Influence of Time Delays on Network-Controlled Diesel Generator Performance

Data transmission over the Ethernet network does not occur instantly, but with some non-constant delay. The time delay τ is the time interval between the moment of sending a data packet (in this case, the error value – the difference between the required and actual values of the controlled parameter), and the moment they are received and fed to the input of the remote PID controller. The time delay of a signal while passing from sender to recipient consists of several parts: the time required for processing at each node, the time required for transfer, as well as the time required for signal propagation. Each stage may differ in duration, which is determined by the network integrity, as well as by various uncontrolled random reasons. In this paper, one assumes that the time delay τ is a continuous random variable having a certain distribution law. Statistical methods of analysis allow the empirical build of a distribution curve for a given random variable. Once it is built, one may find an approximate time delay as the area under the distribution curve is within a given range. Unless one manages to analytically describe the probability density of a given distribution, then the problem of finding out the probability of a delay hitting a given range can be solved using a specified integral:

$P(\alpha <\tau <\beta )=\int\limits_{a}^{b}{f(\tau )d\tau }$,

where, τ is the time delay, f(τ) is the probability density of a random variable τ. The control object covers both the field winding of the synchronous generator and a diesel unit equipped with a speed regulator. The diesel generator is controlled via the network interface by the automation regulator, where the control algorithm is implemented in software. Figure 1 shows a block diagram of a distributed control system along with a remote regulator.

The time-delay line introduces a signal delay to a variable T, which depends both on the sampling interval ν (the distance in units of time between which measurements are taken) and the intensity of requests λ:

$T=f(\nu ,\lambda )$.

The intensity of requests in this case is defined as the number of data packets sent over the network within a certain time interval, for instance 1 sec.

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Figure 1. Structure of a distributed control system equipped with a PID regulator

The paper [26] describes the reasons why digital control sampling intervals cannot be considered the same. It is also substantiated here that in this event, the digital control system's behavior would be rather difficult to analyze analytically, thus the simulation modeling is acceptable. Modeling of the control systems operation is done by replacing a continuous regulator with a digital one, which is implemented in software as a digital state machine in a micro regulator, as well as by varying the time intervals between the variation of the state machine states. Models of the synchronous generator and diesel engine excitation system developed in the Matlab Simulink dataset [27, 28] are presented in Figure 2.

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Figure 2. Models of synchronous generator excitation system and diesel engine speed control systems

The reference value of the excitation voltage is set at the input V_ref of the adder, and the actual value of the generator voltage is input to the input V_sdq, after which a positive sequence voltage is formed, which is fed to the input of the low-pass filter, which is represented by a 1^st-order aperiodic link with a time constant of 0.02 s, which corresponds to the period of mains voltage. Signal filtering is necessary to eliminate the influence of high-frequency interference on the generator excitation control process. At the output of the adder, an error signal is generated, which is fed to the input of the excitation controller. Physically, the excitation of a synchronous generator occurs by changing the voltage on the excitation winding, which is an inductance with a negligibly small active resistance and this part of the model is described by an aperiodic link. The saturation block limits the maximum excitation value of the synchronous generator.

The input of the diesel unit speed control system receives the reference speed value and the actually measured one. At the output of the adder, an error signal is generated, equal to the difference between the required and actual values of the diesel engine speed. This signal is fed to the input of the speed controller. At the output of the regulator, a control signal is generated, which is fed to the actuator, which controls the position of the diesel fuel rail. A DC motor is used as an actuator.

A simplified model of a diesel unit is represented by an aperiodic link and an integrator connected in series with it, which considers the inertia of the masses. In addition, the model also has a delay link, which takes into account the delay in the speed control loop, due to the fact that when the position of the fuel rail changes, the combustible mixture enters the cylinders and ignites not immediately, but with some delay, which is due to the design features of diesel engines. The mechanical power value at the output of the diesel model is obtained by multiplying the actual value of the diesel revolutions by the torque.

When replacing continuous regulators with digital ones, the models presented in Figure 2 will be hybrid [29] and characterized by a variable delay value in control signals. That is, the control system will have a digital regulator along with a variable sampling period. The transfer function of the synchronous generator excitation system is as follows:

${{W}_{EX}}(s)=\frac{{{K}_{p}}}{{{T}_{d}}s+1}=\frac{10}{0.01s+1}$

where, K_p is the gain factor, T_d is the integration time constant.

To obtain an adequate model of a discrete system, it is necessary to discretize a continuous system. The most common sampling method is sampling using a zero-order extrapolator (Zero order hold device). The zero-order extrapolator generates a continuous signal by fixing the previous discrete value before the impact of the next, i.e.

$u(t)=u[k],k{{T}_{s}}\le t\le (k+1){{T}_{s}}$

The transfer function of the discrete model:

$H(z)=(1-{{z}^{-1}})Z\left\{ \frac{H(p)}{p} \right\}$

where, Z means the operation of the Z-transform of the original continuous system transition function.

After performing the Z-transform, the transfer function W_EX(z) of the digital controller of the generator excitation system is:

${{W}_{EX}}(z)=Z\left\{ \frac{1-{{e}^{-s{{T}_{s}}}}}{s}\frac{{{K}_{p}}}{1+{{T}_{d}}s} \right\}=\frac{b{{z}^{-1}}}{1-a{{z}^{-1}}}=\frac{8.65{{z}^{-1}}}{1-0.135{{z}^{-1}}}$

where, the coefficients a and b are calculated using equations:

$a={{e}^{-\frac{{{T}_{s}}}{{{T}_{d}}}}}$

$b={{K}_{p}}(1-a)$

The implementation of a generator excitation digital control system in the form of the digital state machine states diagram is presented in Figure 3(a).

Figure 3(b) shows transition graphs of a digital state machine that implements the given control law alongside the modeling of the control signal time delay.

3a.png

(a) Speed control state machine of diesel unit

3b.png

(b) Control state machine for the excitation system

Figure 3. Transition graphs of digital state machines for the generator excitation

The transfer function of the diesel unit control system W_DG(s) is as follows:

${{W}_{DG}}(s)=\frac{K({{T}_{0}}s+1)}{{{T}_{1}}{{s}^{2}}+{{T}_{2}}s+1}=\frac{40(0.2s+1)}{0.0002{{s}^{2}}+0.01s+1}$

where, K is the gain factor, T₁, T₂, T₃ are the regulator time constants.

Upon performing the Z-transform, the transfer function of the digital regulator at a sampling period of 0.01 seconds is as follows:

${{W}_{DG}}(z)=40\cdot \frac{7.437-7.06{{z}^{-1}}}{1-1.229{{z}^{-1}}+0.606{{z}^{-2}}}$

When the sampling period is reduced by 10 times, the transfer function of the diesel speed regulator is as follows:

${{W}_{DG}}(z)=40\cdot \frac{0.9771-0.9722{{z}^{-1}}}{1-1.946{{z}^{-1}}+0.9512{{z}^{-2}}}$

In Figure 4 a model for researching the synchronous generator (SG) excitation system regulator is shown. The sampling frequency for the digital state machine is set using the clock signal generator Clk.

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Figure 4. A model for researching a digital regulator of the synchronous generator excitation system

In Figure 5 a model for researching of the diesel engine control system is shown.

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Figure 5. A model for researching of the diesel engine digital control system

The average delay variable τ is calculated by the equation:

$\overline{\tau }=\frac{1}{n}\sum\limits_{i=1}^{k}{{{\tau }_{i}}{{n}_{i}}}$

The standard deviation S (a measure of the amount of variation or dispersion of a set of values) and the coefficient of variation V_R (a statistical measure of the relative dispersion of data points in a data series around the mean) are taken as the variance features of this parameter. The variables are calculated by the following formulas:

$S=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{k}{{{({{\tau }_{i}}-\overline{\tau })}^{2}}{{n}_{i}}}}$

$V_R=\left|\frac{S}{\bar{\tau}}\right| \cdot 100 \%$

If the coefficient of variation is less than 30%, it means that the sample is quite compact, i.e., the time delay varies within a small range.

