STABILIZATION OF THE ROTATIONAL SPEED OF A DC MOTOR UNDER CYCLIC LOAD VARIATION

L. S. Udut, A. S. Bobrovskii, V. A. Beinarovich

(Presented by the scientific and technical seminar of the Research Institute of AEM)

To determine the rational structure of the controller of the electric-drive system of a telegraph apparatus with cyclic load variation [1], it is expedient to use the methods of optimal-control theory, taking speed of response as the optimality criterion; this ensures both the absence of overshoot in the electric drive and minimal deviations of the instantaneous rotational speed.

The optimal-control problem is formulated as follows [1]: for the system described by the equations

\[ \frac{d\omega(t)}{dt}=i(t)-i_c(t), \]

\[ \frac{di(t)}{dt}=\frac{1}{\tau_e}\,[u(t)-\omega(t)-i(t)], \tag{1} \]

accelerate from rest to the specified speed \(\omega_{\mathrm{set}}\) and ensure the maintenance of \(\omega_{\mathrm{set}}\) under cyclic changes of the load torque from \(\mu_{c1}\) to \(\mu_{c2}\), with the shortest transient-process time and bounded control \(u(t)\le u\).

According to the maximum principle, for (1) we have

\[ H=\psi_1(t)[i(t)-i_c(t)]+\psi_2(t)\frac{1}{\tau_e}[u(t)-\omega(t)-i(t)]; \tag{2} \]

\[ \frac{d\psi_1(t)}{dt}=\frac{1}{\tau_e}\psi_2(t); \]

\[ \frac{d\psi_2(t)}{dt}=-\psi_1(t)+\frac{1}{\tau_e}\psi_2(t). \tag{3} \]

From the \(H\)-max condition it follows that \(u(t)=\operatorname{sign}\psi_2(t)\).

In accordance with [1], the roots \(p_1, p_2\) of the characteristic equations of systems (1) and (3) are real; consequently,

\[ \psi_2(t)=C_1e^{p_1t}+C_2e^{p_2t}. \]

It is obvious that \(\psi_2(t)\) can change sign no more than once; then the control \(u(t)\) is of relay type, has no more than one switching, and assumes the limiting values \(\pm u\) on individual control intervals.

To carry out the synthesis on the phase plane of the parameters \(\omega\) and \(i\), let us find the solutions of system (1):

\[ \omega(t)=b_1e^{p_1t}+b_2e^{p_2t}+(u-i_{c.\,\mathrm{fin}.}); \tag{4} \]

\[ i(t)=p_1b_1e^{p_1t}+p_2b_2e^{p_2t}+i_{c.\,\mathrm{fin}.}, \tag{5} \]

the time of motion on individual segments

\[ t=p_2\ln \frac{p_1(\omega_{\mathrm{init}.}-u+i_{c.\,\mathrm{fin}.})-(i_{\mathrm{fin}.}-i_{c.\,\mathrm{fin}.})} {p_1(\omega_{\mathrm{init}.}-u+i_{c.\,\mathrm{fin}.})-(i_{\mathrm{init}.}-i_{c.\,\mathrm{fin}.})} \tag{6} \]

and the equation of the trajectory of motion of the system

\[ \begin{aligned} &(i(t)-i_{c.\,\mathrm{fin}.})-p_1(\omega(t)-u+i_{c.\,\mathrm{fin}.}) \\ &=\big[(i(t)-i_{c.\,\mathrm{fin}.})-p_2(\omega(t)-u+i_{c.\,\mathrm{fin}.})\big]^{\frac{p_1}{p_2}} \times \\ &\quad \times \big[-p_1(\omega_{\mathrm{init}.}-u+i_{c.\,\mathrm{fin}.})+(i_{\mathrm{init}.}-i_{c.\,\mathrm{fin}.})\big] \big[p_2(\omega_{\mathrm{init}.}-u+i_{c.\,\mathrm{fin}.})-(i_{\mathrm{init}.}-i_{c.\,\mathrm{fin}.})\big]^{-\frac{p_1}{p_2}}, \end{aligned} \tag{7} \]

where \(b_1\) and \(b_2\) are constant coefficients.

The expressions obtained, (4)—(7), for a known optimal control \(u(t)\), make it possible to construct the phase portrait of the system and determine its parameters (Fig. 1). From the coordinates of the switching points on the

Fig. 1. Limit cycle of the system during braking of the motor by coasting (solid lines) and dynamic

Fig. 2. Calculated dependences \(\Delta\omega=f(\omega_{\mathrm{set}})\) for the parameters \(\tau_e=0.2,\ i_{c1}=0.05,\ i_{c2}=0.25\)

basis of (6), the control time is found; the deviations of the instantaneous speed \(\Delta\omega\) are determined on the basis of (4)

\[ \Delta\omega=(u-i_{c.\,\mathrm{fin}.})+b_1e^{p_1t}+b_2e^{p_2t}-\omega_{\mathrm{set}}, \tag{8} \]

where

\[ t=\frac{1}{p_2-p_1}\ln\left(-\frac{p_1b_1}{p_2b_2}\right). \]

The magnitude of the deviations of the instantaneous speed \(\Delta\omega\) depends on the parameters of the actuating element, the amount of energy supplied to the motor over the control intervals, and on the selected value of \(\omega_{\text{set}}\) (Fig. 2, where the notation is as follows: \(\Delta\omega_{1\text{p}}\) is braking by counter-switching; \(\Delta\omega_{1\text{d}}\) is dynamic braking or coasting). The desire to make fuller use of the motor with respect to speed gives the quantity \(\Delta\omega_2\) the principal role in choosing the value of \(\omega_{\text{set}}\).

For preliminary calculations, the following approximate expression is recommended for the deviations of the instantaneous speed

\[ \Delta\omega=\tau_{\text{e}}\frac{\left(i_{c.\,\text{end}}-i_{c.\,\text{beg}}\right)^2}{\omega_{\text{set}}-u+i_{c.\,\text{beg}}} \tag{9} \]

and for the optimal value of \(\omega_{\text{set}}\),

\[ \omega_{\text{set.opt}}=\frac{(u_1+u_2)-(i_{c1}+i_{c2})}{2}. \tag{10} \]

According to (7), the switching lines \(L_1, L_2\) are nonstationary and depend on the parameters of the disturbance and of the actuating element. In this case it is expedient to confine oneself to the use of a quasi-optimal regulator, which in the case under consideration contains a relay element and nonlinear feedbacks with respect to the speed and current of the actuating motor.

The regulator parameters obtained as a result of the calculation are limiting values, toward which one should strive in its implementation, while the degree of closeness of the actual parameters to the limiting ones will serve as a criterion of the quality of the designed system.

REFERENCES

1. A. S. Bobrovskii, E. V. Shestakov, V. S. Smirnov. A series of thyristor electric drives for telegraph communication devices. Reports of the VI Scientific and Technical Conference on Problems of Production Automation. Vol. III, Tomsk, 1969.