑤(𝑡)∈ℝ^𝑝 is the exogenous input. 𝐴_𝑝,𝐵_𝑝,𝐵_𝑝𝑤, and 𝐶_𝑝 are known real constant matrices with appropriate dimensions.

The sensors are periodically sampled with the sampling interval 𝑇_𝑠. The sensor-to-controller transmission delay is described as  𝜏^sc, and the controller-to-actuator transmission delay is described as 𝜏^ca. We make the following assumptions about NCSs.

Assumption 1. 𝜏=𝜏^sc+𝜏^ca,0<𝜏<𝑇_𝑠.

Assumption 2. The controller is event driven; the sensors and actuators are time driven.

Assumption 3. The updating period of actuators is 𝑇_𝑎, 𝑇_𝑠=𝑁𝑇_𝑎, 𝑁 is positive integer, and 𝑁>1.

First, the discrete-time model of multirate NCSs without transmission deadband is proposed.

The controller output 𝑢(𝑘𝑇) arrives at the actuator at the instants 𝑘𝑇+𝜏_𝑘; if  𝜏_𝑘∈((𝑛−1)𝑇_𝑎,𝑛𝑇_𝑎],(𝑛∈{1,2,…,𝑁}), then the actuator will update at the instances 𝑘𝑇_𝑠+𝑛𝑇_𝑎. Therefore, in the sample interval [𝑘𝑇_𝑠,(𝑘+1)𝑇_𝑠], system (1) is described as 𝑥̇(𝑡)=𝐴_𝑝𝑥(𝑡)+𝐵_𝑝𝐾𝑥(𝑘−1)𝑇_𝑠+𝐵_𝑝𝑤𝑤(𝑡),𝑡∈𝑘𝑇_𝑠,𝑘𝑇_𝑠+𝑛𝑇_𝑎,𝑥̇(𝑡)=𝐴_𝑝𝑥(𝑡)+𝐵_𝑝𝐾𝑥𝑘𝑇_𝑠+𝐵_𝑝𝑤𝑤(𝑡),𝑡∈𝑘𝑇_𝑠+𝑛𝑇_𝑎,(𝑘+1)𝑇_𝑠,𝑦(𝑡)=𝐶_𝑝𝑥(𝑡). (2)(3) Let 𝐴=𝑒^{𝐴𝑝𝑇𝑠}, system (1) with sampling period 𝑇_𝑠 is discretized to 𝑥(𝑘+1)=𝐴𝑥(𝑘)+^𝑛𝑇𝑎₀𝑒^{𝐴𝑝(𝑇𝑠−𝑠)}𝐵_𝑝()+𝑑𝑠⋅𝑢𝑘−1_{𝑇𝑠𝑛𝑇𝑎}𝑒^{𝐴𝑝(𝑇𝑠−𝑠)}𝐵_𝑝𝑑𝑠⋅𝑢(𝑘)+𝐵_𝜔𝜔(𝑘)=𝐴𝑥(𝑘)+^𝑛𝑇𝑎₀𝑒^{𝐴𝑝(𝑁𝑇𝑎−𝑠)}𝐵_𝑝()+𝑑𝑠⋅𝑢𝑘−1_{𝑁𝑇𝑎𝑛𝑇𝑎}𝑒^{𝐴𝑝(𝑁𝑇𝑎−𝑠)}𝐵_𝑝𝑑𝑠⋅𝑢(𝑘)+𝐵_𝜔𝜔(𝑘). (4) Let 𝐵_1𝑛=^𝑁−1_𝛼∑_=𝑛∫^{(𝛼+1)𝑇𝑎}_𝛼𝑇𝑎𝑒^{𝐴𝑝(𝑇𝑠−𝑠)}𝐵_𝑝𝐵𝑑𝑠,𝑛<𝑁0𝑛=𝑁,_2𝑛=^𝑛−1_𝛼=0^{(𝛼+1)𝑇𝑎}_𝛼𝑇𝑎𝑒^{𝐴𝑝(𝑇𝑠−𝑠)}𝐵_𝑝𝑑𝑠. (5) Then 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵_1𝑛𝑢(𝑘)+𝐵_2𝑛𝑢(𝑘−1)+𝐵_𝜔𝜔(𝑘). (6) Next, we introduce the deadband scheduling policy. The transmission deadbands located at the sensor node and controller node are based on such a deadband scheduling policy: if the difference between the present signal and the last transmitted signal exceeds the threshold of deadband, the present signal will be transmitted, otherwise it will not be transmitted for the purpose of reducing the network traffic and saving the network bandwidth.

