# Remote Fault-Tolerant Control for Industrial Smart Surveillance System

Authors: Atif Mahmood; Abdul Qayyum Khan; Ghulam Mustafa; Nasim Ullah; Muhammad Abid; Aadil Sarwar Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

We design a remote fault-tolerant control for an industrial surveillance system. The designed controller simultaneously tolerates the effects of local faults of a node, the propagated undesired effects of neighboring connected nodes, and the effects of network-induced uncertainties from a remote location. The uncertain network-induced time delays of communication links from the sensor to the controller and from the controller to the actuator are modeled using two separate Markov chains and packet dropouts using the Bernoulli process. Based on linear matrix inequalities, we derive sufficient conditions for output feedback-based control law, such that the controller does not directly depend on output, for stochastic stability of the system. The simulation study shows the effectiveness of the proposed approach.


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## Body

## 1. Introduction

Surveillance of processes is generally defined as monitoring of activities and behavior to collect information to influence, manage, or direct. In industries, the surveillance is done via industrial drones that have applications in defense, news-gathering, telecommunication, humanitarian needs, public and private safety, traffic monitoring, security, and process industry [1]. These drones are equipped with a gimbal camera sensor that collects information and sends it to a remote control room through a communication network. A smart surveillance system makes use of a remote control mechanism that can handle the surveillance operation. The remote control of geographically distributed networked systems is widely used [2–4] for cost reduction, simple installation, and maintenance and for increasing efficiency and reliability in the recent industrial revolution. Recent advances in industrial control mechanisms call for more sophisticated schemes for remote surveillance. Such surveillance schemes are desired to take care of detecting and diagnosing faults and tolerating them if needed.In controlling plants from a remote location over communication networks, controllers face two main issues that are network-induced uncertainties and faults. The network lines induce random time delays and data packet dropouts during plant-controller communication over a network. These communication imperfections degrade the system performance. To achieve a prescribed performance, control engineers are trying to tolerate its effects. For this purpose, the researchers modeled the time delays as constant, time-varying, or random variable [5, 6]. Markov chain is suitable to model for these network-induced effects as it depends on the current state only. However, in practice, especially in NCSs, the exact values of Transition Probabilities (TPs) are either hard to determine or costly. Consequently, for a practical design, it is necessary to consider more complex TPs, such as uncertain TPs or partly unknown TPs [7–9]. Similarly, packet dropouts are modeled with probability-based switches for networked systems [10–13].Parallel with network uncertainties faults in the sensor, the actuator, or in the process itself may occur in networked systems. A lot of work has been done in the fault diagnosis and tolerance of such industrial processes [14] as maintenance and repair cannot always be achieved immediately in automated systems. For remotely distributed connected systems, these faults become more uncertain due to the propagation of fault over network links to neighboring connected nodes [15]. The propagated faulty effects may be nonlinear that may add to the already existing faults of other connected nodes [16–18]. The objective is to tolerate the effects of local simultaneous faults, propagated effects of faults that occur at other connected nodes, and network-induced uncertainties.Researchers have put their efforts into the diagnosis of faults in such geographically distributed connected systems [19, 20]. Most of the researchers addressed either the actuator faults or sensor faults with network-induced effects especially with time delays [14]. In real-time applications, both the sensor and actuator faults may occur simultaneously. That is why it is important to address both types of faults at a time. For this purpose, the authors of [21] addressed both sensor and actuator simultaneous faults but with relatively conservative fault models. The authors’ designed a state feedback-based controller that has limitations in practical applications due to the rare accessibility of fully known state vectors. One other source of motivation is that the existing literature addressed the components’ faults only without the propagated effects [15–17] of faults of other nodes. The authors of [18] presented a new fault model for actuators and tolerated its effects using a state feedback controller. The model addressed the nonlinear effects too that may occur due to faults or other technical limitations. In this approach, the following things are important to note. Firstly, we need tolerance of simultaneous sensors and actuators faults in practical applications while the authors addressed only actuator faults. Secondly, the designed state feedback controller has limitations from a practical point of view due to the rare accessibility to all states. The third one is that only time delays are considered in the sensor-to-controller link that means the controller receives packets over a link and command the actuator directly without network link. Furthermore, packet dropout, which may occur in links due to network traffic, is not taken into account in the same link.To address the aforementioned issues, we have designed a remote fault-tolerant control for an industrial smart surveillance system with the following contributions.(1)
The random time delay and the data packet dropouts are modeled in sensor-to-controller (S/C) links and controller-to-actuator (C/A) links separately. The time delays are modeled using two separate Markov chains with partly unknown transition probabilities and packet dropouts with two separate Bernoulli processes. The formulation helps us to establish a new framework to analyze a remote fault-tolerant control problem for industrial systems over a network.(2)
The mode-dependent output feedback-based remote controller is designed such that the controller does not directly depend on the sensor’s output. Such dynamic remote fault-tolerant controllers are significant for industrial processes where the system’s states are unknown.(3)
Sufficient conditions are derived in terms of linear matrix inequality (LMI) for a closed-loop Markovian Jump nonlinear system. The derived control law ensures the stochastic stability of the system and tolerates the sensors and actuator simultaneous faults to maintain the prescribed performance in a networked environment.We organized the subsequent sections of the paper in the following way. We dedicate Section2 for system description and problem formulation, where uncertain network-induced imperfections are modeled as a Markov chain, and then present a new fault model for networked systems. The symbols used throughout the paper are given in Table 1. In Section 3, remote fault-tolerant control is designed for the industrial surveillance system using a new LMI. In Section 4, simulation results are presented and compared with a conventional remote control for the effectiveness of our proposed control design, and then, the paper is concluded.Table 1
Notations with description used throughout the manuscript.
Notations used for mathematical modelingParameterDescriptionτkLoop delayτkscTime delay from the sensor to the controllerτkcaTime delay from the controller to the actuatorukController’s outputu∗kActuator’s output with faults and network effectsykSensor’s outputy∗kController’s input with faults and network effectsΘActuator fault matrixΘ1Markov chain for time delays in sensor-to-controller linksΘ2Markov chain for time delays in controller-to-actuator links∗Symmetric entriesPr⋅ProbabilitiesΘTP…Quadratic forms


