# Chattering-Free Sliding Mode Control for Networked Control System with Time Delay and Packet Dropout Based on a New Compensation Strategy

Authors: Yu Zhang; Shousheng Xie; Ledi Zhang; Litong Ren; Bin Zhou; Hao Wang
Journal: Complexity
 (2019)

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## Abstract

This paper addresses the sliding mode control problem for a class of networked control systems with long time delay and consecutive packet dropout. A new modeling method is proposed, through which time delay and packet dropout are modeled in a unified model described by one Markov chain. To avoid the chattering problem of classic reaching law, a new chattering-free reaching law is proposed. Then with a focus on the problem that controller-actuator channel network condition cannot be foreseen by the controller, a new compensation strategy is proposed, which can effectively compensate for the effect of time delay and packet dropout in controller-actuator channel. Finally, a simulation example is presented to demonstrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

In recent years, a lot of interest and concern have been focused on networked control systems (NCSs) where communication channels interconnect sensors with controllers and/or controllers with actuators [1]. Compared with conventional point-to-point interconnected feedback control systems, NCSs have advantages in many aspects: higher system operability, more efficient resource utilization and sharing, lower cost, and reduced weight, as well as simplicity for system installation and maintenance [2–4]. Despite these distinctive features provided by NCSs, the insertion of communication network in the feedback control loops also brings about new challenges such as bandwidth-limited channel, network-induced delay, packet dropout, and packet disordering [5].Time delay and packet dropout serve as two of the most common and crucial problems of NCSs that can directly degrade the performance of the closed-loop system or even lead to instability [6]. More specifically, long time delay refers to the time delay larger than the sampling period which, compared with short time delay or “one step” time delay, can cause more serious problems like packet disordering. Therefore, for discrete-time systems, it is more valuable to carry out research on long time delay systems. Numerous effective methods have been reported in the literature aiming to tackle the time delay and packet dropout problems arising in NCSs (see, for example, [7–10]). It can be concluded that the solution of the problem consists of two main parts, one of which is to establish an appropriate model to describe time delay and packet dropout and the other is to design a controller that can guarantee system stability despite the existence of time delay and packet dropout. This paper is also organized based on this structure.Firstly, since time delay and packet dropout usually exist simultaneously in the real network, it is important to establish a model to handle them in a common framework. Various modeling methods have been proposed including time delay system model [11], switched system model [12–14], asynchronous dynamical system model [15, 16], and stochastic system model [6, 8, 9, 17–25]. Among them, stochastic system model becomes popular in recent years and one of the most used ways is to adopt a Markov discrete-time linear system to describe random time delay or packet dropouts [9, 20–22, 24, 25]. It is found that the transition from one time delay/packet dropout state to another usually occurs with a certain probability and thus a Markov chain can be used to describe such relation with a transition probability matrix. Therefore, long time delay and consecutive packet dropout can be modeled as a Markov chain. In [24], both sensor-to-controller and controller-to-actuator delays are considered and described by Markov chains and the resulting closed-loop systems are written as jump linear systems with two modes. Similarly, in [25], the characteristics of network communication delays and packet dropouts in both sensor-to-controller and controller-to-actuator channels are modeled and their history behavior is described by three Markov chains. However, it is discovered that most of the existing Markov chain based modeling methods have a common problem that the time delay and packet dropout are usually considered on the controller side, which means they do not reflect the condition of the packets actually received by the actuator. This could lead to the problem of the use of outdated signals and packet disordering. To avoid this problem, buffer or zero-order-holder (ZOH) are introduced [26], but pitifully, the model used has not changed in [26]. Similarly, ZOH is also used in [27] and a corresponding modeling method is proposed, but [27] considers only the problem of packet dropout. Motivated by the above discussions, this paper focuses on proposing a new model that considers both long time delay and consecutive packet dropout in a unified model, and the situation of actual received control signals on the actuator side will be considered to avoid problems like packet disordering and the use of obsolete data.When modeling method is determined, the next key problem is to design an appropriate controller that can guarantee system stability and performance of the NCSs despite the existence of time delay and packet dropout. Among various methods, sliding mode control (SMC) method stands out for its unique feature that system states can be driven onto a specially designed sliding surface and then become totally insensitive to uncertainties [28–30]. However, the well-known chattering problem limits it wide application especially in discrete systems. With a focus on this problem, this paper proposes a chattering-free sliding mode reaching law for a type of multiple-input NCSs, and it needs to be pointed out that, to the best of the author’s knowledge, the use of such chattering-free reaching law in multiple-input systems has rarely been reported.Moreover, another critical problem usually ignored is that the network condition of the controller-actuator channel cannot be foreseen by the controller when calculating the control signal, which makes the calculated control signal not suitable for the systems when there exists time delay or packet dropout. Some researchers have noticed this problem but they tried to solve the problem only by using the time delay and packet dropout information of the former sampling period as compensation [31]. Therefore, another novelty of this paper is that it proposes a new compensation strategy, where all possible time delay and packet dropout states in the controller-actuator channel are considered and a multiple-model based controller is designed which can output a sequence of control signals, and then a compensator on the actuator side will decide which signal of the control signal sequence to use according to the time stamp information. In this way, the time delay and packet dropout can be properly compensated for even when the network condition of controller-actuation channel cannot be foreseen.The main contributions of this paper can be concluded as follows: (i) a new modeling method is proposed for NCSs with long time delay and consecutive packet dropout; (ii) a chattering-free sliding mode reaching law is proposed and first used for multiple-input NCSs; (iii) a multiple-model based sliding mode controller is designed together with a new compensation strategy which can compensate for time delay and packet dropout without the advanced knowledge of controller-actuator channel network condition.The remainder of this paper is organized as follows. Section2 introduces the newly proposed NCSs modeling method. The design of chattering-free sliding mode controller as well as the new compensation strategy is presented in Section 3. Numerical simulation results are shown in Section 4. Section 5 presents some conclusions.