Let 𝑥(𝑘)=[𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)]^𝑇, 𝑥_𝑖(𝑘) is the component of 𝑥(𝑘) and sampled separately by each senor at the sampling instant. If 𝑥_𝑖(𝑘) enters into the deadband, according to the proposed scheduling policy, 𝑥_𝑖(𝑘) will not be transmitted and the controller will hold the previous transmitted one, which is equivalent to transmitting the previous transmitted one again. Therefore, the deadband 1 is described as 𝑥_𝑖(𝑘)=𝑓_𝑖𝑥_𝑖(𝑘)=𝑥_𝑖(𝑘)𝑥_𝑖(𝑘−1)−𝑥_𝑖(𝑘)>𝛿_1𝑖𝑥_𝑖(𝑘)𝑥_𝑖(𝑘−1)𝑥_𝑖(𝑘−1)−𝑥_𝑖(𝑘)≤𝛿_1𝑖𝑥_𝑖(𝑘), (7) where 𝛿_1𝑖 is the deadband threshold coefficient, 𝛿_1𝑖∈(0,1).

According to (7), 𝑥_𝑖(𝑘)−𝛿_1𝑖𝑥_𝑖(𝑘)≤𝑥_𝑖(𝑘)≤𝑥_𝑖(𝑘)+𝛿_1𝑖𝑥_𝑖(𝑘). (8) Define Δ₁=diag{Δ₁₁,Δ₁₂,…,Δ_1𝑛},Δ_1𝑗∈[−𝛿_1𝑗,𝛿_1𝑗],(𝑗=1,2,…,𝑛) and 𝛿₁=diag{𝛿₁₁,𝛿₁₂,…,𝛿_1𝑛}; therefore, one can obtain 𝑥(𝑘)=𝐼+Δ₁𝑥(𝑘). (9)

Similarly, deadband 2 is described as 𝑢_𝑖(𝑘)=𝑔_𝑖𝑢_𝑖(𝑘)=𝑢_𝑖(𝑘)𝑢_𝑖(𝑘−1)−𝑢_𝑖(𝑘)>𝛿_2𝑖𝑢_𝑖(𝑘)𝑢_𝑖(𝑘−1)𝑢_𝑖(𝑘−1)−𝑢_𝑖(𝑘)≤𝛿_2𝑖𝑢_𝑖(𝑘), (10) where 𝛿_2𝑖 is the deadband threshold coefficient, 𝛿_2𝑖∈(0,1).

Define Δ₂=diag{Δ₂₁,Δ₂₂,…,Δ_2𝑚},Δ_2𝑗∈[−𝛿_2𝑗,𝛿_2𝑗],(𝑗=1,2,…,𝑚) and 𝛿₂=diag{𝛿₂₁,𝛿₂₂,…,𝛿_2𝑚}; therefore, one can obtain 𝑢(𝑘)=𝐼+Δ₂𝑢(𝑘). (11)

Combining (6), (9), and (11), the discrete-time model of multirate NCSs is described as 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵_1𝑛𝐼+Δ₁𝐾𝐼+Δ₂𝑥(𝑘)+𝐵_2𝑛𝐼+Δ₃𝐾𝐼+Δ₄𝑥(𝑘−1)+𝐵_𝜔𝜔(𝑘)𝑦(𝑘)=𝐶𝑥(𝑘). (12)

The following definition and lemmas will be essential for the proofs in the next section.

