Distributed control system

Abstract

A system includes a plurality of control devices that respectively control the states of a plurality of apparatuses and are connected to each other via communication lines. When each of the control devices determines a state target value of its own apparatus using the current state indicator value of the own apparatus, and the distributed controller input which is a function of the state indicator value of an apparatus adjacent to the own apparatus and the state indicator value of the own apparatus, the control gain which adjusts contribution of the distributed controller input to the state target value is determined based on a communication delay time between the control devices.

Claims

1. A distributed control system comprising: a plurality of controllers configured to respectively control states of a plurality of apparatuses; and a communication network composed of a plurality of communication lines that connect the plurality of controllers, wherein: a state indicator value indicating a selected state is measured in each of the apparatuses, and is transmitted from a controller of a corresponding one of the apparatuses via the communication lines to a controller of an apparatus adjacent to the corresponding apparatus, from among the plurality of apparatuses connected to the communication lines; the controller of each of the apparatuses is configured to control the state of an own apparatus from among the plurality of apparatuses by referring to the state indicator value of the own apparatus and the state indicator value of the adjacent apparatus, such that the state indicator value of the own apparatus matches a state target value that is determined according to a control protocol of a multi-agent system, the own apparatus controlling the selected state by oneself; each of the plurality of controllers determines, according to the control protocol, the state target value of the own apparatus, using a current state indicator value of the own apparatus, and a distributed controller input that is a function of the state indicator value of the adjacent apparatus and the state indicator value of the own apparatus; each of the plurality of controllers is configured to determine a control gain which adjusts contribution of the distributed controller input to the state target value based on at least one of a communication delay time when the state indicator value of the adjacent apparatus is transmitted from the controller of the adjacent apparatus to each of the plurality of controllers and the communication delay time when the state indicator value of the own apparatus is transmitted from each of the plurality of controllers to the controller of the adjacent apparatus; each of the plurality of controllers is configured to set, when the distributed controller input is a sum of functions of the state indicator values of a plurality of apparatuses adjacent to the own apparatus and the state indicator value of the own apparatus, the control gain for each of functions corresponding to the controllers of the adjacent apparatuses connected to each of the plurality of controllers; and the control gain set for each of the functions corresponding to the controllers of the adjacent apparatuses connected to each of the plurality of controllers is indicated as G.sub.ij, G.sub.ij is given as G.sub.ij=Γ.sup.max(Δij,Δji), where an integer Γ smaller than 1, the communication delay time Δ.sub.ij when the state indicator value of the adjacent apparatus is transmitted from the controller of the adjacent apparatus to each of the plurality of controllers, and the communication delay time Δ.sub.ji when the state indicator value of the own apparatus is transmitted from each of the plurality of controllers to the controller of the adjacent apparatus.

2. The system according to claim 1, wherein each of the plurality of controllers is configured to determine the control gain based on a longer time from among the communication delay time when the state indicator value of the adjacent apparatus is transmitted from the controller of the adjacent apparatus to each of the plurality of controllers and the communication delay time when the state indicator value of the own apparatus is transmitted from each of the plurality of controllers to the controller of the adjacent apparatus.

3. The system according to claim 1, wherein each of the plurality of controllers is configured to reduce the control gain when the communication delay time is a longer delay time compared to when the communication delay time is a shorter delay time, wherein the longer delay time is longer than the shorter delay time.

4. The system according to claim 1, wherein: each of the plurality of controllers transmits the state indicator value of the own apparatus to the controller of the adjacent apparatus; the controller is configured to transmit a latest state indicator value of the own apparatus after the state indicator value of the own apparatus is transmitted and arrives at the controller of the adjacent apparatus; and the function used by each of the plurality of controllers is a function of the latest state indicator value of the adjacent apparatus received from the controller of the adjacent apparatus and the latest state indicator value of the own apparatus received by the controller of the adjacent apparatus.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Features, advantages, and technical and industrial significance of exemplary embodiments of the disclosure will be described below with reference to the accompanying drawings, in which like signs denote like elements, and wherein:

(2) FIG. 1A is a diagram schematically illustrating a configuration of a distributed control system according to an embodiment;

(3) FIG. 1B is a block diagram illustrating a configuration of a control device (agent) of each apparatus in the system according to the embodiment;

(4) FIG. 1C illustrates an example of a communication delay time observed in a communication network used in an actual distributed control system;

(5) FIG. 1D is a histogram illustrating a frequency for each length of communication delay times that occur between agents of the distributed control system;

(6) FIG. 2 illustrates time charts describing each timing of measurement of a state indicator value, transmission and reception of the state indicator value, and calculation of a state target value when the state indicator value is intermittently transmitted from each agent (a transmission-side) to its adjacent agent (a reception-side) in the distributed control system as illustrated in FIGS. 1A to 1D (intermittent transmission correction). Here, it is assumed that a measurement time matches a calculation time;

(7) FIG. 3 is a diagram schematically illustrating transmission and reception of state indicator values x.sub.i, x.sub.j and communication delay times Δ.sub.ij, Δ.sub.ji between agents i, j, which are adjacent to each other, and a sequence (a flow) of values referred to in the state indicator value x.sub.i term, x.sub.j term in the distributed controllers u.sub.i, u.sub.j, in a distributed control system as illustrated in FIGS. 1A to 1D;

(8) FIG. 4 is an example of the communication delay time, which changes randomly for each transmission and is used in the simulation for calculating the time change of the state indicator value of each agent when the communication delay time changes randomly, in the distribution control system illustrated in FIGS. 1A to 1D;