## 2. A New Modeling Method for NCSs with Time Delay and Packet Dropout

### 2.1. Modeling of Time Delay and Packet Dropout

Consider the discrete NCSs with time delay and packet dropout, which is described by the following state space model:(1)xk+1=Axk+Buk+dk,where x(k)∈Rn is system state variable; uk∈Rm is control input; d(k)∈Rn is external disturbance; and A, B, and C are matrices with appropriate dimensions.To make it easier for analysis, the following assumptions are made for system (1).Assumption 1.
System (1) is controllable and all system state variables are observable.Assumption 2.
Time delay and packet dropout exist in the controller-actuator channel only and each data packet is sent with a time stamp.Assumption 3.
The long time delayτ is bounded and the upper bound is known, which is denoted as τ-; consecutive packet dropout is also bounded and the largest number of consecutive packet dropouts is known, which is denoted as ρ-.Assumption 4.
External disturbanced(k) is unknown but bounded and satisfies sliding mode matching condition, i.e.,(2)dk=Bωk.Now we are going to consider long time delay and consecutive packet dropout for system (1). We will first establish a time delay model based on Markov chain, then we will take packet dropout as a special type of time delay, and the obtained transition probability matrix will be extended. Such modeling method considers the connection between time delay and packet dropout on the actuator side, which makes the model established better reflect the real condition in practical use.Inspired by [27], a ZOH is adopted for the NCS. Therefore, due to the existence of ZOH, only the newest received control signal will be used. Here we assume that time delay τk takes values in the set Ωτ=0,1,2,...τ-. It is noticed from the value space of τk that time delay can be zero in our model because, for a normal NCS, it is not possible that there is no on-time arriving signal. Therefore, it is more reasonable to add zero to the value space of time delay. First, let us think about an example, in which τk=0. It means controller output signal u-(k) arrives at time k, so the actuator input u(k)=u-(k). Then, due to the existence of ZOH, there are only two possible conditions for u(k+1). If u-(k+1) also arrives on time, we have u(k+1)=u-(k+1) and τk+1=0. If  u-(k+1) does not arrive on time, then u-(k) will still be used as an actuator input and we have u(k+1)=u-(k) and τk+1=1. In the same way, if we assume the time delay of time instant k is τk=i(i<τ-), there could be only i+2 possible conditions for the next time instant; i.e., if the control signal of k+1 arrives on time, then τk+1=0; if the control signal of k+1 does not arrive but the signal sent out at time k arrives, then we have τk+1=1; if control signal of k+1 or k does not arrive but that of time k-1 arrives, then we have τk+1=2; the same way can explain the conditions all the way to τk+1=i; however, if there is no newly arriving signal at k+1, then, due to the existence of ZOH, the control signal of time k is still used, so the value of time delay becomes τk+1=i+1. In particular, when τk=τ-, which is the upper bound of time delay, there could only be τ-+1 conditions. The existence of ZOH guarantees the use of newest signal, which means, for example, if control signal u-(k+1) arrives at time instance k+1, then all signals sent before time instant k+1 will never be used afterwards. The time delay state transition can be described as(3)πij=Pr⁡τk+1=j∣τk=i∑j=0τ-πij=1,where i∈Ωτ,j∈0,1,…,i+1∩Ωτ, and define Πτ-=πij as system time delay transition possibility matrix, which is given as follows:(4)Πτ-=π00π0100⋯0π10π11π120⋯0⋮⋮⋮⋱⋱⋮πτ--20πτ--21πτ--22⋯πτ--2τ--10πτ--10πτ--11πτ--12⋯πτ--1τ--1πτ--1τ-πτ-0πτ-1πτ-2⋯πτ-τ--1πτ-τ-.Then let us take consecutive packet dropout into consideration. As mentioned above, if there is only bounded time delay, then when τk=τ-, there could be only τ-+1 possible conditions. However, if packet dropout is also considered, the time delay condition of time k+1 is different but can also be included and described in a similar way.Firstly, an example withτmax=2 and ρmax=2 is shown in Figure 1 as follows to illustrate the situation when consecutive packet dropout is included.Figure 1
Signal transmission timing diagram withτmax=2, ρmax=2.It is shown thatu-k is transmitted to the actuator at time instant k without time delay, thus giving τ(k)=0. Then at next time instant k+1, as illustrated above, τ(k+1) could only be τ(k+1)=0or1. If τ(k+1)=1, then at time instant k+2, we have τ(k+2)=0, or 1, or 2. If τ(k+2)=2, since time delay is bounded, then, at time instant k+3, the late arriving signal could only be u-k+1 or u-k+2, and if there is still no newly arrived signal, it is certain that packet u-k+1 and the signals sent before k+1, if still not received, are already lost. On this occasion, ZOH outputs control signal u-k as actuator input and the equivalent time delay is τ(k+3)=3. If τ(k+3)=3, since it is sure that u-k+1 is already lost, then, at time instant k+4, the late arriving signal could only be u-k+2 or u-k+3 and if there is still no newly arrived signal, u-k will still be used by the actuator; thus the equivalent time delay is τ(k+4)=4. If τ(k+4)=4, since consecutive packet dropout is also bounded, τ(k+5) could only take values in the set 0,1,2. Extend the above discussion to τmax=τ-,ρmax=ρ- and define the value space of equivalent time delay as Ωτ=0,1,2,...,(τ-+ρ-); then the following equivalent time delay transition is given:(5)πij=Pr⁡τk+1=j∣τk=i∑j=0τ-+ρ-πij=1,where i∈Ωτ,j∈0,1,…,i+1∩Ωτ∪i+1, and define Πτ-,ρ-=πij as system equivalent time delay transition possibility matrix, which is given as follows:(6)Πτ-,ρ-=π00π010…0π10π11π120⋯0⋮⋮⋱⋱⋱⋮⋮πτ-0πτ-1⋯πτ-τ-πτ-τ-+10⋯0πτ-+10πτ-+11⋯πτ-+1τ-0πτ-+1τ-+20⋯0πτ-+20πτ-+21⋯πτ-+2τ-00πτ-+2τ-+30⋯0⋮⋮⋮⋮⋮⋮⋱⋱⋱⋮πτ-+ρ--20πτ-+ρ--21⋯πτ-+ρ--2τ-00⋯0πτ-+ρ--2τ-+ρ--10πτ-+ρ--10πτ-+ρ--11⋯πτ-+ρ--1τ-00⋯00πτ-+ρ--1τ-+ρ-πτ-+ρ-0πτ-+ρ-1⋯πτ-+ρ-τ-00⋯000.As a result, the NCSs with long time delay and consecutive packet dropout can be expressed as(7)xk+1=Axk+Buk-τk+dk,where τ(k) is equivalent time delay.Remark 5.
The proposed new modeling method stands out from other traditional stochastic system modeling methods in the following three aspects:
(i) It innovatively models long time delay and consecutive packet dropout in a unified model described by only one Markov chain.
(ii) The time scale adopted in our Markov chain is linear with the physical time; i.e., the state transition in our Markov chain always happens over one physical time instant.
(iii) Compared with traditional models, the transition probability matrix of our model is not a full matrix and thus it requires less work in obtaining the transition probability matrix.


