The mathematical instrument describing the operation of the PID regulator consists of the equations of the items:

proportionate:


differential:


integral:


      The description of the PID-regulator is the sum of the three equations:



      For the thermal flow:

      The simulations are carried out on a three-dimensional model of a furnace with three-layer insulation and a body heated in a chamber, where the equations are of matrix kind. In order to check the program and obtain results under extreme conditions, a decision has been accepted for an installation of great capacity, and the regulation is in relation to low temperature.

    A particular case of disturbing influence has been considered - six heaters with equal capacities and regulation of the temperature at the level of 200^oC.
    Figure 1 - shows the charts of a P, I, ID, PD, ID and PID regulator.
    Figure 2 - shows the alteration of the consumed capacity in the mode of regulation (chart 1) of one of the heaters. Chart 2 in the same figure represents the wrapping chart of the rectangular impulses of the capacity. The latter decreases, and with the establishing of the heating process the comprising impulses become steady - at equal distances from each other with equal amplitude. In this state, ERF operates at idle move, i.e. the losses are compensated.
    The adjustment of the PID regulator is made through correction of the coefficients: integral ki, proportionate kp and differential kd.
    Figure 3 - shows the charts of the three items at various adjustments.
    Figure 4 - presents the charts of alteration of the temperature when changing the coefficients of the regulator as given in Table 1. Adjustment 1 corresponds to a tentative adjustment of the regulator, the precise adjustment being obtained and after making a correction: number 5.

    In MatLab the model of a PID regulator is realized as the basic equations of the items are used with the relevant algorithms created by the conditions of regulation. The basic way of operation of the program in this case is regulation of the temperature as a function of the time.

Figure 1
Impulse control of the temperature of the heater 6,8 -controlling impulse; 5,7 - the heater's temperature; 2,4 -the heater's temperature according to the capacity amplitude - 1,3



Figure 2
Impulse control of the temperature of the heater 1,3,4 - the heater's temperature; 2 temperature of the surface of the body; A - influence of the differentiating item.



Figure 3
Operating space of the thermal complex furnace-detail-regulator. 1-establishing of the process without regulation; 2,3 upper and lower limits respectively

Figure 4 Mode of regulation in relation to the temperature of the heater in a determined assignment 1 - heater, 2 - surface of the detail, 3,4 - controlling impulses.	Figure 5 Mode of regulation in relation to the temperature of the surface of the body in a determined assignment; 1 - heater, 2 - surface of the detail, 3,4 - controlling impulses.

    Figure 1 - presents the control of the ERF realized according to the analysis as set forth here. The formation of rectangular controlling impulses (charts 6,8) leads to a correspondent alteration of the temperature of the heater (charts 5,7 respectively) where: the amplitude of the peaks in the upper and in the lower front depends on the amplitude of the impulses, which is smaller in the second case. The same figure gives information about the form of the impulses of the temperature (2, 4) in relation to the impulses of the capacity (1, 3) under various conditions of regulation.
    The combination of the adjustment of the PID regulator in relation to the suggested variant of impulse control, gives the opportunity to realize a wide range of laws of control. Figure 2 shows the alteration of the temperature of the heater within a certain range through a successive assignment of impulses with different amplitudes. The law of control, according to chart 1, describes three successive impulses at intervals of three grades, after which the temperature decreases, and chart 2 shows the temperature of the surface of the body.
    Through alteration of the law of regulation in charts 3 and 4, symmetric impulses can be obtained. The impulse of the highest amplitude (charts 1,2) is characterized by a certain peak in the left front followed by a vacillation of the PID regulator (point A). This process is due to the coefficient of the differential item, and after its elimination (chart 4), the vacillations disappear.
    The time needed to reach a given temperature depends on the thermal parameters of the system furnace - detail and the installed capacity. In the mode of regulation the possible assignment of the temperature - time is determined by the parameters of the regulator in relation to the technological process - minimum and maximum temperature.
    Figure 3 - shows a particular case in which the transitional process without a regulator develops as in chart 1. The technological criteria for operation of the installation have been assigned - temperature 700^oC (for the time t2) and minimum 200^oC (for the time t1). It should be noted that the operating space is closed between charts 2 and 3, i.e. each point lying between the latter two can be assigned as a temperature - time adjustment of the regulator. The next simulation realizes reaching the point A (figure 3), determining the heater's temperature from 200^oC to 500^oC in 60 minutes through impulse control.
    The way of controlling is similar to that of the systems with automatic optimization. The relation dT/dt gives the coefficient determining the increase of the temperature per unit of time.
    Chart 1 (figure 4) shows the temperature of the heater, and charts 3 and 4 show the impulses of control. According to the performed simulations, the vacillation peaks depend on the step of control, where in a smaller discretization (chart 4) they are smaller. This question is closely linked to the assigned step of integration when solving differential equations, and thence to the necessary frequency of analog/digital converter operation at the regulating device.
    The same approach, when applied to the regulation in relation to the temperature of the surface of the body, gives the result shown in figure 5. On reaching its maximum value, the temperature of the heater (chart 1) begins to decrease and inclines towards chart 2, which is due to establishing of the temperature within the body.
    A characteristic particularity of the described regulator is the necessity of information about the regulated value and also taking into account the system time. In case we consider a program mode in which the temperature is not measured, i.e. a model has been made in advance and the process has been computer simulated, it is technologically possible to implement the regulation without measuring items. This requires that the regulator operate by following a determined algorithm.

    The considered processes of operation and control concern a particular case of an ERF construction with a cubic form and heaters having equal capacities in all walls of the furnace.

Figure 1
1 - heaters, 2 - body, 3 - walls

Figure 2
Heating of a furnace in the form of a parallelepiped when the heaters have equal capacities 1,2-temperature of the heater and of the body's surface placed at walls S_2-4, respectively 3,4 and 5,6 placed at S_1-3, S_5-6



Figure 3
Heating of a furnace in the form of a parallelepiped when the surface loading of the walls is equal; 1,2,3 - temperature of the heaters at the walls respectively: S_1-3, S_2-4, S_5-6.

    The constructions which exist in practice are of a system furnace-detail in the form of a parallelepiped. For the last, figure 1 shows the uneven temperature field in heaters (1), body (2) and walls (3) of an ERF. In order to achieve an even distribution of the field, it is necessary to perform a relevant distribution of the capacity of the heaters. This particular case is about a three-layer ERF with areas respectively: left-right side: S_1-3=0.32 [m²]; bottom-door: S_2-4=0,24 [m²], floor-ceiling: S_5-6=0,48 [m²].
    In each wall of the operative chamber are placed heaters having equal capacities of 2[kW] (summary capacity 12[kW]) and density of the thermal flow (W/m²): left-right side: 6250[W], bottom-door: 833[W], floor-ceiling: 4167[W].
    The conditions of regulation are:


    Figure 2 - presents the running of the process.
    As can be seen in the last figure, the assigned temperature can be reached at the fastest by the wall which has the smallest area. The even heating of the body is possible through equalizing the density of the thermal flow: S_1-3=2X2000[kW], S_2-4=2X1550[kW], S_5-6=2X2950[kW]. A precise regulation requires three separate regulators on each pair of walls, or installment of heaters with relevant capacity.
    Charts 1, 2, 3 in figure 3 show the establishing of the temperature for the same furnace.


    The process of regulation in relation to the surface, oriented towards a wall without a heater (chart 5) is shown in figure 4.
    In the beginning of the process of regulation, before establishing the process of heating, the temperature difference between the surfaces is up to 300^oC.
    Figure 5 shows regulation of the temperature in relation to the center of the body (chart 7), under conditions T_reg = T_ct: