On the Optimal Control of a Separately Excited DC Motor

I. P. Trofimuk

(Presented by the scientific seminar of the Electromechanical Faculty)

Many mechanisms in metallurgical production are driven by a separately excited DC motor. In almost all cases in which such a drive is used, it is necessary to ensure the process of following a command signal or a disturbance, optimal in speed of response, or close to optimal. Examples of such systems may include the drive of a pressure device, the drive of electrodes of an electric arc steelmaking furnace, etc.

Questions of the synthesis of control devices for such drives are covered in a number of works 2, 3.

In the present article the problem of synthesizing an optimal system is considered in greater detail for the case when, in addition to dry friction, there is a load due to the excess weight of the unbalanced parts of lifting mechanisms, as occurs in the examples cited. In addition, we assume that the command \(X_0\), or a disturbance reduced to it, and the current coordinate \(X\) are applied to the control device with a transmission coefficient \(\alpha\), which may vary during certain periods of operation. We shall assume that the reducer has no backlash.

In considering the problem, we shall temporarily abandon the usual representation of a closed automatic-control system. We assume that the system consists of an invariant part \(C\) and a regulator \(P\), as shown in Fig. 1. In our case we shall consider the regulator to be inertialess, which is close to reality, for example, in an ionic drive. The inertia of the regulator can be taken into account. However, this is not part of the problem of the present article. In addition, as a number

Fig. 1

of the authors [2, 5], satisfactory results are sometimes obtained even without such allowance, by selecting the parameters during tuning.

Let us first consider the case in which, while executing the command, the motor does not reach the maximum permissible speed. The cases in which the motor reaches the maximum speed, as well as the cases of small deviations, will be considered later.

The remaining natural constraint is that the motor current cannot exceed the maximum permissible value.

\[ i \leq \sigma I_m, \qquad \text{where } \sigma=\pm 1 . \tag{1} \]

Obviously, condition (1) can be rewritten as

\[ \frac{U}{r}-\frac{k_e}{r}\omega \leq \sigma I_m, \tag{2} \]

where

\(U\) — voltage at the regulator output;
\(r\) — total resistance of the armature circuits;
\(\omega\) — angular velocity of the motor;
\(e_d\) — back e.m.f. of the motor;

\[ k_e=\frac{e_d}{\omega}. \]

The equation of motion for the motor is

\[ J\frac{d\omega}{dt}=M_d-M_c\operatorname{sign}\omega, \tag{3} \]

where \(J\) is the moment of inertia on the motor shaft;
\(M_d\) — motor torque;
\(M_c\) — resisting torque on the motor shaft, which varies depending on the direction of motion.

Introducing

\[ k_m=\frac{M_d}{i}, \]

(3) can be rewritten in the form

\[ J\frac{d\omega}{dt}=k_m i-M_c\operatorname{sign}\omega. \tag{4} \]

It is known that the process will be time-optimal if the value of the maximum constrained quantity from (1) is introduced into the equation of motion. Carrying out this substitution, we obtain

\[ J\frac{d\omega}{dt}=k_m\sigma I_m-M_c\operatorname{sign}\omega. \tag{5} \]

Introducing the notation:

\(\xi_0\) — gear ratio of the reducer;

\[ R_0=\frac{X}{2\pi n} \]

— coefficient of conversion from rotary motion to translational motion;

\(X\) — travel distance;

we can write the equation of the optimal process in the form:

\[ \frac{d^{2}x}{dt^{2}}=\sigma\,\frac{K_{M}R_{0}I_{M}}{J\xi_{0}}-\frac{M_{c}\cdot R_{0}}{J\xi_{0}}\operatorname{sign}\frac{dx}{dt} \tag{6} \]

or

\[ \frac{d^{2}x}{dt^{2}}=\sigma A-B\operatorname{sign}\frac{dx}{dt} \tag{7} \]

Obviously \(A>B\).

\[ \tag{8} \]

Transform (7) to the form:

\[ \frac{d^{2}X}{dt^{2}} = \sigma\left|A-B\,\frac{\operatorname{sign}\dfrac{dX}{dt}}{\sigma}\right| = \sigma I, \tag{9} \]

where

\[ I=\left|A-B\,\frac{\operatorname{sign}\dfrac{dX}{dt}}{\sigma}\right| \tag{9a} \]

Consider the form of the optimal process in the phase plane

\[ (x,y),\qquad \text{where } x=X_{0}-X,\quad y=\frac{dx}{dt}. \tag{10} \]

\(X_{0}\) is the equivalent change in the regulator setpoint;
\(X\) is the current coordinate.

