Transient and Steady-State Processes in a Discrete Control System for the Rotational Speed of a DC Micromotor

I. V. Bulin-Sokolov, S. M. Mironov

(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)

At present, pulse-width control systems have found wide application for regulating the rotational speed of separately excited DC micromotors.

The main requirements imposed on systems for regulating the rotational speed of an electric drive are the quality of transient processes and the accuracy of the systems under various disturbing influences. Therefore, when designing control systems it is necessary to strive to optimize these quality characteristics. This can be achieved by knowing the nature of the influence of various parameters on the quality of transient processes and on system accuracy, and by selecting optimal values of these parameters. Treating pulse-width systems for regulating the rotational speed of an electric drive as continuous systems in their analysis, even when the condition of Kotelnikov’s theorem is satisfied, does not make it possible to investigate the behavior of the system with sufficient completeness. For example, it is in principle impossible to take into account the influence of changes in the relative duration of the control pulses in the nominal regime on the quality characteristics of interest. Therefore, in the investigation the system was considered as a pulse system. The apparatus of the discrete Laplace transform was used as the mathematical basis of the study.

In carrying out the study, the following assumptions were adopted:

1) electromagnetic processes in the armature circuit of the electric motor may be neglected;

2) the relative duration of the control pulses changes insignificantly;

The block diagram of the pulse-width control system is shown in Fig. 1a.

The feedback signal, produced by the measuring element (ME)—a tachogenerator—and having a voltage frequency proportional to the rotational speed of the electric motor, is applied to the input of the sensing element (SE), which is a frequency-phase discriminator. The latter converts the alternating voltage into a DC voltage, the polarity and magnitude of which are a function of the deviation of the frequency of the input signal from the prescribed value. The device forming the control signal is a pulse-width modulator (PWM). Control of the rotational speed of the electric motor is effected through the armature circuit by varying the relative duration of the control pulses.

Fig. 1

To determine the transfer function of the closed pulse system, it is more convenient to reduce the external actions acting on the system to the input of the pulse element [2].

Introduce the dimensionless quantities

\[ \bar t=\frac{t}{T}; \qquad q=p\cdot T, \]

where \(T\) is the PWM commutation period, and denote

\[ \beta_1=\frac{T}{T_{\mathrm{dv}}}; \qquad \beta_2=\frac{T}{T_{\mathrm{dm}}}. \]

Then the block diagram of the system, in increments and dimensionless quantities, for a disturbing action of the load torque on the motor shaft in the form of a unit step, takes the form shown in Fig. 1b.

Using the apparatus of the discrete Laplace transform, we obtain the transfer function of the closed system \(k_3^*(q,\varepsilon)\):

\[ k_2^*(q,\varepsilon)= \frac{ k_{\mathrm{dv}}\cdot k_{\mathrm{u}} \left( 1-\frac{e^q-e^{-\beta_1(1-\gamma)}}{e^q-e^{-\beta_1}}\,e^{-\beta_1\varepsilon} \right) (\beta_2-\beta_1)(e^q-e^{-\beta_1})\times }{ (1+k_0)(\beta_2-\beta_1)(e^q-e^{-\beta_1})(e^q-e^{-\beta_2})- } \to \]

\[ = \frac{ (e^q-e^{-\beta_2}) }{ -k_0\beta_2(e^q-e^{-\beta_2})(e^q-e^{-\beta_1(1-\gamma)}) +k_0\beta_1(e^q-e^{-\beta_1})(e^q-e^{\beta_2(1-\gamma)}) } \tag{1} \]

for \(0\leq \varepsilon\leq \gamma\),

\[ k_3^*(q,\varepsilon)= \frac{ k_{\mathrm{u}}\cdot k_{\mathrm{dv}}\cdot e^q (e^{-\beta_1\gamma}-1) (e^{-\beta_1(\varepsilon-\gamma)})(e^q-e^{-\beta_2})(\beta_2-\beta_1) }{ (1+k_0)(\beta_2-\beta_1)(e^q-e^{-\beta_1})(e^q-e^{-\beta_2}) -k_0\beta_2(e^q-e^{-\beta_2})\times } \to \]

\[ \to \frac{1}{ \times (e^q-e^{\beta_1(1-\gamma)}) +k_0\beta_1(e^q-e^{-\beta_1})(e^q-e^{-\beta_2(1-\gamma)}) } \tag{2} \]

for \(\gamma\leq \varepsilon\leq 1\).

where \(k_{\mathrm{i}}\) is the gain coefficient of the open-loop pulse system.