(9) FIG. 5A is a simulation for calculating the time change of the state indicator value of each agent, obtained according to a control protocol (reference correction) that compensates for the communication delay of the state indicator value between agents according to the teachings of the present embodiment in the distributed control system illustrated in FIGS. 1A to 1D. In the calculation simulation illustrated in the figure, it is assumed that the initial values of the state indicator values of the agents are randomly given such that the average value thereof becomes 10, and a measurement (sampling) cycle and an operation cycle of the state indicator value of each agent are 1 second (one step). Here, the measurement time is assumed to match the calculation time. The communication delay time occurring in transmission of the state indicator value between the agents is set to occur randomly from 0 to 5 seconds. In addition, each agent is set to transmit the next state indicator value after receiving the reception notification of the transmitted state indicator value from the agent which is the transmission destination (after confirming that the transmitted state indicator value has arrived) (intermittent transmission correction). It is assumed that the state indicator value referred to in the distributed controller u.sub.i of each agent is the latest state indicator value of the adjacent apparatus received from the control device of the adjacent apparatus and the latest state indicator value of the own apparatus received by the control device of the adjacent apparatus. In this case, an overall control gain γ.sub.i of the distributed controller with respect to the state target value is one, and a constant coefficient Γ, which gives control gain G.sub.ij determined for each adjacent agent, is one;

(10) FIG. 5B is a simulation when the overall control gain γ.sub.i of the distributed controller with respect to the state target value of each agent is 0.9, and the constant coefficient Γ, which gives the control gain G.sub.ij determined for each adjacent agent, is one, in the calculation simulation of FIG. 5A;

(11) FIG. 6A is a simulation for calculating the time change of the state indicator value of each agent, obtained according to a control protocol (reference correction) that compensates for the communication delay of the state indicator value between agents according to the teachings of the present embodiment in the distributed control system illustrated in FIGS. 1A to 1D, similar to FIGS. 5A and 5B. In this case, the overall control gain γ.sub.i of the distributed controller with respect to the state target value is 0.5, and the constant coefficient Γ, which gives the control gain G.sub.ij determined for each adjacent agent is, one (without control gain correction);

(12) FIG. 6B is a simulation when the overall control gain γ.sub.i of the distributed controller with respect to the state target value of each agent is 1.0, and the constant coefficient F, which gives the control gain G.sub.ij determined for each adjacent agent, is 0.9 (with the control gain correction), in the calculation simulation of FIG. 6A;

(13) FIG. 7 illustrates time charts describing each timing of measurement of a state indicator value, reception of the state indicator value, and calculation of a state target value in the transmission-side agent (the adjacent apparatus) and the reception-side agent (the own apparatus) when the transmission of the state indicator value is sequentially performed according to the existing control protocol in the distributed control system as illustrated in FIGS. 1A to 1D (intermittent transmission correction). Here it is assumed that a measurement time match a calculation time;

(14) FIG. 8A is a simulation for calculating the time change of the state indicator value of each agent, obtained according to the existing control protocol (without correction) described in FIG. 7, in the distributed control system as illustrated in FIGS. 1A to 1D. It is the calculation simulation in which the measurement time (sampling time) interval is 1.0 second and there is no communication delay time;

(15) FIG. 8B is a simulation when the measurement time (sampling time) interval is 1.0 second and the communication delay time of 1.25 seconds occurs symmetrically in the signal transmission between the agent 6 and the agents 2, 5, and 7 in FIGS. 1A to 1D, in the calculation simulation in FIG. 8A. The initial values of the state indicator values of the agents are given randomly such that the average value thereof becomes 10. “Convergence determination” indicates a point of time when a difference between the state indicator values of the agents falls within ±0.01%; and

(16) FIG. 8C is a simulation when the communication delay time occurring in transmission of the state indicator value between the agents is set to occur randomly from 0 to 5 seconds, in the calculation simulation illustrated in FIGS. 8A and 8B. Here, each agent is set to transmit the state indicator value after receiving the reception notification of the transmitted state indicator value from the agent which is the transmission destination (after confirming that the transmitted state indicator value has arrived) (intermittent transmission correction). It is assumed that the state indicator value referred to in the distributed controller u.sub.i of each agent is the latest state indicator value of the adjacent apparatus received from the control device of the adjacent apparatus and the latest state indicator value of the own apparatus obtained in the calculation time (without reference correction). Further, in this case, an overall control gain γ.sub.i of the distributed controller with respect to the state target value is set to 0.5.

DETAILED DESCRIPTION OF EMBODIMENTS

(17) Embodiments of the present disclosure will be described in detail with reference to the drawings hereinafter. In the drawings, the same reference numerals indicate the same parts. In addition, in the following embodiments, a case in which the state indicator value of each agent of a distributed control system is controlled by the average consensus control of a multi-agent system will be described as an example. However, configurations of “an intermittent transmission correction” and “a control gain correction” to be described below are also applied to other control forms, such as consensus control other than average consensus control, covering control, and distributed optimization control, such that convergence of the state indicator value of each agent can be further improved. It should be understood that such cases also fall within the scope of the present disclosure.

(18) With reference to FIG. 1A, in a distributed control system 1 according to the present embodiment, a communication network is formed by control devices E.sub.s (agents) of a plurality of apparatuses connected to each other via a communication line I.sub.e. Each agent E.sub.s is configured to acquire a state indicator value indicating a selected state of an apparatus (adjacent apparatus) adjacent to each agent E.sub.s via the communication network. In such a system, the apparatus may be any apparatus of which an operation state is controlled, for example, an energy source, a mobile object, various pieces of manufacturing machinery, various sensors, and the like, as already described in the “Technical Field”. Frequently, the selected state of each apparatus is any measurable physical quantity and/or rate of change or rate of fluctuation thereof, as described in the “Technical Field”, and the selected state may be an arbitrarily selected state. The communication network may be configured in any type, such as wired communication, wireless communication, and optical communication. According to a control protocol of the multi-agent system, each agent E.sub.s is controlled such that its state indicator value in the system matches a control target value that has been determined using the state indicator values of other agents acquired via the communication network. Particularly, in the examples of FIGS. 1A to 1D, the system is composed of an undirected graph in which all the agents are connected to each other. In this case, when the consensus control is performed as the control protocol, the “average consensus control”, in which the consensus value is the average value of initial values of the state indicator values of all the agents, is performed. In the present specification, the average consensus control is performed as the consensus control, as in the illustrated examples. However, the present embodiment may be applied to cases in which leader and follower consensus control or other consensus control is performed depending on a configuration of graphs in the system. It should be understood that such a case also falls within the scope of the present disclosure.