### 2.2. Elimination of Time Delay Term in the Form

First, according to Assumption4, system (7) can be rewritten as(8)xk+1=Axk+Buk-τk+ωk.The following linear transformation can be adopted to obtain a new expression of system (8) which does not have time delay term in the form. When τ(k)≥1, by defining(9)x-k=Aτxk+∑i=0τ-1Aτ-i-1Buk-τ+i+ωk,system (8) can be transformed as(10)x-k+1=Ax-k+Buk+ωk.According to Assumption 1, system (10) is also controllable.Proof.
The termBu(k-τ(k))+ω(k) can be decomposed as follows:(11)Buk-τ+ωk=Buk-τ+ωk-A-1Buk-τ+1+ωk+A-1Buk-τ+1+ωk-⋯-A-τ+1Buk-1+ωk+A-τ+1Buk-1+ωk-A-τBuk+ωk+A-τBuk+ωk=A-τBuk+ωk+∑i=0τ-1A-iBuk-τ+i+ωk-∑i=0τ-1A-i-1Buk-τ+1+i+ωk.Substituting (11) into (8), we have(12)xk+1+∑i=0τ-1A-i-1Buk-τ+1+i+ωk=Axk+∑i=0τ-1A-i-1Buk-τ+i+ωk+A-τBuk+ωk.Premultiplying both sides of (12) by Aτ, we have (13)Aτxk+1+∑i=0τ-1Aτ-i-1Buk-τ+1+i+ωk=AAτxk+∑i=0τ-1Aτ-i-1Buk-τ+i+ωk+Buk+ωk.Finally let x-(k)=Aτx(k)+∑i=0τ-1Aτ-i-1Bu(k-τ+i)+ω(k); system (10) can obtained. In particular, when τ(k)=0, it is defined that x-(k)=x(k).


## 2.1. Modeling of Time Delay and Packet Dropout

Consider the discrete NCSs with time delay and packet dropout, which is described by the following state space model:(1)xk+1=Axk+Buk+dk,where x(k)∈Rn is system state variable; uk∈Rm is control input; d(k)∈Rn is external disturbance; and A, B, and C are matrices with appropriate dimensions.To make it easier for analysis, the following assumptions are made for system (1).Assumption 1.
System (1) is controllable and all system state variables are observable.Assumption 2.
Time delay and packet dropout exist in the controller-actuator channel only and each data packet is sent with a time stamp.Assumption 3.
The long time delayτ is bounded and the upper bound is known, which is denoted as τ-; consecutive packet dropout is also bounded and the largest number of consecutive packet dropouts is known, which is denoted as ρ-.Assumption 4.
External disturbanced(k) is unknown but bounded and satisfies sliding mode matching condition, i.e.,(2)dk=Bωk.Now we are going to consider long time delay and consecutive packet dropout for system (1). We will first establish a time delay model based on Markov chain, then we will take packet dropout as a special type of time delay, and the obtained transition probability matrix will be extended. Such modeling method considers the connection between time delay and packet dropout on the actuator side, which makes the model established better reflect the real condition in practical use.Inspired by [27], a ZOH is adopted for the NCS. Therefore, due to the existence of ZOH, only the newest received control signal will be used. Here we assume that time delay τk takes values in the set Ωτ=0,1,2,...τ-. It is noticed from the value space of τk that time delay can be zero in our model because, for a normal NCS, it is not possible that there is no on-time arriving signal. Therefore, it is more reasonable to add zero to the value space of time delay. First, let us think about an example, in which τk=0. It means controller output signal u-(k) arrives at time k, so the actuator input u(k)=u-(k). Then, due to the existence of ZOH, there are only two possible conditions for u(k+1). If u-(k+1) also arrives on time, we have u(k+1)=u-(k+1) and τk+1=0. If  u-(k+1) does not arrive on time, then u-(k) will still be used as an actuator input and we have u(k+1)=u-(k) and τk+1=1. In the same way, if we assume the time delay of time instant k is τk=i(i<τ-), there could be only i+2 possible conditions for the next time instant; i.e., if the control signal of k+1 arrives on time, then τk+1=0; if the control signal of k+1 does not arrive but the signal sent out at time k arrives, then we have τk+1=1; if control signal of k+1 or k does not arrive but that of time k-1 arrives, then we have τk+1=2; the same way can explain the conditions all the way to τk+1=i; however, if there is no newly arriving signal at k+1, then, due to the existence of ZOH, the control signal of time k is still used, so the value of time delay becomes τk+1=i+1. In particular, when τk=τ-, which is the upper bound of time delay, there could only be τ-+1 conditions. The existence of ZOH guarantees the use of newest signal, which means, for example, if control signal u-(k+1) arrives at time instance k+1, then all signals sent before time instant k+1 will never be used afterwards. The time delay state transition can be described as(3)πij=Pr⁡τk+1=j∣τk=i∑j=0τ-πij=1,where i∈Ωτ,j∈0,1,…,i+1∩Ωτ, and define Πτ-=πij as system time delay transition possibility matrix, which is given as follows:(4)Πτ-=π00π0100⋯0π10π11π120⋯0⋮⋮⋮⋱⋱⋮πτ--20πτ--21πτ--22⋯πτ--2τ--10πτ--10πτ--11πτ--12⋯πτ--1τ--1πτ--1τ-πτ-0πτ-1πτ-2⋯πτ-τ--1πτ-τ-.Then let us take consecutive packet dropout into consideration. As mentioned above, if there is only bounded time delay, then when τk=τ-, there could be only τ-+1 possible conditions. However, if packet dropout is also considered, the time delay condition of time k+1 is different but can also be included and described in a similar way.Firstly, an example withτmax=2 and ρmax=2 is shown in Figure 1 as follows to illustrate the situation when consecutive packet dropout is included.Figure 1
Signal transmission timing diagram withτmax=2, ρmax=2.It is shown thatu-k is transmitted to the actuator at time instant k without time delay, thus giving τ(k)=0. Then at next time instant k+1, as illustrated above, τ(k+1) could only be τ(k+1)=0or1. If τ(k+1)=1, then at time instant k+2, we have τ(k+2)=0, or 1, or 2. If τ(k+2)=2, since time delay is bounded, then, at time instant k+3, the late arriving signal could only be u-k+1 or u-k+2, and if there is still no newly arrived signal, it is certain that packet u-k+1 and the signals sent before k+1, if still not received, are already lost. On this occasion, ZOH outputs control signal u-k as actuator input and the equivalent time delay is τ(k+3)=3. If τ(k+3)=3, since it is sure that u-k+1 is already lost, then, at time instant k+4, the late arriving signal could only be u-k+2 or u-k+3 and if there is still no newly arrived signal, u-k will still be used by the actuator; thus the equivalent time delay is τ(k+4)=4. If τ(k+4)=4, since consecutive packet dropout is also bounded, τ(k+5) could only take values in the set 0,1,2. Extend the above discussion to τmax=τ-,ρmax=ρ- and define the value space of equivalent time delay as Ωτ=0,1,2,...,(τ-+ρ-); then the following equivalent time delay transition is given:(5)πij=Pr⁡τk+1=j∣τk=i∑j=0τ-+ρ-πij=1,where i∈Ωτ,j∈0,1,…,i+1∩Ωτ∪i+1, and define Πτ-,ρ-=πij as system equivalent time delay transition possibility matrix, which is given as follows:(6)Πτ-,ρ-=π00π010…0π10π11π120⋯0⋮⋮⋱⋱⋱⋮⋮πτ-0πτ-1⋯πτ-τ-πτ-τ-+10⋯0πτ-+10πτ-+11⋯πτ-+1τ-0πτ-+1τ-+20⋯0πτ-+20πτ-+21⋯πτ-+2τ-00πτ-+2τ-+30⋯0⋮⋮⋮⋮⋮⋮⋱⋱⋱⋮πτ-+ρ--20πτ-+ρ--21⋯πτ-+ρ--2τ-00⋯0πτ-+ρ--2τ-+ρ--10πτ-+ρ--10πτ-+ρ--11⋯πτ-+ρ--1τ-00⋯00πτ-+ρ--1τ-+ρ-πτ-+ρ-0πτ-+ρ-1⋯πτ-+ρ-τ-00⋯000.As a result, the NCSs with long time delay and consecutive packet dropout can be expressed as(7)xk+1=Axk+Buk-τk+dk,where τ(k) is equivalent time delay.Remark 5.
The proposed new modeling method stands out from other traditional stochastic system modeling methods in the following three aspects:
(i) It innovatively models long time delay and consecutive packet dropout in a unified model described by only one Markov chain.
(ii) The time scale adopted in our Markov chain is linear with the physical time; i.e., the state transition in our Markov chain always happens over one physical time instant.
(iii) Compared with traditional models, the transition probability matrix of our model is not a full matrix and thus it requires less work in obtaining the transition probability matrix.