## 2. System’s Description and Problem Formulation

### 2.1. Smart Surveillance System

The smart surveillance system consists of multiple drones connected via a shared communication network. Each drone is equipped with a gimbal camera sensor and is controlled from a remote location, as shown in Figure1. The drone is used to carry the gimbal-based camera sensor to different locations for monitoring and surveillance. The collected information via a camera sensor or from other connected drones is then sent to a remote control room. The camera sensor is pivoted in the inner channel of the gimbal system that decouples it from the drone body. The decoupling helps the camera sensor to collect information with minimum viewpoint changes [22]. The camera sensor inside the gimbal system is actuated using BLDC motors according to the commands of the controller. This gimbal-based smart surveillance system for industries has to be controlled from remote locations for safe and easy use.Figure 1
Remote control of the industry surveillance system.To make the smart monitoring and surveillance system, we consider two things. Firstly, the controller can stabilize and control the camera system from a remote location for safe and easy use. Secondly, the designed controller can tolerate the effects of local faults, propagated or external faults that may be nonlinear, and network-induced uncertainties simultaneously. The local faults are sensors and actuators faults in our case. This smart surveillance system has applications in industries where we need information from remote locations with safety and ease.


### 2.2. Problem Formulation

Consider the remote control scenario of a smart surveillance system in Figure2 where a remote controller controls the gimbal camera sensor through a shared communication network. The system modeling with related detail can be seen in [23,  24]. The state dynamics of the system are given as(1)xk+1=Axk+Bu∗k,yk=Cxk,where xk∈ℝn is the state vector, u∗k∈ℝm is the input signal, and yk∈ℝp is the output. A, B, and C are system constant matrices of compatible dimensions. Besides, the matrix C has a full-row rank.Figure 2
Control over network.