Definition 1. System (12) with a given matrix 𝐾 is said to be asymptotically stable with an 𝐻_∞ norm bound 𝛾 if the following conditions hold:(1)system (12) is asymptotically stable with 𝜔(𝑘)=0; (2)under the zero initial condition, the controlled output 𝑦(𝑘) satisfies ‖𝑦(𝑘)‖₂<𝛾‖𝜔(𝑘)‖₂, where 𝑤(𝑡)∈𝐿₂[0,∞).

Lemma 2 (see [24] Schur complement). For matrices 𝐴,𝑄=𝑄^𝑇,𝑃=𝑃^𝑇>0, the matrix inequality 𝐴^𝑇𝑃𝐴+𝑄<0 holds, if and only if the following matrix inequalities 𝑄𝐴^𝑇𝐴−𝑃⁻¹<0𝑜𝑟−𝑃⁻¹𝐴𝐴^𝑇𝑄<0 (13) hold.

Lemma 3 (see [25]). For some given matrices 𝑊, 𝑀, and 𝑁 of appropriate dimensions and with 𝑊 being symmetric, then 𝑊+𝑁^𝑇𝐹^𝑇(𝑡)𝑀^𝑇+𝑀𝐹(𝑡)𝑁<0. (14) holds, where 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼, if and only if there exists a scalar 𝜀>0 such that 𝑊+𝜀𝑀𝑀^𝑇+𝜀⁻¹𝑁^𝑇𝑁<0. (15)

3. Main Results

In this section, we will find a state feedback control matrix 𝐾 such that the system is asymptotically stable with an 𝐻_∞ norm bound 𝛾.

Theorem 4. For given state feedback control matrix 𝐾, system (12) is asymptotically stable with an 𝐻_∞ norm bound 𝛾, if there exist symmetric positive definite matrices 𝑃 and 𝑄, such that the following condition holds: 𝑄−𝑃00𝐶^𝑇𝐴+𝐵_1𝑛𝐼+Δ₁𝐾𝐼+Δ₂^𝑇∗−𝑄00𝐵_2𝑛𝐼+Δ₃𝐾𝐼+Δ₄^𝑇∗∗−𝛾²𝐼0𝐵_𝜔∗∗∗−𝐼∗∗∗∗∗−𝑃⁻¹<0. (16)

Proof. Consider the following Lyapunov candidate functional: 𝑉(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥^𝑇(𝑘−1)𝑄𝑥(𝑘−1). (17)
First, we proof that system (12) is asymptotically stable with 𝜔(𝑘)=0 under (16).
Let 𝐴=𝐴+𝐵_1𝑛𝐼+Δ₁𝐾𝐼+Δ₂,𝐵=𝐵_2𝑛𝐼+Δ₃𝐾𝐼+Δ₄, (18) along the trajectory of the system (12), we have Δ𝑉(𝑘)=𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)+𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)−𝑥^𝑇()()=𝑥𝑘−1𝑄𝑥𝑘−1_𝑇(𝑘+1)𝑥^𝑇(𝑘)𝑄−𝑃+𝐴^𝑇𝑃𝐴∗𝐵^𝑇𝑃𝐴𝐵𝑃×𝑥𝐵−𝑄_𝑇(𝑘+1)𝑥^𝑇(𝑘)^𝑇, (19) using Lemma 2 (Schur complement), we have Δ𝑉(𝑘)<0 if and only if 𝑄−𝑃0𝐴^𝑇∗−𝑄𝐵^𝑇∗∗−𝑃⁻¹<0. (20)
By Lemma 2 (Schur complement), we know that (16) implies (20); Therefore, system (12) is asymptotically stable with 𝜔(𝑘)=0 being proved.
Now, we will consider the 𝐻_∞ performance of system (12).
When 𝜔(𝑘)≠0, define 𝐽=𝑦(𝑘)²₂−𝛾²‖𝜔(𝑘)‖²₂=^∞_𝑘=0𝑦^𝑇(𝑘)𝑦(𝑘)−𝛾²𝜔^𝑇(𝑘)𝜔(𝑘)(𝑘(∞)(0);+Δ𝑉)−𝑉+𝑉 (21) under zero condition and V(∞)≥0, we have 𝐽≤^∞_𝑘=0𝑦^𝑇(𝑘)𝑦(𝑘)−𝛾²𝜔^𝑇(𝑘)𝜔(𝑘)+Δ𝑉(𝑘). (22)
Let 𝜂(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−1)𝜔^𝑇(𝑘)^𝑇, we obtain 𝐽≤^∞_𝑘=0𝜂^𝑇(𝑘)𝐻𝜂(𝑘), (23) where 𝐻=𝑄−𝑃+𝐴^𝑇𝑃𝐴+𝐶^𝑇𝐶∗∗𝐵^𝑇𝑃𝐴𝐵𝑃𝐵𝐵−𝑄∗_𝑇𝜔𝑃𝐴𝐵^𝑇_𝜔𝑃𝐵𝐵^𝑇_𝜔𝑃𝐵_𝜔−𝛾². (24)
By Lemma 2 (Schur complement), (16) is equivalent to 𝐻<0, which implies that 𝐽<0; thus, we have ‖𝑦(𝑘)‖₂<𝛾‖𝜔(𝑘)‖₂. This completes the proof.