(19) In the above system, the control device (i, j, and the like, are numerals of the agents) of each agent i may be typically composed of a primary controller that controls an object to be controlled, that is, a selected state and an output of an apparatus, and a secondary controller that determines a target value of the selected state of the apparatus, as schematically illustrated in FIG. 1B. The control device of each agent may typically be a computer device, and usually is provided with a CPU, a storage device, and an input/output device (I/O) connected to each other via a two-way common bus (not shown) and the operation of each part in the device is performed by executing a program on the CPU. More specifically, the primary controller receives an indicator value (a state indicator value) x.sub.i[k] of a selected state measured in the controlled object and a target value (a state target value) x.sub.i[k+1] of a state transmitted by the secondary controller, and controls the state of the controlled object such that a difference e between the state target value and the state indicator value becomes zero (state feedback control). Alternatively, the primary controller may control an output y.sub.i of the controlled object such that a difference ε between the output y.sub.i of the controlled object and a target value yt.sub.i of the output of the controlled object, which is determined by using a transmit function G from the state target value received from the secondary controller, becomes zero. Output feedback control and servo control do not have to be required when the output is controlled by the selected state. Moreover, control processing of the state and/or the output in the controlled object may be performed according to the selected state and a type of the output. A detailed type of the control processing may be appropriately determined by those skilled in the art according to the selected state. On the other hand, the secondary controller determines the state target value x.sub.i of its controlled object using the state indicator value x.sub.i of its own controlled object and the state indicator value x.sub.j of the controlled object of an adjacent agent connected via a communication line in a manner to be described below in detail, such that the state indicator value x.sub.i of its own controlled object matches or converges to the consensus value.

(20) In the distributed control system as illustrated in FIG. 1A, as described above, the state indicator value indicating the selected state of each agent is transmitted as a signal to the adjacent agent via the communication line I.sub.e. In other words, each agent receives, from the adjacent agent, the signal of the state indicator value thereof via the communication line I.sub.e. Regarding the signal communication of the state indicator value between the agents, in an actual communication network, as already described in the “Summary”, an occurrence of the delay in the signal communication is inevitable since a finite time is required until the signal of the state indicator value from the adjacent agent arrives at each agent, that is, from the time when the adjacent agent transmits the signal of the state indicator value to the time when each agent receives the signal due to various factors. In addition, generally, the delays in a two-way signal communication between any two agents are not necessarily symmetric, and the time required for signal communication from an agent i to an agent j (a communication delay time Δ.sub.ji) does not always match the communication delay time Δ.sub.ij, which is the time required for signal communication from the agent j to the agent i. As illustrated in FIGS. 1C and 1D, it is observed that the communication delay time randomly varies in length within a certain range.

(21) However, as described in the “Summary”, in arithmetic processing of the control protocol of an existing multi-agent system using a differential equation of the above equation (1), the communication delay time occurring in the communication network is not taken into consideration. In addition, when the differential equation of the above equation (1) is used as it is for control in an environment in which the communication delay time occurs in the communication network, phenomena such as the state indicator value of each agent failing to converge to the consensus value, and the like, an error occurring in the consensus value, and the like, and a fluctuation in the consensus value, and the like are observed. In other words, when the arithmetic processing of the control protocol of the existing multi-agent system is applied as it is to an actual distributed control system as described above, a situation in which stable control cannot be achieved may occur. Therefore, in the present embodiment, as to be described below in detail, the configuration of the secondary controller of the control device of each agent is improved such that the signal communication and the arithmetic processing are performed using a new control protocol that can stably converge the state indicator value of each agent to the consensus value, and the like even in an environment in which the communication delay time occurs in the communication network, especially even when the communication delays between agents are not symmetric.

(22) Arithmetic Processing of Consensus Control in Existing Multi-Agent System

(23) Before description of the control configuration according to the present embodiment, a phenomenon that occurs in the existing control configuration will be briefly described. With reference to FIG. 7, in each agent of the system, typically, the state indicator value indicating the selected state of each apparatus (each controlled object) is sequentially measured at every predetermined time interval (a measurement time or a sampling time) which is any time interval that may be set by any sensor, and the measured state indicator values are transmitted to the adjacent agent as a signal via the communication line, for determination of the state indicator value of the apparatus of the adjacent agent. In the consensus control of the multi-agent system, in each agent (the secondary controller), in the case of the existing system, generally, the state target value x.sub.i[k+1] indicating the state to be taken by the own apparatus at a next measurement time is calculated by the equations (1), (2) using the state indicator value x.sub.i[k] measured in the own apparatus and the state indicator value x.sub.j[k] measured in the adjacent agent, and the calculated value is given to an adder for the feedback control of the state of the primary controller. Here, assuming that the state indicator value x.sub.j[k] of the adjacent agent arrives at each agent instantaneously, the state indicator values of all agents in the system converge to the consensus value (in this case, the average value of the initial state indicator values of all agents, as illustrated in equation (3)) by calculating the state target value according to the equations (1), (2). Here, it is assumed that the measurement time and the calculation time of each agent are substantially the same, and the same applies hereinafter. FIG. 8A illustrates an example of a simulation for calculating a time change until the state indicator values of all agents converge to the consensus value. In the actual distributed control system, as described above, since the state target value is calculated by the secondary controller and then servo control of the state of the controlled object is performed by the primary controller such that the state indicator value of the controlled object matches the state target value, the state target value may not match the state indicator value. However, since the time change of the state indicator value of each agent in the drawings (see FIGS. 8A to 8C, FIGS. 5A and 5B, and FIGS. 6A and 6B) attached to the present specification is the calculation simulation, the state indicator values are illustrated as matching the state target values.