## 2.2. Elimination of Time Delay Term in the Form

First, according to Assumption4, system (7) can be rewritten as(8)xk+1=Axk+Buk-τk+ωk.The following linear transformation can be adopted to obtain a new expression of system (8) which does not have time delay term in the form. When τ(k)≥1, by defining(9)x-k=Aτxk+∑i=0τ-1Aτ-i-1Buk-τ+i+ωk,system (8) can be transformed as(10)x-k+1=Ax-k+Buk+ωk.According to Assumption 1, system (10) is also controllable.Proof.
The termBu(k-τ(k))+ω(k) can be decomposed as follows:(11)Buk-τ+ωk=Buk-τ+ωk-A-1Buk-τ+1+ωk+A-1Buk-τ+1+ωk-⋯-A-τ+1Buk-1+ωk+A-τ+1Buk-1+ωk-A-τBuk+ωk+A-τBuk+ωk=A-τBuk+ωk+∑i=0τ-1A-iBuk-τ+i+ωk-∑i=0τ-1A-i-1Buk-τ+1+i+ωk.Substituting (11) into (8), we have(12)xk+1+∑i=0τ-1A-i-1Buk-τ+1+i+ωk=Axk+∑i=0τ-1A-i-1Buk-τ+i+ωk+A-τBuk+ωk.Premultiplying both sides of (12) by Aτ, we have (13)Aτxk+1+∑i=0τ-1Aτ-i-1Buk-τ+1+i+ωk=AAτxk+∑i=0τ-1Aτ-i-1Buk-τ+i+ωk+Buk+ωk.Finally let x-(k)=Aτx(k)+∑i=0τ-1Aτ-i-1Bu(k-τ+i)+ω(k); system (10) can obtained. In particular, when τ(k)=0, it is defined that x-(k)=x(k).


## 3. Design of Chattering-Free Sliding Mode Controller

### 3.1. Design of Linear Sliding Surface

Since disturbance satisfies the matching condition, then when designing the sliding surface, the system can be transformed to a sliding mode regular form so that the sliding mode dynamic is totally free from the disturbance. To obtain the sliding mode regular form of (10), we can define a nonsingular matrix T∈Rn×n, which makes(14)TB=0n-m×mBm,where Bm∈Rm×m is nonsingular. Choose T=T2T1T, where T1∈Rn×m, T2∈Rn×(n-m) are two subblocks of a unitary matrix resulting from the singular value decomposition of B, i.e.,(15)B=T1T2Σm×m0n-m×mJT,where Σ is a diagonal positive-definite matrix and J is a unitary matrix.Then through linear transformationz=Tx-, system (10) can be decomposed into two subsystems:(16a)z1k+1=A^11z1k+A^12z2k,(16b)z2k+1=A^21z1k+A^22z2k+Bmuk+ωk,where z1(k)∈Rn-m, z2(k)∈Rm; A^=TAT-1.It has been proved that (16a) is the sliding mode dynamics of system (10).Choose the following classic linear sliding surface:(17)sk=Czk=C1Izk=C1z1k+z2k.In the sliding mode, we have(18)sk=C1z1k+z2k=0⇒z2k=-C1z1k.Substituting (18) into (16a), we have (19)z1k+1=A^11-A^12C1z1k,which is a classic sliding surface design problem, and the sliding surface parameter C can be obtained through pole placement.