### 2.3. Sensor-to-Controller Communication

From the sensor to the controller, the output signals propagate via network links and then are available to the controller. Output of the camera sensor available to the controller may be with faulty effects in the camera sensor and with communication uncertainties. To model the outputyk with faults, we introduce(2)y∗k=Ωyk+f⋅.The sensors faults are modeled as(3)Ω=diagω1,ω2,…,ωp,where Ω denotes the faults’ matrix and the functions ωl∈0,1, for l=1,…,p, indicate the status of each individual actuator as(4)ωl=1,the sensor is working normally,0,the sensor is failed completely,otherwise,the sensor is partially working.For the gimbal-based surveillance system,p=1 because of only one camera sensor that is in the inner channel of the gimbal system. f⋅ is a nonlinear term that may add due to occurrence of fault or other technical limitation [18].The aforementioned faulty output data may or may not reach to the controller due to network traffic. The scenario can be modeled with random events as(5)Event1:Succesful transmission and output is available,Event2:Link fail and output is not available.To model the random events, we introduceak which is a binary random vector. A value “1” of aik indicates that the packet was delivered successfully in the ith channel and “0” indicates that the packet was dropped. Here, i∈1,2,…,m. We assume the following probabilities for aik:(6)Praik=1=Eaik=a¯ik,Praik=0=1−a¯ik,where Praik=1 is the probability of packet transmission in S/C links and E is the mathematical expectation. When the probability of aik=1, it means maximum probability or successful transmission of data packet through S/C links, else the packet is lost.The data packets that reach from the sensor to the controller face random time delaysτksc. We assume that τksc varies randomly between 0 and τ1, where τ1 is the maximum time delay of a packet. We model the time delays τksc with a Markov chain Θ1 with N number of stats, i.e., Θ1∈1,2,…,N1. Let ρ1 be the transition probability matrix of the Markov chain that is given as(7)ρ1=ρij,where ρij=ℙΘ1=j|Θ1=i=ℙτksc+1=j|τksc=i≥0, for all i,j∈Γ1 and Σj=0τ1ρij=1,ρij≥0. For example, if τ1=3, the complex transition probability matrix ρ is given as(8)ρ=ρ00ρ01?ρ03?ρ11ρ12?ρ20ρ21?ρ23ρ30?ρ32?,where ? in the matrix are unknown state transition probabilities of time delays and the rest of entries are known transition probabilities.The stochastic faulty output that is available to the controller is(9)y∗k=Ωa¯ikCxk−τksc+f⋅.Due to the network-induced probabilistic packet dropouts and time delays in the network links, the controller mode may not synchronize with the system modes. For this networked system, we introduce a dynamic output feedback-based remote controlFτksc as follows:(10)uk=Fk−τkscy∗k,uk=Fk−τkscΩa¯ikCxk−τksc+f⋅.Remark 1.
The direct output of the camera sensor is not available to the controller due to network impurities. The network uncertainties are stochastic, that is why the output may or may not be available. The available output is the delayed version with a time delay ofτksc and maybe with the effects of faults.


### 2.4. Controller-to-Actuator Communication

The controller outputuk propagates via network links from the controller to the actuator, as shown in (2). The data face packet dropout and random time delay in the controller-to-actuator links. It means the data packets of the controller may reach the actuator or may drop due to network traffic. This communication is modeled with random events as(11)Event1:Succesful transmission andukis available,Event2:Link fail andukis not available.For the random events that occur in the controller-to-actuator links, we introducebk that is a binary random vector. bik=1 indicates that the packet delivered successfully in the ith channel while bik=0 indicates that the packet dropped. Here, i∈1,2,…,m. We assume the following probabilities for bik:(12)Prbik=1=Ebik=b¯ik,Prbik=0=1−b¯ik,where Prbik=1 is the probability of packet transmission in C/A links and E is the mathematical expectation. When the probability of bik=1, it means maximum probability or successful transmission of data packet through C/A links, else the packet is lost.The data packets that successfully transmit via C/A link face random time delaysτkca. We assume that τkca varies randomly between 0 and τ2, where τ2 is the maximum time delay. We model the time delays τkca with a Markov chain Θ2 with N2 number of stats, i.e., Θ2∈1,2,…,N2. Let ρ2 be the transition probability matrix of the Markov chain that is given as(13)ρ2=ρi2j2,where ρi2j2=ℙΘ=j2|Θ=i2=ℙτkca+1=j2|τkca=i2≥0, for all i2,j2∈Γ2 and Σj2=0τ1ρi2j2=1,ρi2j2≥0. The transition probability matrix is similar to the sensor-to-controller link. We assume that ρ=ρ1+ρ2.The industrial surveillance system may experience local faults in the actuators. The effects of actuators’ faults can be modeled as(14)Θ=diagθ1,θ2,…,θm,where Θ denotes the faults matrix and functions θl∈0,1, for l=1,…,m2, indicate the status of each individual actuator, that is,(15)θl=1,the actuator is working normally,0,the actuator is failed completely,otherwise,the actuator is partially working.The plant’s inputu∗k with the communication link impurities can be written as(16)u∗k=Θbi¯kuk−τkca+g⋅,where g⋅ is the nonlinear term that may add due to technical limitations [18].Remark 2.
Equation (16) demonstrates that the control signal may or may not reach the actuator. If bi¯k=1, then the delayed version of uk with time delay of τkca will be available to the actuator. The packet dropouts and time delays in C/A links and S/C links are considered independent random events. We consider time delays for those packets which are successfully transmitted via network links. The lump delay is τk=τksc+τkca.
Consequently, considering the augmented state vector asηk=xTk,xTk−1,xTk−2,…,xTk−τk−1,xTk−τkT∈Rτk+1n. Substituting (10) and (16) in (1) leads us to the closed-loop dynamics of the networked system as follows:(17)ηk+1=A+BΘbi¯kFτkΩai¯kCηk+BΘbi¯kFτkf⋅+Bg⋅.Remark 3.
The output feedback-based closed-loop networked system derived in (17) is a nonlinear stochastic system with two data packet dropout modes ai¯k and bi¯k and two time-delay modes τksc and τkca.
The system in (17) can be written in a compact form as follows:(18)ηk+1=A¯ηk+f^⋅+g^⋅,where(19)A¯=A00⋯0BΘbi¯kFτkΩai¯kCI00⋯000I0⋯00⋮⋮⋱⋯⋮⋮000⋯I0,f^⋅=BΘbi¯kFτkf⋅00⋮0,g^⋅=Bg⋅00⋮0.