(24) However, as described above, since a finite time is required for the signal communication of the state indicator value in the actual communication network, as illustrated in FIG. 7, the time when the signal (⋄) of the state indicator value of the agent (the transmission-side agent) of the transmission-side arrives at the receiver of the agent (the reception-side agent) of the reception-side is delayed by the amount of a communication delay time Δ from the measurement time of the state indicator value. Therefore, when the reception-side agent sequentially calculates the state target value using the latest state indicator value without considering the communication delay, the calculation is performed according to the following equation:
x.sub.i[k+1]=x.sub.i[k]+Σ.sub.j∈N.sub.i.sup.na.sub.ij(x.sub.j[k−δk]−x.sub.i[k])  (8)

(25) Here, k−δk is the measurement time immediately before the point in time traced back by the amount of the communication delay time Δ from the current time k. δk is the number of sampling time intervals corresponding to the sum of the communication delay time Δ and the standby time Δ.sub.w after reception (see FIG. 7). Then, as represented by an arrow a in FIG. 7, a difference occurs in the measurement time of the state indicator value between the transmission-side agent and the reception-side agent, in the distributed controller (the second term on the right side of the equation (7)). In this case, the state indicator values of all the agents converge to a certain consensus value, phenomena are observed, such as a phenomenon in which when the state target value is calculated, the state indicator values of all the agents converge to a certain consensus value but the consensus value deviates from the expected consensus value (a case in which the communication delay time is not zero but equal to or less than the sampling time interval), a phenomenon in which the state indicator values of all the agents do not converge (a case in which the communication delay time exceeds the sampling time interval, as illustrated in FIG. 8B), or a phenomenon in which the state indicator values of all the agents converge to a certain consensus value, but the consensus value oscillates over time (not shown).

(26) In addition, according to another aspect, in measurement of the state indicator value of each agent, the measurement time of the state indicator value of each agent is simultaneously recorded, and data of the measurement time is transmitted together with the state indicator value to the adjacent agent. In a case in which the state target value is calculated using an equation which is modified such that the measurement time of the state indicator value of the transmission-side agent matches that of the reception agent, in the distributed controller (the second term on the right side of the equation (1)), as represented by an arrow b in FIG. 7, that is, the following equation (a time stamp correction):
x.sub.i[k+1]=x.sub.i[k]+Σ.sub.j∈N.sub.i.sup.na.sub.ij(x.sub.j[k−δk]−x.sub.i[k−δk])  (9)
when the communication delay time is equal to or less than the sampling time interval (not zero), the state indicator values of all the agents converge to the expected consensus value, but when the communication delay time slightly exceeds the sampling time interval, it is observed that the state indicator values of all the agents do not even show a tendency to converge (see Japanese Patent Application No. 2019-010040).

(27) Improvement of Arithmetic Processing of Consensus Control of Multi-Agent System

(28) (A) Intermittent Transmission Correction

(29) As described above, in an environment in which a finite delay time occurs in the signal communication between the agents in the distributed control system, the consensus control might not be able to be stably achieved depending on the condition of the delay of the signal communication in the control protocol using the widely known existing equation (2) (or equations (8) and (9)). Therefore, as described in the “Summary”, the inventors of the present disclosure have proposed in Japanese Patent Application No. 2019-010040, regarding processing in which each agent transmits the state indicator value to the adjacent agent, a configuration for improving the convergence of the state indicator value of each agent by changing the control protocol so as to intermittently transmit the state indicator values instead of transmitting all the state indicator values measured in each agent (intermittent transmission correction), as described below.

(30) With reference to FIG. 2, specifically, (1) once transmitting, as a transmission-side agent, a state indicator value to the adjacent agent (reception-side agent), each agent stands by transmission processing even when the state indicator value is sequentially measured. When each agent receives, from the adjacent agent of the transmission destination, the notification that the transmitted state indicator value has arrived thereto, the agent transmits the latest measured state indicator value in response to the notification (see FIG. 2). In other words, each agent does not transmit the state indicator values measured after the transmission of the state indicator value until the transmission completion notification is received. In addition, after the transmission of the state indicator value, when the reception notification from the reception-side agent does not arrive even after a predetermined time, which may be set to any time, each agent may transmit the latest measured state indicator value at that point in time (see “TO” in FIG. 2, time-out processing). (2) When receiving, as a reception-side agent, the state indicator value transmitted from the adjacent agent (transmission-side agent), each agent transmits the reception notification to the adjacent agent of the transmission source (see FIG. 2). Since the time required for transmission of the reception notification from the reception-side agent to the transmission-side agent is generally shorter than the time or the sampling time interval required for the communication of the state indicator value, the time width from transmission to reception of the reception notification is omitted from the figure.