The key regulator operation quality indicators, that are calculated from the transition feature, are overcorrection, oscillation, as well as integral assessment [30]. In this paper an integral pointer of the control quality criterion A is used to assess the control quality:

$A=\int\limits_{0}^{\infty }{{{(h(t)-{{h}_{st}}(t))}^{2}}}dt$

where, h_st(t) is the parameter value in the steady state, after the end of transients, h(t) deviation of the parameter from the value in the steady state.

The value of the quality criterion integral indicator of the generator excitation control at the discretization period fluctuations different values are presented in Tables 1 and 2.

Table 1. The value of the integral indicator of the the SG excitation system control quality criterion

Table 2. The value of the integral indicator of the diesel engine control quality criterion

The dependence of the integral pointer of the control quality criterion on the delay in transferring information and control datasets towards the remote regulator of the synchronous generator excitation system is presented in Figure 6(a). The dependence of the integral control quality criterion on the delay in transferring information and control datasets towards the remote diesel unit speed regulator is presented in Figure 6(b).

As one can see from the diagrams, an increase in the delay variable deteriorates control quality. Therefore, to control automation objects in real-time mode one should minimize delays in transferring datasets via the network. This messages, optimizing thread processing algorithms within network nodes, as well as dividing threads into service classes to specify the most priority ones [31]. The paper is focused on reducing the time of information and control datasets to be transferred via the network.

The presence of an Ethernet network in the control loop, which can be considered as a delay line, introduces a signal delay by an amount that depends on the sampling interval and the intensity of requests (i.e., network congestion):

$T=f(\nu ,\lambda )$

The developed Matlab model for researching diesel generator control processes over the network includes a minimal set of objects that are interconnected and allow external control (Figure 7).

The generator is represented by a model of a synchronous machine with a damper winding. Machine parameters are set in the system of absolute units. The power of the generator is 400 kW, the linear voltage on the stator is 380V. The generator has 2 pairs of poles (nominal frequency of rotation is 1500 rpm). Two computers connected via a local Ethernet network are used to simulate the process.

6a.png

(a) Synchronous generator excitation system

6b.png

(b) Diesel unit speed

Figure 6. Dependencies of integral control quality criterion values on the time delays

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Figure 7. Matlab model of the studied system

In Figure 8(a), a model of a remote controller is presented, which contains the controller itself, as well as blocks for organizing communication over a local network. In Figure 8(b), the simulation results are presented.

8a.png

(a) Remote PID controller model

8b.png

(b) Simulation results

Figure 8. Results of simulation two transition process curves

When creating the model, the CommStr6.5 Blockset libraries were used. This model uses blocks: "LAN Setup", "LAN Write" and "LAN Read". The "STR WriteBin" and "STR RdBin" blocks were used to format the transmitted data. The "LAN Setup" module performs LAN connection setup. In the settings window that opens, it is necessary to set the port number and IP address of the remote computer. The output of this module is a handle to an open port used by other blocks. The "LAN Write" block is used for data transfer. The input of the block receives the data that will be transmitted. In the dialog box for setting the parameters of the "STR WriteBin" block, the mask is specified, according to which the packet for transmission will be generated.

As a result of simulation, two transition process curves were obtained. Curve (2) in Figure 8 is a graph of the transition process when modeling a remote PID controller.

When analyzing the network of a distributed control system, the network topology and communication channels bandwidth are well-known. The objective is to reduce the delay time T of message flow λ_ij under the specified restrictions. Sum of rates in i-th node therefore equals to the following:

$\sum\limits_{j}{{{\lambda }_{ij}}}={{\lambda }^{i}}$

where, $i,j=\overline{1,n}$, $i\ne j,$ n is a number of nodes.