We assume that the equivalent change in the setpoint \(X_{0}\) belongs to the class of admissible disturbances of the form \(X_{0}=A_{0}+A_{1}t\). In what follows we assume \(X_{0}=\mathrm{const}\), which does not change the result.

Obviously,

\[ \left. \begin{aligned} y&=\frac{dx}{dt}=-\frac{dX}{dt};\qquad \operatorname{sign}y=-\operatorname{sign}\frac{dX}{dt};\\ \frac{d^{2}x}{dt^{2}}&=-\frac{d^{2}X}{dt^{2}}; \end{aligned} \right\} \tag{11} \]

Then

\[ \frac{d^{2}x}{dt^{2}} = -\sigma I = -\sigma\left|A+B\,\frac{\operatorname{sign}y}{\sigma}\right| \tag{12} \]

To determine the switching line \(\sigma\) in the phase plane, consider the optimal trajectories passing through the origin. To study them, we reverse the counting of time from the instant \(t_{0}\) at which the setpoint is completed, introducing \(\tau=t_{0}-t\).

Then

\[ \frac{dx}{dt} = \frac{dx}{d\tau}\cdot\frac{d\tau}{dt} = -\frac{dx}{d\tau}; \qquad \frac{d^{2}x}{dt^{2}} = \frac{d^{2}x}{d\tau^{2}} \tag{13} \]

and (12) is written in the form:

\[ \frac{d^{2}x}{dt^{2}}=-\sigma I. \tag{14} \]

Solving (14) with zero initial conditions, we obtain:

\[ \left. \begin{aligned} y &= -\frac{dx}{d\tau}=\sigma I\tau,\\ x &= -\frac{\sigma I}{2}\cdot \tau^{2} \end{aligned} \right\} \tag{15} \]

Thus

\[ \left. \begin{aligned} \operatorname{sign} y &= \operatorname{sign}\sigma=\sigma,\\ \operatorname{sign} x &= -\operatorname{sign}\sigma \end{aligned} \right\} \tag{16} \]

and (9, a) can be written in the form:

\[ I=A+B \tag{17} \]

Eliminating \(\tau\) from (15), we obtain the equation of the optimal trajectory

\[ x=-\sigma\frac{y^{2}}{2I}=-\sigma\frac{y^{2}}{2(A+B)}. \tag{18} \]

Let us consider the quadrants of the phase plane \((x,y)\), see Fig. 2. In the first and second quadrants \(y>0\), which corresponds to downward motion. In this case

\[ M_c=M_g=M_{\mathrm{tr}}-M_v, \tag{19} \]

where \(M_{\mathrm{tr}}\) is the dry-friction torque;
\(M_v\) is the torque of the excess weight.

Fig. 2

Accordingly, for \(y>0\) we set \(B=B''\). In the third and fourth quadrants \(y<0\), which corresponds to upward motion. In this case

\[ M_c=M_1=M_{\mathrm{fr}}+M_w \tag{20} \]

and

\[ B=B',\quad \text{for } y<0 . \]

Let us investigate the optimal trajectory for \(\sigma=-1\); in this case \(y<0;\ x>0\), i.e., the optimal trajectory lies in the fourth quadrant. Substituting the value \(B=B'\) into (18), we obtain:

\[ x=-\sigma \frac{y^2}{2(A+B')}=-\frac{y^2\operatorname{sign} y}{2(A+B')} \tag{21} \]

The second branch of the optimal trajectory, for \(\sigma=+1\), is located in the second quadrant, and its equation will evidently be

\[ x=-\sigma \frac{y^2}{2(A+B'')}=-\frac{y^2\operatorname{sign} y}{2(A+B'')} \tag{22} \]

Substituting the quantity \(B\) in the form

\[ B=\frac{B'+B''}{2}+\frac{B''-B'}{2}\operatorname{sign} y \tag{23} \]

we can combine (21) and (22) and write the equation of the optimal trajectory:

\[ x=-\frac{y^2\operatorname{sign} y}{2\left(A+\frac{B'+B''}{2}+\frac{B''-B'}{2}\operatorname{sign} y\right)} . \tag{24} \]