(31) As described above, when each agent, as a transmission-side agent, changes the type of transmission of its state indicator value, each agent, as a reception-side agent, uses the latest state indicator value that has arrived from the transmission source as the state indicator value of the adjacent agent in the distributed controller (corresponding to the second term on the right side of the equation (1)) for calculation of the state target value (see FIG. 2). In other words, the equation (1) for calculation of the state target value is modified as follows:
x.sub.i[k+1]=x.sub.i[k]+Σ.sub.j∈N.sub.i.sup.na.sub.ij(x.sub.j[k.sub.aj]−x.sub.i[k])  (10)

(32) Here, k.sub.aj is the measurement time of the state indicator value transmitted from the transmission-side agent j, and is expressed as k.sub.aj=l.sub.aj−δk . . . (10a), δk=Δ.sub.s+Δ.sub.ij+Δ.sub.r . . . (10b), using a first measurement time l.sub.aj (<k [current time]) after the reception of the state indicator value by the reception-side agent i. Here, Δ.sub.s is the standby time from the measurement time k.sub.a immediately before the transmission time of the transmission-side agent to the transmission time, and Δ.sub.r is the standby time from when the state indicator value of the transmission-side agent arrives at the reception-side agent to the calculation time. In addition, Δ.sub.ij is the communication delay time, that is, the time required for signal transmission from the transmission-side agent j to the reception-side agent i. In a configuration in which the reception notification is transmitted from the reception-side agent to the transmission-side agent, Δ.sub.ij may include the time until the reception notification arrives. According to this protocol, once receiving the state indicator value of the transmission-side agent (adjacent agent), the reception-side agent continuously uses the received state indicator value in the distributed controller until receiving the next state indicator value of the transmission-side agent. Moreover, the state indicator value of the adjacent agent used in the distributed controller of each agent may be updated for each adjacent agent. For example, measurement times k.sub.aj of the state indicator values of agents 2, 5, and 7 used in the distributed controller of an agent 6 in FIG. 1A may be different.

(33) Thus, in a case in which the state target value is calculated by applying the intermittent transmission correction using the above equation (10), as illustrated in the result of the calculation simulation in the upper part of FIG. 8C, in time change of the state indicator values, even when the communication delay time changes randomly within a range exceeding the sampling time interval, it is possible to greatly improve the convergence of the state indicator values of all agents to the consensus value. Further, the example of FIG. 8C is a result obtained by multiplying the second term on the right side of the equation (10), which is the distributed controller input, by the control gain γi=0.5 so as to accelerate the convergence of the state indicator value. However, in the case of the average consensus control, simply by applying the above intermittent transmission correction, as illustrated in the lower part of FIG. 8C, it is not possible to resolve an issue in which the average value of the state indicator values of all agents is changed and the consensus value to which the state indicator values converge does not match an expected value (in this case, initial values of the state indicator values of all agents), that is, an issue of a deviation of the consensus value. Indeed, as disclosed in Japanese Patent Application No. 2019-010040, it is observed that as the communication delay time becomes longer, the deviation of the consensus value and the convergence time also become longer.

(34) (B) Reference Correction

(35) As described above, when the state indicator value is controlled using the above equations (1), (2), or (8) to (10) in an environment in which a finite delay time in signal communication between the agents in the distributed control system occurs, the average value of the state indicator values of all agents which are connected to form the undirected graph, is not maintained. As a result, a phenomenon occurs in which even when the state indicator values converge to the consensus value by applying, for example, the intermittent transmission correction, the consensus value deviates from the expected average value of the initial values of the state indicator values of all agents. On the other hand, when the communication delay time in signal transmission between the agents in the system does not occur, for any two agents i, j, a difference between terms on the two agents in the respective distributed controllers u.sub.i, u.sub.j are (x.sub.j[k]−x.sub.i[k]), (x.sub.i[k]−x.sub.j[k]), respectively. In other words, the state indicator values referred to in the terms are the same x.sub.i[k], x.sub.j[k]. In short, since the state indicator value referred to in the distributed controller of each agent in the system is common to that of the adjacent agent, the average value of the state indicator values of all agents is maintained. However, in the existing control protocol of the state indicator value of the agent, when the communication delay time occurs in the signal transmission between the agents in the system, the state indicator value of the own apparatus referred to in the distributed controller of each agent may be different from that transmitted to the adjacent agent. Accordingly, the state indicator value of the own apparatus does not necessarily match the state indicator value of the own apparatus referred to as the state indicator value of an adjacent agent in the distributed controller of the adjacent agent, and as a result, the average value of the state indicator values of all the agents is not maintained. Therefore, according to the present embodiment, the control protocol is modified such that the state indicator value of the own apparatus referred to in the distributed controller of each agent is the same as that transmitted to the adjacent agent. As such, the average value of the state indicator values of all agents can be maintained. Hereinafter, this modification of the control protocol is referred to as a “reference correction”.

(36) Theoretically, the distributed controller u.sub.i in the equation (2) is modified as follows:
u.sub.i[k]=Σ.sub.j∈N.sub.i.sup.na.sub.ij(x.sub.j[k−Δ.sub.ij[k]]−x.sub.i[k−Δ.sub.ji[k]]  (11)

(37) Here, Δ.sub.ij [k] is the communication delay time required for signal transmission from the agent j, which is adjacent to the agent i, to the agent i, and x.sub.j[k−Δ.sub.ij[k]] is the state indicator value of the agent j at a time traced back by the amount of Δ.sub.ij[k] from the current time k, which is received from the agent i, and Δ.sub.ji[k] is the communication delay time required for signal transmission from the agent i to the agent j, and x.sub.i[k−Δ.sub.ji[k]] is the state indicator value of the agent i at a time traced back by the amount of Δ.sub.ji from the current time k, which is received by the agent j. Alternatively, Δ.sub.ij[k] and Δ.sub.ji[k] do not have to be constant but may change every moment.