## 2.1. Smart Surveillance System

The smart surveillance system consists of multiple drones connected via a shared communication network. Each drone is equipped with a gimbal camera sensor and is controlled from a remote location, as shown in Figure1. The drone is used to carry the gimbal-based camera sensor to different locations for monitoring and surveillance. The collected information via a camera sensor or from other connected drones is then sent to a remote control room. The camera sensor is pivoted in the inner channel of the gimbal system that decouples it from the drone body. The decoupling helps the camera sensor to collect information with minimum viewpoint changes [22]. The camera sensor inside the gimbal system is actuated using BLDC motors according to the commands of the controller. This gimbal-based smart surveillance system for industries has to be controlled from remote locations for safe and easy use.Figure 1
Remote control of the industry surveillance system.To make the smart monitoring and surveillance system, we consider two things. Firstly, the controller can stabilize and control the camera system from a remote location for safe and easy use. Secondly, the designed controller can tolerate the effects of local faults, propagated or external faults that may be nonlinear, and network-induced uncertainties simultaneously. The local faults are sensors and actuators faults in our case. This smart surveillance system has applications in industries where we need information from remote locations with safety and ease.


## 2.2. Problem Formulation

Consider the remote control scenario of a smart surveillance system in Figure2 where a remote controller controls the gimbal camera sensor through a shared communication network. The system modeling with related detail can be seen in [23,  24]. The state dynamics of the system are given as(1)xk+1=Axk+Bu∗k,yk=Cxk,where xk∈ℝn is the state vector, u∗k∈ℝm is the input signal, and yk∈ℝp is the output. A, B, and C are system constant matrices of compatible dimensions. Besides, the matrix C has a full-row rank.Figure 2
Control over network.