To determine the steady-state error, we shall use the equation of the system in terms of images:

\[ \omega^{*}(q,\varepsilon)=k_{z}^{*}(q,\varepsilon)\cdot F_{\mathrm{in}}^{*}(q,0)-z_{2}^{*}(q,\varepsilon). \tag{3} \]

Since, in the system under consideration, a change in the load torque is a disturbing action, the steady-state error will be the steady-state response of the system to a step change in the load torque.

Then, by the final-value theorem for the discrete Laplace transform,

\[ \Delta\omega_{\mathrm{st}} = \lim_{q\to 0}(e^{q}-1) \left[ F_{\mathrm{in}}^{*}(q,0)\cdot k_{z}^{*}(q,\varepsilon)-z_{2}^{*}(q,\varepsilon) \right]. \tag{4} \]

Passing from the image of continuous functions to the image of dimensionless lattice functions, we obtain:

\[ F_{\mathrm{in}}(q,0) = D\left\{ \frac{ k_{\mathrm{M}}\cdot k_{\mathrm{TD}}\cdot k_{\mathrm{DM}}\cdot \beta_{1}\cdot \beta_{2} }{ (q+\beta_{1})(q+\beta_{2})q } \right\} = \]

\[ = k_{\mathrm{M}}\cdot k_{\mathrm{TD}}\cdot k_{\mathrm{DM}} \left[ \frac{e^{q}}{e^{q}-1} - \frac{\beta_{2}e^{q}}{(\beta_{2}-\beta_{1})(e^{q}-e^{-\beta_{1}})} + \frac{\beta_{1}e^{q}}{(\beta_{2}-\beta_{1})(e^{q}-e^{-\beta_{2}})} \right], \tag{5} \]

\[ z_{2}^{*}(q,\varepsilon) = D\left\{ \frac{k_{\mathrm{M}}\cdot \beta_{1}}{(q+\beta_{1})q} \right\} = k_{\mathrm{M}} \left( \frac{e^{q}}{e^{q}-1} - \frac{e^{q}}{e^{q}-e^{-\beta_{1}}} \right). \tag{6} \]

Taking (5) and (6) into account, the expression for the steady-state error takes the form

\[ \Delta\omega_{\mathrm{st}} = \left[ k_{\mathrm{DM}}\cdot k_{\mathrm{TD}}\cdot k_{z}^{*}(0,\varepsilon)-1 \right]k_{\mathrm{M}}. \tag{7} \]

Substituting in (7) the value of \(k_{z}^{*}(q,\varepsilon)\) at \(\varepsilon=0\), we obtain

\[ \Delta\omega_{\mathrm{st}} = \left\{ \frac{ k_{0}(\beta_{2}-\beta_{1})(e^{-\beta_{1}(1-\gamma)}-e^{-\beta_{1}})(1-e^{-\beta_{2}}) }{ (\beta_{2}-\beta_{1})(1-e^{-\beta_{1}})(1-e^{-\beta_{2}}) + k_{0}\left[ \beta_{2}(1-e^{-\beta_{2}})(e^{-\beta_{1}(1-\gamma)}-e^{-\beta_{1}}) \right. } \right. \]

\[ \left. \left. \frac{ \phantom{k_{0}(\beta_{2}-\beta_{1})(e^{-\beta_{1}(1-\gamma)}-e^{-\beta_{1}})(1-e^{-\beta_{2}})} }{ {}-\beta_{1}(1-e^{-\beta_{1}})(e^{-\beta_{2}(1-\gamma)}-e^{-\beta_{2}})] } -1 \right\}k_{\mathrm{M}}. \tag{8} \]

The dependences of the steady-state error of the pulse system under consideration on the coefficient of the open-loop system \(k_{0}\) and \(\psi=\dfrac{\beta_{2}}{\beta_{1}}\), for various values of \(\gamma\), are presented in Fig. 2.