The equation for the switching line \(\psi_1\) will have the form \(\psi_1=0\), where from (24):

\[ \psi_1=x+\frac{y^2\operatorname{sign} y}{2(A+B_1+B_2\operatorname{sign} y)};\quad \text{where }\frac{B'+B''}{2}=B_1;\quad \frac{B''-B'}{2}=B_2 . \tag{25} \]

Instead of the control quantity \(w=\psi_1\), one may feed to the input of the switching block a quantity \(w_1\) equal to it in sign:

\[ w_1=2\omega=2(A+B_1+B_2\operatorname{sign}y)x+y^2\operatorname{sign}y . \tag{26} \]

Temporarily taking \(\alpha=1\) (see Fig. 1), one can synthesize the regulator structure. A variant of a block diagram satisfying conditions (1) and (26) is given in Fig. 3.

To eliminate disturbances, the differentiators D1 and D2 must be provided with filters. To eliminate excessive oscillations of the representative point near the equilibrium position, the relay elements H2, H3, and \(У\) must have either a dead zone or characteristics of the form shown in Fig. 3. The circuit contains a multiplying element M. The quantity \(v\), determined by the sign of \(w_1\) and by condition (1), is applied to the input of the switching device \(У\).

Let us consider the operation of the circuit under small disturbances. In this case the representative point has a slight deviation from the origin of coordinates, and the quantities \(x\) and \(y\) are small. Then the quantities \(y^2\) and \(2B_2x\), as second-order small quantities, may be neglected, and the equivalent circuit for small deviations will have the form shown in Fig. 4. The dif-

The differentiator \(D2\) is introduced to improve the transient characteristics of the given closed-loop control system.

Fig. 3

Now let us consider the operation of the system for such deviations, when the motor reaches a steady-state speed. In this case the optimal trajectories of the representative point in the phase plane will have horizontal segments, as shown in Fig. 2.

Fig. 4

The equation of the switching line for these segments shall be written in the form \(\psi_2=0\), where

\[ \psi_2 = y + y_m \operatorname{sign} x, \tag{27} \]

and the value \(y_m\) of the maximum permissible speed is established from certain considerations, for example, design or technological ones.

Since the permissible raising and lowering speeds may be different, we assume

\[ \left. \begin{aligned} y_m &= y_m' \quad &&\text{for } \operatorname{sign} x = +1,\\ y_m &= y_m'' \quad &&\text{for } \operatorname{sign} x = -1 \end{aligned} \right\} \tag{28} \]

Then (27) can be written in the form

\[ \psi_2 = y + y_1 + y_2 \operatorname{sign} x, \qquad \text{where}\quad y_1=\frac{y'+y''}{2};\qquad y_2=\frac{y'-y''}{2}. \tag{29} \]

Instead of the control quantity \(w=\psi_2\), one may feed to the input of the switching element H3 a quantity \(w_2\) equal to it in sign:

\[ w_2 = 2l w - 2l y + 2l\left(y_1+y_2\operatorname{sign}x\right). \tag{30} \]

Thus, the element H3 must be controlled either by the quantity \(w_1\) or by the quantity \(w_2\), depending on their relation. From the phase diagram of Fig. 2 it is seen that for \(x<x_k''\) and for \(x>x_k'\) the quantity \(w_2\) must be fed to the input of element H3, while for \(x_k'' \le x \le x_k'\) the quantity \(w_1\) must be fed. For this purpose it is necessary to introduce an additional switching device into the block diagram. To the input of this device one may, for example, feed the quantity

\[ \sigma = x_1 - \frac{x_k' + x_k''}{2} - \frac{x_k' - x_k''}{2}\operatorname{sign}x. \tag{31} \]

Then the block diagram will take the form shown in Fig. 5.

Fig. 5

Let us now determine the quantity \(U_m\). For this purpose introduce the notation

\(U_{\mathrm{n}}\) — the rated voltage of the motor;

\(I_{\mathrm{n}}\) — the rated current of the motor;

\(\omega_{\mathrm{n}}\) — the rated angular velocity of the motor;

\[ R_{\mathrm{n}}=\frac{U_{\mathrm{n}}}{I_{\mathrm{n}}} \]
— the rated resistance of the motor;

\[ \rho=\frac{r}{R_{\mathrm{n}}} \]
— the relative resistance of the armature circuit.