Next, the design method of the state feedback controller is proposed.

Theorem 5. For given diagonal matrices 𝛿₁, 𝛿₂ and a scalar  𝜆, if there exist positive definite matrices 𝑄, 𝑋, and 𝑊, scalars 𝜀_𝑖>0(𝑖=1,2,…,6) and a matrix 𝑌, such that the following linear matrix inequalities −𝑊𝑌^𝑇∗−𝜆𝐼<0,𝑊<2𝑋−𝐼,Γ₁₁Γ₁₂∗Γ₂₂<0, (25)(26)(27) hold, then the system is asymptotically stable with an 𝐻_∞ norm bound 𝛾; the desired controllers are given as follows: 𝐾=𝑌𝑋⁻¹, (28) where

Γ₁₁=𝑄−𝑋00(𝐶𝑋)^𝑇𝐴𝑋+𝐵_1𝑛𝑌^𝑇∗−𝑄00𝐵_2𝑛𝑌^𝑇∗∗−𝛾²𝐼0𝐵^𝑇_𝜔,Γ∗∗∗−𝐼0∗∗∗∗−𝑋₂₂=diag−𝜀₁,−𝜀₁,−𝜀₂,−𝜆𝜀₂,−𝜀₃,−𝜆𝜀₃,−𝜀₄,−𝜀₄,−𝜀₅,−𝜆𝜀₅,−𝜀₆,−𝜆𝜀₆,Γ₁₂=𝜀₁𝐿₁𝛿₁𝐿₂𝑌^𝑇𝜀₂𝐿₁𝜆𝐿₂𝑋𝛿₂𝜀₃𝐿₁𝛿₁𝜆𝐿₂𝑋𝛿₂𝜀₄𝐿₃𝛿₁𝐿₄𝑌^𝑇𝜀₅𝐿₃𝜆𝐿₄𝑋𝛿₂𝜀₆𝐿₃𝛿₁𝜆𝐿₄𝑋𝛿₂,𝐿₁=0000𝐵^𝑇_1𝑛^𝑇,𝐿₂=𝐼0000^𝑇,𝐿₃=0000𝐵^𝑇_2𝑛^𝑇,𝐿₄=0𝐼000^𝑇. (29)