Fig. 2.

It is seen from Fig. 2 that, as \(k_0\) increases up to the value \(k_0=0.9K_{\mathrm{lim}}\) (\(K_{\mathrm{lim}}\) is the value of \(k_0\) corresponding to the stability boundary*), the steady-state error decreases monotonically; for \(k_0>k_0'\) the dependences \(\Delta\omega_{\mathrm{st}}=f(k_0)\) have no meaning, since in this case the system is either unstable or does not possess the required stability margin in modulus and phase.

Increasing the relative duration of the control pulses leads to a decrease in \(\Delta\omega_{\mathrm{st}}\), and the decrease is achieved at a smaller \(K_{\mathrm{lim}}\). When \(\gamma\) is decreased, the dependence \(\Delta\omega_{\mathrm{st}}=f(k_0)\) becomes more stringent. With a decrease in \(\beta_1=\dfrac{T}{T_M}\) (with a decrease in the commutation period \(T\) for \(T_M=\mathrm{const}\)), the steady-state error decreases. The most optimal values from the point of view of the minimum \(\Delta\omega_{\mathrm{st}}\) are

\[ \psi=\frac{T_M}{T_{dM}}=1\div 2. \]

To determine the transient process in the pulse system under consideration, we represent \(k_3^*(\varepsilon,q)\) in the form

\[ k_3^*(q,\varepsilon)= \frac{m_2(\varepsilon)e^{2q}+m_1(\varepsilon)e^q+m_0(\varepsilon)} {p_2e^{2q}+p_1e^q+p_0} = \frac{m_2(\varepsilon)+m_1(\varepsilon)e^{-q}+m_0e^{-2q}} {p_2+p_1e^{-q}+p_0e^{-2q}} . \tag{9} \]

Then expression (3), taking (9) into account, is written as:

\[ \omega^*(q,\varepsilon)= \left(p_2+p_1e^{-q}+p_0e^{-2q}\right) = \left[m_2(\varepsilon)+m_1(\varepsilon)e^{-q}+m_0e^{-2q}\right]F_{\mathrm{in}}^*(q,0) - z_2(q,\varepsilon)\left(p_2+p_1e^{-q}+p_0e^{-2q}\right), \tag{10} \]

where

\[ z_2(q,\varepsilon)= D\left\{\frac{k_M\cdot\beta_1}{(q+\beta_1)q}\right\}; \qquad F_{\mathrm{in}}^*(q,0)= D\left\{\frac{k_i\cdot k_{Td}\cdot k_{dM}\cdot \beta_1\cdot \beta_2} {(q+\beta_1)(q+\beta_2)q}\right\}. \]

Passing in (10) from images to originals, we obtain a recurrence formula determining the transient process in the pulse system when a unit step of torque acts on it:

\[ \omega[n,\varepsilon]\cdot p_2 +p_1\omega[(n-1),\varepsilon] +p_0\omega[(n-2),\varepsilon] = m_2(\varepsilon)f_{\mathrm{in}}[n,0] + m_1(\varepsilon)f_{\mathrm{in}}[(n-1),0] + m_0(\varepsilon)f_{\mathrm{in}}[(n-2),0] - p_2z_2[n,\varepsilon] - p_1z_2[(n-1),\varepsilon] + p_0z_2[(n-2),\varepsilon], \tag{11} \]

where:

\[ z_2[n,\varepsilon]=k_M\left[1-e^{-\beta_1(n-\varepsilon)}\right]; \tag{12} \]

\[ f_{\mathrm{in}}[n,0]= k_M\cdot k_{T,d}\cdot k_{dM} \left[ 1-\frac{\beta_2}{\beta_2-\beta_1}e^{-\beta_1 n} + \frac{\beta_1}{\beta_2-\beta_1}e^{-\beta_2 n} \right]. \tag{13} \]

The general nature of the transient process can be estimated from the change in the envelopes corresponding to the maximum and minimum values of the process. The speed of the electric motor reaches maximum and minimum values, respectively, at \(\varepsilon=\gamma\) and \(\varepsilon=0\).

