1 Introduction

Currently, the pulp and paper industry is one of the leading dynamically developing industries in the world [1]. The task of further development of this industry is associated with an increase in the efficiency of paper production and the quality of finished products with economical and rational use of raw materials, fuel and energy and other material resources [2]. However, without modern means of automation and control of protection, the solution of this problem is difficult. One of the promising directions in the development of modern automation tools is advanced control.

The wide application of this control method in automated paper production systems is limited by the presence of a time delay in the control object, which occurs due to the finite propagation time of disturbances in the environment where the controlled process takes place; the emergence in the control object of modes excited by non-stationary external disturbances and poorly controlled by the input [3, 4].

During the experiment, an extreme dependence was obtained between the pressure (static pressure of the paper pulp) of the inlet device and the standard deviation of the weight of the paper web, shown in the Fig. 1. This graph is in good agreement with the results given in [4].

Fig. 1
figure 1

Extreme dependence of the standard deviation on the pressure of the paper pulp

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The obtained experimental dependence of STANDARD DEVIATION of the paper web weight on the pressure of the paper mass allows formulating the problem of extreme combined regulation of the unevenness of the clearance of the paper web: implementing a system for stabilizing the weight of the paper web and extreme regulation of its STANDARD DEVIATION according to the current measured values of the pressure of the paper mass and the weight of the paper web on output of the paper-making machine.

Existing systems of extreme control [5,6,7,8,9,10] do not provide the required accuracy of the technological process due to strong disturbances in the control channel and the presence of delay. Therefore, the task was set to develop an extreme control system for an inertial object with a delay, operating under conditions of tangible interference.

It should be noted that the processes of forming the weight of the paper web are subject to the influence of random factors, the effect of which on the technological characteristics is multidirectional. Management of such a process by traditional systems is extremely inefficient, and these systems are configured only during development.

An analysis of the extreme dependence of the standard deviation of the weight of the paper web on the pressure of the inlet device allows stating that the use of traditional methods for finding the extremum used in extreme systems for the case under consideration is very difficult due to the high signal-to-noise ratio.

Extremal control of inertial objects with delay operating under conditions of strong noise, based on traditional methods of optimal filtering [11], stochastic approximation [12], methods of statistical decision theory and dynamic programming (dual control) [13,14,15] and requires certain restrictions on characteristics of the object and disturbances acting on it.

In this regard, the problem of effective separation of the useful signal against the background of strong interference comes to the fore, and the problem of extremal control is relegated to the background.

In order not to suffer the quality of the paper, it is also necessary to stabilize the parameters of this technological process [16, 17]: the coordinate of the jet falling on the grid of the paper machine and the change in the speed of the paper mass inlet from the pressure device. It should be noted that this problem was solved in [18], but this algorithm can be simplified without loss of quality using dynamic compensators [8, 19].

2 Theoretical basis

One of the effective methods of useful signal filtering is the accumulation (averaging) method [20]. When using this method, the greatest effect is achieved if the desired signal is a harmonic signal with a known period. The frequency of this signal must be within the system bandwidth and, to increase performance, must approach its upper limit. Then its efficient extraction is achieved using either the correlation method [21] or the synchronous accumulation method [22].

The most preferable is the use of the synchronous accumulation method, as it is simpler to implement and less demanding on the shape of the input and output signals of the extremal system [23].

The use of the synchronous accumulation method predetermines the use of periodic search signal methods to search for an extremum. In the event that the time delay in the extremal object is fixed, to search for the extremum, you can use the method of synchronous detection in advance, taking into account the delay, synchronizing the phases of the reference and output signals of the extremal system using a phase shifter. Otherwise, with a changing time delay, the use of special methods for searching for an extremum is required.

We will assume that the object is located in the vicinity of the extremum. At a considerable distance from it, the transition to the vicinity of the extremum point can be carried out by combining the methods of synchronous accumulation and detection, choosing the period of the reference signal several times longer than the delay time.

When it is in the extremum region, due to the parity of the extremal characteristic, even harmonics will appear in the output signal of the extremal object, the input of which is modulated by a harmonic signal. Then the extreme value of the amplitude of the second harmonic will indicate that the system is at the extremum point. The time delay in this case will not affect the accuracy of determining the extremum [12]. In addition, it is possible to increase the frequency of the harmonic signal at the input of the extremal object by choosing it, as already noted, near the upper limit of the inertial object’s bandwidth.

Based on the foregoing, the following noise-proof algorithm for searching for the extremum of an inertial object with delay is proposed:

  1. 1.

    A harmonic signal is formed, the frequency of which is chosen several times greater than the time delay in the object, and the amplitude does not disturb its operation;

  2. 2.