Let's note

${{\lambda }_{k}}=\sum\limits_{\begin{smallmatrix} i=1 \\ j=1\end{smallmatrix}}^{a}{{{\lambda }_{ij}}}k$

as the flow rate within k-th channel. The delay time is expressed by the following formula [2]:

$T=\sum\limits_{k=1}^{m}{\frac{{{\lambda }_{k}}}{U}\cdot \left( \frac{1}{\mu {{C}_{k}}-{{\lambda }_{k}}} \right)},$    (1)

where, m is a number of channels; U is an aggregate rate of receiving messages; C_k is a bandwidth of k-th channel; 1/μ is an average message length (in bits).

Let's put the value λ_k into Eq. (1):

$T=\sum\limits_{k=1}^{m}{\frac{\sum\limits_{i,j=1}^{n}{{{\lambda }_{ijk}}}}{U}}\cdot \left( \frac{1}{\mu {{C}_{k}}-\sum\limits_{i,j=1}^{n}{{{\lambda }_{ijk}}}} \right)$

Let's provide a restriction:

$\sum\limits_{k\in {{S}_{ij}}}{{{\lambda }_{ijk}}}-{{\lambda }_{ij}}=0$

where, S_ij is any cross-section of the graph that divides i-th and j-th nodes. Where λ_ijk >0. By using a well-known method of Lagrange undetermined multipliers, we have the Lagrange function as follows:

${{T}^{'}}=\sum\limits_{k=1}^{m}{\frac{\sum\limits_{i,j=1}^{n}{{{\lambda }_{ijk}}}}{U}}\cdot \left( \frac{1}{\mu {{C}_{k}}-\sum\limits_{i,j=1}^{n}{{{\lambda }_{ijk}}}} \right)+\sum\limits_{i,j=1,i\ne j}^{n}{{}}\sum\limits_{{{S}_{ij}}}{{{\gamma }_{Sij}}}\sum\limits_{K\in {{S}_{ij}}}{{{\lambda }_{ijk}}}-{{\lambda }_{ij}},$    (2)

where, γ_Sij is the Lagrange multiplier.

To search for the optimum, we should differentiate an Eq. (2) and equate it to zero.

The equation is solved with regard to λ_ijk when using a numerical method. Another approach to solving the problem is as follows. A dataset switching network is considered. The network topology is described by a matrix C sized n×n, the elements of which have the variables of the communication channels bandwidth between i-th and j-th nodes. A matrix is symmetric. It has zero nodes and non-negative non-diagonal elements. If C_ij – C_ji > 0, then there is a communication channel having bandwidth C_ij bit/s that binds i-th and j-th network nodes. Message flows are assumed to be of Poisson along with rates of r_i messages per second. The average length of messages (datasets) within the network is 1/μ. The average message flow rate is known for each node. When transferring datasets via the communication channel between two nodes the average delay time in queue for processing is determined by the following expression:

${{\bar{t}}_{wait}}=\frac{\mu }{\mu C-u}\cdot \frac{1}{\mu C}$,

where, u is a message flow rate [1/s]; C is an effective bandwidth of the communication channel [bps]; 1/μ is an average message length [bits].

If i-th node is connected by communication channels with other nodes, then the problem of an optimal flow distribution is formulated as follows. Let's note by u_ij a message flow rate from i-th node to j-th one in [1/s]. Here u_ij = 0 if C_ij (no connection). A stationary mode can only exist within the network given the solution of the following equality:

${{r}_{i}}+\sum\limits_{\begin{smallmatrix} j=1 \\ j\ne i\end{smallmatrix}}^{n}{({{u}_{ji}}-{{u}_{ij}})}={{y}_{i}}$      (3)

and inequality

$\mu {{C}_{ij}}>{{u}_{ij}}\ge 0$     (4)

where, r_i  is a rate of messages received from i-th node; y_i is a rate of messages transferred from one node to another, meaning that all messages must reach the recipients.

