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During last decades, networked control systems have been widely explored in several fields, namely teleoperation, smart grids, intelligent systems, and aerospace systems. The concept of networked control systems (NCS) is a very subjective one. On one hand, significant benefits provided by NCS include less cost, more reliability, and easy maintenance. In particular, networked control systems are becoming necessary to interconnect distributed units of the control system via appropriate digital networks. Transmission over networked control systems are confronted to several constraints namely, time delay, packets loss, quantization effects, and bandwidth limited[1, 2]. Although the interaction between control and computing theories is fruitful, it is somehow restricted. Recent advances in control methodologies make it feasible to used in several subjects like congestion control, control through networks, managing power, smart grids, and cloud computing applications.
The objective of this work is to make a review about the interaction between control and computing theories. We will consider particular control technologies that can be used to improve the reliability and the performance of the cloud computing[3-5]. Cloud control technology has been recently developed by means of networked control systems. Several researchers direct special attention to internet of things (IOT) as well as distributed sensors networking. Within this frame, a huge information needs to transit through very large and complex systems.
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A typical networked control system layout is depicted in Fig. 1. In general, the research of NCS can be divided into two distinct categories: control of network and control over network. The former is to study the problem about network including routing control, congestion reduction, efficient data communication, networking protocol. The latter, of our interest herein, is to achieve the desired performance of control systems with shared-digital network being the transmission media. This category includes network control systems and remote control systems.
Figure 1. A typical networked control system[6]
From a control viewpoint, the research of NCS is focused on two aspects: the quality of service (QoS) and the quality of control (QoC). QoS is related to the performance of the network, such as transmission rates and error rates. QoC generally refers to the stability/stabilization of the system. One of the most important issues is to maintain QoS and QoC at the same time in the research of NCS[2, 7].
Due to the existence of the network in NCS, some signal processing issues may degrade the performance of the system. They are presented as follows:
1) Network-induced delay. This delay is inevitable in NCS since the information is exchanged through a network among system components. It can be constant (up to jitter) or random and expressed as
$ \begin{eqnarray} \tau(t_k) &=& \tau_{sc}(t_k) + \tau_{ca}(t_k) \label{eqS1} \end{eqnarray} $
(1) where
$t_k$ is the$k$ -th sampling time,$\tau(t_k)$ is the total transmission delay,$\tau_{sc}(t_k)$ and$\tau_{ca}(t_k)$ are the network delays from the sensor to the controller and from the controller to the actuator nodes, respectively. Moreover, in NCS, there are three delay components arising from finite computations: one in the sensor node$\tau_s(t_k)$ , the second in the controller node$\tau_c(t_k)$ and the third in the actuator node$\tau_a(t_k)$ , respectively[8].2) Major delay models. There are four delay models treated in the literature, namely constant delay model, mutually independent stochastic delay model, Markov chain model, and hidden Markov model[9].
3) Limited channel capacity. This refers to the limited bandwidth and the limited transmission energy, which both influence the transmission quality.
4) Packet dropout. Due to several factors including data traffic congestion, data collision or interference, packet loss is an inherent problem in most communication networks. Effectively, the dropout process is often modeled as a Bernoulli process or a Markov process.
5) Network security. Any network medium, particularly wireless medium, is susceptible to easy intercepting. Network security is therefore receiving more and more concern.
At this stage, consider the dynamical system in Fig. 2 modeled by
Figure 2. Time delays in NCS
$ \begin{eqnarray} \dot{x}(t) &=& A x(t)+ B u(t) \nonumber\\ y(t) &=& C x(t)+ D u(t) \label{eqS2} \end{eqnarray} $
(2) where
$x(t) \in {\bf R}^n$ is the state vector,$u(t) \in {\bf R}^m$ is the control input,$y(t) \in {\bf R}^p$ is the output, and the matrices$A \in {\bf R}^{n \times n}$ ,$B \in {\bf R}^{n \times m}$ ,$C \in {\bf R}^{p \times n}$ and$D \in {\bf R}^{p \times m}$ are constant with appropriate dimensions. Due to the presence of a digital network in the feedback loop, the control signal is expressed as$ \begin{eqnarray} {u}(t) = K x(t - \tau_{sc} - \tau_c - \tau_{ca}) \label{eqS3} \end{eqnarray} $
(3) where
$K$ represents the feedback control gain matrix.Recalling that the time delay may be constant, variable, or even random and considering (1), the network time delay is given by
$ \begin{eqnarray} \tau = \tau_{sc} + \tau_c + \tau_{ca}. \label{eqS4} \end{eqnarray} $
(4) For a general formulation, the packet dropouts can be incorporated in (4) to yield
$ \begin{eqnarray} \tau = \tau_{sc} + \tau_c + \tau_{ca} + d h \label{eqS5} \end{eqnarray} $
(5) where
$d$ is the number of dropouts and$h$ is the sampling period. By (5), the data dropouts can be considered as a special case of time delay.Factors affecting system performance
Due to the digital transmission of the data in communication networks, NCS are viewed as special class of sampled-data systems with limited bandwidth. This leads to the situation in which a smaller sampling period produces huge sensing data thereby overloading the network and causing congestion. In turn, more data packet dropouts occur and longer delays, and finally degrading the system performance. A schematic interlink among the sampling period, network loads, and system performance in NCS is illustrated in Fig. 3. The interlink emphasizes a trade-off between the period of sampling of the plant data and the system performance in NCS. Hence, an optimal sampling period exists which offers the best system performance (point "
$b$ " in Fig. 3.
Figure 3. Factors governing system performance in NCS
Next, we turn attention to the fundamental issues and current approaches of cloud control systems.
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A basic approach to cloud control systems with a significant computing potential has been investigated[6]. Cloud computing systems are increasingly used as a computing service in which more and more people prefer to utilize it in view the possibility of eliminating full installation of the software on the computers. On the other hand, cloud computing infrastructure for control application has a variety of advantages including[10]:
1) zero start-up cost
2) less effort
3) avoiding resource over/under provisioning.
It is demonstrated that subscribers of IT computing environment have access to cloud services using desktop-hosted web browsers as clients. One potential area for which cloud computing can play an important function is enhancing the electric power scheduling namely smart-grid. Implementation of cloud computing can sustain the stability and reliability of the grid with reasonable price[11].
In cloud computing, everything is typically treated as a service (i.e., XaaS), e.g., SaaS (software as a service), PaaS (platform as a service) and IaaS (infrastructure as a service). These services define a sort of layered system structure for cloud computing, see Fig. 4. At the infrastructure layer, processing, storage, networks, and other fundamental computing resources are defined as "standardized services over the network"[12].
Figure 4. Cloud computing services
In cloud of the fast growing deployment of internet of things, networked control systems are going to perform as an effective role in IOT technologies. NCS mainly comprise of sensors, actuators, and controllers connected through digital networks. In the past years, the controller sensors, and the actuators are located in the area where the communication is peer to peer. This principle is called a centralized control in which there is practically no time delay result due to the short distance among the NCS elements, and no data loss as well. With the technology advancement, manufactures turn to separate their large plants to subsystems with their own control system. Consequently, there is no direct communication between the subsystems thereby leading to decentralized systems. After that, manufactures began to use hybrid systems containing centralized and decentralized systems. From this perspective, internet of things is a rising research area in academia as well as in industrial. It establishes the future interaction between the computing and communications[6].
IOT is a prototype that acquires of sensor networks. It becomes more challenging to employ in cloud manufacturing. From control systems viewpoint, it is not easy to model the IOT systems due to their complexity and therefore, the data collection process is crucial. The progressive advancement in computing systems provides enhanced capabilities to compute, store and process IOT data with high quality and reliable measurements[6]. The basic issues confronting practical control systems are:
1) severe data time delayed
2) irregular, time-varying, and packet dropout
3) data transmitted might be lost due to traffic congestion, unreliable nature of the link or protocol malfunctions.
These constraints affect the stability and the overall performance of the system and can cause instability.
Expansion of the controlled systems might lead to a massive information shared, it becomes complicated and render the control task intricate. Hence, increased effort was called for to find appropriate control and computing strategies for transporting and managing information and decision of distributed or parallel subsystems. Alternative software and hardware methodologies have commanded the interest of considerable research to achieve reliable computing performance. In this sense, the cloud control was successfully developed to enable virtual operation and control of distributed plants. These subsystems can easily access data centers through digital communication links. Recent development in the cloud manufacturing provides rapidly changing and enhances computing systems through sharing visualized resources, remote control architectures, and remote computing via networks[6].
As the cloud control systems embrace the cloud computing and NCS, it offers incredible potential applications in industrial and other related areas. Fig. 4 exhibits the architecture of cloud based on networked visual servo control. In fact, a massive mount of measurements information from the distributed sensors transfers to the central/decentralized of cloud controllers. The cloud processes the incoming information to make an effective decision by sending a proper control signal. For instance, model predictive control method has been widely utilized to estimate the data and to predict future situations. Predictive control strategy is a powerful tool to take proactive action to avoid faults and unexpected events. In this sense, cloud control systems will contribute effective functions to regulate the complex and distrusted systems[6]. The existence of cloud technology improves the methodology that enterprises perform their business efficiently, especially over internet. With in these developments, cloud control systems are promoting to involve several manufacturing such as networked, smart grid, distributed process and virtual systems[13]. According to computation and communication developments, parallel or distributed computation methodologies have been dynamically accomplished over networked resources. These facilities direct to employ the concept networked computation into classical control systems. Thus, cloud control systems are designed to provide more flexibly and reliability to add/remove any node remotely. Moreover, it has the potential of monitoring, control, communication, and maintenance[14]. To cope with the weakness of transmission time delay in communication links, a fast virtual feedback scheme has been broadly used to tackle this problem. Fig. 4 demonstrates a image processing principles within cloud technology. Images processing are carried out in parallel or distributed manner by real-time image transmission protocol. Cloud computing enables a fast virtual feedback to enhance the system performance, to overcome load and communication constraints[14].
Some basic principles
A significant development of computational theories and methodologies causes an urgent need for reliable and effective computing resources. Cloud computing environment plays a vital key in business pattern since cloud systems have been basically recognized as high virtual computing for distributed sensors via internet with application programming interfaces. With evolving power methodology, the conventional grid has been gradually transferred to future smart-grid. Thus, smart--grid is born as integration of conventional grid with internet technology (IT). It is interesting to record that the utilization of cloud computing in electric power grid has been extremely increased in which:
1) Cloud enhances the multi-level control systems of the power grid
2) Cloud computing sustains the reliability
3) Cloud computing supports highly scalable and shares neighbor parameters.
An attractive attribute of grid-cloud technology (GCT) is that there are relevant incorporated technologies namely, GridStat, Isis2, TCP-R, and GridSim. TCP-R is a new version of TCP that provides a support to the nominal transmission control protocol (TCP). The well known TCP protocol is a reliable internet protocol that enables receiving acknowledgment. GridSim provides real time simulation and control for wide electric grid systems. While GridStat offers a simple interface between the substations with the center station. Also, it has capability to investigate communication channels for transmit redundant power grid data in high speed and synchronize manners. Isis2 is a powerful tool that facilitates holding data in a specific node and distributing to the neighbor nodes. This technology can implement for fault tolerant, and security[11].
Similar to NCS infrastructure, the cloud control systems are designed as distributed systems connected through communication networks. In this framework, the control units assumed to access directly their subsystems and neighbor units using broadcast method. The control units are attractive with smart programming to handle the assigned task with the computation abilities. However, the transmission data are subject to communication constraints such as time delay, packet loss, bandwidth limitation, and quantization effects. These constraints can decline the system performance and further might cause unstable[6, 11]. Mostly, the controlled dynamic NCS systems are considered to be linear discrete time systems. Particularly, an elementary idea is that cloud control problems consist of two parts: a networked control system part and a cloud control part. With in the former networked control, the sensors send their reading and the measurement to the control/estimation unit. The networked control takes decisions in terms of the analysis data. These control signals are transmitted back to the actuator at the plant side. In the later cloud control systems, reliable and robust communication protocols have been widely implemented. The acknowledgment plays vital key to enhance the reliability of large scale systems exemplify TCP protocol. Using the acknowledgment, each controller (NCS or cloud) sorts the willing nodes by priority. Besides, the controller may remove any node from the list when the acknowledgment get lost. The control can schedule and manage when the subsystems allow to send their data. Usually, the system may receive a variety of control decisions from some of the cloud controller in surrounding. The local controller of the certain system can select the most recent control signal.
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A multi-layered structure of an industrial automation system is shown in Fig. 5, which is used to control a wide class of industrial plants. The following summarize the main points:
Figure 5. Multi-layers architecture
1) For the plant to operate as desired, several physical quantities need to be maintained at constant values (including oil temperature) in spite of any disturbance, and other quantities need to follow a certain profile (flow levels).
2) To maintain automatically the desired performance, we need a system to keep monitoring the physical quantities through sensors, and make necessary adjustments through actuators. These are termed field devices and constitute the filed-level network in Fig. 5.
3) Given monitored data and to produce control actions, controllers are needed. These are special-purpose microprocessor-based computer systems running control algorithms on top of special-purpose operating systems. Typically, a single controller can interact with several control loops, depending on the processing speed as well as the sampling rates and complexity of the control algorithms. Controllers are hosted in or next to a control room. Such controllers belong to layer L1 in Fig. 5.
4) In modern industrial systems, communication between field devices and controllers takes place over a special network called field-level network.
a) Operators in the control room need to continuously monitor the process variables of the plant, such as oil flow rate and water temperature. This needs human machine interface (HMI).
b) Also, operators need to supervise the operation of individual control loops, like changing the desired oil temperature, through supervisory control and data acquisition (SCADA).
c) In addition, they need to log the process variables constantly to review their trends later and assess how the system condition changes with time. L2 in Fig. 5 is for HMI/SCADA and logging.
d) Other functionalities such as alarm management (for example when steam temperature dangerously goes beyond a certain threshold) and control software update to improve a control algorithm also belong to L2.
e) Communication among controllers and HMI/SCADA machines take place over a special network called control network.
5) To optimize the overall operation of the industrial plant (like across different seasons or weather conditions), higher level optimization software (L3 in Fig. 5) is needed to coordinate the operating conditions of control loops so that the optimization objectives are met. For example, the objectives could be minimizing natural gas consumption, or maintaining a certain production rate/quality.
6) Finally, linking the plant to the outside world is sought to tie the plant production to market demands as well as different assets, such as material and labor prices. L4 in Fig. 5 is there to perform such enterprise level management. Plant optimization objectives in L3 are decided based on the analysis performed in L4.
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In the absence of any commercially available system that offers direct digital control through the cloud, a proposed cloud control system layout is presented in Fig. 6. In the proposed approach, we do the following things:
Figure 6. Cloud control system layout
1) We move all computing functions of the automation system into the cloud in order to provide full automation as a service. This makes it easier, faster and less costly for users to deploy, maintain, and upgrade their automation systems.
2) It is to design supports switching to a different cloud automation providers since all virtual machines (VMs) can be group-migrated to a different provider.
3) Some components are not movable to the cloud, such as sensors, actuators, and safety/emergency shutdown control functions.
4) To connect sensors and actuators to the cloud, we use field-level protocols that run on top of TCP, such as Modbus/TCP and Profibus/TCP, which are either built in the devices or provided through separate I/O modules. For cases where advanced functions, such as security and message scheduling are required, we dedicate a gateway server, which could be replicated for more reliability.
Fig. 6 illustrates a proposed automation architecture, which essentially relaxes the existing systems layers and reflects the relationship between each component and the layers shown in Fig. 5.
Herein, direct digital control algorithms (L1) run on cloud VMs instead of real hardware in the control room. Also, in existing automation systems (like Fig. 5), controllers communicate with sensors/actuators over a network with mostly deterministic communication delays that are negligibly. Whereas in our design, communication occurs over the internet, which adds large and variable delays to the control loop. Therefore, straightforward migration of direct digital control algorithms to the cloud may affect the functionality of the control loop or even make the system unstable, and thus jeopardize the theoretical performance guarantees offered by traditional controllers. As a result, more components are needed to mitigate the variable internet delays and the lack of reliability of internet links and virtual machines (VMs).