Proof. Rewrite (16) as 𝐹=𝐹₁+𝐹₂<0, (30) where 𝐹₁=𝑄−𝑃00𝐶^𝑇𝐴+𝐵_1𝑛𝐾^𝑇∗−𝑄00𝐵_2𝑛𝐾^𝑇∗∗−𝛾²𝐼0𝐵_𝜔∗∗∗−𝐼∗∗∗∗∗−𝑃⁻¹,𝐹₂=𝐿₁Δ₁𝐾𝐿^𝑇₂+𝐿₁Δ₁𝐾𝐿^𝑇₂^𝑇+𝐿₁𝐾Δ₂𝐿^𝑇₂+𝐿₁𝐾Δ₂𝐿^𝑇₂^𝑇+𝐿₁Δ₁𝐾Δ₂𝐿^𝑇₂+𝐿₁Δ₁𝐾Δ₂𝐿^𝑇₂^𝑇+𝐿₃Δ₃𝐾𝐿^𝑇₄+𝐿₃Δ₃𝐾𝐿^𝑇₄^𝑇+𝐿₃𝐾Δ₄𝐿^𝑇₄+𝐿₃𝐾Δ₄𝐿^𝑇₄^𝑇+𝐿₃Δ₃𝐾Δ₄𝐿^𝑇₄+𝐿₃Δ₃𝐾Δ₄𝐿^𝑇₄^𝑇, (31)𝐿₁,𝐿₂,𝐿₃, and 𝐿₄ are defined above in Theorem 5.
By Lemma 3, 𝐹<0 is equivalent to 𝐹=𝐹₁+𝐹₂<0, (32) where 𝐹₂=𝜀₁𝐿₁Δ₁Δ^𝑇₁𝐿^𝑇₁+𝜀₁⁻¹𝐿₂𝐾^𝑇𝐾𝐿^𝑇₂+𝜀₂𝐿₁𝐿^𝑇₁+𝜀₂⁻¹𝐿₂Δ^𝑇₂𝐾^𝑇𝐾Δ₂𝐿^𝑇₂+𝜀₃𝐿₁Δ₁Δ^𝑇₁𝐿^𝑇₁+𝜀₃⁻¹𝐿₂Δ^𝑇₂𝐾^𝑇𝐾Δ₂𝐿^𝑇₂+𝜀₄𝐿₃Δ₃Δ^𝑇₃𝐿^𝑇₃+𝜀₄⁻¹𝐿₄𝐾^𝑇𝐾𝐿^𝑇₄+𝜀₅𝐿₃𝐿^𝑇₃+𝜀₅⁻¹𝐿₄Δ^𝑇₄𝐾^𝑇𝐾Δ₄𝐿^𝑇₄+𝜀₆𝐿₃Δ₃Δ^𝑇₃𝐿^𝑇₃+𝜀₆⁻¹𝐿₄Δ^𝑇₄𝐾^𝑇𝐾Δ₄𝐿^𝑇₄. (33) By the definitions of Δ₁,Δ₂,Δ₃, and Δ₄, we have Δ₁Δ^𝑇₁=Δ^𝑇₁Δ₁≤𝛿^𝑇₁𝛿₁,Δ₃Δ^𝑇₃=Δ^𝑇₃Δ₃≤𝛿^𝑇₁𝛿₁,Δ₂Δ^𝑇₂=Δ^𝑇₂Δ₂≤𝛿^𝑇₂𝛿₂,Δ₄Δ^𝑇₄=Δ^𝑇₄Δ₄≤𝛿^𝑇₁𝛿₁, (34) using Lemma 2 (Schur complement), (25) is equivalent to 𝑌^𝑇𝑌≤𝜆𝑊. Define 𝐾=𝑌𝑋⁻¹, (26) implies that 𝐾^𝑇𝐾≤𝜆𝐼. (35)
Combining (33), (34), and (35), we have 𝐹₂≤𝐹₂=𝜀₁𝐿₁𝛿₁𝛿^𝑇₁𝐿^𝑇₁+𝜀₁⁻¹𝐿₂𝐾^𝑇𝐾𝐿^𝑇₂+𝜀₂𝐿₁𝐿^𝑇₁+𝜆𝜀₂⁻¹𝐿₂𝛿^𝑇₂𝛿₂𝐿^𝑇₂+𝜀₃𝐿₁𝛿₁𝛿^𝑇₁𝐿^𝑇₁+𝜆𝜀₃⁻¹𝐿₂𝛿^𝑇₂𝛿₂𝐿^𝑇₂+𝜀₄𝐿₃𝛿₃𝛿^𝑇₃𝐿^𝑇₃+𝜀₄⁻¹𝐿₄𝐾^𝑇𝐾𝐿^𝑇₄+𝜀₅𝐿₃𝐿^𝑇₃+𝜆𝜀₅⁻¹𝐿₄𝛿^𝑇₄𝛿₄𝐿^𝑇₄+𝜀₆𝐿₃𝛿₃𝛿^𝑇₃𝐿^𝑇₃+𝜆𝜀₆⁻¹𝐿₄𝛿^𝑇₄𝛿₄𝐿^𝑇₄. (36) Therefore, 𝐹=𝐹₁+𝐹₂<0 (37) Guarantees (16).
By Lemma 2 (Schur complement), (37) is equivalent to 𝐹𝐹=¹𝐹₁₂∗𝐹₂₂<0, (38) where 𝐹₁₂=𝜀₁𝐿₁𝛿₁𝐿₂𝐾^𝑇𝜀₂𝐿₁𝜆𝐿₂𝛿₂𝜀₃𝐿₁𝛿₁𝜆𝐿₂𝛿₂𝜀₄𝐿₃𝛿₁𝐿₄𝐾^𝑇𝜀₅𝐿₃𝜆𝐿₄𝛿₂𝜀₆𝐿₃𝛿₁𝜆𝐿₄𝛿₂,𝐹₂₂=diag−𝜀₁,−𝜀₁,−𝜀₂,−𝜆𝜀₂,−𝜀₃,−𝜆𝜀₃,−𝜀₄,−𝜀₄,−𝜀₅,−𝜆𝜀₅,−𝜀₆,−𝜆𝜀₆, (39)
Let 𝑋=𝑃⁻¹, we pre- and postmultiply both sides of (38) with diag{𝑋,𝑋,𝐼,𝐼,…¹⁵,𝐼} we have (27). This completes the proof.