A total time delay in queues for transfer in the stationary network mode is determined by the following expression:

$T=\sum\limits_{i=1}^{n}{\sum\limits_{\begin{smallmatrix} j=1 \\ j\ne i\end{smallmatrix}}^{n}{\left( \frac{{{u}_{ij}}}{\mu {{C}_{ij}}-{{u}_{ij}}}\cdot \frac{1}{\mu {{C}_{ij}}} \right)\to \min }}$       (5)

The optimization objective is to specify such variables u_ij, which satisfy restrictions of Eq. (3) and Eq. (4), as well as reduce functionality to a minimum Eq. (5). The problem is solved analytically using a method of Lagrange multiplier. By differentiating those u_ij, for which C_ji > 0, we get a following system of algebraic equations as a required condition of extremum:

$\left\{ \begin{matrix}   \begin{matrix}   \frac{1}{(\mu {{C}_{ij}}-{{u}_{ij}})}+{{\lambda }_{i}}=0  \\   \sum\limits_{i=1}^{n}{\left( {{r}_{i}}+\sum\limits_{\begin{smallmatrix} j=1 \\ j\ne i\end{smallmatrix}}^{n}{({{u}_{ji}}-{{u}_{ij}})} \right)=0}  \\\end{matrix} & \begin{matrix}   i=\overline{1,n}  \\   j=\overline{1,n}  \\   j\ne i  \\   {{C}_{ij}}>0  \\\end{matrix}  \\\end{matrix} \right.$       (6)

Eq. (6) is solved using a numerical method, namely Newton's method. The search for a solution is carried out by constructing successive approximations and is based on the principles of simple iteration. The method has quadratic convergence. Attributes associated with data and resources transfer channels, as well as parameters related to routing, together represent a set of control variables that can be modified to ensure proper state of the network.

Another way to reduce the time delay in data transmission is to implement an algorithm for reducing collisions in the Ethernet network, which in this case is used as a data transmission channel between the control system and the control object. Ethernet networks use one of the most efficient methods for managing a local area network - Carrier Sense Multiple Access with Collision Detection (CSMA/CD) [32].

Networks with impact control (with collisions) are used as peer-to-peer non-priority systems [33]. This approach is widely used in computer networks and, in particular, applies in the context of Ethernet and the IEEE 802.3 standard. In a discovery-protected network, everyone is considered equal. However, it is possible to set a priority system based on different approaches. Before transmitting, the station needs to "check" the channel and determine if the channel is active. If the channel is idle, then any station that has data to transmit can send its frames on the channel.

In carrier sense networks, several capture methods are used. One method is the “non-persistent” carrier sensing method. Non-persistent Carrier Sense Multiple Access (CMSA) is a non-aggressive version of protocol that operates in the Medium Access Control (MAC) layer. Using CMSA protocols, more than one user or nodes send and receive data through a shared medium that may be a single cable or optical fiber connecting multiple nodes, or a portion of the wireless spectrum. In non-persistent CSMA, when a transmitting station has a frame to send and it senses a busy channel, it waits for a random period of time without sensing the channel in the interim and repeats the algorithm again. So, this method provides all nodes with the opportunity to start transmitting immediately after it turns out that the channel is free.

Another method used in time-slicing systems is the “p-hard” carrier sensing method. It assumes for each node some waiting algorithm (p means probability). For example, nodes A and B do not immediately start transmission after they detect that the channel is free. In this case, each station calls the program for generating a random number – the waiting time (several microseconds). If the node detects that the channel is busy, it waits for a certain period of time and makes a new attempt. It will send to the freed channel with probability p and delay the transmission with probability 1-p. Both considered methods can only transmit at the start of the procedure (if the channel is idle), but their behavior on a busy channel differs: non-persistent CSMA doesn't attempt to sense the channel and restarts its logical cycle, whilst p=0 necessarily gets stuck in an infinite loop of waiting (since it has zero probability of transmission even if the channel goes back to being idle).