Internet communication features
Distinct from other private transmission media, the internet is a public transmission media which could be used by many end-users for different purposes. The major obstacle of internet-based control systems is how to overcome uncertain transmitting time delay and data loss problems. According to a study of internet transmission[15], the performance associated with time delay and data loss shows large temporal and spatial variation. As the results show, the performance decreases with distance as well as the number of nodes traversed. The performance also depends on the processing speed and the load on the network nodes. Uncertain transmission time delay and data loss problems are not avoidable for any internet-based application. The reasons why the variable time delay occurs are as follows[16]:
1) Network traffic changes all the time because multiple users share the same computer network.
2) Routes or paths of data transmission decided by internet protocol (IP) are not certain. Data is delivered through different paths, gateways, and networks whose distances vary.
3) Large data is separated into smaller units such as packets. Moreover, data may also be compressed and extracted before sending and after receiving.
4) Using TCP/IP protocols, when error in data transmission occurs, data will be retransmitted until the correct data is received. Therefore, data loss and time delay in data transmission can be treated identically.
Strictly speaking, the internet time delay (ITD) is characterized by the processing speed of nodes, the load of nodes, the connection bandwidth, the amount of data, the transmission speed, etc. The internet time delay
$T_d(k)$ at the instant$k$ can be described as follows[17]:$ \begin{eqnarray} T_d &=& \sum^{n}_{j=0} \Bigg (\frac{l_j}{C} + t^R_j + \frac{M}{b_j} \Bigg ) + \sum^{n}_{j=0} t^L_j(k) = \nonumber\\ && d_N + d_L(k)\label{eqZAG1} \end{eqnarray} $
(6) where
$l_j$ is the$j$ -th length of the network link,$C$ is the speed of light,$t^R_j$ is the routing speed of the$j$ -th node,$t^L_j(k)$ is the delay caused by the$j$ -th nodes load,$M$ is the amount of data, and$b_j$ is the bandwidth of the$j$ -th link.$d_N$ is a term, which is independent of time, and$d_L(k)$ is a time-dependent term. Because of the time-dependent term$d_L(k)$ , it is somewhat unreasonable to model the internet time delay for accurate prediction at every instant. Internet time delay must be explicitly handled through control system architecture and/or control algorithms. Eventually, a suitable control structure is required to deal with the uncertain execution time, time delay and data loss for internet-based control systems.
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This section explores the predictive control concept to systems analysis and design.
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Extending on [18], an internet-based control system structure is proposed in Fig. 7, in which a tolerant time
$\Delta t$ and two compensators are introduced to handle the unpredictable information transmission delay. For practical purposes, the dynamic matrix controller (DMC) is found to be suitable in the design of the two time delay compensators shown in Fig. 7 because DMC is widely accepted in industry. The compensator in the feedback channel with a data buffer is designed to overcome the time delay occurring in the transmission from the local site to a remote site. The compensator in the feedforward channel is designed to overcome the time delay occurring in the transmission from a remote site to the local site. All the data were sent over the internet with a time stamp generated by a global timer in the internet-based control system. The receiving time will be compared with the time stamp for each data to see if a delay has occurred or the transmission is normal. Given the step response model:
Figure 7. Internet-based control structure with delay compensation
$ y(t) = \sum\limits_{j = 1}^\infty {{g_j}} \Delta u(t - j) $
(7) where
$y$ is the process output variable,$\Delta u$ is the increment of the control action,$ g_j $ is the coefficient of the step response,$t$ is the current time instant. The DMC general control law can be given by$ \begin{equation} u(t) = (\lambda I + G^t G)^{-1} G^t (w - f) \label{DMC2} \end{equation} $
(8) in which
$\lambda$ is the penalization factor for the control costs and$I$ is the unit matrix. The system dynamic matrix$G \in {\bf R}^{ p \times m}$ defined by the elements are taken from the coefficients of the step response model of the process,$g_1, \cdots, g_p$ :$ \begin{equation} G = \left [\begin{array} {ccccc} g_1&0&\dots&0& 0 \\ g_2&g_1&0 &0 &0 \\ \vdots& \vdots& \ddots&\vdots&0 \\ g_m&g_{m-1}& \dots&g_2&g_1 \\ \vdots& \vdots & \ddots&\vdots&0 \\ g_p&g_{p-1}& \dots&g_{p-m+2} &g_{p-m+1} \end{array} \right] \label{DMC3} \end{equation} $
(9) where
$m$ is the control horizon,$p$ is the prediction horizon and the control action$u$ is defined as$ \begin{equation} u(t) = [\Delta u(t) \Delta u(t+1) \cdots \Delta u(t+m-1)]^t. \label{DMC4} \end{equation} $
(10) The reference trajectory
$w$ and free response$f$ vectors are expressed by$ \begin{eqnarray} w(t) &=& [w(t+1) w(t+2) \cdots w(t+p)]^t \nonumber\\ f(t) &=& [f(t+1) f(t+2) \cdots f(t+p)]^t. \label{DMC5} \end{eqnarray} $
(11) The free response
$f(t+k)$ is computed as$ f\left( t+k \right)={{y}_{m}}\left( t \right)+\sum\limits_{j=1}^{N}{\left[ {{g}_{k+j}}-{{g}_{j}} \right]\Delta u\left( t-j \right)} $
(12) with
$y_m(t)$ is the measurement value of the process output variable,$N$ is the process horizon. The reference trajectory$w(t+k)$ is computed as$ \begin{align} w(t) =&\, y_m(t) w(t+k) =\nonumber\\&\, \alpha w(t+k-1) + (1-\alpha) r(t + k) \label{DMC7} \end{align} $
(13) where 0 < a < 1 is a parameter and
$r(t+k)$ is the setpoint of the controller.Remark 1.
1) The objective of the compensation at the feedback channel is to reduce the effect of the control signal blanks caused by the transmission delay.
2) It can be observed that the time delay occurring in the transmission from the local site to a remote site causes the controller in the remote site to be unable to receive the feedback signal
$y_m(t)$ from the local site on time. Once the time delay occurs,$y_m(t)$ in (12) and (13) will be replaced with the predictive value$\hat{y}(t|t)$ which is calculated from the step response model in (7) based on the available measures at the instant$t$ .3) When the transmission recoveries, the accumulated error
$e(t)$ during the period of the time delay is used to compensate the effect.4) The control vector
$u$ is composed of the$m-1$ future control increments. The increment at the current instant,$\Delta u(t)$ , is normally taken into action and the future control increments from$\Delta u(t+1) \rightarrow \Delta u(t+m-1)$ are simply not in use. Therefore, it is possible to use these available future control increments in the situation where the time delay occurs.Further elaboration on this approach deserves future research.
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Consider NCS under the following operational assumption:
Assumption 1.The sensors are clock driven, the controllers and the actuators are event driven, the data are transmitted as a single packet, the old packets are discarded, all the states are available for measurements and hence for transmission and finally, the time delay
$\tau$ is small to be less than one unit of its measurement.From (2) and (3), it follows that the closed-loop system can be expressed as
$ \begin{eqnarray} \dot{x}(t) &=& A x(t)+ B K x(t - \tau) = \nonumber\\ && [A+BK] x(t)+ B K [x(t-\tau)-x(t)] . \label{eqS6} \end{eqnarray} $
(14) In what follows, let
${\cal C}^{-}$ denote the left potion of the complex plane and$\lambda_j(S)$ represent the$j$ -th eigenvalue of matrix$S$ .Theorem 1.For system (2) with the feedback control of (3), the closed-loop system is globally asymptotically stable subject to Assumption 1 if the following condition:
$ \begin{eqnarray} \lambda_j([(I + \tau BK)^{-1} (A + BK)]) \in {\cal C}^{-}, j =1, 2, \cdots, n \label{eqS65} \end{eqnarray} $
(15) is satisfied
Proof.Subject to Assumption 1 for small delay, a Taylor series expansion yield
$ \begin{eqnarray} x(t - \tau) &\approx& x(t) - \tau \dot{x} (t). \label{eqS7} \end{eqnarray} $
(16) Using (16) into (14) and manipulating the result, it yields
$ \begin{eqnarray} \dot{x}(t) &\approx& [A+BK] x(t) - \tau B K \dot{x}(t) \Rightarrow \nonumber\\ && [(I + \tau BK)^{-1} (A + BK)] x(t). \label{eqS8} \end{eqnarray} $
(17) System (17) will be globally asymptotically stable if condition (15) is met.