## 2.3. Sensor-to-Controller Communication

From the sensor to the controller, the output signals propagate via network links and then are available to the controller. Output of the camera sensor available to the controller may be with faulty effects in the camera sensor and with communication uncertainties. To model the outputyk with faults, we introduce(2)y∗k=Ωyk+f⋅.The sensors faults are modeled as(3)Ω=diagω1,ω2,…,ωp,where Ω denotes the faults’ matrix and the functions ωl∈0,1, for l=1,…,p, indicate the status of each individual actuator as(4)ωl=1,the sensor is working normally,0,the sensor is failed completely,otherwise,the sensor is partially working.For the gimbal-based surveillance system,p=1 because of only one camera sensor that is in the inner channel of the gimbal system. f⋅ is a nonlinear term that may add due to occurrence of fault or other technical limitation [18].The aforementioned faulty output data may or may not reach to the controller due to network traffic. The scenario can be modeled with random events as(5)Event1:Succesful transmission and output is available,Event2:Link fail and output is not available.To model the random events, we introduceak which is a binary random vector. A value “1” of aik indicates that the packet was delivered successfully in the ith channel and “0” indicates that the packet was dropped. Here, i∈1,2,…,m. We assume the following probabilities for aik:(6)Praik=1=Eaik=a¯ik,Praik=0=1−a¯ik,where Praik=1 is the probability of packet transmission in S/C links and E is the mathematical expectation. When the probability of aik=1, it means maximum probability or successful transmission of data packet through S/C links, else the packet is lost.The data packets that reach from the sensor to the controller face random time delaysτksc. We assume that τksc varies randomly between 0 and τ1, where τ1 is the maximum time delay of a packet. We model the time delays τksc with a Markov chain Θ1 with N number of stats, i.e., Θ1∈1,2,…,N1. Let ρ1 be the transition probability matrix of the Markov chain that is given as(7)ρ1=ρij,where ρij=ℙΘ1=j|Θ1=i=ℙτksc+1=j|τksc=i≥0, for all i,j∈Γ1 and Σj=0τ1ρij=1,ρij≥0. For example, if τ1=3, the complex transition probability matrix ρ is given as(8)ρ=ρ00ρ01?ρ03?ρ11ρ12?ρ20ρ21?ρ23ρ30?ρ32?,where ? in the matrix are unknown state transition probabilities of time delays and the rest of entries are known transition probabilities.The stochastic faulty output that is available to the controller is(9)y∗k=Ωa¯ikCxk−τksc+f⋅.Due to the network-induced probabilistic packet dropouts and time delays in the network links, the controller mode may not synchronize with the system modes. For this networked system, we introduce a dynamic output feedback-based remote controlFτksc as follows:(10)uk=Fk−τkscy∗k,uk=Fk−τkscΩa¯ikCxk−τksc+f⋅.Remark 1.
The direct output of the camera sensor is not available to the controller due to network impurities. The network uncertainties are stochastic, that is why the output may or may not be available. The available output is the delayed version with a time delay ofτksc and maybe with the effects of faults.


## 2.4. Controller-to-Actuator Communication

The controller outputuk propagates via network links from the controller to the actuator, as shown in (2). The data face packet dropout and random time delay in the controller-to-actuator links. It means the data packets of the controller may reach the actuator or may drop due to network traffic. This communication is modeled with random events as(11)Event1:Succesful transmission andukis available,Event2:Link fail andukis not available.For the random events that occur in the controller-to-actuator links, we introducebk that is a binary random vector. bik=1 indicates that the packet delivered successfully in the ith channel while bik=0 indicates that the packet dropped. Here, i∈1,2,…,m. We assume the following probabilities for bik:(12)Prbik=1=Ebik=b¯ik,Prbik=0=1−b¯ik,where Prbik=1 is the probability of packet transmission in C/A links and E is the mathematical expectation. When the probability of bik=1, it means maximum probability or successful transmission of data packet through C/A links, else the packet is lost.The data packets that successfully transmit via C/A link face random time delaysτkca. We assume that τkca varies randomly between 0 and τ2, where τ2 is the maximum time delay. We model the time delays τkca with a Markov chain Θ2 with N2 number of stats, i.e., Θ2∈1,2,…,N2. Let ρ2 be the transition probability matrix of the Markov chain that is given as(13)ρ2=ρi2j2,where ρi2j2=ℙΘ=j2|Θ=i2=ℙτkca+1=j2|τkca=i2≥0, for all i2,j2∈Γ2 and Σj2=0τ1ρi2j2=1,ρi2j2≥0. The transition probability matrix is similar to the sensor-to-controller link. We assume that ρ=ρ1+ρ2.The industrial surveillance system may experience local faults in the actuators. The effects of actuators’ faults can be modeled as(14)Θ=diagθ1,θ2,…,θm,where Θ denotes the faults matrix and functions θl∈0,1, for l=1,…,m2, indicate the status of each individual actuator, that is,(15)θl=1,the actuator is working normally,0,the actuator is failed completely,otherwise,the actuator is partially working.The plant’s inputu∗k with the communication link impurities can be written as(16)u∗k=Θbi¯kuk−τkca+g⋅,where g⋅ is the nonlinear term that may add due to technical limitations [18].Remark 2.
Equation (16) demonstrates that the control signal may or may not reach the actuator. If bi¯k=1, then the delayed version of uk with time delay of τkca will be available to the actuator. The packet dropouts and time delays in C/A links and S/C links are considered independent random events. We consider time delays for those packets which are successfully transmitted via network links. The lump delay is τk=τksc+τkca.
Consequently, considering the augmented state vector asηk=xTk,xTk−1,xTk−2,…,xTk−τk−1,xTk−τkT∈Rτk+1n. Substituting (10) and (16) in (1) leads us to the closed-loop dynamics of the networked system as follows:(17)ηk+1=A+BΘbi¯kFτkΩai¯kCηk+BΘbi¯kFτkf⋅+Bg⋅.Remark 3.
The output feedback-based closed-loop networked system derived in (17) is a nonlinear stochastic system with two data packet dropout modes ai¯k and bi¯k and two time-delay modes τksc and τkca.
The system in (17) can be written in a compact form as follows:(18)ηk+1=A¯ηk+f^⋅+g^⋅,where(19)A¯=A00⋯0BΘbi¯kFτkΩai¯kCI00⋯000I0⋯00⋮⋮⋱⋯⋮⋮000⋯I0,f^⋅=BΘbi¯kFτkf⋅00⋮0,g^⋅=Bg⋅00⋮0.


## 3. Fault-Tolerant Control Design

Assumption 1.
We assume that there is no packet dropouts and time delays in the bounded nonlinear functions that we add to the sensor and actuator faultsfyk and guk, respectively, which satisfy the following local Lipschitz conditions:(20)fTykfyk≤ν2yTkyk,gTykguk≤γ2yTkuk,where ν>0 and γ>0 are the bounding parameters for nonlinear functions fyk and guk.Definition 1.
The Markovian jump nonlinear stochastic system in (18) is robust stochastic stable with any initial condition (η0, τk0, ai¯0, bi¯0) if(21)E∑k=0∞ηk2|η0,τk0,ai0,bi0≤∞,where ηk0 denotes the solution of (18) at time k under the initial conditions (ζ0, τk0, ai0, and bi0).
Here, we present our LMI for the proposed remotely located output feedback controller design to ensure stochastic stability.Theorem 1.
Consider the closed-loop system for the industrial surveillance systems as in (18), and let ν>0, γ>0, ϵ1>0, and ϵ2>0 be the given scalars. A desired output feedback controller gain of form (10) exists if there exist matrices Xi>0, Ri, Gi, Qi, and Miτk, satisfying(22)ξ1100ξ14ξ15ξ16ξ17ξ18⋆−ϵ1I0ξ24ξ25ξ2600⋆⋆−ϵ2Iξ34ξ35ξ3600⋆ξ42ξ43ξ440000ξ51⋆⋆0ξ55000ξ61⋆⋆00ξ6600⋆⋆⋆⋆⋆⋆−Λ¯10⋆⋆⋆⋆⋆⋆⋆−Λ¯2I<0,where ξ11=−GiTXi−1Gi,ξ44=ξ55=ξ66=ξ99=diagXk1i,…,−Xkmi,−Xj,(23)ξ14=ξ15=ξ16=ρik1iA¯GiT⋮ρikmiA¯GiT1−ρkiA¯GiT,ξ24=ξ25=ξ26=ρik1iMTτk⋮ρikmiMTτk1−ρkiMTτk,ξ34=ξ35=ξ36=ρik1iB¯T⋮ρikmiB¯T1−ρkiB¯T,ξ51=ξ61=ρik1iA¯T⋮ρikmiA¯T1−ρkiA¯T,ξ42=ρik1iMτkGi⋮ρikmiMτkGi1−ρkiMτkGi,ξ43=ρik1iB¯Gi⋮ρikmiB¯Gi1−ρkiB¯Gi,ξ49=ρik1iH¯⋮ρikmiH¯1−ρkiH¯,Λ¯1=ϵ1ν2−1IΛ¯2=ϵ2γ2−1I. According to the definition FiτkC¯Gi=RiC¯, it is derived that(24)Fiτk=RiQ−1.Proof.
We choose the Lyapunov function candidate as(25)Vτk,aik,bik=ηTkPiτk,aik,bikηk,EΔVτk,aik,bik=EηTk+1∑k=0NρijPjτk+1,aik+1,bik+1ηk+1−ηTkPiτk,aik,bikηk,Pj¯=∑k=0NρijPjτk+1,a¯k+1,b¯k+1,and then, we have(26)EΔVτk,aik,bik≤ηTkA+BΘbi¯kFτkΩai¯kCTPj¯…ηk+ηTkA+BΘbi¯kFτkΩai¯kCTPj¯BΘbi¯kFτkf⋅+ηTkA+BΘbi¯kFτkΩai¯kCTPj¯Bg⋅+fT⋅BΘbi¯kFτkTPj¯A+BΘbi¯kFτkΩai¯kCηk+fT⋅BΘbi¯kFτkTPj¯…+fT⋅BΘbi¯kFτkTPj¯Bg⋅+gT⋅BTPj¯A+BΘbi¯kFτkΩai¯kCηk+gT⋅BTPj¯BΘbi¯kFτkf⋅+gT⋅BTPj¯…−ε1fT⋅f⋅+ε1ν2yTkyk+ε2γ2uTkuk−ε2gT⋅g⋅−ζTkPiζk.