Consider the case of one indivisible resource - a communication channel (Figure 9). The state of the i-th process, in which the i-th process is waiting for the provision of a resource required by several processes, is called the active state and is denoted by $A_{i}^{i}$.

The state in which the i-th process does not require the provision of a resource is called the passive state and is denoted by $A_{0}^{i}$. The state space of the i-th process is denoted by${{L}_{i}}=\{A_{0}^{i},A_{i}^{i}\}$. A conflict occurs when multiple processes are active.

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Figure 9. Model OSI for networks CSMA/CD

Control object modelling involves using asynchronous information exchange between applications running on different computers. In this case a client-server-based architecture is used. To control automation objects in real-time mode one should minimize delays in transferring datasets via the network. This can be achieved by searching for the optimal route of transferring messages, optimizing thread processing algorithms within network nodes, as well as dividing threads into service classes to specify the most priority ones. When solving problems of optimizing information flows, one should identify optimization criteria. Optimization criteria may include a variable of dataset transfer delay between two network nodes, the transfer rate, the amount of data transferred, the load of the information channel, as well as the reliability of data transfer. However, optimization of a single parameter may cause a deterioration in the AEPS automation equipment control quality. Optimizing time for dataset transfer via the network requires an increase in the transfer rate and usage of channels with a wide bandwidth along with high-speed routers. Therefore, the load factor of the transfer channel will be minor. Reducing the amount of data transferred may again cause a deterioration in the control quality. Thus, when solving optimization problems, one should consider all factors listed.

Conflict resolution for one process is the provision of a shared resource to one of the conflicting processes and the delay of other conflicting processes that are in the active state. The choice of such a process is determined by rule R of conflict resolution. The probability of exit of the i-th process from the active state is denoted by:

${{P}_{i}}=P_{00}^{i}=P_{10}^{i}$

and in active state:

$1-{{P}_{i}}=P_{01}^{i}=P_{11}^{i}$

where, i is the process number, index 0 means the passive state, and index 1 means the active state.

The rule R₀ is optimal if it minimizes the average frequency of conflicts. Let’s find the optimal conflict resolution rule for reducing the average frequency of conflicts for an infinite system operation time. In the study [3], an optimization criterion is given, which has the form:

${{\psi }^{R}}(S,K)=\underset{T\to \infty }{\mathop{\lim }}\,\frac{1}{T}\sum\limits_{t=0}^{T}{M_{S}^{R}\chi ({{S}^{t}},K)\to \underset{\{R\}}{\mathop{\min }}\,}$

where, $M_{S}^{R}$ – expected value at initial state $S\in L$;

R – conflict resolution rule;

S^t – current state of a set of processes;

$\chi (S,K)=\left\{ \begin{align}  & 1,S\in K \\ & 0,S\notin K \\\end{align} \right.$ – characteristic function; K is the set of states in the space L.

To reduce the time delay of packet transmission, optimization of packet routing in the network can also be used, but this approach is applicable for networks with an extensive system of communication channels, if there are alternative data transmission routes. In the problem being solved in this paper, there is only one communication channel, which can be accessed by several resources. Another optimization criterion is to minimize the cost of network equipment for the implementation of control functions. However, the use of the cost optimization criterion will not affect the reduction of time delays in the transmission of data packets and, accordingly, the improvement of the control quality indicators. Therefore, the optimal conflict resolution rule is proposed to use in order to decrease the time delay when using remote controller in a distributed control system with a single communication channel, which is shared between different resources.

