Time-Optimal Control of an Asynchronous Gyroelectric Motor

I. F. Kalinin, B. P. Soustin

(Presented by the scientific and technical seminar of the Department of Electrical Equipment)

The readiness of a gyrosystem for operation is determined primarily by the acceleration time of the gyromotor. This circumstance is taken into account in the design of gyromotors by selecting the appropriate design parameters for a prescribed value of the kinetic moment. In addition, special electrical devices are used that force the start-up; these are based in principle on two methods [1]:

1) start-up at a constant elevated voltage;
2) start-up at an elevated voltage varying smoothly in time.

In both cases the stator frequency remains unchanged. The time required for the rotational speed of the gyromotor rotor to change from zero to the steady-state value is, in the general case, determined from the equation of electric-drive dynamics

\[ t_p = I \int_{0}^{\omega_y} \frac{d\omega}{M_\text{э} - M_\text{с}} . \tag{1} \]

Since the moment of inertia \(I\) and the resisting torque \(M_\text{с}\) are determined by the parameters of the gyromotor, a reduction in acceleration time can be achieved only by increasing the electromagnetic torque of the motor. The electromagnetic torque of an asynchronous gyromotor is written [1] as

\[ M_\text{э} = \frac{10200\, m_1 s_1 U_1^2} {\left[(1+s_1)^2+\left(s_1\frac{x_k}{R_1}\right)^2\right] z_1 R_1 \omega}, \tag{2} \]

where \(R_1, s_1, z_1, x_k, m_1\) are motor parameters.

Having specified an admissible value of the stator voltage \(U_1\) or its law of variation, one can determine the magnitude of the motor torque and, consequently, the minimum acceleration time of the motor for this case. However, such a start-up cannot be regarded as optimal, since at the beginning of acceleration the motor torque increases, reaching its maximum value at the critical slip, and then decreases as the rotor speed \(\omega\) increases.

To reduce the acceleration time of the gyromotor, starting the motor with variation of the frequency and vol—

voltage, since in this case conditions are created for obtaining the maximum constant motor torque over the entire acceleration time.

The starting process of an asynchronous gyromotor under frequency control is described by the following system of equations:

\[ \left. \begin{aligned} M_\mathrm{э} &= M_c + I \frac{d\omega}{dt},\\ \omega &= \omega_{1N}(\alpha - \beta),\\ M_\mathrm{э} &= f(\alpha,\beta),\\ \alpha &= \varphi(t). \end{aligned} \right\} \tag{3} \]

Here

\[ \alpha=\frac{f_1}{f_{1N}} \]
is the stator frequency referred to the nominal frequency;

\[ \beta=\frac{f_2}{f_{1N}} \]
is the parameter of the absolute slip of the rotor;

\(\omega_{1N}\) is the angular velocity of rotation of the stator field at the nominal frequency \(f_{1N}\).

For a known law of variation of the stator frequency and a starting regime close to critical, the system of equations (3) is written in the form [2]

\[ \kappa_\beta \beta - M_c = I\omega_{1N}\left(\frac{d\alpha}{dt}-\frac{d\beta}{dt}\right), \]

where

\[ \kappa_\beta=\frac{M_N}{\beta_N}. \tag{4} \]

If it is assumed that starting is carried out in the regime of constancy of the magnetic flux, then it is not difficult to see that the torque of the electric motor depends both on the change in rotor speed and on the acceleration of the stator field. Integrating expression (4) with the slip parameter \(\beta\) constant, we find the dependence of the change in stator frequency while ensuring \(M=M_{\max}=\mathrm{const}\):

\[ \alpha=\alpha_0+\frac{\kappa_\beta \beta-M_c}{I\omega_{1N}}\,t. \tag{5} \]

Here \(\alpha_0\) is the stator frequency at the instant the motor is switched on, numerically equal to the critical absolute slip. The acceleration time in this case will be determined by the formula

\[ t_p=T_N \ln \frac{M_k-M_c-I\varepsilon_k}{I\varepsilon_k}. \tag{6} \]

Here

\[ T_N=I\omega_{1N}\frac{s_N}{M_N} \]
is the electromechanical time constant,

\[ \varepsilon_k=\frac{M_k-M_c}{I\omega_{1N}} \]
is the critical rate of change of the stator-field frequency and is equal to the sum of the times for \(\alpha=\varphi(t)\) and for \(\alpha=\mathrm{const}\).

If it is possible to vary the stator frequency above the nominal value, then the acceleration time can be reduced. For this purpose the stator frequency is increased according to (5) until the rotor speed reaches its steady-state value at the nominal frequency, and then it abruptly takes the value \(\alpha=1\). In this case the quantity \(t_p\) pri-

approaches \(T_N\). To determine the limiting value of the frequency, the conditions

\[ \left. \begin{aligned} \beta_N &= \frac{\omega_{1N}-\omega}{\omega_{1N}},\\[4pt] \beta_{\mathrm{gr}} &= \frac{\omega_{\mathrm{gr}}-\omega}{\omega_{1N}} . \end{aligned} \right\} \tag{7} \]

are used. Hence

\[ \alpha_{\mathrm{gr}}=\alpha_0+(1+\beta_{\mathrm{gr}}), \tag{8} \]

where \(\beta_N\) is the nominal slip parameter in the steady-state regime.