### 3.2. Design of Chattering-Free SMC

To suppress chattering, a new chattering-free sliding mode reaching law is proposed, inspired by [32]. However, the reaching law proposed in this paper is for multiple-input systems, which means sliding surface s(k) is not a scalar but a vector, and the reaching law should be newly defined. The vector form reaching law is given bellow:(20)sk+1=sk-qTsk-εTsigα⁡sk,where 0<qT<1, 0<εT<1, 0<α<1 and,(21)sigα⁡sk≜diag⁡s1kα,s2kα,…,smkα·sgn⁡sk.Therefore, the sliding mode controller can be designed by comparing (16a), (16b), and (20):(22)uk=-Bm-1C1A^11z1k+C1A^12z2k+A^21z1k+A^22z2k-1-qTsk+εTsigα⁡sk+Bmωk.It can be seen that the unknown disturbance term is included in the expression of u(k), so u(k) cannot be used directly as a control signal. To tackle this problem we introduce the following disturbance estimation strategy:(23)ω^k=z2k-A^21z1k-1-A^22z2k-1-Bmuk-1=ωk-1.Substituting (23) into (22) yields the final SMC law:(24)uk=-Bm-1C1A^11z1k+C1A^12z2k+A^21z1k+A^22z2k-1-qTsk+εTsigα⁡sk+Bmω^k.With the SMC law (24), the new sliding mode reaching law considering disturbance is(25)sk+1=1-qTsk-εTsigα⁡sk+Bmωk-ω^k=1-qTsk-εTsigα⁡sk+Bmωk-ωk-1.Since s(k+1) is a vector with m sliding mode functions, then we write (25) into the following componentwise form and we will prove that each sliding mode state satisfies the reaching condition. The componentwise form of (25) is(26)sik+1=1-qTsik-εTsikαsgn⁡sik+wiωik-ωik-1i=1,2,…,m,where wi is the ith row of Bm, i.e.,(27)Bm=w1⋮wm.Assumption 6.
The change rate of disturbanceδi(k)=ωi(k)-ωi(k-1) is limited, and there exists a constant δ∗ such that maxwiδi(k)≤δ∗,(i=1,2,…,m).Theorem 7.
With the constant matrixC and the linear sliding surface given by (17), the trajectory of system sliding mode state si(k)i=1,2,…,m can be driven by controller (25) into the sliding surface neighborhood region Ω within at most mi∗ steps, where(28)Ω=sik∣sik≤η=ξα·max⁡δ∗εT1/α,εT1-qT1/1-αξα=1+αα/1-α-α1/1-αmi∗=mi+1withmi=si20-η2μ2μ=qTη+1-qT.Proof.
The proof of the theory consists of two problems:
(i) we need to prove that system sliding mode state will enter the regionΩ within mi∗ steps.
(ii) we will prove that ifsik∈Ω, then sik+1∈Ω.
Proof of Problem (i). Define a Lyapunov function Vik=si2k; then it is obtained that(29)ΔVik=si2k+1-si2k=-qTsik+εTsikαsgn⁡sik-wiδik·2sik-qTsik-εTsikαsgn⁡sik+wiδik.Then the following two conditions are discussed.
① Ifsik>η, there is(30)ΔVik=-qTsik+εTsiαk-wiδik·2sik-qTsik-εTsiαk+wiδik.Since sik>η, there is no doubt that sik>ξα·δ∗/εT1/α. Thus, we have(31)qTsik+εTsiαk-wiδik≥qTsik+ξααδ∗-wiδik≥qT·η+ξαα-1δ∗.Define μ=qT·η+ξαα-1δ∗, and then since 0<qT<1, 1<ξαα<2, it is sure that μ is a positive constant.
Similarly, sincesik>η, there is no doubt that sik>η=ξα·εT/1-qT1/1-α. Thus, we have(32)1-qTsik>ξα1-α·εT.Therefore,(33)1-qTsik>ξ1-αα·εTsiαk>εTsiαk.By further calculation, we have(34)sik>qTsik+εTsiαk.Then it can be obtained that(35)2sik-qTsik-εTsiαk+wiδik≥qTsik+εTsiαk+wiδik≥qTsik+εTsiαk-wiδik≥μ.Consequently, there is(36)ΔVik=-qTsik+εT-wiδik·2sik-qTsik-εT+wiδik≤-μ2<0.② If sik<-η, by a similar proof procedure, it can be shown that the relation ΔVik≤-μ2 still holds. Therefore, when sik∉Ω, ΔVik=si2k+1-si2k≤-μ2<0.
Moreover, it can be obtained from (36) that(37)ΔVik=si2k+1-si2k≤-μ2⇔si2k+1≤si2k-ε2T2≤si2k-1-2μ2≤⋯≤si20-k+1μ2.It means that if(38)si20-k+1μ2=η2,then sik+1∈Ω.
However, the solution of (38) may not be an integer, so we denote mi=si20-η2/μ2 as the real number solution of (38). Then we can say that after at most mi∗=mi+1 steps, the ith sliding mode state will enter the sliding surface neighborhood region Ω.
Proposition  8. Denote φ=max⁡δ∗/εT1/α,εT/1-qT1/1-α, and then there is δ∗≤εTφα≤1-qTφ.
Lemma  9. For functions in the form:(39)ησ=1+σσ/1-σ-σ1/1-σ,where 0<σ<1, there exists 1<ησ<2, and for any parameter x∈0,1, the following relation holds [33]:(40)zησ-zσησσ+ησ-1≥0.Proof of Problem (ii). When sik∈Ω, it means -η≤sik≤η. We can define a new parameter -1≤θ≤1 such that sik=θ·η=θ·ξαφ. The following two cases are discussed.
① When0≤θ≤1, according to (26), we have(41)sik+1=1-qTθ·ξαφ-εTθαξααφα+wiδi.First, we will prove that si(k+1)≤η.
According to Proposition 8 and Assumption6, there is(42)sik+1≤1-qTθ·ξαφ+δ∗1-θαξαα.If θξα≥1, we have(43)sik+1≤1-qTθ·ξαφ≤ξαφ=η.If 0≤θξα<1, we have(44)sik+1≤1-qTφ1+θ·ξα-θαξαα≤1-qTφ≤ξαφ=η.Second, we will prove that si(k+1)≥-η.
According to Proposition 8, we have(45)sik+1≥1-qTθ·ξαφ-εTθαξααφα-δ∗≥1-qTφθξα-θαξαα-1.If θξα≥1, since 0<α<1, we have(46)sik+1≥-1-qTφ≥-ξαφ=-η.If 0≤θξα<1, we have(47)sik+1≥1-qTφθξα-θαξαα-1≥-ξα1-qTφ≥-ξαφ=-η.② When -1≤θ≤0, according to (26), we have(48)sik+1=-1-qTθ·ξαφ+εTθξααφα+wiδi.Firstly, we will prove that si(k+1)≤η.
According to Proposition 8, we have(49)sik+1=-1-qTθ·ξαφ+εTθξααφα+wiδi≤-1-qTθ·ξαφ+1-qTφθξαα+δ∗≤1-qTφ-θ·ξα+θξαα+1.If θξα≥1, since 0<α<1, we have(50)sik+1≤1-qTφ≤ξαφ=η.If 0≤θξα<1, according to Lemma 9, we have(51)sik+1≤1-qTφξα≤φξα=η.Second, we will prove that si(k+1)≥-η.
According to Proposition 8, we have(52)sik+1=-1-qTθ·ξαφ+εTθξααφα+wiδi≥-1-qTθ·ξαφ+θξααδ∗-wiδi=-1-qTθ·ξαφ+δ∗θξαα-1.If θξα≥1, we have(53)sik+1≥-1-qTθ·ξαφ≥-ξαφ=-η.If 0≤θξα<1, we have(54)sik+1≥-1-qTφθ·ξα-θξαα+1≥-1-qTφ≥ξαφ=-η.To this end, we can conclude that when -η≤sik≤η, we have -η≤sik+1≤η, which means sik+1∈Ω.Remark 10.
For discrete-time systems, the accurate arrival of sliding mode state on the sliding surface is hard to realize, so to totally eliminate chattering is actually not possible in the discrete-time systems. This paper usessigα⁡sk to replace the sign function to avoid the sharp switching of control input to suppress chattering. The closer the sliding mode state to the sliding surface, the lower the chattering that will be obtained. It needs to be pointed out that the saying “chattering-free” here is just a conventional way to describe such chattering suppression methods [32, 34, 35].


