Special requirements on capacity for work are made to manufacturing control systems. First of all such systems must have certain stability factor, e. g. occurring in them transient processes must damp intensively enough. Continuous reheating furnaces operate in transient conditions considerable part of time [1]. In this connection stability factor estimation for control systems of heating process in furnaces has great significance. Such estimation has been carried out for automatic control system of stocks heating in continuous furnaces of rolling mill 2000 of Volgograd metallurgical centre “Red October”. This system has been realized with integral controllers [2].

Processing of furnaces heating records, plotting of transient response graph for object by experimental points and drawing of tangent in the field of its maximum inclination must precede estimation procedure. As a result transfer function of first heating zone has been obtained:

                                (1)

Stability factor of system can be determined not only by arrangement of roots of its characteristic equation but by shape of unit-impulse response of its closed loop as well. Stability factor evaluates by subsidence ratio of oscillations which can emerge in loop. Appearance of resonance peak in plot of complex frequency response (CFR) absolute magnitude corresponds to oscillatory character of unit-impulse response. Height of resonance peak depends on degree of open loop CFR approach to “dangerous” point -1, j0. The nearer response of open loop W_o.l(jω) approaches to this point, the greater this peak turns out. Stability condition of closed loop for control systems with integral component in algorithm of controller performance looks in the following way [3]:

                                                  (2)

where |Φ(jω_res)| – absolute magnitude of open loop CFR; M_ext – assigned beforehand permissible value of frequency index for loop oscillation, namely, relative value of CFR resonance peak.

At determination of maximum permissible value for transfer constant of controller k_t, which is situated in closed loop of system, it is necessary to plot response W_o.l(jω) at arbitrary value of this constant and draw circle of radius of  at the same plane. Center of this circle is located on negative real semiaxis at distance of . Required ratio between radius of M-circle and coordinate of its center u looks in the following way:

                                                    (3)

If calculations are automatized, expression (3) can be included in calculation program and task is reduced only to matching of u value at which circle will be tangent to CFR of open loop. If it turns out that response W_o.l(jω) has hit inside “forbidden” area which is confined by this circle, it is necessary to replot it choosing greater value of u. If response passes outside the circle, it is necessary to decrease value of u [3].

In figure 1 CFR of open loop at unitary transfer constant of I controller (solid-line curve) and M-circle (dashed curve) which is tangent to it are plotted at value of M = 1,8 what corresponds to subsidence ratio of oscillations ψ = 0,85 in oscillatory link. q(ω) and q_m(ω, u) are imaginary components of CFR and M-circle respectively, and p(ω) and p_m(ω, u) are their real components. As a result of position matching of circle center value u = 6,628 has been obtained.

Figure 1 – CFR of open loop and М-circle

 

Then value of maximum permissible transfer constant of I controller for first heating zone is calculated by formula [3]:

                                                                                                 (4)

Value k_t = 0,218 has been obtained from expression (4). Similar procedures have been executed for controllers of second heating and soaking zones. As a result of calculations limit values of tuning coefficients equal to 0,211 and 0,172 respectively have been obtained. Calculations have been carried out in MathCAD package.

Proposed stability factor estimation for I controllers will allow choosing tuning coefficients at stage of control systems designing. At these coefficients occurring in closed loops transient processes will damp most intensively.