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Centralized model predictive controller has been established to guarantee the subsystems to work at nominal values using the end-to-end utilization control (EUCON) algorithm. In this framework, a single center control is ultimately implemented to regulate the other subsystems such as the industrial process. These subsystems are counted on center control to meet their requirements and interaction among them. Indeed, the centralized multimedia personal computer (MPC) enables a predefined networked channel to perform all control and measurement tasks. A major shortcoming of centralized structure is a single point of failure in which the entire system will fail when the center controller breaks down. Also, when the controlled plant expands, it becomes hard to control and diagnostic using only one controller[19, 20]. Theoretically, MPC controller is employed to minimize the utility cost function for the future time expectations[10, 21]. Additionally, a smart centralized controller can be remotely investigated with object to control and to maintain different nodes. In Fig. 8, a schematic of structure of centralized MPC is provided and further illustrated in Fig. 9 for the case of Alkylation of benzene process. The dynamics of a discrete system can be defined in the state-space form as
Figure 8. Structure of centralized MPC[22]
Figure 9. Structure of centralized MPC for the Alkylation of benzene process[22]
$ \begin{eqnarray} x(t+1)&=& A x (t)+B u (t) \nonumber\\ y(t)&=&C x (t) \label{Cen} \end{eqnarray} $
(18) where
$x (t) \in {\bf R}^{n}$ is the state variable,$u (t) \in {\bf R}^{ m}$ is the control input, and$y (t) \in {\bf R}^{p}$ is the output signal. The matrices of dynamic system can be qualified as$A \in {\bf R}^{n\times n}$ and the control input matrix$B \in {\bf R}^{n \times m}$ and the output matrix$C \in {\bf R}^{p \times n}$ . We denote a real vector space with$n$ -dimensional by$ {\bf R}^{ n} \in {\bf R}^{ +}$ , and$ {\bf R}^{ m} \in {\bf R}^{ +}$ with$m$ -dimensional,$ {\bf R}^{ p} \in {\bf R}^{ +}$ with$p$ -dimensional.$ \begin{align} &A=\left[ \begin{matrix} {{A}_{11}}&{{A}_{12}}&\cdots &{{A}_{1n}} \\ {{A}_{21}}&\cdots &\cdots &{{A}_{2n}} \\ \cdots &\cdots &\cdots &\cdots \\ {{A}_{n1}}&{{A}_{n2}}&\cdots &{{A}_{nn}} \\ \end{matrix} \right] \\ &B=\left[ \begin{matrix} {{B}_{11}}&{{B}_{12}}&\cdots &{{B}_{1m}} \\ {{B}_{21}}&\cdots &\cdots &{{B}_{2m}} \\ \cdots &\cdots &\cdots &\cdots \\ {{B}_{n1}}&{{B}_{n2}}&\cdots &{{B}_{nm}} \\ \end{matrix} \right] \\ &C={{\left[ \begin{matrix} {{C}_{11}}&{{C}_{12}}&\cdots &{{C}_{1n}} \\ {{C}_{21}}&\cdots &\cdots &{{C}_{2n}} \\ \cdots &\cdots &\cdots &\cdots \\ {{C}_{p1}}&{{C}_{p2}}&\cdots &{{C}_{pn}} \\ \end{matrix} \right]}^{\text{T}}}. \\ \end{align} $
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Complexity of large scale systems (LSS) has attracted the attention of researches in systems and control from both theory and applications. It appears that the time delay and uncertainty are the main source of complexity among the LSS components. In most of the times, the scope of LSS is in general subject to centrality fails because of the shortfall of centralized observations. The operational principles of large-scale systems have been widely implemented to tackle complex and coupled problems using various techniques. In these senses, the large problem can be decomposed into independent controllable subproblems. The group of decoupled subproblems can satisfy the system performance and guarantee stability of the overall system. The control units can interchange only limited measurements, input decisions and the dynamic of the states with other controllers. Decentralized control methodologies have been studied actively to induce more flexibility and reduce cost of communication processing[23, 24]. In Fig. 10, a schematic layout of decentralized MPC is presented and its demonstration to the case of Alkylation of benzene process is given in Fig. 11.
Figure 10. Structure of decentralized MPC[22]
Figure 11. Structure of decentralized MPC for the alkylation of benzene process[22]
Decentralized MPC is proposed to overcome single point of failure. Within this architecture, there are some local independent controller in which the interactive among them is ignored. Typically, the autonomous controller can receive an observations from the related subsystems and carry out suitable decisions regarding that. The decentralized strategy divides the main problem to sub-problems regulated by local MP controllers. The local controllers do not trend to interconnect with the most of surrounded controllers. It is only interesting with willing neighbor control units. In particular, the influence of the remote nodes on the local node is ignored[19, 20]. In [25], a decentralized control algorithm is investigated using distributed CPU controllers. The decentralized control strategies have been widely used and studied for microgrid systems, smart grids, cloud control, and aerospace vehicles. Decentralized control is strongly promoted to provide highly performance systems and more reliable system than centralized MPC. While this paradigm to process control has been successful, there is an increasing interest in developing distributed model predictive control schemes, where agents share information in order to improve closed-loop performance, robustness and fault-tolerance, see Fig. 12 for an example of this control scheme. The performance of the closed-loop system depends on the decisions that all the agents take, hence, cooperation and communication policies become very important issues. A demonstration of distributed MPC to the case of Alkylation of benzene process is given in Fig. 13. In this section, a constrained discrete time invariant system will consider:
Figure 12. Structure of distributed MPC[22]
Figure 13. Structure of distributed MPC for the alkylation of benzene process[22]
$ \begin{eqnarray} x_{j}(t+1)&=&A_{jj} x_{j}(t) +\alpha_{j} B_{jj} u_{j}(t) \nonumber\\ y_{j}(t)&=& \beta_{j} C_{jj} x_{j}(t) \end{eqnarray} $
(19) where
$\alpha_{j}$ and$\beta_{j}$ represent the networked channel constraints. -
Time-varying delays are mainly due to limited network bandwidth and nodes waiting in a busy channel[26]. The use of Markov chain has dominated the representation of network-induced time delay as a time-varying parameter[27-34]. It is recognized that this approach cannot guarantee the closed-loop stability unless the network delay is very small. In [35, 36], a networked-model based predictive control (NMBPC) approach is presented. In this approach, the focus is the ideas of packet-based networked control combined with the conventional variable sampling period, see Fig. 14. Recent studies are reported in [37-50], yet the need for excessively high bandwidth networks still remains as an unresolved issue. This has motivated the development of new control algorithms that can cope with longer time delays in actual networks.
Figure 14. Model of improved predictive control
An improved predictive control strategy is proposed with the packet-based control methodology. In particular, event-driven sensors are employed, i.e., the plant output sampling is performed only when a new control input signal is received by the actuator. In order to deal with the network delay and packet loss, which are associated with a range of pre-specified time delays, appropriate discrete-time (DT) models of the networked plant are calculated offline, and, based on them, some stabilizing control signals are constructed online. Similar to the traditional packet-based control methodology, these control signals are then packed in the control-side packet and are transmitted back to the plant side. To cope with the packet dropout issue, a simple, yet effective, algorithm is adopted in the controller. In the light of this formulation, the NCS can be considered as a switched linear system, and therefore the important theoretical results that were previously extended to such systems can be used for the stability analysis of the improved prediction method.
Effectively, the improved prediction method has the following advantages:
1) The actual plant input is known to the controller at each time step, and therefore it can cope with relatively large network delays and packet dropouts.
2) The sampling period is adapted on the basis of the network condition.
3) Unlike other methods, the stability analysis is proposed using matrices with a fixed size for different network conditions.
-
Consider a typical NCS as shown in Fig. 14 and described by
$ \begin{eqnarray} \dot{x}(t) &=& A x(t)+ B u(t) \nonumber\\ y(t) &=& C x(t) \label{eqR1} \end{eqnarray} $
(20) where
$x(t) \in {\bf R}^n$ is the state vector,$u(t) \in {\bf R}^m$ is the control input,$y(t) \in {\bf R}^p$ is the system output, and$A \in {\bf R}^{n \times n}$ ,$B \in {\bf R}^{n \times m}$ and$C \in {\bf R}^{p \times n}$ are constant matrices with appropriate dimensions. Herein,$u(t)$ is the control input generated in the closed-loop system, and no delay was present in the network. As depicted in Fig. 14, the control input is generated and transmitted to the plant with some unknown and time varying delay. The dynamics of the closed-loop system can be then represented as$ \begin{eqnarray} \dot{x}(t) &=& A x(t)+ B u(t - \tau(t)) \nonumber\\ y(t) &=& C x(t) \label{eqR2} \end{eqnarray} $
(21) where
$\tau(t)$ is the control loop delay at time$t$ satisfying 0 <$\tau(t) \leq \tau_M$ , where$\tau_M$ is a prescribed upper bound for the possible range of time delays.Assumption 2.