### 3.3. Design of Multiple-Model Based Compensator

Traditionally, if time delay information cannot be foreseen by the controller, there can only be two control input compensation strategies, i.e., zero-input or hold-input. For zero-input strategy, it means if the controller outputu-(k) does not arrive on time, then we let control input u(k)=0. However, when time delay happens frequently, this compensation strategy will result in very poor dynamic performance. And for hold-input strategy, it means control signal will remain unchanged unless new control signal arrives, but this may cause serious overshoot when long time delay exists. With a focus on the above problems, a new compensation strategy is proposed, which also comes from the idea of hold-input strategy but, compared with traditional hold-input strategy, it uses different system models to calculate control input signal for different equivalent time delay conditions. And to overcome the problem that controller cannot foresee the controller-actuator equivalent time delay, all possible models relating to different equivalent time delay τ(k) are established in the controller. Then, at each controlling period, instead of outputting one control signal, the controller will calculate and output a sequence of different control signals U-(k)=[u-(k)τ=0,u-(k)τ=1,…,u-(k)τ=τ-+ρ-] based on different models and the controller output U-(k) will be put in one data packet with a time stamp. Also at each controlling period, the ZOH will output a signal sequence packet U(k) based on the time stamp information, which means U(k)=U-(k) for τ(k)=0, or U(k)=U-(k-1) for τ(k)=1, etc. Finally, the compensator will decide which control signal of U(k) to use according to the time stamp information; i.e., if U(k)=U-(k), it will use the first signal of U(k), and if U(k)=U-(k-1), it will use the second signal of U(k), etc.In this way, even though there is no newly received control signal by the ZOH, the actuator can still get an updated control signal from the compensator and, unlike the traditional hold-input strategy, the updated control input signal has already taken into account the state changes of the former sampling period.Based on the above discussion, the algorithm of the compensator can be described as(55)uk=Ukgτk=U-k-τkgτk,where the compensation signal choosing function g(x) is defined as(56)gx=g0,g1,…,gi,…,gτ-+ρ-T,gi=1,i=x0,i≠x.To show the relationship between actuator input and controller output more clearly, the following illustrations are given:(57)uk=U-k·10…0T=u-kτ=0,τ=0U-k-1·010…0T=u-k-1τ=1,τ=1⋮⋮U-k-τk·0…0︸τk10…0T=u-k-τkτ=τk,τ=τk⋮⋮U-k-τ--ρ-·0…01T=u-k-τ--ρ-τ=τ-+ρ-,τ=τ-+ρ-.In conclusion, the steps of designing this chattering-fee sliding mode controller are given below.Step 1.
Obtain the system’s time delay and packet dropout characteristics through experiments and then get the transition probability matrix (6) based on the proposed modeling method.Step 2.
Obtain the system sliding mode regular form (16a) and (16b) through linear transformation z=Tx-, where the definition of nonsingular matrix T∈Rn×n is given by (14) and (15).Step 3.
Design the classic linear sliding surface (17), whose parameter C=[C1I] can be determined by pole placement method.Step 4.
Design the chattering-free reaching law (20) and choose appropriate reaching law parameters q, ε, and α that satisfy 0<qT<1, 0<εT<1, 0<α<1.Step 5.
Design the sliding mode controller (22) according to reaching law (20).Step 6.
Design the multiple-model based compensator (57) based on the upper bound of long time delay and consecutive packet dropout.


## 3.1. Design of Linear Sliding Surface

Since disturbance satisfies the matching condition, then when designing the sliding surface, the system can be transformed to a sliding mode regular form so that the sliding mode dynamic is totally free from the disturbance. To obtain the sliding mode regular form of (10), we can define a nonsingular matrix T∈Rn×n, which makes(14)TB=0n-m×mBm,where Bm∈Rm×m is nonsingular. Choose T=T2T1T, where T1∈Rn×m, T2∈Rn×(n-m) are two subblocks of a unitary matrix resulting from the singular value decomposition of B, i.e.,(15)B=T1T2Σm×m0n-m×mJT,where Σ is a diagonal positive-definite matrix and J is a unitary matrix.Then through linear transformationz=Tx-, system (10) can be decomposed into two subsystems:(16a)z1k+1=A^11z1k+A^12z2k,(16b)z2k+1=A^21z1k+A^22z2k+Bmuk+ωk,where z1(k)∈Rn-m, z2(k)∈Rm; A^=TAT-1.It has been proved that (16a) is the sliding mode dynamics of system (10).Choose the following classic linear sliding surface:(17)sk=Czk=C1Izk=C1z1k+z2k.In the sliding mode, we have(18)sk=C1z1k+z2k=0⇒z2k=-C1z1k.Substituting (18) into (16a), we have (19)z1k+1=A^11-A^12C1z1k,which is a classic sliding surface design problem, and the sliding surface parameter C can be obtained through pole placement.