The optimal conflict resolution rule R₀, when several processes claim the same resource, is the rule according to which, when a conflict occurs, the resource is allocated to the process conflict that has the minimum probability of getting into the active state. This result has the following physical content: in the event of a conflict, preference should be given to one of the conflicting processes, which is less likely to conflict with other processes.

Consider the case of several indivisible resources for two parallel processes. Suppose two processes are running in parallel, but there are several resources due to which they conflict. This situation arises in widespread dual-processor computing systems with shared memory.

Thus, in this case n=2, r=1÷m (number of resources), the i-th process enters the state $A_{r}^{i}$ in the absence of another process with a probability:

$P_{r}^{i},\sum\limits_{r=0}^{m}{P_{r}^{i}}=1$, i=1,2.

The set of conflicts looks like:

$K=\{S\in L,\,\,{{S}_{1}}=A_{r}^{1}\,\,\wedge \,\,{{S}_{2}}=A_{r}^{2},\,\,1\le r\le m\}$,

denote $\alpha _{i}^{n}(A_{r}^{1},A_{r}^{2})$ – the average number of conflicts after n steps based on the states $(A_{r}^{1},A_{r}^{2})$, if at the first step preference is given to the i-th process, and then the optimal conflict resolution rule is used $\alpha _{0}^{n}(A_{r}^{1},A_{r}^{2})$; is the average number of conflicts after n steps, based on the state $(A_{r}^{1},A_{r}^{2})$, with the optimal conflict resolution rule. In accordance with the Bellman optimality rule [34]:

$\begin{align}  & \alpha _{0}^{n}(A_{r}^{1},A_{r}^{2})=\min (\alpha _{1}^{n}(A_{r}^{1},A_{r}^{2}),\alpha _{2}^{n}(A_{r}^{1},A_{r}^{2})) \\ & \alpha _{1}^{n}(A_{r}^{1},A_{r}^{2})=1+P_{r}^{1}\alpha _{0}^{n-1}(A_{r}^{1},A_{r}^{2})+(1-P_{r}^{1})\alpha _{0}^{n-1}(N) \\ & \alpha _{2}^{n}(A_{r}^{1},A_{r}^{2})=1+P_{r}^{2}\alpha _{0}^{n-1}(A_{r}^{1},A_{r}^{2})+(1-P_{r}^{2})\alpha _{0}^{n-1}(N) \\ & \alpha _{0}^{n-1}(N)=\sum\limits_{j=1}^{m}{P_{j}^{1}}P_{j}^{2}\alpha _{0}^{n-2}(A_{j}^{1},A_{j}^{2})+(1-\sum\limits_{j=1}^{m}{P_{j}^{1}P_{j}^{2}})\alpha _{0}^{n-2}(N) \\\end{align}$

In the study [34] it is shown that:

$n\alpha _{0}^{n}(A_{r}^{1},A_{r}^{2})\ge \alpha _{0}^{n}(N)$,

from where at $P_{r}^{2}\ge P_{r}^{1}$ we get:

$\alpha _{0}^{n}(A_{r}^{1},A_{r}^{2})=\alpha _{1}^{n}(A_{r}^{1},A_{r}^{2})$.

The optimal conflict resolution rule R for the case where two processes claim multiple resources is the above rule. The effectiveness of the optimal rule can be evaluated analytically. For this, it is necessary to find the average frequency of conflicts with the optimal conflict resolution rule. In the study [35] it is shown that

${{\psi }^{R}}_{0}(S,K)=\sum\limits_{i=1}^{m}{{{\pi }_{{{R}_{0}}}}}(A_{r}^{1},A_{r}^{2})$,

where, ${{\pi }_{{{R}_{0}}}}(A_{r}^{1},A_{r}^{2})$ - stationary state probability with a fixed rule R₀.

