EFFECT OF ELECTROMAGNETIC TIME CONSTANTS ON THE NATURE OF TIME-OPTIMAL CONTROL OF A DC MOTOR

L. S. Udut, A. E. Alekhin, V. A. Beinarovich

(Presented by the scientific seminar of the Research Institute of Electrical Machines and the Department of Electric Drive of TPI)

In the synthesis of optimal systems, the use of exact mathematical descriptions of the object being optimized leads to a complication of the synthesis, sometimes making it practically impossible. In connection with this, it becomes necessary to obtain simplified mathematical models of objects and to evaluate the influence of the assumptions adopted.

In the synthesis of optimal controls for electric drives, the time constants of motors have a significant influence. The present article investigates the influence of electromagnetic time constants on the nature of time-optimal control of a DC motor, with the aim of determining the conditions under which these constants may be neglected. Suppose that a time-optimal electric drive must, in minimum time, execute a prescribed value of the angle of rotation \(\alpha_{\mathrm{zad}}\), or provide a prescribed rotational speed \(v_{\mathrm{zad}}\) of the system output shaft, for arbitrary initial conditions and the final state \(\alpha_{\mathrm{kon}}=\alpha_{\mathrm{zad}}\), \(v_{\mathrm{kon}}=v_{\mathrm{zad}}\), \(i_{\mathrm{ya\,kon}}=i'_c\). In this case, the final value of the magnetic flux may have either an arbitrary value, or \(\varphi_{\mathrm{kon}}=\mu_c/i'_c\), when the requirement of matching the speeds at the end of control is imposed, \(\dot v=0\). The control must take into account the constraints:

\[ |U_1|\leq 1;\quad |i_{\mathrm{ya}}|\leq i_{\mathrm{ya\,dop}};\quad |v|\leq v_{\mathrm{dop}};\quad |U_2|\leq 1. \]

Taking into account that \(\tau_v>\tau_{\mathrm{ya}}\), we assume that reversal is carried out from the armature-winding side. In this case, the unchanged direction of the flux \(\varphi\) will be considered positive.

We shall determine the control through two channels (by armature voltage and by excitation voltage) for an independently excited motor on the basis of the use of the maximum principle.

The system of equations in relative units (see Appendix) is represented in the form:

\[ \begin{aligned} \dot{\alpha} &= v;\\ \dot{v} &= i_{\mathrm{ya}}\cdot \varphi(i_v)-\mu_c;\\ \dot{i}_{\mathrm{ya}} &= \beta_{\mathrm{ya}}\left[U_1-v\varphi(i_v)-i_{\mathrm{ya}}\right];\\ \dot{i}_v &= \beta_v(U_2-i_v). \end{aligned} \tag{1} \]

In accordance with the maximum principle,

\[ H=\psi_1v+\psi_2\bigl(i_{\mathrm{ya}}\varphi[i_v]-\mu_c\bigr) +\psi_3\beta_{\mathrm{ya}}\bigl[U_1-v\varphi(i_v)-i_{\mathrm{ya}}\bigr] +\psi_4\beta_v(U_2-i_v); \tag{2} \]

\[ \begin{aligned} \dot{\psi}_1 &= 0;\\ \dot{\psi}_2 &= -\psi_1+\psi_3\beta_{\mathrm{ya}}\varphi(i_v);\\ \dot{\psi}_3 &= -\psi_2\varphi(i_v)+\psi_3\beta_{\mathrm{ya}};\\ \dot{\psi}_4 &= \left(-\psi_2 i_{\mathrm{ya}}+\psi_3\beta_{\mathrm{ya}}v\right) \frac{\partial\varphi(i_v)}{\partial i_v}+\psi_4\beta_v. \end{aligned} \tag{3} \]

We take as the control parameters the armature voltage \(I_1\) and the magnetic flux \(\varphi\).

\(H(U_1)=\max\) when \(U_1(t)=\operatorname{sgn}\psi_3\), which, for possible initial conditions in practical systems under the prescribed constraints, provides three or two segments of the control \(U_1(t)\), respectively, when processing \(\alpha_{\mathrm{set}}\) or \(v_{\mathrm{set}}\). In this case, for \(\tau_{\mathrm{a}}<(0.5\div 1)\) and with the current limitation \(i_{\mathrm{a\,adm}}\le 0.5\), the transient processes in the system differ little from aperiodic ones.

The optimal control \(\varphi(t)\), from the condition \(\dfrac{dH}{d\varphi}=0\) when \(H(\varphi)=\max\), is determined in the following form:

\[ \varphi = \frac{ \dfrac{u_1\mu_c^2}{v^2\beta_{\mathrm{a}}^2} - U_1\left(2-\dfrac{\mu_c}{\beta_{\mathrm{a}}v}\right)^2 + U_1\left(2-\dfrac{\mu_c}{\beta_{\mathrm{a}}v}\right) }{ \dfrac{u_1\mu_c}{\beta_{\mathrm{a}}^2v^2\left(2-\dfrac{\mu_c}{\beta_{\mathrm{a}}v}\right)} - v\left(2-\dfrac{\mu_c}{\beta_{\mathrm{a}}v}\right)^2 }, \tag{4} \]

which ensures motion of the system with current

\[ i_{\mathrm{a\,opt}}= \frac{U_1}{2-\dfrac{\mu_c}{\beta_{\mathrm{a}}v}} \]

and acceleration to the steady-state speed

\[ v_{\mathrm{opt}}= \frac{(U_1-i_{\mathrm{a\,opt}})\,i_{\mathrm{a\,opt}}}{\mu_c} = \frac{0.25}{\mu_c}. \]

For \(\mu_c=0\) or \(\tau_{\mathrm{a}}=0\)

\[ \tag{6} \]

\[ \varphi=\frac{U_1}{2v},\qquad i_{\mathrm{a\,opt}}=\frac{U_1}{2}=0.5\,i_{\mathrm{a\,nom}}. \tag{7} \]

The control found (field weakening) is possible only on the final acceleration segment when \(|v|\ge 0.5\). Flux control in processing \(v_{\mathrm{set}}\le v_c\) proves ineffective, especially in systems with a rigid limitation of \(i_{\mathrm{a}}\) and small \(\tau_{\mathrm{a}}\).

The character of \(i_{\mathrm{a\,opt}}\) depends on the ratio \(\mu_c/\beta_{\mathrm{a}}\), and already for \(\mu_c/\beta_{\mathrm{a}}=\mu_c\tau_{\mathrm{a}}\le 0.25\), which is usually satisfied for DC servo motors, \(i_{\mathrm{a\,opt}}\) differs little from the value 0.5 corresponding to systems with \(\mu_c=0\) or \(\tau_{\mathrm{a}}=0\). When the armature-current limitation is at the level \(i_{\mathrm{a\,adm}}<0.5\), flux control does not depend on \(\tau_{\mathrm{a}}\) or \(\mu_c\).

Thus, with a rigid limitation of the armature current on the flux-control segments, the inertia of the armature winding may be neglected.

Let us find the required control \(I_2(t)\), which ensures the required character of variation of \(\varphi(t)\).

We take \(\tau_{\mathrm{a}}=0\) and a piecewise-linear approximation of the magnetization curve

\[ \varphi=\varphi_0\pm k i_{\mathrm{b}}, \tag{8} \]

then

\[ U_2= \frac{U_1-i_{\mathrm{a}}}{kv} - \frac{\varphi_0}{k} - \frac{(U_1-i_{\mathrm{a}})^2 i_{\mathrm{a}}}{k\beta_{\mathrm{b}}v^3} + \frac{U_1-i_{\mathrm{a}}}{k\beta_{\mathrm{b}}v^2}\,\mu_c, \tag{9} \]

where \(i_{\mathrm{a}}=i_{\mathrm{a\,opt}}\), or \(i_{\mathrm{a}}=i_{\mathrm{a\,adm}}\) if \(i_{\mathrm{a\,adm}}<i_{\mathrm{a\,opt}}\). Figure 1 gives the curves of variation of \(I_2\) for the SL-262 motor (connected according to the separately excited scheme), which ensure the attainment of the extremal regime for different \(\beta_{\mathrm{b}}\). Taking into account \(U_2\le 1\) and \(U_{2\,\mathrm{init}}=-1\), the required law \(\varphi(t)\) (4) is fulfilled for \(\beta_{\mathrm{b}}\le \beta_{\mathrm{b\,crit}}\), where

\[ \beta_{\mathrm{b\,crit}}= \frac{ -(U_1-i_{\mathrm{a\,init}}) \left[(U_1-v_{\mathrm{init}}\varphi_{\mathrm{init}})\varphi_{\mathrm{init}}-\mu_c\right] }{ (U_{2\,\mathrm{init}}-i_{\mathrm{b\,init}})\,k\,v_{\mathrm{init}}^2 }. \tag{10} \]

Analysis of expressions (9) and (10) shows that limiting \(i_{\mathrm{a}}\) facilitates the realization of the required control law.