To determine the minimum acceleration time under frequency starting, it is necessary to find an analytical optimum. Expression (3) is written as

\[ \left. \begin{aligned} M-\mu &= \frac{d\nu}{d\tau},\\[4pt] \nu &= \alpha-\beta,\\[4pt] M &= \frac{2\Phi^{*2}\beta}{1+\beta^2},\\[4pt] \alpha &= \alpha(t). \end{aligned} \right\} \tag{9} \]

Here

\[ \nu=\frac{\omega}{\omega_c}, \qquad \Phi^*=\frac{\Phi}{\Phi_{\mathrm{opt}}}. \]

Restrictions are imposed on (9) with respect to the amount of heat released in the rotor, \(Q_{\mathrm{dop}}\), and with respect to the magnetic flux \(\Phi\):

\[ \left. \begin{aligned} \int_0^\tau \frac{2\Phi^{*2}\beta}{1+\beta^2}\,d\tau &\leq Q_{\mathrm{dop}},\\[4pt] \Phi^* &\leq 1 . \end{aligned} \right\} \tag{10} \]

To ensure the maximum change in the relative speed of the gyromotor rotor, \(\Delta\nu\), per unit time while satisfying (10),

\[ \Delta\nu=\int_0^\tau \nu\,d\tau \tag{11} \]

it is necessary to find the dependences \(\beta(\tau)\) and \(\Phi^*(\tau)\). They must satisfy the following Euler equations \([3, 5]\):

\[ \left. \begin{aligned} \frac{\partial F}{\partial \Phi^{*2}} -\frac{d}{d\tau}\frac{\partial F}{\partial \dot{\Phi}^{*2}} &= 0,\\[6pt] \frac{\partial F}{\partial \nu} -\frac{d}{d\tau}\frac{\partial F}{\partial \dot{\nu}} &= 0,\\[6pt] \frac{\partial F}{\partial \beta} -\frac{d}{d\tau}\frac{\partial F}{\partial \dot{\beta}} &= 0, \end{aligned} \right\} \tag{12} \]

where

\[ F=\frac{2\Phi^{*2}\beta}{1+\beta^2} -\lambda_1\nu +\lambda\left(\nu+\mu-\frac{2\Phi^{*2}\beta}{1+\beta^2}\right). \]

\(\lambda,\ \lambda_1\) are Lagrange multipliers.

Extremals are found as the solution of (12)

\[ \left. \begin{aligned} \frac{\partial F}{\partial \Phi_2^*} &= \frac{2\beta}{1+\beta^2} - \lambda \frac{2\beta}{1+\beta^2},\\[4pt] \frac{d\lambda}{d\tau} &= \lambda \frac{d\mu}{d\nu},\\[4pt] \lambda &= \frac{1}{1-\beta^2}. \end{aligned} \right\} \tag{13} \]

From the system of equations (12) one determines the dependence of the optimal variation of the rotor frequency for the minimum acceleration time

Fig. 1. Acceleration characteristics of the gyromotor under ordinary and frequency control

of the motor for different dependences of the load torque on the rotor speed

for \(\mu=\mathrm{const.}\)

\[ \lambda = c_1 - \lambda_1 \tau, \tag{14} \]

for \(\mu=\kappa_1 \nu\)

\[ \lambda = c_2 - \kappa_1 \lambda_2 \exp(\kappa_1 \tau), \tag{15} \]

for \(\mu=\kappa_2 \nu^2\)

\[ \lambda = c_3 - \kappa_2 \lambda_3 \exp(\kappa_2 \nu t). \tag{16} \]

For a gyroscopic motor with \(M_{\mathrm{nom}} = 26000\ \text{g·cm/sec}\), experimental investigations were carried out for ordinary and frequency starting (Fig. 1). Under amplitude-frequency control, the stator frequency was varied according to a law close to (5). The acceleration time was reduced by a factor of 4 in comparison with starting at increased voltage and \(\alpha=\text{const.}\)

Fig. 2. Block diagram of the system for optimal control of the speed of a gyromotor

In the practical implementation of the optimal-control system according to the scheme in Fig. 2, the frequency converter (\(FC\)) is implemented on the basis of a thyristor current inverter. The moment at which acceleration ends and braking begins is effected by a switching device (\(SD\)), controlled either by the signal \(I_1\), or by the frequency sensor (\(FS\)) and the polling generator (\(PG\)).

Similar investigations are carried out in [4].

In conclusion, it should be noted that under frequency control of asynchronous motors, in order to obtain large accelerations, current peaks are absent, which is especially important when starting, braking, and reversing a motor with a source of limited power. This circumstance makes it possible to use a lower-power supply source, or, with the selected source, to obtain maximum speed of response.

REFERENCES

1. Collection of articles edited by Doctor of Technical Sciences Prof. A. I. Bertinov. Contactless and Unipolar Electric Machines. Mashinostroenie, 1966.

2. A. A. Bul’gakov. Frequency Control of Asynchronous Electric Motors. Moscow, “Nauka,” 1966.

3. A. A. Fel’dbaum. Foundations of the Theory of Optimal Automatic Systems. Fizmatgiz, 1963.

4. P. L. Lawrenson, I. M. Stephenson. Note on induction-machine performance with a variable-frequency supply. Proc. Instn. Electr. Engrs. 1966, 111, No. 10.

5. Yu. P. Petrov. Optimal Control of an Electric Drive. GEI, 1961.