PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER OF THE RED BANNER OF LABOR
named after S. M. Kirov

Vol. 285　　　　　　　　　　　　　　　　　　　　　　　　　　　　1975

ON THE FORMATION OF TRANSIENT PROCESSES IN NONLINEAR AUTOMATIC CONTROL SYSTEMS FOR ELECTRIC DRIVES WITH A REFERENCE MODEL

V. A. Beinarovich, A. E. Alekhin

(Submitted by the Scientific and Technical Seminar of the Research Institute of Automated Electric Drives)

In the development of automatic control systems (ACS), problems arise of forming transient processes in accordance with specified technological requirements, which may vary under production conditions when an installation operates in different technological operations or even at different stages of a single technological operation. The required changes in the dynamic properties of an ACS may be obtained by changing the parameters of the ACS according to an appropriate program, or by using self-adjusting systems (SAS) with a reference model, in which only the parameters of the model are changed. Obviously, the second approach is more expedient, since changing the dynamic properties of the reference model is achieved by simpler means and with greater accuracy than changing the properties of the ACS. In addition, the dynamic properties of an ACS during operation tend to change spontaneously under the action of fluctuations in load, supply voltages, temperature, and other factors. The use of an SAS with a reference model makes it possible to reduce and even practically eliminate the influence of limited changes in the dynamic properties of the ACS on the transient process of the output coordinate [1].

One of the most important requirements for an ACS, determining the productivity and accuracy of operation of many production installations, is speed of response. However, increasing the speed of response of an ACS leads to the occurrence and increase of overshoots and oscillations, which usually adversely affects the technological process. Moreover, in production installations, as a rule, the rate of change of the output coordinate of the ACS (the first derivative) and the jerk (the second derivative) are limited by physical conditions. For example, in feed-drive systems of metal-cutting machine tools, overshoots and oscillations of the feed are inadmissible; the feed rate is limited by the maximum cutting speed, and the jerk is limited by the maximum permissible loads of the kinematic links and the feed motor.

It is known that in linear SAS with a reference model, with a large increase in the gain in the feedback circuit, one can obtain a transfer function of the system practically equal to the transfer function of the reference model [1]. In this case, formation of any transient process specified by the reference model is possible. However, all real electric-drive systems always contain nonlinearities, and this leads to the fact that the possibilities of forming trans—

transient processes are limited by the maximum permissible dynamic properties of the links entering into the main control loop. In the present article, as applied to nonlinear automatic-control systems for electric drives (in particular, using as an example a nonlinear system of a tracking dc electric drive), questions are considered concerning the formation of transient processes with high speed of response and absence of overshoot, achieved through the use of a self-tuning loop with a reference model [1].

In tracking dc electric drives, the most significant nonlinearities are saturation of the power amplifier, current limiting, and loading by a reactive static torque (friction torque). Figure 1 shows the structural diagram of an STS of a nonlinear tracking electric drive. The structural diagram of the tracking electric drive is represented by an inertia-free saturating power amplifier 1, a current-limiting loop 8, a linearized motor structure 2—3—4—5—6 with reactive load \(I_c\), a displacement-processing device 7, and a correcting device 8. The self-tuning loop contains the reference model M and the amplifier K.

Fig. 1.

In the system under consideration (Fig. 1), three nonlinear elements are separated by linear elements, and therefore an analytical investigation of such a system is associated with very great difficulties and cannot be carried out in general form [2]. For the analysis and synthesis of such systems it is recommended to use the method of structural modeling on an analog computer [3]. Figure 2 shows the circuit of the electronic model of the STS under consideration. Here block I models the tracking electric drive with the principal nonlinearities (saturating amplifier — 1, current-limiting node — 6, a link modeling the reactive static torque — 3); block II represents the self-tuning loop with the reference model M and amplifiers 8, 9, 10; block III models the correcting device KU (correction is carried out according to the first derivative of the output coordinate by amplifier 7). It was assumed that in the tracking electric drive a motor of type MI-11 is used.

Figure 3 shows the transient processes for working out step input actions of unit and double magnitude (analogous to starting and reversing processes). Curve 1 depicts the form of the transient process at the output of the reference model, prescribing the working out of the input action with the maximum permissible constant velocity, independent of the magnitude of the input action. Curves 2 and 3 depict the transient processes in the tracking electric drive without self-tuning (\(K=0\) in Fig. 1), respectively at no load (\(I_c=0\) in Fig. 1) and under nominal load by reactive static torque. Tran-

Fig. 2.

Fig. 3.

transient processes are characterized by the presence of overshoots and oscillatory behavior, especially increasing as the load is reduced. Curve 4 depicts the optimal transient process in a tracking electric drive with a reference model, corresponding to maximum speed of response in the absence of overshoot (in this case the overall gain of amplifiers 9 and 10 of the model in Fig. 2 should have a value of about 100). Curves 5 and 6 show transient processes in a tracking drive without self-tuning under the condition of absence of overshoot, respectively at no load (\(I_c = 0\)) and at the nominal load by friction torque.

As a result of the studies it was established that the use of a self-tuning loop with a reference model in a tracking nonlinear electric drive makes it possible to ensure elimination of mismatches without overshoot, while increasing the speed of response by approximately a factor of 4 in comparison with an ordinary tracking electric drive (curves 4 and 6 in Fig. 3). Moreover, the study showed that, in a system of a nonlinear tracking electric drive without a self-tuning loop, it is impossible, by sufficiently simple means, to form a prescribed transient process of the type of curve 1 (Fig. 3). The simplicity of implementation of an STS of the type considered is evident: adding a reference model and a summing amplifier to any nonlinear automatic-control-system model transforms the system into a self-tuning one, in which the formation of transient processes is accomplished by a simple change in the dynamic properties of the reference model.

Thus, self-tuning systems with a reference model (Fig. 1) are highly promising for forming prescribed dynamic properties of nonlinear ACSs with electric drives, while the method of structural electronic modeling provides broad possibilities for the analysis and synthesis of such nonlinear ACSs.

REFERENCES

1. Theory of Automatic Control. Part II. Edited by A. V. Netushil. Moscow, “Vysshaya shkola,” 1972.

2. E. I. Khlypalo. Nonlinear Systems of Automatic Regulation. (Calculation and Design). Leningrad, “Energiya,” 1967.

3. E. A. Arkhangel’skii, A. A. Znamenskii et al. Modeling on Analog Computing Machines. Leningrad, “Energiya,” 1972.