    The input signal of the extreme system is modulated by this harmonic signal;

  3. 3.

    The output signal is filtered by the synchronous accumulation method;

  4. 4.

    Harmonic analysis extracts the second harmonic of the spectrum of the filtered output signal;

  5. 5.

    The direction of movement to the extremum is determined by the phase shift;

  6. 6.

    The input signal of the object is varied until the second harmonic appears in the output signal;

  7. 7.

    Increases the frequency of the modulating signal within the bandwidth of the extreme object;

  8. 8.

    Traditional methods of extremal control implement the search for the maximum amplitude of the second harmonic.

To test the developed algorithm, a Simulink model of an extremal object was built. The object of the study was a paper-making machine.

3 Methodology

An extreme relationship is known between the ratio of the speeds of the paper pulp coming from the inlet device and the conveyor mesh of the paper machine and the non-uniformity of the paper gap [24]. Since a change in the speed of a paper machine leads to a change in the technological mode and affects all its systems, we will set the ratio of speeds by changing the speed of the outflow of the paper mass, adjusting it by changing the pressure in the inlet device. Data on the unevenness of the gap will be indirectly calculated by the weight of the paper web measured by the scanning device. It is obvious that the smaller the change in weight relative to the average value, the less the unevenness of the paper web and, therefore, the change in the dispersion of the weight of a square meter will be an estimate of the unevenness of the gap.

According to experimental data, using the System Identification Toolbox package, a model was obtained that establishes the relationship between the pressure and the standard deviation of the weight of the paper web. The model includes two submodels: a Hammerstein-type submodel that reproduces the nonlinear extremal dependence (Fig. 1) and linear perturbation submodels acting at the input and output of the plant. Simulink - a model of such an object is shown in the Fig. 2.

Fig. 2
figure 2

Simulink model of an inertial extremal object

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The Saturation block limits the input signal of the non-linear (extreme) element to the range of 660–710 mm. The nonlinear block Fcn reproduces the extremal dependence shown in the Fig. 2, the delay in the object is 30 s.

With the help of the developed Simulink model of an inertial extremal object, an analysis of the noise-protected extremal control, built on the basis of the proposed algorithm, was carried out. To do this, a harmonic signal with an amplitude of 10 mm and a period of 125 s was applied to the input of the adder Add (Fig. 1), which is approximately four times the delay time.

During the simulation, the constant input signal of the extreme object, specified by the Constant block, was chosen so that the operating point of the object was to the right of the extremum point (Fig. 3), at the extremum point (Fig. 4) and to the left of the extremum point (Fig. 5).

On the Figs. 3a, 4a and 5a show the input harmonic and output noisy signals of an inertial extremal object, in the Figs. 3b, 4b and 5b show results of synchronous accumulation of input and output signals for 23 periods of the input signal, in the Figs. 3c, 4c and 5c show results of harmonic analysis of input and output signals filtered by the synchronous accumulation method.

The simulation results confirm the efficiency of the proposed algorithm. When you are at the extremum point, you can see that the amplitude of the second harmonic reaches its maximum value.

The operation of the algorithm was tested on a real object for an optimal head value of 550 mm (Fig. 6), which also confirmed the effectiveness of the proposed algorithm.

The extreme dependence between the ratio of the speeds of the paper pulp and the paper machine and the standard deviation of the weight of the paper web, which determines the degree of non-uniformity of the paper gap, has been experimentally confirmed.

Fig. 3
figure 3

Simulation results with an average value of the input signal greater than the optimal head (690 mm): a input harmonic and output noisy signals of an inertial extremal object; b results of synchronous accumulation of input and output signals; c results of harmonic analysis of input and output signals

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Fig. 4
figure 4

Simulation results with an average value of the input signal equal to the optimal pressure (685 mm): (a) input harmonic and output noisy signals of an inertial extreme object; (b) results of synchronous accumulation of input and output signals; (c) results of harmonic analysis of input and output signals

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Fig. 5
figure 5

Simulation results with the average value of the input signal less than the optimal head (680 mm): a input harmonic and output noisy signals of an inertial extremal object; b results of synchronous accumulation of input and output signals; c results of harmonic analysis of input and output signals

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Fig. 6
figure 6

Simulation results with an average head value equal to the optimal one (550 mm): a input harmonic and output noisy signals of an inertial extremal object; b results of synchronous accumulation of input and output signals; c results of harmonic analysis of input and output signals

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4 Results

A noise-proof algorithm for searching for the extremum of an inertial object with a delay has been developed, based on the method of a periodic search signal with noise filtering by synchronous accumulation and subsequent search for the maximum of the second harmonic extracted from the output signal. On the basis of modeling and experimental verification on a real object, the effectiveness of the proposed algorithm is confirmed.