4. Simulations

In this section, we will give an example to show the usefulness of the derived results and the effectiveness of the proposed methods.

The controlled plant parameters are given as follows: 𝐴_𝑝=−30.50.6−2,𝐵_𝑝=,𝐵10.50.31_𝑝𝑤=0.1000.1,𝐶_𝑝=.1001 (40)

Given the sampling period 𝑇_𝑠=50ms, and the actuator updating period 𝑇_𝑎=𝑇_𝑠/3, following (6), we have 𝐴=0.8610.02210.02660.9052,𝐵_𝑤=,𝐵0.00460.00010.00010.0048₁₁=0.03180.01610.01000.0324,𝐵₂₁=,𝐵0.01480.00770.00500.0155₁₂=0.01630.00820.00500.0164,𝐵₂₂=,𝐵0.03030.01560.01000.0315₁₃=0000,𝐵₂₃=.0.04660.02380.01500.0479 (41)

The deadband threshold coefficients of deadbands 1 and 2 are given as 𝛿₁=diag{0.3,0.3}, 𝛿₂=diag{0.2,0.2}, and the disturbance attenuation 𝛾=0.3. According to Theorem 5, using the LMI toolbox of MATLAB, the controller parameters can be calculated as follows: 𝐾=.−2.726−0.8501−0.8501−3.897 (42)

The exogenous input is chosen as 𝑤(𝑡)=5sin(𝑡), the states responses of the closed-loop multirate NCSs are shown in Figure 2, which confirms that the closed-loop multirate NCSs are robustly asymptotically stable under the exogenous input.

Figure 2 

State trajectories of the closed-loop system.

The packet dropout rates of the S-to-C channel and the C-to-A channel are shown in Figure 3. The packet dropout mentioned here is the data packets which are lost “actively” when the multirate NCSs enter into the transmission deadband; thus, Figure 3 confirms that the proposed deadband scheduling policy could reduce the network traffic remarkably.

Figure 3 

Packet dropout rate curve of the closed-loop system.

5. Conclusions

In this paper, 𝐻_∞ control problem for multirate networked control systems with deadband scheduling policy and transmission delay is considered. The transmission deadbands are set at the sensor node and controller node to reduce the network traffic. The systems are modeled as the discrete switch system with uncertain parameters and the design method of 𝐻_∞ controller is established by using the Lyapunov theorem. Finally, the simulation example is presented to show the advantage of the deadband scheduling policy and the efficiency of the proposed theoretical results.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants nos. 60975075 and 61074023 and the innovation project of graduate students’ scientific research of Jiangsu Province no. CXLX11_0256.