It can be shown thatΔV≤0 implies EΣk=0∞ηk2|η0,τk0,a0,bI0≤∞ that guaranteed stochastic stability for the augmented state vector given in (18). By defining the augmented state vectors,(27)ξ=ηkf⋅g⋅,we can write equation (26) as(28)ξ0=ξ11−λξ12ξ13ξ21ξ22−ϵ1I+ϵ2γ2FτkT…ξ23ξ31ξ32ξ33−ϵ2I,ξ11=A¯TPj¯A¯, where A¯=A+BΘbi¯kFτkΩai¯kC, λ=C¯ϵ1ν2C¯+ϵ2γ2FτkΩαkC¯T…−ζTkPiζk, and ξ12=A¯TPj¯Mτk, where Mτk=B¯Θβk, ξ13=A¯TPj¯B¯, where B¯=B, ξ21=MτkTPj¯A¯, ξ22=MτkTPj¯Mτk, ξ23=MτkTPj¯B¯, ξ31=B¯TPj¯A¯, ξ32=B¯TPj¯Mτk, and ξ33=B¯TPj¯B¯.
Multiplication ofdiagGiT,I from left and diagGi,I from right to ξ0 and defining Xi=Pi−1, FτkC¯Gi=RiC¯, then, by using the Schur complement and the introduction of network-induced uncertainties as in [18] results the following is obtained:(29)ξ1100ξ14ξ15ξ16ξ17ξ18⋆−ϵ1I0ξ24ξ25ξ2600⋆⋆−ϵ2Iξ34ξ35ξ3600⋆ξ42ξ43ξ440000ξ51⋆⋆0ξ55000ξ61⋆⋆00ξ6600⋆⋆⋆⋆⋆⋆−Λ¯10⋆⋆⋆⋆⋆⋆⋆−Λ¯2I<0,where ξ11=−GiTXi−1Gi,(30)ξ44=ξ55=ξ66=diagXk1i,…,−Xkmi,−Xj,ξ14=ξ15=ξ16=ρik1iA+MτkRiC¯T⋮ρikmiA+MτkRiC¯T1−ρkiA+MτkRiC¯T,ξ24=ξ25=ξ26=ρik1iMTτk⋮ρikmiMTτk1−ρkiMTτk,ξ34=ξ35=ξ36=ρik1iB¯T⋮ρikmiB¯T1−ρkiB¯T,ξ51=ξ61=ρik1iA+MτkRiC¯T⋮ρikmiA+MτkRiC¯T1−ρkiA+MτkRiC¯Tξ42=ρik1iMτkGi⋮ρikmiMτkGi1−ρkiMτkGi,ξ43=ξ17=ρik1iB¯Gi⋮ρikmiB¯Gi1−ρkiB¯Gi,ξ18=ρik1iB¯⋮ρikmiB¯1−ρkiB¯,Λ¯1=ϵ1ν2−1IΛ¯2=ϵ2γ2−1I. According to the definition FiτkC¯Gi=RiC¯, it is derived that(31)Fiτk=RiQ−1.Lemma 2.
ForC¯ with rank p, there exist nonsingular matrices Qi such that QiC¯=C¯Gi for which there exist a full-rank matrices Gi satisfying(32)Gi=ζGi10Gi2Gi3ζT=ζi1Gi1ζi1T+ζi2Gi2ζi1T+ζi2Gi3ζi2T,where Qi=UTσGi1−1σ−1U. Here, U is find by singular value decomposition of matrix C¯ that is C¯=UC¯ζi=UC¯ζi1ζi2=UC¯σ0, where σ is the nonzero singular value of C¯.Remark 4.
The output matrixC¯ is assumed as a full-row rank. The bilinear matrix inequalities are transformed under this assumption into a set of linear matrix inequalities. The assumption is removed in [10]. As it can be satisfied for many practical applications, that is, why it is not a major constrain. In [18], a similar problem is solved for the fault-tolerant control design of networked systems with the state feedback control design. The state feedback control is not significant for real-time applications due to the rare accessibility of all states.