1) The number of consecutive dropouts and the network time delay are bounded.
2) The pairs
$(A, B)$ and$(A, C)$ are controllable and observable, respectively.The objectives herein are:
1) To construct a switching feedback controller that can asymptotically stabilize the closed-loop NCS described by (21) for time delays in the prescribed range with an assumed upper bound.
2) To develop a method that can transform the packet loss and packet disordering problems into a fictitious time delay, so that a unified control approach could be used for transmission time delay, packet loss, and packet disordering.
-
Fig. 15 illustrate the different delay element at the
$k$ -th sampling instant$t_k, k = \{0, 1, 2, \cdots, \infty \}$ . It can be easily seen from Fig. 15 that the next sampling period at the$t_{k+1}$ , i.e.,$T_k = t_{k+1} -t_{k}$ , will be such that the property$T_k = \tau_{k}$ holds. This property greatly simplifies the mathematical derivations that follow. In the light of this consideration, for a shift-varying sampling period$T_k$ , the zero-order-hold (ZOH) equivalent model of the continuous-time plant (21) is of the form
Figure 15. Signal diagram
$ \begin{eqnarray} {x}(t_{k+1}) &=& A_k x(t_{k})+ B_k u(t_{k-1}) \nonumber\\ y(t_{k+1}) &=& C x(t_{k+1}) \label{eqR3} \end{eqnarray} $
(22) where
$ $$A_k = {\rm e}^{A T_k}, B_k = \int^{t_k+T_k}_{t_k} {\rm e}^{A (T_k - \lambda)} {\rm d} \lambda B. $$ $
For simplicity, we use the notation
$x(k+1), x(k)$ instead of${x}(t_{k+1}), x(t_{k})$ , respectively and simply write$ \begin{eqnarray} x(k+1) &=& A_k x(k)+ B_k u(k-1) \nonumber\\ y(k+1) &=& C x(k+1). \label{eqR4} \end{eqnarray} $
(23) -
Time-delay compensator (TDC) is an event-driven device in the plant-side that continuously memorizes the latest received control-side packet. Suppose that the current time shown by the TDC timer is
$t_k$ , the time difference$\tau_e = t_k -t_{k'}$ is then computed, where$t_{k'}$ denotes the time stamp of the received packet, which is already latched in the TDC buffer. In fact,$\tau_e$ is a directly measured overall network time delay at the outset of$t_k$ . The TDC then selects the entry associated with$\tau_e$ of the control-side packet, which is available from the TDC buffer. There is high possibility that none of the entries of$\tau^v$ equals$\tau_e$ , i.e.,$\tau_e \neq \tau_j, j = 1, \cdots, \beta$ .In order to overcome such an issue, TDC chooses
$k_s \in \{1, 2, \cdots, \tau_{M}\}$ such that$\tau_{k_{s-1}} < \tau_e \leq \tau_{k_{s}}$ for$\tau_j \in \tau^v$ and then waits for$\tau_{k_{s}} -\tau_e$ seconds before applying the correct control input to the plant (Fig. 16). The TDC buffer is updated only when the received control-side packet includes a more recent time stamp.
Figure 16. Time delay compensator structure
-
A benchmark networked control problem of a cart and inverted pendulum as the controlled plant is considered[49] where the random behavior of the network is simulated using the TrueTime toolbox, which is a Matlab/Simulink-based network simulator. In this example, the linear-quadratic control is used to design stabilizing controllers/observers.
The resulting state and control trajectories of the closed-loop system are shown in Figs. 17-19.
Figure 17. Inverted pendulum model
Figure 18. Initial state response of the cart-mounted inverted pendulum
Figure 19. Control input response
As mentioned, the TDC may add an extra delay to the plant input signal if the actually occurring time delay differs from the time stamps in
$\tau^v$ . This intentional delay will be large for$\tau^v = \tau_\beta = 0.9$ s. Fig. 20 shows the state results obtained for such a case. As expected, it took twice the time to settle down and hence the performance is considerably degraded.
Figure 20. Initial state response of the cart-mounted inverted pendulum with
$\tau^v = [0.9]$
6.1. Predictive control with time delay compensation
6.2. A basic result for small delay
6.3. Centralized predictive control
6.4. Decentralized predictive control
6.5. Improved predictive control
6.6. Model development
6.7. Signal diagram
6.8. Time-delay compensator
6.9. Simulation experiment
-
A typical design diagram for a networked system is shown in Fig. 21.
Figure 21. A possible control architecture for networked/distributed systems
Such systems consist of the same components as a conventional system, dynamic processes, sensors, controllers, etc., but there are many of them. Thus, there are many processes that need to be controlled. Their states are being sensed by many sensors. There is no one-to-one correspondence between processes and sensors. Many sensors might observe the same process and sensors may even switch from one process to another. The sensors may share information among them and also transmit it to the controllers. There might be many controllers with varying information and goals.
Consensus problems have a long history in computer science and form the foundation of the field of distributed computing[51]. Formal study of consensus problems in groups of experts originated in management science and statistics in 1960s (see [52] and references therein). The ideas of statistical consensus theory by DeGroot reappeared two decades later in aggregation of information with uncertainty obtained from multiple sensors[53] and medical experts[54]. This is known as sensor fusion and is an important application of modern consensus algorithms.
Distributed computation over networks has a tradition in systems and control theory starting with the pioneering work of [55-57] on asynchronous asymptotic agreement problem for distributed decision-making systems and parallel computing[58].
In networks of agents (or dynamic systems), "consensus" means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents. A "consensus algorithm" (or protocol) is an interaction rule that specifies the information exchange between an agent and all of its neighbors on the network. The term "nearest neighbors" is more commonly used in physics than "neighbors" when applied to particle/spin interactions over a lattice.
The theoretical framework for posing and solving consensus problems for networked dynamic systems was introduced in [59] and [60] building on the earlier work of [61, 62]. The study of the alignment problem involving reaching an agreement "without computing any objective functions" appeared in the work of [63]. Further theoretical extensions of this work were presented in [64] and [65] with a look toward treatment of directed information flow in networks as shown in Fig. 22. Note that Fig. 22(a) presents a network of integrator agents in which agent
$i$ receives the state$x_j$ of its neighbor, agent$j$ , if there is a link$(i, j)$ connecting the two nodes. On the other hand, Fig. 22(b) provides the block diagram for a network of interconnected dynamic systems all with identical transfer functions$P(s) = \frac{1}{s}$ . The collective networked system has a diagonal transfer function and is a multiple-input multiple-output (MIMO) linear system.
Figure 22. Two equivalent forms of consensus algorithms
Recent results on consensus of mobile agent server (MAS) can be found in [66, 67]. The common motivation behind the work in [55, 56, 60] is the rich history of consensus protocols in computer science[51], whereas Jadbabaie et al.[63] attempted to provide a formal analysis of emergence of alignment in the simplified model of flocking by Vicsek et al.[68] The setup in [60] was originally created with the vision of designing agent-based amorphous computers[69, 70] for collaborative information processing in networks. Later, [60] was used in development of flocking algorithms with guaranteed convergence and the capability to deal with obstacles and adversarial agents[71]. Recent studies on network-based
$H_\infty$ stabilization are included in [72-75].
-
This section provides a system-theoretic framework for addressing the problem of cooperative control of networked multi-vehicle systems using distributed controllers. On one hand, a multi-vehicle system represents a collection of decision-making agents that each has limited knowledge of both the environment and the state of the other agents. On the other hand, the vehicles can influence their own state and interact with their environment according to their dynamics which determines their behavior.
The design goal is to execute tasks cooperatively exercising both the decision-making and control capabilities of the vehicles. In real-life networked multi-vehicle systems, there are a number of limitations including limited sensing capabilities of the vehicles, network bandwidth limitations, as well as interruptions in communications due to packet-loss[76-78] and physical disruptions to the communication devices of the vehicle.