## 3.2. Design of Chattering-Free SMC

To suppress chattering, a new chattering-free sliding mode reaching law is proposed, inspired by [32]. However, the reaching law proposed in this paper is for multiple-input systems, which means sliding surface s(k) is not a scalar but a vector, and the reaching law should be newly defined. The vector form reaching law is given bellow:(20)sk+1=sk-qTsk-εTsigα⁡sk,where 0<qT<1, 0<εT<1, 0<α<1 and,(21)sigα⁡sk≜diag⁡s1kα,s2kα,…,smkα·sgn⁡sk.Therefore, the sliding mode controller can be designed by comparing (16a), (16b), and (20):(22)uk=-Bm-1C1A^11z1k+C1A^12z2k+A^21z1k+A^22z2k-1-qTsk+εTsigα⁡sk+Bmωk.It can be seen that the unknown disturbance term is included in the expression of u(k), so u(k) cannot be used directly as a control signal. To tackle this problem we introduce the following disturbance estimation strategy:(23)ω^k=z2k-A^21z1k-1-A^22z2k-1-Bmuk-1=ωk-1.Substituting (23) into (22) yields the final SMC law:(24)uk=-Bm-1C1A^11z1k+C1A^12z2k+A^21z1k+A^22z2k-1-qTsk+εTsigα⁡sk+Bmω^k.With the SMC law (24), the new sliding mode reaching law considering disturbance is(25)sk+1=1-qTsk-εTsigα⁡sk+Bmωk-ω^k=1-qTsk-εTsigα⁡sk+Bmωk-ωk-1.Since s(k+1) is a vector with m sliding mode functions, then we write (25) into the following componentwise form and we will prove that each sliding mode state satisfies the reaching condition. The componentwise form of (25) is(26)sik+1=1-qTsik-εTsikαsgn⁡sik+wiωik-ωik-1i=1,2,…,m,where wi is the ith row of Bm, i.e.,(27)Bm=w1⋮wm.Assumption 6.
The change rate of disturbanceδi(k)=ωi(k)-ωi(k-1) is limited, and there exists a constant δ∗ such that maxwiδi(k)≤δ∗,(i=1,2,…,m).Theorem 7.
With the constant matrixC and the linear sliding surface given by (17), the trajectory of system sliding mode state si(k)i=1,2,…,m can be driven by controller (25) into the sliding surface neighborhood region Ω within at most mi∗ steps, where(28)Ω=sik∣sik≤η=ξα·max⁡δ∗εT1/α,εT1-qT1/1-αξα=1+αα/1-α-α1/1-αmi∗=mi+1withmi=si20-η2μ2μ=qTη+1-qT.Proof.
The proof of the theory consists of two problems:
(i) we need to prove that system sliding mode state will enter the regionΩ within mi∗ steps.
(ii) we will prove that ifsik∈Ω, then sik+1∈Ω.
Proof of Problem (i). Define a Lyapunov function Vik=si2k; then it is obtained that(29)ΔVik=si2k+1-si2k=-qTsik+εTsikαsgn⁡sik-wiδik·2sik-qTsik-εTsikαsgn⁡sik+wiδik.Then the following two conditions are discussed.
① Ifsik>η, there is(30)ΔVik=-qTsik+εTsiαk-wiδik·2sik-qTsik-εTsiαk+wiδik.Since sik>η, there is no doubt that sik>ξα·δ∗/εT1/α. Thus, we have(31)qTsik+εTsiαk-wiδik≥qTsik+ξααδ∗-wiδik≥qT·η+ξαα-1δ∗.Define μ=qT·η+ξαα-1δ∗, and then since 0<qT<1, 1<ξαα<2, it is sure that μ is a positive constant.
Similarly, sincesik>η, there is no doubt that sik>η=ξα·εT/1-qT1/1-α. Thus, we have(32)1-qTsik>ξα1-α·εT.Therefore,(33)1-qTsik>ξ1-αα·εTsiαk>εTsiαk.By further calculation, we have(34)sik>qTsik+εTsiαk.Then it can be obtained that(35)2sik-qTsik-εTsiαk+wiδik≥qTsik+εTsiαk+wiδik≥qTsik+εTsiαk-wiδik≥μ.Consequently, there is(36)ΔVik=-qTsik+εT-wiδik·2sik-qTsik-εT+wiδik≤-μ2<0.② If sik<-η, by a similar proof procedure, it can be shown that the relation ΔVik≤-μ2 still holds. Therefore, when sik∉Ω, ΔVik=si2k+1-si2k≤-μ2<0.
Moreover, it can be obtained from (36) that(37)ΔVik=si2k+1-si2k≤-μ2⇔si2k+1≤si2k-ε2T2≤si2k-1-2μ2≤⋯≤si20-k+1μ2.It means that if(38)si20-k+1μ2=η2,then sik+1∈Ω.
However, the solution of (38) may not be an integer, so we denote mi=si20-η2/μ2 as the real number solution of (38). Then we can say that after at most mi∗=mi+1 steps, the ith sliding mode state will enter the sliding surface neighborhood region Ω.
Proposition  8. Denote φ=max⁡δ∗/εT1/α,εT/1-qT1/1-α, and then there is δ∗≤εTφα≤1-qTφ.
Lemma  9. For functions in the form:(39)ησ=1+σσ/1-σ-σ1/1-σ,where 0<σ<1, there exists 1<ησ<2, and for any parameter x∈0,1, the following relation holds [33]:(40)zησ-zσησσ+ησ-1≥0.Proof of Problem (ii). When sik∈Ω, it means -η≤sik≤η. We can define a new parameter -1≤θ≤1 such that sik=θ·η=θ·ξαφ. The following two cases are discussed.
① When0≤θ≤1, according to (26), we have(41)sik+1=1-qTθ·ξαφ-εTθαξααφα+wiδi.First, we will prove that si(k+1)≤η.
According to Proposition 8 and Assumption6, there is(42)sik+1≤1-qTθ·ξαφ+δ∗1-θαξαα.If θξα≥1, we have(43)sik+1≤1-qTθ·ξαφ≤ξαφ=η.If 0≤θξα<1, we have(44)sik+1≤1-qTφ1+θ·ξα-θαξαα≤1-qTφ≤ξαφ=η.Second, we will prove that si(k+1)≥-η.
According to Proposition 8, we have(45)sik+1≥1-qTθ·ξαφ-εTθαξααφα-δ∗≥1-qTφθξα-θαξαα-1.If θξα≥1, since 0<α<1, we have(46)sik+1≥-1-qTφ≥-ξαφ=-η.If 0≤θξα<1, we have(47)sik+1≥1-qTφθξα-θαξαα-1≥-ξα1-qTφ≥-ξαφ=-η.② When -1≤θ≤0, according to (26), we have(48)sik+1=-1-qTθ·ξαφ+εTθξααφα+wiδi.Firstly, we will prove that si(k+1)≤η.
According to Proposition 8, we have(49)sik+1=-1-qTθ·ξαφ+εTθξααφα+wiδi≤-1-qTθ·ξαφ+1-qTφθξαα+δ∗≤1-qTφ-θ·ξα+θξαα+1.If θξα≥1, since 0<α<1, we have(50)sik+1≤1-qTφ≤ξαφ=η.If 0≤θξα<1, according to Lemma 9, we have(51)sik+1≤1-qTφξα≤φξα=η.Second, we will prove that si(k+1)≥-η.
According to Proposition 8, we have(52)sik+1=-1-qTθ·ξαφ+εTθξααφα+wiδi≥-1-qTθ·ξαφ+θξααδ∗-wiδi=-1-qTθ·ξαφ+δ∗θξαα-1.If θξα≥1, we have(53)sik+1≥-1-qTθ·ξαφ≥-ξαφ=-η.If 0≤θξα<1, we have(54)sik+1≥-1-qTφθ·ξα-θξαα+1≥-1-qTφ≥ξαφ=-η.To this end, we can conclude that when -η≤sik≤η, we have -η≤sik+1≤η, which means sik+1∈Ω.Remark 10.
For discrete-time systems, the accurate arrival of sliding mode state on the sliding surface is hard to realize, so to totally eliminate chattering is actually not possible in the discrete-time systems. This paper usessigα⁡sk to replace the sign function to avoid the sharp switching of control input to suppress chattering. The closer the sliding mode state to the sliding surface, the lower the chattering that will be obtained. It needs to be pointed out that the saying “chattering-free” here is just a conventional way to describe such chattering suppression methods [32, 34, 35].