Taking \(I_m \approx 2 I_{\mathrm{n}}\) and \(U_{\mathrm{n}}\approx K_{\mathrm{en}}\omega_{\mathrm{n}}\), and using the notation introduced, we obtain:

\[ U_m \approx (K_v K_\phi + 2\rho), \tag{32} \]

where

\[ K_v=\frac{\omega_{\max}}{\omega_{\mathrm{n}}} \]
is the speed-excess coefficient;

\[ K_\phi=\frac{K_e}{K_{\mathrm{en}}} \]
is the field-weakening coefficient.

Since usually \(\rho \leq 0.3\) and \(K_\phi K_v \leq 1.4\), it follows that \(U_m \leq 2U_{\mathrm{n}}\).

Very often the equivalent setpoint \(X_0\) and the coordinate \(X\) are fed to the regulator input with a scale-conversion coefficient \(\alpha\), that is, instead of the mismatch \(x\), the quantity \(\alpha x\) is supplied. In this case, to ensure regulation along the optimal trajectory, the control quantities \(w_1\) and \(w_2\) must have the form:

\[ w_1=\frac{2}{\alpha}(\alpha x)+(\alpha y)^2 \operatorname{sign} y, \tag{33a} \]

\[ w_2=(\alpha \dot y)+\alpha (y_1+y_2 \operatorname{sign} x). \tag{33b} \]

For \(\alpha=\text{const}\), (26) and (30) differ from (33) only by a constant factor, that is, the block diagram of the regulator remains the same. But sometimes the coefficient \(\alpha\) undergoes slow changes over very wide limits. Then, obviously, in order to maintain the optimal process it is necessary by some means to determine the coefficient \(\alpha\) and introduce it into the regulator.

For calculating \(\alpha\) and introducing it into the regulator, the following method may be used. Suppose that we can obtain, either from a tachogenerator or by some other means, the quantity

\[ \beta \frac{dx}{dt}=-\beta y; \]

where \(\beta\) is a constant scale coefficient.

Then

\[ \alpha=\left|\frac{(\alpha y)}{(\beta y)}\right|\cdot \beta . \tag{34} \]

From (33a) and (34) we find a new expression for the control quantity:

\[ w_1=2\beta |(\alpha x)(\alpha y)|+(\alpha y)^2\cdot(\beta y)\operatorname{sign} y \tag{35a} \]

or

\[ w_1=2\beta |(\alpha x)|+(\alpha y)\cdot(\beta y)\operatorname{sign} y. \tag{35b} \]

In this case we shall construct the control quantity \(w_2\) directly from the signal \((\beta y)\).

\[ w_2=(\beta y)+\beta(\dot y_1+y_2 \operatorname{sign} x). \tag{36} \]

The control quantity for the switching block — \(\delta\) — will have the form:

\[ \delta = |\alpha x|\cdot |\beta y| - \beta |\alpha y|\cdot \left( \frac{x'_k+x''_k}{2}+\frac{x'_k-x''_k}{2}\operatorname{sign}x \right) \tag{37} \]

From (35b), (36), and (37) one can construct the structural diagram of the regulator. A drawback of this regulator is that, when moving with steady-state speed, self-oscillations will arise near the switching line \(\psi_2\) in the phase plane. These self-oscillations may be undesirable.

Fig. 6

If the maximum admissible speed is not strictly fixed, then, to eliminate self-oscillations, the control device \(U\) must operate in two modes: one for the period of acceleration and braking, and the other for motion with steady-state speed. In the second mode, the control device \(U\) must supply the voltage \(|y|=\mathrm{const}\), determined only by the required speed of displacement. Switching of the operating modes must be carried out depending on the magnitude of the speed and on the sign of the quantity \(\delta\). An example of the structural diagram of such a regulator is given in Fig. 6.