-
According to [60], a simple consensus algorithm to reach an agreement regarding the state of n integrator agents with dynamics
$ $$\dot{x}_i = u_i$$ $
can be expressed as an
$n$ -th-order linear system on a graph$ \begin{eqnarray} \dot{x}_i &=&\sum_{j \in {\bf N}_i} (x_j(t) - x_i(t)) + b_i \nonumber\\ x_i(0)&=&x_{io} \in {\bf R}, \, \, b_i(t)= 0. \label{Req11} \end{eqnarray} $
(24) The collective dynamics of the group of agents following protocol (24) can be written as
$ \begin{eqnarray} \dot{x} = - L x \label{Req12} \end{eqnarray} $
(25) where
$L = \{l_{ij}\}$ is the graph Laplacian of the network and its elements are defined as follows:$ \begin{eqnarray} L = \left \{ \begin{array}{c} -1, \, \, j \in {\bf N}_i \\ \frac{1}{|{\bf N}_i|}, \, \, j = i. \label{Req13} \end{array} \right. \end{eqnarray} $
(26) Here,
$|{\bf N}_i|$ denotes the number of neighbors of node$i$ (or out-degree of node$i$ ). Fig. 22 shows two equivalent forms of the consensus algorithm in (24) and (25) for agents with a scalar state. The role of the input bias$b$ in Fig. 22(b) is defined later. -
In the sequel, we present two basic consensus protocols that solve agreement problems in a network of continuous-time (CT) integrator agents with dynamics
$ \begin{equation} \dot{x}_i = u_i \label{Req20C} \end{equation} $
(27) or agents with discrete-time (DT) model
$ \begin{equation} {x}_i(k+1) = x_i(k) + \epsilon u_i \label{Req20D} \end{equation} $
(28) and step-size
$\epsilon > 0$ . We consider two scenarios.1) Linear consensus protocol:
$ {u_i} = \sum\limits_{{v_i} \in {{\bf{N}}_i}} {{a_{ij}}} ({x_j} - {x_i}). $
(29) 2) Linear time-delayed consensus protocol:
$ \begin{eqnarray} u_i = \sum_{v_i \in {\bf N}_i} a_{ij} [x_j(t-\tau_{ij})- x_i(t-\tau_{ij})].\label{Req23} \end{eqnarray} $
(30) Of interest is to perform analysis of protocols (29) and (30) for the aforementioned scenarios and show how in each case the consensus is asymptotically reached.
-
We consider a network of decision-making agents with dynamics of the form $\dot{x}_i = u_i$ who are interested in reaching a consensus via local communication with their neighbors on a graph $G=(V, E, A)$. By reaching a consensus, we mean asymptotically converging to a one-dimensional agreement space characterized by the following equation:
$ \begin{equation} x_1 = x_2 = \cdots = x_n. \label{Req200C} \end{equation} $
(31) This agreement space can be expressed as
$x = \alpha {\bf 1}$ where${\bf 1} = [1, 1, \cdots, 1]^{\rm T}$ and$\alpha \in {\bf R}$ is the collective decision of the group of agents.
8.1. Consensus in networks
8.2. Consensus protocols
8.3. Information consensus
-
A deterministic event-triggered strategy introduced in [79] and similar results on deterministic event--triggered feedback control have appeared in [80, 81]. In [79], the approach involves triggering of the control actuation whenever a certain error becomes large enough with respect to the norm of the state. The main results are derived assuming that some kinds of asymptotic stability hold for the nominal system and tools from perturbation analysis of nonlinear systems are used to analyze the convergence of the event-driven system. In particular, it is assumed that the nominal system is input-to-state stable (ISS) with respect to measurement errors.
In the sequel, we follow [82] to show that this approach is suitable for a class of cooperative control algorithms, namely those that can be reduced to a first order agreement problem[60], which has been proven to be ISS. Both the cases of centralized and decentralized event-triggered control are considered. In the first case, it is assumed that there exists a "global embedded microprocessor" that collects information about the whole system and triggers the feedback events for each agent. Results on distributed/decentralized event-triggered estimation/control are reported in [30-33].
In what follows, we will demonstrate how cooperative control schemes of multi-agent systems under event-triggered actuation update rules can be effectively analyzed. We focus on the stability of such schemes considering actuation updates are event-driven, depending on the ratio of a certain measurement error with respect to the norm of the state.
-
We consider
$N$ agents operating in$ {\bf R}$ , although the results are easily extendable to arbitrary dimensions. We assume that agents' motion obeys the single integrator model where$x_i \in {\bf R}$ denote the state of agent$i$ and$u_i \in {\bf R}$ denotes the control input for each agent. We further assume that each agent has "limited information" on the other group members. This means that, each agent is assigned a subset${\bf N}_i \subset \{1, \cdots, {\bf N}\}$ of the rest of the team, called agent is communication set, that includes the agents with which it can communicate.The limited communication capabilities can be encoded in terms of an undirected communication graph, see Section 3,
$G = \{V, E\}$ , which consists of a set of vertices$V = \{1, \cdots, {\bf N} \}$ indexed by the team members, and a set of edges,$E = \{(i, j) \in V \times |i \in {\bf N}_j\}$ containing pairs of vertices that represent inter-agent communication specifications. To facilitate further development, In the sequel, we assume that each agent is equipped with one or more embedded microprocessors which are responsible for collecting information from neighboring agents and triggering the control actuation for each individual agent at discrete time instants. -
In this case, we consider that for each
$i \in {\bf N}, t \geq 0$ , we define a state measurement error$e_i(t)$ . Denote the stack vector$e(t) = [e^{\rm T}_1(t), \cdots, e^{\rm T}_{N}(t)]^{\rm T}$ . The discrete time instants where the events are triggered are defined when a condition$f(e(t)) = 0$ holds. Then, the sequence of event-triggered executions is denoted by$ $$t_0, t_1, \cdots$$ $
and each
$t_i$ is defined by$f(e(t_i)) = 0, \textrm{for}~ i = 0, 1, \cdots$ .To the sequence of events
$t_0, t_1, \cdots$ , there corresponds a sequence of control updates$u(t_0), u(t_1), \cdots$ . Between control updates, the value of the input$u$ is held constant and equal to the last control update:$ \begin{equation} u (t) = u(t_i), \forall t \in [t_i, t_{i+1}]\label{Req25} \end{equation} $
(32) which means that the control law is piecewise constant between the event time
$t_0, t_1, \cdots$ .The centralized cooperative control (CCC) problem can then be stated as follows: drive control laws of the form (32) and event times
$t_0, t_1, \cdots$ that steer the single-integrator system to an agreement point (31). In what follows, it is assumed that the control law is actuated at instants triggered by events, and in particular, at time when the measurement error of the state variable reaches a certain threshold. Also, The control scheme is centralized and it is assumed that their exists a global microprocessor that collects information about the whole system and triggers the control actuation events for the whole team.Proceeding further, we let the state measurement error be defined by
$ \begin{equation} e(t) = x(t_i) - x(t), i = 0, 1, \cdots, \textrm{for} ~t \in [t_i, t_{i+1}). \label{Req26} \end{equation} $