(38) According to the above equation (11), it is proved that the average value of the state indicator values of all the agents which are connected to form the undirected graph is maintained as below. First, the equation (11) is expressed as follows by performing z-transform:

(39) $\begin{array}{cc}{U}_i\left[z\right]=\underset{j\in N_i}{\overset{n}{.Math.}}{a}_{\mathrm{ij}}\left(\frac{1}{z^{{\Delta }_{\mathrm{ij}}\left[k\right]}}{Q}_j\left[z\right]-\frac{1}{z^{{\Delta }_{\mathrm{ji}}\left[k\right]}}{Q}_i\left[z\right]\right)& \left(12\right)\end{array}$

(40) Here, U.sub.i[z], Q.sub.i[z], and Q.sub.j[z] are z-transforms of u.sub.i[k], x.sub.i[k], and x.sub.j[k], respectively. Therefore, the distributed controller U of all agents is expressed as U[z]=−L.sup.d.sub.a[k]Q[z] . . . (13), using Graph Laplacian L.sup.d.sub.a[k] and the vector Q[z] having z-transforms of the state indicator values of all agents as a component. Here, the Graph Laplacian L.sup.d.sub.a[k] is as follows:

(41) $\begin{array}{cc}{L}_a^d\left[k\right]=\left[\begin{array}{cccc}\underset{j\in N_1}{.Math.}\frac{1}{z^{{\Delta }_{j1}\left[k\right]}}{a}_{1j}& -\frac{1}{z^{{\Delta }_{12}\left[k\right]}}{a}_{12}& \mathrm{.Math.}& -\frac{1}{z^{{\Delta }_{1n}\left[k\right]}}{a}_{1n}\\ -\frac{1}{z^{{\Delta }_{21}\left[k\right]}}{a}_{21}& \underset{j\in N_2}{.Math.}\frac{1}{z^{{\Delta }_{j2}\left[k\right]}}{a}_{2j}& \ddots & -\frac{1}{z^{{\Delta }_{2n}\left[k\right]}}{a}_{2n}\\ \mathrm{.Math.}& \ddots & \ddots & \mathrm{.Math.}\\ -\frac{1}{z^{{\Delta }_{n1}\left[k\right]}}{a}_{n1}& -\frac{1}{z^{{\Delta }_{n2}\left[k\right]}}{a}_{n2}& \mathrm{.Math.}& \underset{j\in N_n}{.Math.}\frac{1}{z^{{\Delta }_{\mathrm{jn}}\left[k\right]}}{a}_{\mathrm{nj}}\end{array}\right]& \left(14\right)\end{array}$

(42) Then, when the Graph Laplacian L.sup.d.sub.a[k] is multiplied by a row vector 1.sup.T.sub.n in which all components are one, from the left, 1.sup.T.sub.nL.sup.d.sub.a[k]=0.sup.T . . . (15) is obtained (0.sup.T is a row vector in which all components are zero). As such, the change in the sum of the state indicator values of all agents can be zero, and the average value of the state indicator values of all agents is maintained.

(43) In the above modification of the control protocol, when the above-described intermittent transmission correction is applied to the state indicator value of each agent, the distributed controller u.sub.i (of a time area) is expressed as follows:

(44) $\begin{array}{cc}{u}_i\left[k\right]=\underset{j\in N_i}{\overset{n}{.Math.}}{a}_{\mathrm{ij}}\left(x_j\left[k_{\mathrm{aj}}\right]-x_i\left[k_{\mathrm{bi}}\right]\right)& \left(16\right)\\ a_{\mathrm{ij}}=\{\begin{array}{cc}\frac{1}{1+\max \left(.Math.N_i.Math.,.Math.N_j.Math.\right)}& :j\in N_i\\ 0& :j.Math.N_i\end{array}& \left(16a\right)\end{array}$

(45) Here, k.sub.aj is the measurement time of the latest state indicator value transmitted from the adjacent agent j and received by the agent i, and k.sub.bi is the measurement time of the latest state indicator value transmitted from the adjacent agent i and received by the agent j. In addition, when the intermittent transmission correction is applied in the reference correction in which the value referred to as the state indicator value of own apparatus in the distributed controller of each agent is set to that transmitted to the adjacent agent, it should be understood that the state indicator value x.sub.i of the own apparatus referred to in the distributed controller u.sub.i of the agent i is also a value intermittently transmitted to and received by the adjacent agent from among the state indicator values measured in time series. Moreover, in this regard, each agent cannot notice whether the state indicator value transmitted to the adjacent agent has arrived at the adjacent agent just by transmitting the state indicator value. Therefore, according to the present embodiment, each agent may use the transmitted state indicator value in the distributed controller after receiving, from the adjacent agent of the transmission destination, the notification that the transmitted state indicator value has arrived at the adjacent agent. In other words, the state indicator value of the own apparatus referred to by each agent in the distributed controller may be the state indicator value of the own apparatus, which is confirmed to have been received by the adjacent agent. In addition, for this purpose, each agent is appropriately configured to, upon receiving the state indicator value from the adjacent agent, notify the adjacent agent of the transmission source of that fact.

(46) In the above reference correction, the respective measurement times of the state indicator value of the own apparatus and the state indicator value of the adjacent apparatus referred to in the distributed controller of each agent do not have to match each other. Therefore, it should be understood that an advantageous effect in which the average value of the state indicator values of all agents in the system is maintained by the reference correction can be achieved even when the time delay of the signal communication between any two agents is not symmetric.