## 4. Simulation Study

A gimbal-based camera sensor communicates with a remote controller through a communication link and the control controls the actuator over a network link, as shown in Figure2. State variables of the surveillance system are the angular positions ϕ and ψ of inner and outer channels and angular velocities of inner and outer channels ϕ˙ and ψ˙ of the gimbal system, respectively. To actuate the system, two brushless DC motors are used with torques Uτ1,Uτ2, respectively. Angular velocity of the inner channel, which holds the camera sensor, is taken as output of the surveillance system. The system dynamics [23, 25] for the surveillance system with initial conditions are(33)ϕ˙ψ˙ϕ¨ψ¨=01008.96002.4300012.0602.40.2ϕψϕ˙ψ˙+000.15000011.3Uτ1Uτ2,C=00100000.There are two actuators and one sensor. The first actuator is working normally and the second actuator and the sensor both are working partially. The fault matrices are given as follows:Θ=1000.7 and Ω=0.6. The nonlinearities that may add to faults are g⋅=0.6ukcosuk and f⋅=0.7ykcosyk. The complex transition probability matrices for the sensor-to-controller link and the controller-to-actuator link, respectively, are given as (34)ρ1=0.30.4?0.3?0.80.8?0.50.1?0.70.9?0.5?,ρ2=0.30.4?0.3?0.70.4?0.50.6?0.70.1?0.5?,if τkca=τksc=3.The Bernoulli stochastic variables for sensor-to-controller and controller-to-actuator links areai¯k=bi¯k=0.7. The data packet dropouts and random time delay for the sensor-to-controller link and controller-to-actuator links are given in Figures 3 and 4, respectively. Similarly, the Markov evaluation of the controller modes and the system modes are given in Figure 5.Figure 3
Data packet dropouts and random time delays in the sensor-to-controller communication link.
(a)(b)Figure 4
Data packet dropouts and random time delays in the controller-to-actuator communication link.
(a)(b)Figure 5
Markov chain evaluation of controller and system modes.
(a)(b)The simulation results of a smart surveillance system for the tolerance of simultaneous local sensors and actuator faults propagated effects of faults of other nodes, and network-induced uncertainties are shown in Figures6 to 8. In Figure 6, we plot the tolerance of faults with and without the nonlinear effects. The blue solid line is for tracking without nonlinear effects and the red dotted line is for tracking with nonlinearities. Similarly, Figure 7 demonstrates the tracking performance with and without network-induced uncertainties. For comparison in tracking response and fault tolerance, we plot the output of a conventional controller and the proposed controller in presence of network-induced uncertainties without faults in Figure 8. Oscillation in the tracking response of the conventional controller will further increase if we add faults, especially, if we add simultaneous faults. This proves the superiority of the proposed controller in fault tolerance and in tracking targets. In Figure 9, we plot the tracking response with states to show that the states are stable even in the presence of the sensor fault. Starting with different initial conditions, all the states are stable and converge to zero with output tracking the reference input. The output feedback controller gain of the remote controller for the industrial smart surveillance system are calculated using MATLAB LMI toolbox and is given as(35)F1=−0.2169−0.56820.02170.0284,F2=−0.9865−2.58460.09870.1292,F3=0.7612−1.99430.07610.0997,F4=−0.2473−0.64790.02470.0324.Figure 6
Tracking performance with and without nonlinearities.Figure 7
Tracking response with and without network-induced uncertainties.Figure 8
Tracking response comparison with the conventional remote controller without faults.Figure 9
Tracking performance with stable state response in the presence of sensor and actuator simultaneous faults.


## 5. Conclusion

We presented two new fault models for sensors and actuators that are more general than the existing fault models. Random time delays and uncertain data packet dropouts of the network links are modeled in S/C links and C/A network lines separately by using Markov chains and Bernoulli processes, respectively. The remote fault-tolerant controller successfully tracks the reference input and tolerates the effects of simultaneous faults in the presence of network-induced uncertainties. Simulation results for the surveillance system verify the stochastic stability of the systems and the applicability of the controller design.



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*Source: 9927113-2021-06-10.xml*