(33) Indeed, the choice of
$t_i$ is linked by the function$f(\cdot)$ as explained shortly. The proposed control law in this case has the form (32) and is defined as the event-triggered analog of the ideal control law (29):$ \begin{equation} u(t) =- L x(t_i), t \in [t_i, t_{i+1}). \label{Req27} \end{equation} $
(34) The closed loop system is then given by
$ \begin{equation} \dot{x}_i (t) =-L x(t_i) =-L (x(t) + e(t)). \label{Req28} \end{equation} $
(35) Similarly to [60], the state vector
$x$ can be decomposed as$ \begin{eqnarray} x(t) = a(t) {\bf 1} + \delta(t)= \frac{1}{\pmb N} \sum_{i} x_i(t) + \delta(t) \label{Req29} \end{eqnarray} $
(36) where
$a(t)$ denotes the average of the agents' states and$\delta(t)$ is called the "disagreement vector" in [60] with$\sum_{j \in {\pmb N}_i} \delta_i = 0$ and${\bf 1} = [1, 1, 1, \cdots, 1]^{\rm T}$ . The vector$\delta$ is orthogonal to$ {\bf 1}$ and belongs to an$n-1$ -dimensional subspace called the disagreement eigen-space of$L$ provided that$G$ is strongly connected. Now, a simple computation shows that$ \begin{eqnarray} \dot{a}(t) = \frac{1}{\pmb N} \sum_{i} \dot{x}_i (t) = -\frac{1}{\pmb N} \sum_{i} \sum_{j \in {\pmb N}_i} (x_j - x_i) = 0 \label{Req30} \end{eqnarray} $
(37) so that
$ $$a(t) = a(0) = \frac{1}{\pmb N} \sum_{i} x_i(0) = a.$$ $
If follows from (35) and (37) that
$ $$\dot{x} (t) = \dot{\delta} (t) = - L (x(t) + e(t)) = - L (a(t) {\bf 1} + \delta(t) + e(t))$$ $
so that
$ \begin{equation} \dot{\delta} (t) = - L (\delta(t) + e(t)). \label{Req31} \end{equation} $
(38) Now, we examine the behavior of the disagreement dynamics (38). A candidate ISS Lyapunov function for this purpose would be
$V(\delta) = \frac{1}{2} ||\delta||^2$ . Simple algebra yields:$ \begin{eqnarray} &\dot{V}(\delta) = \delta^t \dot{\delta} = - \delta^t L (\delta(t) + e(t)) =\nonumber\\ & \!\!\!\!\!\!\!\!- \delta^t L \delta - \delta^t L e \leq\nonumber\\ &\qquad \quad - \lambda_2(G) ||\delta||^2 + ||\delta|| ||L|| ||e||. \label{Req40} \end{eqnarray} $
(39) For some scalar
$\sigma$ satisfying 0 <$\sigma < 1$ , let the following inequality holds$ \begin{eqnarray} ||e|| &\leq& \sigma \frac{\lambda_2(G) ||\delta||} {||L||}. \label{Req41} \end{eqnarray} $
(40) It yields
$ \begin{eqnarray} \dot{V}(\delta) &\leq& (\sigma - 1) \lambda_2(G) ||\delta||^2 < 0 \label{Req42} \end{eqnarray} $
(41) which suggests that the events are triggered when
$ \begin{eqnarray} f(e) = ||e|| - \sigma \frac{\lambda_2(G) ||\delta||} {||L||} = 0.\label{Req43} \end{eqnarray} $
(42) Thus, the event time are defined by
$f(e(t_i)) = 0, ~i = 0, 1, \cdots $ . At each$t_i$ , the control is updated according to (34) for all$t \in [t_i, t_{i+1})$ . Once the control task is executed, the error is reset to zero, since at that point we have$e(t_i) = x(t_i)-x(t_i) = 0$ for the specific event time so that (40) is enforced.Similarly to [79], this control policy attains a strictly positive lower bound on the inter-event time. This is proven in the following theorem.
Theorem 2.Consider system
$\dot{x} = u$ with the control law (34) and the triggering condition (42). Assume that$G$ is connected. Then, for any initial condition in$ {\bf R}^N, $ the inter-event time$\{t_{i+1} -t_i\}$ implicitly defined by the event rule (30) are lower bounded by a strictly positive time$\tau$ which is given by$ \begin{equation} \tau = \frac{\sigma \lambda_2(G)}{||L|| (||L|| + \sigma \lambda_2(G))}. \label{Req44} \end{equation} $
(43) Proof. Motivated by [77], we proceed to estimate the quantity
$ \frac{\textrm{d}}{\textrm{d}t} {\frac{|| e ||}{ || \delta ||}}$ .$ \begin{eqnarray} \frac{\rm d}{{\rm d}t} \frac{||e||}{||\delta||} &=& - \frac{e^t \dot{\delta}}{|| e || || \delta ||} - \frac{\delta^t \dot {\delta} || e ||}{|| \delta ||^2 ||\delta||} \leq \nonumber\\ && \frac{||e^t|| ||\dot {\delta}||}{|| e || || \delta||} + \frac{||\dot {\delta}||}{|| \delta ||} \frac{||e||}{||\delta||}= \nonumber\\ && \Big[1 + \frac{|| e ||} {|| \delta ||} \Big] \frac{|| \dot{\delta} ||}{|| \delta ||}\leq\nonumber\\ && \Big[1 + \frac{|| e ||} {|| \delta ||} \Big] \frac{{|| L ||} (|| \delta ||+ || e ||)}{|| \delta ||}=\nonumber\\ && {|| L ||} \Big[1 + \frac{|| e ||} {|| \delta ||} \Big]^2. \label{Req2LA} \end{eqnarray} $
(44) In terms of
$\xi = \frac{||e||} {||\delta||}$ , then relation (44) amounts to$\dot{\xi} \leq ||L|| (1 + \xi)^2$ . Hence,$\xi$ satisfies the bound$\xi \leq z(t, z_o)$ where$z(t, z_o)$ is the solution of$ $$\dot{z} = ||L|| (1 + z )^2, z(0, z_0) = z_0.$$ $
Hence, the inter-event time is bounded from below by the time
$\tau$ that satisfies$ $$z(\tau, 0) = \sigma \frac{\lambda_2(G)}{||L||}.$$ $
On the other hand, it is not difficult to show that
$ $$z(\tau, 0) = \frac{\tau ||L||}{1 - \tau ||L||}.$$ $
Combining both relations, we get (43) as desired.
$\square$ Corollary 1.Consider system
$\dot{x} = u$ with the control law (34) and the triggering condition (42). Assume that$G$ is connected. Then, all agents converge to their initial average, i.e.,$ \mathop {\lim }\limits_{t \to \infty } {x_i} = a = \frac{1}{N}\sum\limits_i {{x_i}} ,\forall i \in {\bf{N}}. $
Proof.It follows from Theorem 2 and the closed-loop switched system does not exhibit Zeno behavior. Moreover,
$V (\delta) > 0, $ and$\dot{V} (\delta) < 0 $ in continuous evolution intervals. By Theorem 1 in [83], we have that$\lim_{t \rightarrow \infty} \delta (t) = 0$ , which in turn is equivalent to$\lim_{t \rightarrow \infty} x_i = a = \frac{1}{N} \sum_{i} x_i, \forall i \in {\bf N} $ .$\square$ -
As discussed in the previous section, the approach was centralized, in the sense that agents had to be aware of the global measurement error
$e$ in order to enforce the constraint (40). In what follows, we examine the decentralized version of the problem in which there is a separate sequence of events$ $$t^i_0, t^i_1, \cdots$$ $
defined for agent
$i$ according to$ $$f_i(e_i(t^i_k)); \{e_j(t^i_k)|j \in {\bf N}_k\}, ~~ {\rm for }~~ i \in {\bf N} \textrm{ and } ~ i =0, 1, \cdots.$$ $
Hence, the separate condition
$ $$f_i \big(e_i(t), \{e_j(t)|j \in {\bf N}_i\}\big) = 0$$ $
triggers the events for agent
$k \in {\bf N}$ .Following parallel development to the centralized case, the measurement error for agent
$i$ is defined as$ \begin{eqnarray} e_i(t) = x_i(t^i_k) - x_i(t), t \in [t^i_k, t^i_{k+1}). \label{Req45} \end{eqnarray} $
(45) The control strategy for agent
$i$ is now given by$ {u_i} = - \sum\limits_{j \in {{\bf{N}}_i}} [ {x_i}(t_k^i) - {x_j}(t_{{k^*}}^j)] $
(46) where
$k^* = \textrm{arg} \min_{l \in {\bf N}:t^i_k \leq t^j_C} \big \{t^i_k -t^j_l \big\}$ . Hence, each agent takes into account the last update value of each of its neighbors in its control law. The control law for$i$ is updated both at its own event time$t^i_0, t^i_1, \cdots$ , as well as at the event time of its neighbors$t^j_0, t^j_1, \cdots, j \in {\bf N}_i$ .$ \begin{array}{l} {{\dot x}_i}(t) = - \sum\nolimits_{j \in {{\bf{N}}_i}} {\left[ {{x_i}(t_k^i) - {x_j}(t_{{k^*}}^j)} \right]} = \\ - \sum\nolimits_{j \in {{\bf{N}}_i}} {\left[ {{x_i}(t) - {x_j}(t)} \right]} - \\ \sum\nolimits_{j \in {{\bf{N}}_i}} {\left[ {{e_i}(t) - {e_j}(t)} \right]} . \end{array} $
(47) Using again the decomposition