In conclusion, let us note that, even with a variable scale factor \(\alpha\), the regulator circuit can be simplified. Such a simplification becomes possible if the switching line \(\psi_1\) in the phase plane is replaced by a straight line passing through the origin. Let us write the equation of this straight line:

\[ x=-ay, \tag{38} \]

where

\[ \begin{aligned} a &= a'' \quad \text{for } y>0,\\ a &= a' \quad \text{for } y<0. \end{aligned} \tag{39} \]

The equation of the straight lines, taking (39) into account, can be written in the form:

\[ x=-\left(\frac{a''+a'}{2}+\frac{a''-a'}{2}\operatorname{sign}y\right)\cdot y \tag{40} \]

or

\[ x=-a_1y-a_2|y|, \tag{40a} \]

where

\[ a_1=\frac{a''+a'}{2};\qquad a_2=\frac{a''-a'}{2}. \]

Then the control quantity \(w\) will take the form:

\[ w=x+a_1y+a_2|y|. \tag{41} \]

Obviously, at the input of the switching unit, instead of the quantity \(w\), one may apply the quantity \(w_1\), equal to it in sign:

\[ w_1=(\sigma x)+a_1(\sigma y)+a_2|\sigma y|. \tag{42} \]

Braking with the control quantity (42) is, of course, slower than with optimal regulation.

In order to clarify the possibilities of replacing the optimal trajectory by a straight line, let us examine in more detail the motions that occur under such a replacement. For definiteness, consider the fourth quadrant of the phase plane in Fig. 2 and Fig. 7. The relations of interest for this case can be obtained from (11), (12), (18), (38), (39), and (42). For simplicity we assume \(I=\mathrm{const}\), \(a=\mathrm{const}\), which will not change the result qualitatively.

From an analysis of the phase diagram in Fig. 7 it is seen that the character of the motion depends on the coordinates of the point \(y1\) at which the phase trajectory of acceleration meets

Fig. 7

with the straight line (38). Under optimal regulation, the subsequent motion from \(Y1\) takes place along the trajectories \(Y1=B\) and \(B=0\), and the time of this motion is minimal. Under control according to (42), the motion from \(Y1\) to \(Y2\) takes place along the “optimal” parabola, and from \(Y2\) to 0 along a certain broken line lying on the straight line \(x=-ay\).

If the switching element \(H3\) has a sufficiently small dead zone, it may be considered approximately that the motion from \(Y2\) to 0 takes place along the straight line \(x=-ay\), that is, is exponential in time. It is clear that, with exponential motion, the point 0 of the phase diagram can be reached only after an infinitely long interval of time. However, striving necessarily to reach the point \((0,0)\) is unnecessary and may even be harmful, since under a relay mode of regulation and with a highly sensitive regulator the occurrence of self-oscillations about this point is inevitable, which is often undesirable. Therefore we shall assume that some mismatch \(\delta x_k=\delta x_k\) is admissible, that is, in the phase plane \((x,y)\) near the point 0 there is some region \(S\) such that, when the representative point enters this region, the regulation process may be regarded as completed. Let this region be bounded by the coordinates:

\[ \begin{aligned} x &= \delta x_k;\quad y = y_k,\\ \text{where}\quad y_k &= -\sqrt{2\delta x_k I} \end{aligned} \tag{43} \]

To determine the relations between the coordinates \(y_1\) and \(y_2\) and to calculate the time relation, we obtain from (11) and (12) the time dependences of the coordinates and the equations of the trajectories of motion in the phase plane:

\[ \begin{aligned} y &= -\sigma I t + y_{\text{init}},\\ x &= -\frac{\sigma I t^2}{2}+y_{\text{init}}t+x_{\text{init}} \end{aligned} \tag{44} \]

\[ x=\sigma\,\frac{y_{\text{init}}^{2}-y^{2}}{2I}+x_{\text{init}} \tag{45} \]

For \(\sigma=+1\), equation (45) defines a family of acceleration parabolas; for \(\sigma=-1\), a family of braking parabolas.

From the simultaneous solution of (18), (38), and (45), for \(\sigma=-1\) we obtain the coordinates of the intersection of the braking parabola with the straight line:

\[ y_{1,2}=\frac{y_M}{2}\pm \sqrt{\left(\frac{y_M}{2}\right)^2+y_{\text{init}}^2-y_My_{\text{init}}} \tag{46} \]

That is, the points \(Y1\) and \(Y2\) are located symmetrically with respect to the point \(A\) with coordinates \(\dfrac{x_k}{2},\ \dfrac{y_M}{2}\). At the point \(A\), the derivative \(\dfrac{dy}{dx}\) and the tangent of the angle of inclination of the straight line \(x=-ay\) coincide, and the points \(Y1\) and \(Y2\) merge into one.