When $\beta_{\mathrm{v}} > \beta_{\mathrm{vkr}}$, in order to achieve maximum speed of response, anticipatory flux weakening is necessary (point a in Fig. 2). For the same purpose, anticipatory flux restoration will also be required (before switching $U_1$) on the braking interval (point г in Fig. 2). Theoretical and experimental studies have shown that the influence of $\tau_{\mathrm{v}}$ on the speed of response when entering the flux-control interval is insignificant (within the limits of experimental error). To simplify implementation, one should restrict oneself to choosing a common point at which flux weakening begins.

Fig. 1.

The influence of $\tau_{\mathrm{v}}$ is manifested to a greater degree on the braking interval. Combining the instants of switching $U_1$ and restoring the flux may cause an increase in braking time of up to 10% for real values $\tau_{\mathrm{v}} \leq (10 \div 20)\tau_{\mathrm{m}}$. For cases in which such a reduction in speed of response is undesirable, anticipatory flux restoration should be applied on the basis of calculating the flux-restoration line (LRF). For preliminary calculations, the method for determining the LRF reduces to the following:

1. On the basis of the system of equations describing the motion on the interval $t_4 - t_5$ ($\tau_{\mathrm{ya}} = 0$) (Fig. 2),

\[ \begin{aligned} \dot{v} &= (U_1 - v\varphi)\varphi - \mu_{\mathrm{c}};\\ \dot{i}_{\mathrm{v}} &= \beta_{\mathrm{v}}(U_2 - i_{\mathrm{v}}) \end{aligned} \tag{11} \]

for $U_2 = -1$ and $U_1 = +1$, the dependence $\tau_1(v_{\mathrm{per}})$ is constructed for various $v_{\mathrm{nach}} = \mathrm{const}$. Here $\tau_1$ is the time of motion on the interval $t_4 - t_5$; $v_{\mathrm{nach}}$ and $v_{\mathrm{per}}$ are the values of the velocity at the instants $t_4$ and $t_5$, respectively.

1. On the basis of the system of equations

\[ \begin{aligned} \dot{v} &= i_{\mathrm{ya}}\varphi - \mu_{\mathrm{c}};\\ \dot{i}_{\mathrm{v}} &= \beta_{\mathrm{v}}(U_2 - i_{\mathrm{v}}), \end{aligned} \tag{12} \]

describing the motion on the segment \(t_5 - t_k\), for \(U_2 = +1\) and \(i_{ya} = -i_{yadop}\), the dependence \(\tau_2(v_{per})\) is constructed, where \(\tau_2\) is the time of motion on the segment \(t_5 - t_k\). The solution of (12) has the form:

\[ v = - ( i_{yadop}\varphi_0 + k i_{yadop} U_2 + \mu_c ) t + k\tau_v i_{yadop} (U_2 - i_{ynach}) \left(1 - \frac{t}{e^{\tau_v}}\right) + v_{nach}. \tag{13} \]

1. On the basis of \(\tau_1(v_{per})\) and \(\tau_2(v_{per})\), the dependence \(\tau_3(v_{per})\) is constructed for different \(v_{nach} = \mathrm{const}\), where \(\tau_3 = \tau_1 + \tau_2\).

2. Choosing \(\tau_3 = \min.\), we find the dependence \(v_{per}(v_{nach})\).

3. To construct the line of braking onset (LBO) and the line of flux restoration in the phase plane \((\alpha, v)\), the value of \(\alpha\) is found on the segments considered. The angle traversed by the system on the segment \(t_5 - t_k\)

Fig. 2

and required for constructing the LBO is determined on the basis of expression (13).

The determination of the angle for the segment \(t_4 - t_5\) is performed from the system of equations (11).

In systems with \(\tau_ya \ne 0\) and \(\tau_v \ne 0\), under the additional condition that after acceleration the motion with \(v_{zad}\) be ensured, the control should be terminated at \(i_{yakon}\varphi_{kon} = \mu_c\). It is most expedient to take \(i_{yakon} = \min.\), which is ensured by the condition \(U_{1kon} = \pm 1\) \((v_{zad} > v_n)\).