$x(t) = a(t) {\bf 1} + \delta(t)$ , we have$\dot{a}(t)=0 $ so that$ $$\dot{x} (t) = \dot{\delta} (t) = - L (x(t) + e(t)) = - L (\delta(t) + e(t))$$ $
as before. To examine the stability, consider
$V(\delta) = \frac{1}{2} ||\delta||^2 = \frac{1}{2} \sum_{i}||\delta_i||^2$ . Simple algebra yields$ \begin{eqnarray} \dot{V}(\delta)&=&\delta^t \dot{\delta} = - \delta^t L (\delta(t) + e(t))= \nonumber\\ && -\delta^t L \delta - \delta^t L e \leq \nonumber\\ &&-\lambda_2(G) ||\delta||^2 - \delta^t L e= \nonumber\\ && -\lambda_2(G) \sum_{i} \delta_i^2 - \nonumber\\ && \sum_{i} \sum_{j \in {\bf N}_i} \delta_i (e_i - e_j).\label{Req48} \end{eqnarray} $
(48) Therefore,
$ \begin{eqnarray*} \dot{V}(\delta) &\leq& - \lambda_2(G) \sum_{i} \delta_i^2 + \sum_{i} \sum_{j \in {\bf N}_i} |\delta_i| |(e_i - e_j)| - \nonumber\\ && \lambda_2(G) \sum_{i} \Bigg [\delta_i^2- \bigg|\frac{\delta_i}{\lambda_2(G)} \bigg| \sum_{j \in {\bf N}_i} |(e_i-e_j)| \Bigg]. \end{eqnarray*} $
For some scalars
$\sigma_j$ satisfying$0 < \sigma_j < 1$ , let the following inequality holds$ \begin{eqnarray} \sum_{j \in {\bf N}_i} (|e_i| + |e_j|) &\leq& \sigma_i \lambda_2(G) |\delta_i|. \label{Req49} \end{eqnarray} $
(49) We get
$ \begin{eqnarray} \sigma_i \delta^2_i &\geq& \bigg|\frac{\delta_i}{\lambda_2(G)} \bigg| \sum_{j \in {\bf N}_i} (|e_i| + |e_j|) \label{Req50} \end{eqnarray} $
(50) so that
$ \begin{eqnarray} \dot{V}(\delta) &\leq& - \lambda_2(G) \sum_{i} \big ( \delta^2_i - \sigma_i \delta_i \big )= \nonumber\\ && \lambda_2(G) \sum_{i} (1 - \sigma_i) \delta^2_i \leq 0. \label{Req51} \end{eqnarray} $
(51) Hence, for each
$i$ , an event is triggered when$ \begin{eqnarray} f_i(e_i, \{e_j|j \in {\bf N}_i \}) \!=\!\sum_{j \in {\bf N}_i} (|e_i| \!+\! |e_j|) \! -\! \sigma_i \lambda_2(G) |\delta_i| \!=\! 0 . \label{Req52} \end{eqnarray} $
(52) The update rule (52) holds at the event time
$t^i_k$ corresponding to agent$i$ :$ $$f_i\big(e_i(t^i_k), \{e_j(t^i_k)|j \in {\bf N}_i \}\big) = 0, k = 0, 1, \cdots, i \in {\bf N}_i.$$ $
At an event time
$t^i_k$ , we have$e_i(t^i_k) = x_i(t^i_k) -x_i(t^i_k) = 0$ , and since$ \sum\limits_{j \in {{\bf{N}}_i}} {(|{e_i}| + |{e_j}|)} \le |{e_j}|,\forall t \ge 0 $
the condition (49) is enforced.
Remark 2.Although (52) is verified by agent
$i$ only based its own and neighboring agents information, it does require some global information, in the sense that agents need to know the values of$\lambda_2(G)$ in order to check (52). We note that this condition has been relaxed in our later work[84], using a different event-triggered formulation. In particular, in the condition derived in [84], each agent only needs to know the sum of relative states and the number of its neighbors to implement it. Hence, knowledge of the aforementioned global parameters is relaxed.For recent survey papers on NCS and event-triggered NCS, see [73, 74]. Theorem 3 regarding the inter-event time holds in the decentralized case.
Theorem 3.Consider system
$\dot{x}_i = u_i, i \in {\bf N} = \{0, 1, \cdots \}$ with the control law (46) and the triggering condition (52). Assume that$G$ is connected. Then, for any initial condition and any time$t \geq 0$ , there exists at least one agent$k \in {\bf N}$ for which the next inter-event interval is strictly positive.Proof.We assume that (52) holds for all
$i \in {\bf N} = \{0, 1, \cdots \}$ at time$t$ . If it does not hold, then continuous evolution is possible since at least one agent can still let its absolute measurement error increase without resetting (45). Hence, assume that at$t$ , all errors are reset to zero. We will show that there exists at least one$k \in {\bf N}$ such that its next inter-event interval is bounded from below by a certain time$\tau_D > 0$ .It can be recalled that the term
$\sum_{j \in {\bf N}_i} (|e_i| + |e_j|)$ is the$i$ -th row of the vector$({ \Delta} + A) e $ , where${ \Delta}$ is the degree matrix of$G$ and$A$ its adjacency matrix. Let$ k = {\rm{arg}}\mathop {\max }\limits_i |{\delta _i}| $
be the maximum element of
$||\delta_i||, $ we have$ \begin{eqnarray*} \frac{\sum_{j \in {\bf N}_i} (|e_i| + |e_j|)}{||\delta||} &\leq& \frac{||{ \Delta} + A|| ||e||}{||\delta||} \Rightarrow \nonumber\\ \frac{\sum_{j \in {\bf N}_k} (|e_k| + |e_j|)}{N|\delta_k|} &\leq& \frac{||{ \Delta} + A|| ||e||}{||\delta||} \Rightarrow \\ \frac{\sum_{j \in {\bf N}_k} (|e_k| + |e_j|)}{|\delta_k|} &\leq& N \frac{||{ \Delta} + A|| ||e||}{||\delta|| }, \nonumber\\ \forall i \in {\bf N}. \end{eqnarray*} $
In follows from the proof of Theorem 2 and (52), the next interevent interval of agent
$k$ is bounded from below by a time$\tau_D$ that satisfies$ $$N \frac{||{ \Delta} + A|| \tau_D ||L||}{\ - \tau_D ||D||} = \sigma_k \lambda_2(G)$$ $
so that
$ $$\tau_D = \frac{\sigma_k \lambda_2(G)}{||L|| || N || { \Delta} + \sigma_k \lambda_2(G)}.$$ $
$\square$ Remark 3.Please note that the result of Theorem 3 is more conservative than the centralized case, since it only guarantees that there are no accumulation points and continuous evolution is viable at all time instants. However, no lower bound similar to the one of Theorem 2 is provided. In that sense, the result on the decentralized case is rather preliminary and more work will be devoted to providing a strictly positive inter-event time of the overall switched system in the future.
On the other hand, the result of Theorem 3 guarantees that the overall switched system does not exhibit Zeno behavior, i.e., there are no infinite switches in finite time. Using now La Salles invariance principle for hybrid systems[83], the following convergence result is straightforward.
9.1. System description
9.2. Centralized event-triggered cooperative control
9.3. Decentralized event-triggered cooperative control
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In this paper, cloud control systems' architectures have been presented by means of networked control systems. In general, several researchers attempted to exemplify a proper definition of cloud control systems. However, the majority of them consider cloud control systems as a combination of networked control technologies as well as cloud computing. Model predictive control has been extremely studied and broadly investigated for large scale complex system especially in industrial applications. In this work, we have introduced networked predictive control methodology in distributed and wide area application as attempted to represent an initial notions of cloud control systems.
Future work could be carried out along the following avenues:
1) The master-slave approach to remote control systems seems to be a productive research direction. One should further pursue investigations into the issues of synchronization of the time clocks, structure of the master, control data processing, structure of the slave(s), and pilot-scale real-time implementations.
2) The industrial automation area deserves collaborative work to build and experiment with robust wide-plant control architecture making use of cloud computing and intelligent control methods.
3) The class of model-predictive control methods deserves additional for networked control systems with particular emphasis on industrial systems.
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