Thus, when the point \(Y1\) falls below the point \(A\) in Fig. 7, under the control law (41) the subsequent motion of the representative point consists of motion along a parabola and time-exponential motion along a straight line in the plane \((x,y)\). When \(Y1\) falls at the point \(A\) or above it, the subsequent motion of the representative point will be only along the straight line.

If the time of optimal motion \(t_0\) from the point \(Y1\) to the origin of coordinates is taken as the base time, then, in motion with the control law

after applying (41), after the time \(t_0\), there remains a certain mismatch \(\delta x_k\)

\[ \delta x_k = \frac{\Delta x_k} {\exp\left\{2\sqrt{2(1-\Delta)^2+2(1-\Delta)+6\Delta-4}\right\}}, \quad \text{for } y_1 \leq \frac{y_m}{2} \tag{47} \]

The value \(\Delta\) for (47) is determined from the relation

\[ y_1=(1-\Delta)y_m, \tag{48} \]

where \(y_1\) is the coordinate of point \(Y1\).

Or:

\[ \delta x_k = \frac{\Delta x_k} {\exp\left\{4\sqrt{\frac{\Delta^2}{2}+\frac{\Delta}{2}-2\Delta}\right\}}, \quad \text{for } y_1 \geq \frac{y_m}{2} \tag{49} \]

The value \(\Delta\) for (49) is determined from the relation:

\[ y_1=\Delta y_m, \tag{50} \]

where \(y_1\) is the coordinate of point \(Y1\). Point \(Y1\) lies on the straight line \(x=-ay\) above point \(A\). The graph of the dependence \(\delta\left(\frac{y_1}{y_m}\right)\) is given in Fig. 8. From

Fig. 8

the graph it is evident that the values of \(\delta\) do not exceed \(\delta=0.124\). If such a residual mismatch is admissible, then replacing the switching line (24) by a straight line seems possible.

It is necessary, however, to note that the same final mismatch \(\delta x_k\) can also be obtained when moving along optimal trajectories in a time \(t_\delta < t_0\). The relation between the time \(t_e\) of motion under the control law (41) up to the mismatch \(\delta x_k\) and the time \(t_\delta\) of motion along optimal trajectories up to the same mismatch is determined by the expression:

\[ \frac{t_e}{t_\delta} = \frac{1}{2}\cdot \frac{ 2(1-2\Delta)+\ln\frac{\Delta}{\delta} }{ \sqrt{2(1-\Delta)^2+2(1-\Delta)}-(1-\Delta)-\sqrt{\delta} }, \quad \text{for } y_1 \leq \frac{y_m}{2} \tag{51} \]

\[ \frac{t_e}{t_\delta} = \frac{1}{2}\cdot \frac{ \ln\frac{\Delta}{\delta} }{ 2\sqrt{\frac{\Delta^2}{2}+\frac{\Delta}{2}-\Delta}-\sqrt{\delta} }, \quad \text{for } y_1 \geq \frac{y_m}{2} \tag{52} \]

The quantities \(\Delta\) for (51) and (52) are determined from (48) and (50), respectively. The graphs of the dependences

\[ \frac{t_c}{t_\delta}=f\left(\lambda,\frac{y_1}{y_m}\right) \]

are shown in Fig. 9. From these graphs it is evident that replacement of the switching line by a straight line is far from always possible and must be justified with sufficient care in each individual case, which to a certain extent contradicts recommendation 2.

Fig. 9

The issues considered in this article concerning the construction of the structure of controllers when the scale factor of the signal and the disturbance vary, under different static loads and at lifting and lowering speeds, provide a basis for designing optimal controllers under these conditions. The creation of such a controller is especially relevant for the drive of the electrodes of an arc steel-melting furnace, where all the listed features are present.

The analysis presented of the phenomena that arise when the optimal switching line is replaced by a straight line makes it possible to assess the feasibility of such a replacement, which is often desirable in order to simplify the structure of the controller circuit.

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