(47) (C) Control Gain Correction

(48) As described above, in an environment in which a finite delay time occurs in the signal communication between the agents in the distributed control system, it is possible to improve the convergence of the state indicator value of each agent to a certain degree, using the intermittent transmission correction. Moreover, as disclosed in Japanese Patent Application No. 2019-010040, when the communication delay time between the agents becomes longer, oscillations occur in the state indicator value due to oscillations of the calculated value by the distributed controller, and it is difficult for the state indicator values to converge. Therefore, it has been found that the convergence of the state indicator value can be further improved by multiplying the distributed controller by a gain γ.sub.i (0<γ<1) as the following equation (17), so as to reduce the contribution of the distributed controller to the target value of the state indicator value:
x.sub.i[k+1]=x.sub.i[k]+T.sub.s.Math.γ.sub.i.Math.u.sub.i[k]  (17)

(49) In this regard, as described above, since the oscillations of the calculated value by the distributed controller depends on the length of the communication delay time, in a general system in which the communication delay time may randomly fluctuate between the agents, it is considered that the oscillations of components (for example, for agents i and j, (x.sub.j[k.sub.aj]−x.sub.i[k.sub.bi]) and (x.sub.i[k.sub.bi]−x.sub.j[k.sub.aj])) of the distributed controller, which are associated with agents having a long communication delay time, become larger. Therefore, the convergence of the state indicator value can be further improved using the control gain which is determined based on the communication delay time so as to adjust the contribution of the components of the distributed controller associated with the agents to the state target value according to the length of the communication delay time between the agents.

(50) Specifically, the distributed controller u.sub.i may be modified as follows:
u.sub.i[k]=Σ.sub.j∈N.sub.i.sup.nG.sub.ij.Math.a.sub.ij(x.sub.j[k.sub.aj]−x.sub.i[k.sub.bi])  (18)

(51) Here, G.sub.ij is the control gain set for each difference corresponding to the control device of the adjacent apparatus connected to each control device, and it may be given by the following equation:
G.sub.ij=g(Δ.sub.ij,Δ.sub.ji)  (19)

(52) Here, g(Δ.sub.ij, Δ.sub.ji) may be a function of a first communication delay time Δ.sub.ij in the transmission of the state indicator value from the agent j to the agent i and a second communication time Δ.sub.ji in the transmission of the state indicator value from the agent i to the agent j. The communication delay time Δ.sub.ij and the communication delay time Δ.sub.ji are generally time variables as described above. Moreover, as described above, generally, since the longer the communication delay time Δ.sub.ij or the communication delay time Δ.sub.ji is, the larger the oscillations of the components of the distributed controller becomes, function g may be a function of which the size is decreased as the communication delay time Δ.sub.ij, or the communication delay time Δ.sub.ji becomes longer, or a monotonically decreasing function. In addition, when the communication delay time between the agents is not symmetric, function g may be determined according to the longer one of the two-way communication delay times, in which case, function g may be a function of max(Δ.sub.ij, Δ.sub.ji). Further, as described in the above description of the reference correction, when there is a request to maintain the average value of the state indicator values of all agents in the system, the contribution of the distributed controller to the state indicator value of the agent i and the contribution of the distributed controller to the state indicator value of the agent j need to be equal to each other. Therefore, the control gain may be set as follows:
G.sub.ij=G.sub.ji  (20)

(53) Assuming that the above requirement is satisfied, the control gain G.sub.ij may be given as, for example, G.sub.ij=Γ.sup.max(Δij,Δji) . . . (21), using an integer Γ smaller than 1, the first communication delay time Δ.sub.ij, and the second communication delay time Δ.sub.ji. Alternatively, the control gain G.sub.ij may be G.sub.ij=1/{c.Math.max(Δ.sub.ij, Δ.sub.ji)} . . . (22). Here, c is a positive coefficient.

(54) In the above configuration, the two-way communication delay time (Δ.sub.ij, Δ.sub.ji) between the agents may be acquired in each agent by any method. According to one embodiment, each agent records a measurement time tm at the time of measuring the state indicator value, and transmits the state indicator value to the transmission destination agent together with the measurement time tm. In addition, the reception time tr when the transmission destination agent receives the state indicator value is recorded, and the communication delay time to the transmission destination agent may be calculated by subtracting the measurement time tm from the reception time tr. Here, the calculated communication delay time may be used for determination of the control gain in the agent of the transmission destination. Then, the communication delay time may be transmitted from the agent which receives the state indicator value to each agent which transmits the state indicator value, together with the reception notification of the state indicator value, and may be used for determination of the control gain in each agent.

(55) (D) Communication Sequence

(56) FIG. 3 illustrates a sequence of measurement, communication, and reference of a state indicator value between any two adjacent agents i, j in the system when the above intermittent transmission correction, reference correction, and control gain correction are applied. In addition, in the illustrated example, the length of time required for signal transmission is schematically represented for the purpose of description, and may be different from the actual length of time.

(57) With reference to FIG. 3, first, it is assumed that state indicator values x.sub.ik, x.sub.jk are measured in time series in agents i, j, respectively (k=1, 2, . . . ). Then, the agents i, j respectively transmit the measured state indicator values x.sub.ik, x.sub.jk, together with their measurement times, to the agents j, i. When the transmitted state indicator values x.sub.ik, x.sub.ik are received by the agents j, i, the reception time is recorded in each agent, the measurement times of the corresponding state indicator values x.sub.ik, x.sub.jk are subtracted from the respective reception time, and the communication delay time Δ.sub.jik is calculated in the agent j and the communication delay time Δ.sub.ijk is calculated in the agent i. Moreover, the state indicator value x.sub.ik received by the agent j is referred to as the x.sub.i term (the first term of the difference) in the distributed controller u.sub.j, and the state indicator value x.sub.jk received by the agent i is referred to as the x.sub.j term (the first term of the difference in equation (18)) in the distributed controller u.sub.i. Further, the agents j, i that have received the state indicator values x.sub.ik, x.sub.ik transmit the reception notification to the transmission source agents i, j together with the communication delay times Δ.sub.jik, Δ.sub.ijk. When the agents i, j receive the reception notification, the agents i, j refer to the transmitted state indicator values x.sub.ik, x.sub.jk corresponding to the reception notification as a term of the own state indicator value (the second term of the difference of the equation (18)) in the distributed controller. Subsequently, the agents i, j transmit, to the agents j, i, the latest measured state indicator values x.sub.ik, x.sub.jk (the values may be values measured until the reception notification is received or values measured immediately after the reception notification is received), and repeat the above operation. Thus, the state indicator value referred to in the distributed controller of each agent is the latest value received for the state indicator value of the adjacent agent, and the latest value, of which the reception notification is received after transmission, for the state indicator value of the own agent. In other words, the distributed controller uses the latest value from among the received values until receiving a new value or receiving a new reception notification.