## 3.3. Design of Multiple-Model Based Compensator

Traditionally, if time delay information cannot be foreseen by the controller, there can only be two control input compensation strategies, i.e., zero-input or hold-input. For zero-input strategy, it means if the controller outputu-(k) does not arrive on time, then we let control input u(k)=0. However, when time delay happens frequently, this compensation strategy will result in very poor dynamic performance. And for hold-input strategy, it means control signal will remain unchanged unless new control signal arrives, but this may cause serious overshoot when long time delay exists. With a focus on the above problems, a new compensation strategy is proposed, which also comes from the idea of hold-input strategy but, compared with traditional hold-input strategy, it uses different system models to calculate control input signal for different equivalent time delay conditions. And to overcome the problem that controller cannot foresee the controller-actuator equivalent time delay, all possible models relating to different equivalent time delay τ(k) are established in the controller. Then, at each controlling period, instead of outputting one control signal, the controller will calculate and output a sequence of different control signals U-(k)=[u-(k)τ=0,u-(k)τ=1,…,u-(k)τ=τ-+ρ-] based on different models and the controller output U-(k) will be put in one data packet with a time stamp. Also at each controlling period, the ZOH will output a signal sequence packet U(k) based on the time stamp information, which means U(k)=U-(k) for τ(k)=0, or U(k)=U-(k-1) for τ(k)=1, etc. Finally, the compensator will decide which control signal of U(k) to use according to the time stamp information; i.e., if U(k)=U-(k), it will use the first signal of U(k), and if U(k)=U-(k-1), it will use the second signal of U(k), etc.In this way, even though there is no newly received control signal by the ZOH, the actuator can still get an updated control signal from the compensator and, unlike the traditional hold-input strategy, the updated control input signal has already taken into account the state changes of the former sampling period.Based on the above discussion, the algorithm of the compensator can be described as(55)uk=Ukgτk=U-k-τkgτk,where the compensation signal choosing function g(x) is defined as(56)gx=g0,g1,…,gi,…,gτ-+ρ-T,gi=1,i=x0,i≠x.To show the relationship between actuator input and controller output more clearly, the following illustrations are given:(57)uk=U-k·10…0T=u-kτ=0,τ=0U-k-1·010…0T=u-k-1τ=1,τ=1⋮⋮U-k-τk·0…0︸τk10…0T=u-k-τkτ=τk,τ=τk⋮⋮U-k-τ--ρ-·0…01T=u-k-τ--ρ-τ=τ-+ρ-,τ=τ-+ρ-.In conclusion, the steps of designing this chattering-fee sliding mode controller are given below.Step 1.
Obtain the system’s time delay and packet dropout characteristics through experiments and then get the transition probability matrix (6) based on the proposed modeling method.Step 2.
Obtain the system sliding mode regular form (16a) and (16b) through linear transformation z=Tx-, where the definition of nonsingular matrix T∈Rn×n is given by (14) and (15).Step 3.
Design the classic linear sliding surface (17), whose parameter C=[C1I] can be determined by pole placement method.Step 4.
Design the chattering-free reaching law (20) and choose appropriate reaching law parameters q, ε, and α that satisfy 0<qT<1, 0<εT<1, 0<α<1.Step 5.
Design the sliding mode controller (22) according to reaching law (20).Step 6.
Design the multiple-model based compensator (57) based on the upper bound of long time delay and consecutive packet dropout.


## 4. Simulation Example

In this section, a simulation example is given to illustrate the effectiveness of the proposed method. Consider the following NCS in the form of (1), which comes from a type of aeroengine control system [36]. System parameters are initialed as follows:(58)A=-0.90.1-0.2-0.70.9-0.50.4-0.80.6,B=0.20.10.10.40.3-0.1,x=nLnHp3T,u=mfA8T,where nL is low pressure rotor speed; nH is high pressure rotor speed; p3 is compressor exit total pressure; mf is fuel flow; and A8 is critical section area of nozzle.System sampling period is set asT=20ms. The initial system state is x0=-0.90.5-0.7T. The disturbance is chosen as(59)dk=Bωk=BNζk,where ζk=1.2sin⁡(0.5π·k) and N=0.50.3T.The sliding function parameterC=[C1I] can be designed based on pole placement method, and by choosing the pole location of closed-loop system (16a) as -0.5, it can be obtained that C1=2.8539-0.0238T, and thus C=2.853910-0.023801.Then choose the reaching law parameters asq=10, ε=0.5, and α=0.5.For long time delay and packet dropout, it is defined thatτ-=2 and ρ-=2, and the probability transition matrix is given below:(60)Πτ-,ρ-=0.70.30000.60.20.2000.40.20.20.200.30.20.200.30.40.40.200.Based on matrix Πτ-,ρ-, a Markov chain typed equivalent time delay distribution is shown in Figure 2, which simulates the actual time delay and packet dropout condition of a networked control system during 100 sampling periods.Figure 2
Distribution of equivalent time delay.It can be seen from Figure2 that τ(k)=3 occurs four times, and τ(k)=4 does not occur. Accordingly, we can see that, due to the use of ZOH and the application of the proposed new modeling method, consecutive packet dropout does not really affect the system in this case even though it does exist. However, the effect of consecutive packet dropout may become noticeable if the upper bound of consecutive packet dropout is larger than two, but there is still no doubt that the proposed method greatly suppresses the effect of it.Simulation results obtained by using the proposed chattering-free SMC are shown in Figures3–5. To show the superiority of the proposed controller in chattering suppression, another controller is applied which is the classic reaching law based SMC proposed by Gao [37]. The same compensation strategy is also used in this controller. The results obtained are shown in Figures 6–8. The comparison of Figure 3 with Figure 6 reveals that the system states converge in less time with lower chattering in the case of the proposed controller as compared to the classic reaching law based SMC. The comparison of Figure 4 with Figure 7, and Figure 5 with Figure 8 gives the same conclusion in control input and sliding function.Figure 3
System state trajectory with the proposed controller.Figure 4
Control input trajectory with the proposed controller.Figure 5
Sliding function trajectory with the proposed controller.Figure 6
System state trajectory with classic reaching law based SMC.Figure 7
Control input trajectory with the classic reaching law based SMC.Figure 8
Sliding function trajectory with the classic reaching law based SMC.Moreover, to testify the effectiveness of the proposed multiple-model based compensator, another controller has been constructed which has the same chattering-free reaching law with the proposed controller but without compensator. Instead, the traditional hold-input compensation strategy is used. First, the condition ofτ-=2 and ρ-=2 is considered and the response of system state x1 with two controllers is shown in Figure 9, where controller 1 is the proposed controller and controller 2 is the proposed controller without compensator. It can be seen that there is no essential difference between the response curves of x1 with two controllers because the upper bound of time delay and packet dropout is small and the hold-input strategy is sufficient for compensation. Then Figures 10 and 11 depict the response curves of x1 on conditions τ-=3, ρ-=3 and τ-=4, ρ-=4, respectively. It is obvious that the performance of controller 2 (proposed controller without compensator) deteriorates sharply when the upper bound of time delay and packet dropout grows, while the proposed controller suffers little from the changes.Figure 9
Trajectory of statex1 with two controllers on condition τ-=2,ρ-=2.Figure 10
Trajectory of statex1 with two controllers on condition τ-=3,ρ-=3.Figure 11
Trajectory of statex1 with two controllers on condition τ-=4,ρ-=4.


## 5. Conclusion

This paper proposes a chattering-free sliding mode controller with multiple-model based delay compensator for NCSs with long time delay and consecutive packet dropout. By proposing a new modeling method, long time delay and consecutive packet dropout can be modeled in a unified model described by one Markov chain, which not only simplifies the system model but also makes the controller designed more suitable for practical use. To the best of the authors’ knowledge, this modeling method for time delay and packet dropout has not been reported by any existing literature and it is the use of this model that makes the proposed compensator function well. A chattering-free sliding mode reaching law is then proposed and it is also the first time that such chattering-free reaching law is used for multiple-input discrete-time systems. To overcome the problem that controller-actuator channel network condition cannot be foreseen by controller, a new compensation strategy is proposed and a multiple-model based compensator is constructed. Finally, a simulation example is given and it shows that the proposed method can effectively overcome the effect of time delay and packet dropout as well as disturbance and can make the system states converge to the origin quickly without noticeable chattering. Therefore, the proposed method serves as a suitable choice for uncertain NCSs with long time delay and consecutive packet dropout. In our future research, we will work on testing the proposed method in a practical system so that the theory proposed can be more completed.



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*Source: 1016381-2019-07-11.xml*