Satisfaction of the condition \(\tau(t) = 0\) when the system reaches \(v_{zad}\) is possible if the condition \(\tau_{ya} < \tau_{yakr}\) is satisfied, where

\[ \tau_{yakr} = -\frac{2\tau_v} {i_{yaksn}\left(1 - \dfrac{i_{yakon}\varphi_n}{\mu_c}\right)} > \tau_v . \tag{14} \]

If \(\tau_{ya} = 0\), then entry onto the segment \(\mu = \mu_c\) occurs instantaneously, and the control \(U_1\) must have the form (in order to achieve \(i_{yakon} = i_{smin}\))

\[ U_1 = v_{zad}\varphi(t) + \frac{\mu_c}{\varphi(t)} . \tag{15} \]

In the general case, the optimal control of a series-excited DC motor has the same character as the control of an independently excited motor. However, for a DC motor in the general case \(i_{\mathrm{a}} \ne i_{\mathrm{f}}\), and the processes in its circuits are determined by the specific implementation schemes, which will require additional studies in each case. For \(i_{\mathrm{a}} = i_{\mathrm{f}}\), all the statements made above prove to be valid. From the standpoint of the speed of response of systems with DC motors, preference should be given to reversal methods in which, during the braking interval, the largest flux value is maintained, while the armature current rapidly decreases in the plugging mode.

APPENDIX

\(i_c\) — value of the armature current in steady-state operation at \(\mu=\mu_c\), in p.u.;

\(u_1\) — voltage on the motor armature, in p.u.;

\(u_2\) — voltage on the motor field winding, in p.u.;

\(i_{\mathrm{a\,adm}}\) — admissible current value, in p.u.;

\[ \alpha=\frac{A}{\omega_\delta T_{\mathrm{m}}} \]
— angle of rotation of the system shaft in relative units (p.u.);

\[ t=\frac{T}{T_{\mathrm{m}}} \]
— time, in p.u.;

\(T_{\mathrm{m}}\) — electromechanical time constant of the motor;

\[ v=\frac{\omega}{\omega_\delta} \]
— rotational speed, in p.u.;

\(\omega_\delta\) — base value of rotational speed;

\[ i_{\mathrm{a}}=\frac{I_{\mathrm{a}}}{I_\delta} \]
— armature current, in p.u.;

\[ \varphi=\frac{\Phi}{\Phi_\delta} \]
— magnetic flux, in p.u.;

\(\mu_c=i_c\varphi_c\) — value of the static torque, in p.u.;

\[ \beta_{\mathrm{a}}=\frac{R_{\mathrm{a}}}{L_{\mathrm{a}}T_{\mathrm{m}}} \]
— quantity reciprocal to the electromagnetic constant of the armature winding, in p.u.;

\(H\) — Hamiltonian;

\(\psi\) — auxiliary function;

\(K\) — proportionality coefficient;

\(\varphi_0\) — initial value of the flux on an individual approximation segment;

\[ \beta_{\mathrm{f}}=\frac{R_{\mathrm{f}}}{L_{\mathrm{f}}T_{\mathrm{m}}} \]
— quantity reciprocal to the electromagnetic constant of the field winding, in p.u.;

\(\tau_{\mathrm{f}}, \tau_{\mathrm{a}}\) — time constants of the field and armature windings, and their values in p.u.;

\[ i_{\mathrm{f}}=\frac{I_{\mathrm{f}}}{I_{\mathrm{fN}}} \]
— current in the field winding, in p.u.;

\(i_{\mathrm{a\,opt}}, i_{\mathrm{a\,sc}}\) — optimal and short-circuit value of the armature current, in p.u.;

\(v_c\) — rotational speed, in p.u., at \(\mu=\mu_c\);

\(\beta_{\mathrm{a\,cr}}, \beta_{\mathrm{f\,cr}}\) — critical values of \(\beta_{\mathrm{a}}\) and \(\beta_{\mathrm{f}}\).

REFERENCES

1. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko. Mathematical Theory of Optimal Processes. Fizmatgiz, 1961.

2. V. P. Chistov, V. I. Bondarenko, V. A. Svyatoslavskii. Optimal Control of Electric Drives. Energiya, 1968.