(58) For example, assuming that the state indicator value x.sub.j4, which are measured at k=4 in agent j and transmitted from the agent j, is received by the agent i after k=7 in the agent i, the state indicator value x.sub.j4 is referred to as the x.sub.j term of the distributed controller u.sub.i from that point of time, the communication delay time Δ.sub.ij4, which is a difference between the measurement time and the reception time of the state indicator value x.sub.j4, is calculated and used for determination of the control gain G.sub.ij. Then, the state indicator value x.sub.j4 and the communication delay time Δ.sub.ij4 are used until the next state indicator value arrives from the agent j. Then, the reception notification of the state indicator value x.sub.j4 together with the communication delay time Δ.sub.ij4 is transmitted from the agent i to the agent j, and when the agent j receives the notification, the state indicator value x.sub.j4 is referred to as the x.sub.j term of the distributed controller u.sub.j of the agent j from that point of time, and the communication delay time Δ.sub.ij4 is used for determination of the control gain G.sub.ji. Subsequently, the latest state indicator value x.sub.j9 in the agent j is transmitted to the agent i. In the agent j, the state indicator value x.sub.j4 and the communication delay time Δ.sub.ij4 are used in the calculation of the distributed controller until the reception notification of the state indicator value x.sub.j9 arrives.

(59) (E) Calculation Simulation

(60) In the system illustrated in FIGS. 1A to 1D, an advantageous effect of the above reference correction and control gain correction has been confirmed by the calculation simulation (see FIGS. 5A to 6B). In the calculation, the initial value is given to each agent, the intermittent transmission correction is applied in transmission of the state indicator value between the agents, the reference correction is applied in the calculation of the distributed controller, and the state indicator value of each agent is calculated using the equation (17) according to the control protocol in which the control gain of the distributed controller is appropriately set in the calculation of the target value of the state indicator value. Further, the communication delay time between the agents is given randomly within the range of 0 to 5 seconds in two ways. The distribution controller of each agent is calculated by the equation (18) obtained by applying the control gain G.sub.ij to the equation (16). The control gain G.sub.ij is calculated by equation (21).

(61) First, FIG. 5A illustrates an example in which the intermittent transmission correction and the reference correction are applied. In this example, since γi=1.0, Γ=1.0 are set, the control gain correction is not applied. As can be understood from FIG. 5A, the state indicator value of each agent changes in a convergent direction, but a convergence condition (in which a difference between the state indicator values of each agent falls within ±0.01%) is not achieved within the test time (120 seconds: 120 steps). However, as illustrated in the lower part of FIG. 5A, since the reference correction is applied, it is confirmed that the average value of the state indicator values of the agents is maintained during the calculation. FIGS. 5B and 6A illustrate examples in which the intermittent transmission correction and the reference correction are applied, and further, the gains γi for reducing the contribution of the entire distributed controller are reduced to 0.9 and 0.5, respectively. With reference to FIGS. 5B and 6A, since the reference correction is applied to FIGS. 5B and 6A, it is confirmed that the average value of the state indicator values of the agents is maintained during the calculation, and the state indicator value of each agent changes in a more convergent direction, compared to the example of FIG. 5A. Moreover, in the example of FIG. 5B, the convergence condition is not achieved within the test time (120 seconds), but in the example of FIG. 6A, the convergence condition is achieved in about 100 seconds.

(62) On the other hand, in the example of FIG. 6B, the gain γ for reducing the contribution of the entire distributed controller is returned to 1.0, Γ is set to 0.9, and the control gain correction for applying the control gain for reducing the contribution of the corresponding component associated with the agent in the distributed controller is performed according to the communication delay time between the agents. In this case, as can be understood from FIG. 6B, the average value of the state indicator values of the agents is maintained, and the state indicator value of each agent changes quickly in a more convergent direction compared to FIGS. 5B and 6A. In the example of FIG. 6B, the convergence condition is achieved in about 50 seconds. As illustrated in FIG. 6B, the state indicator value of each agent converges earlier when the control gain is applied according to the communication delay time between the agents compared to when the gain of the entire distributed controller is adjusted. The reason for this is considered that the state indicator value of the agent associated with components with the short communication delay time can relatively quickly approach the consensus value by making the contribution of components having a long communication delay time relatively smaller and making the contribution of components having a short communication delay time relatively larger.

(63) From the results of the above calculation simulation, it is confirmed that the average value of the state indicator values of the agents is maintained by the control protocol to which the above reference correction is applied and the state indicator value of each agent can converge to the expected consensus value. In addition, it is confirmed that the convergence of the state indicator value of each agent can be accelerated by performing the correction for applying the control gain determined according to the communication delay time between the agents. Further, it should be understood that the above advantageous effect is achieved even when the delay in signal transmission between the agents is not symmetric.

(64) In addition, the above control gain correction is not limited to the illustrated average consensus control, but may be also applied when a communication delay time occurs in other control forms, for example, consensus control, covering control, and distributed optimization control. Further, it should be understood that the effect of compensating for a worsening of the convergence of the state indicator value of each agent caused by the communication delay time can be obtained.

(65) Although the above description has been made related to the embodiments of the present disclosure, many modifications and changes can be easily made by those skilled in the art. It will be clear that the present disclosure is not limited only to the above-exemplified embodiments, but can be applied to various devices without departing from the concept of the disclosure.