Simulink - extreme control system model is shown in the Fig. 7.

Fig. 7
figure 7

Simulink - extreme control system model

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Fig. 8
figure 8

Oscillograms of extremum search: a pressure change; b changing the output signal; c fragment of changing the output signal when the extremum is reached

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The oscillograms in the Fig. 8 clearly show that when moving towards the extremum, the second harmonic gradually appears in the output signal of the extremal system.

The use of dynamic compensators for dynamic channel decoupling due to the nonlinearity of the control object encounters well-known problems of the physical feasibility and stability of such compensators [25, 26]. In this regard, it was proposed to use static compensation of control channels as shown in Simulink - a combined system model (Fig. 9).

The compensation functions for the coordinates of the upper bar of the pressure device X and Y were selected experimentally. It was found that the compensation function for both the X-coordinate and the Y-coordinate turned out to be almost linear. Then, these functions were approximated by a linear polynomial.

Comparison of system operation without compensation and with compensation of parameters is shown in the Fig. 10 and indicates the acceptability of using static compensators.

Fig. 9
figure 9

Simulink model of parameter stabilization

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Fig. 10
figure 10

Comparison of the system operation without compensation and with compensation parameters: a productivity; b coordinates of the fall of the paper pulp jet on the paper-making machine grid; c change in the speed of paper pulp inlet from the pressure device

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5 Discussion

An analysis of the figures shows that with a change in air pressure Pg with a compensator, the performance in the steady state does not differ by more than ± 1.5%. The coordinate of the fall of the paper pulp jet on the mesh varies within ± 3.4%.

The operation of the system using the method of extreme combined control was tested on a real object; in the production of paper grade “KPS-2” (TU 5441-045-00279344-2005), the optimal pressure value of 550 mm was achieved, which confirmed the effectiveness of the proposed algorithm.

The programmable logic controller performs extreme regulation of the standard deviation of the mass per square meter of the paper web. Process data is mapped to a workstation that has Supervisory Control and Data Acquisition (SCADA) WinCC installed. A project of SCADA-control system was developed for an automated control system for the technological process of paper production. A mnemonic diagram of the designed SCADA system for automating the functional subsystems of paper production is shown in the Fig. 11. The Fig. 12 shows the trends in the operation of the automated process control system for paper production. As a result of the work of the designed system, the fiber consumption rates decreased by an average of 2%.

A comparative assessment of the operation of functional subsystems was carried out before and after the implementation of the developed control systems. The results are shown in the Table 1. The change in fiber consumption rates is shown in the Table 2.

From the data in the Table 1 we can see that the time required for the technological adjustment of the paper machine to produce a certain grade of paper has decreased by 33%.

Thus, fiber consumption rates decreased by an average of 2%. The developed system of automated control of paper production has made it possible to increase its accuracy and stability. In particular, the standard deviation of the mass per square meter of the paper web when using the control system decreased by an average of 3–4%, which confirms the increase in the accuracy and capabilities of the process.

Table 1 Evaluation of the effectiveness of automated control system for the technological process of paper production
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Table 2 Estimating Fiber Rates
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Fig. 11
figure 11

Mnemonic diagram of automated process control system for paper production

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Fig. 12
figure 12

Trends in operation of automated control system for the technological process of production with noise-proof extreme control of paper production with control

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The stability of the technological process was estimated by the ratio of the range of changes in the standard deviation of the mass of square meter of paper web for a certain period of time to its average value. During the study period T = 6 h, this indicator decreased by 20% when using the system.

6 Conclusion

The tuning coefficients of the extremal controller for the above coordinates were chosen on the basis of experimental data with subsequent approximation. The linear function gives the maximum approximation accuracy. The analysis of the obtained results shows that with a technological spread of air pressure in a container with paper pulp, the extreme controller allows maintaining a given mass flow rate of paper pulp in a steady state with an error of no more than ± 1.5%. The error of the horizontal positioning of the paper pulp jet on the conveyor mesh varies within ± 3.4%.

It has been established that the use of an automated control system for paper production with an extreme controller makes it possible to increase the accuracy of maintaining the parameters of the paper web. The consumption of raw materials decreased by 2%, and the standard deviation of the weight of the paper web decreased by 1.5-2%.

Through the use of methods of extreme control and stabilization of process parameters, it was possible to increase the accuracy and stability of paper web weight control, decrease the amount of rejects when reconfiguring the system to another type of paper, reducing the time of the transition process and setting up paper production, thereby saving fiber consumption.