For the probabilities ${{\pi }_{{{R}_{0}}}}(A_{r}^{1},A_{r}^{2})$, i=1÷m, the following system of equations can be written considering normalization:

$\sum\limits_{i=1}^{m}{{{\pi }_{{{R}_{0}}}}}(A_{r}^{1},A_{r}^{2})+{{\pi }_{{{R}_{0}}}}(N)=1$,

where, N=L/K, that is, a non-conflict state:

${{\pi }_{{{R}_{0}}}}(A_{r}^{1},A_{r}^{2})=\min (P_{r}^{1},P_{r}^{2}){{\pi }_{{{R}_{0}}}}(A_{r}^{1},A_{r}^{2})+P_{r}^{1},P_{r}^{2}{{\pi }_{{{R}_{0}}}}(N),$

whence it follows:

${{\pi }_{{{R}_{0}}}}(A_{r}^{1},A_{r}^{2})=\frac{P_{i}^{1}P_{i}^{2}}{1-\min (P_{i}^{1}P_{i}^{2})}{{\pi }_{{{R}_{0}}}}(N)$,

and therefore:

$\begin{align}  & \sum\limits_{i=1}^{m}{{{\pi }_{{{R}_{0}}}}}(A_{r}^{1},A_{r}^{2})=1-{{\pi }_{{{R}_{0}}}}(N)=& \left[ \sum\limits_{i=1}^{m}{\frac{P_{i}^{1}P_{i}^{2}}{1-\min (P_{i}^{1}P_{i}^{2})}} \right]/\left[ 1+\sum\limits_{i=1}^{m}{\frac{P_{i}^{1}P_{i}^{2}}{1-\min (P_{i}^{1}P_{i}^{2})}} \right] \\\end{align}$.

In this case:

${{\psi }^{{{R}_{0}}}}(S,K)=\frac{\sum\limits_{i=1}^{m}{P_{i}^{1}P_{i}^{2}/\left[ 1-\min (P_{i}^{1}P_{i}^{2}) \right]}}{1+\sum\limits_{i=1}^{m}{P_{i}^{1}P_{i}^{2}/\left[ 1-\min (P_{i}^{1}P_{i}^{2}) \right]}}$.

In Figure 10 the ratio of the average frequency of conflicts depending on the number of conflicting processes is shown:

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Figure 10. The ratio of the average frequency of conflicts

The use of an optimal rule for conflict resolution reduces the frequency of conflicts by almost half compared to a rule that randomly (equiprobably) selects one of the conflicting processes. The inclusion of the described scheduling algorithm in the functional standards of the application and channel levels, and the construction of a complete machine algorithm will almost halve the frequency of conflicts for the resources of the information-computing system and, in the same way, improve its quality.

Figure 11 shows waveforms of signals at the digital regulator output at different variables of the time delay.

Thus, as can be seen from the Figure 11, an increase in the delay in the transmission of data packets over the network leads to a significant decrease in the quality of control and an increase in the static error. The presence of a positive static error for the case of the absence of time delays in data transmission (curve 1) is similar to the small value of the integral component of the PID controller, and the oscillation is similar to the small value of the differential component of the controller. At the same time, an increase in the time delay leads to the fact that the static control error becomes negative, which, in fact, is similar to a decrease in the coefficient of the proportional component of the PID controller. However, the figure shows that the nature of the transient processes for different time delays is the same and is determined by the control object itself and the settings of the PID controller coefficients. It should be noted that, theoretically, the use of the considered rule of optimal conflict resolution when accessing several resources to a communication channel reduces the frequency of conflicts in almost 2 times, and, consequently, the value of time delay. This will reduce the static control error. However, this also reduces the natural period of the system and the overshoot. With some delay in the control loop, a situation may arise when the static error is completely absent. Therefore, it is important to tune the PID controller of the control object, considering the delay introduced by the network, in order to obtain the best indicators of control quality.

11.png

Figure 11. Waveforms of signals at the digital regulator output at different time delay variables: 1 means there is no time delay; 2 means there is a 40 ms delay; 3 means there is an 80 ms delay