Substituting the values \(\varepsilon=\gamma\) and \(\varepsilon=0\) into (11), we obtain expressions for the envelopes of the transient process:

\[ \omega[n,0]= \frac{ m_2(0)f_{\mathrm{in}}[n,0] + m_1(0)f_{\mathrm{in}}[(n-1),0] + m_0(0)f_{\mathrm{in}}[(n-2),0] - }{p_2} \;\to \]

\[ \to \frac{ p_2z_2[n,0] - p_1z_2[(n-1),0] + p_0z_2[(n-2),0] - p_1\cdot\omega[(n-1),0] }{p_2} \;\to \]

\[ p_0\omega[(n-2),0] \tag{14} \]

\[ \underline{\hspace{2.5cm}} \]

\[ {}^*\ \text{The stability boundaries of the pulse system of the structure under consideration were determined in [1].} \]

\[ \omega[u,\gamma]= \frac{m_2(\gamma)f_{\mathrm{in}}[n,0]+m_1(\gamma)f_{\mathrm{in}}[(n-1),0]+m_0(\gamma)f_{\mathrm{in}}[(n-2),0]}{p_2} \;\to \]

\[ \to \frac{p_2 z[n,\gamma]-p_1 z[(n-1),\gamma]-p_0 z[(n-2),\gamma]-p_1\omega[(n-1),\gamma]}{p_2} \;\to \]

\[ \to -\frac{p_0\omega[(n-2),\gamma]}{p_2} \tag{15} \]

From expressions (1) and (2) we obtain:

\[ \left. \begin{aligned} m_2(0)&=0,\\ m_1(0)&=k_{\mathrm{i}}\cdot k_{\mathrm{dv}}(\beta_2-\beta_1)e^{-\beta_1}(e^{\beta_1\gamma}-1),\\ m_0(0)&=-k_{\mathrm{i}}\cdot k_{\mathrm{dv}}(\beta_2-\beta_1)e^{-(\beta_1+\beta_2)}(e^{\beta_1\gamma}-1) \end{aligned} \right\} \tag{16} \]

\[ \left. \begin{aligned} m_2(\gamma)&=-k_{\mathrm{i}}\cdot k_{\mathrm{dv}}(e^{-\beta_1\gamma}-1)(\beta_2-\beta_1),\\ m_1(\gamma)&=k_{\mathrm{i}}\cdot k_{\mathrm{dv}}(e^{-\beta_1\gamma}-1)(\beta_2-\beta_1)e^{-\beta_2},\\ m_0(\gamma)&=0 \end{aligned} \right\} \tag{17} \]

\[ \left. \begin{aligned} p_2&=\beta_2-\beta_1,\\ p_1&=-\{(\beta_2-\beta_1)(e^{-\beta_1}+e^{-\beta_2})+k_0[\beta_2 e^{-\beta_1}-\beta_1 e^{-\beta_2}+\\ &\qquad+\beta_1e^{-\beta_2(1-\gamma)}-\beta_2e^{-\beta_1(1-\gamma)}]\},\\ p_0&=(\beta_2-\beta_1)e^{-(\beta_1+\beta_2)}+k_0[(\beta_2-\beta_1)e^{-(\beta_1+\beta_2)}-\beta_2e^{-[\beta_2+\beta_1(1-\gamma)]}+\\ &\qquad+\beta_1e^{-[\beta_1+\beta_2(1-\gamma)]}] \end{aligned} \right\} \tag{18} \]

Using the recurrence expressions (14) and (15), it is possible to construct the transient process in the system under a disturbing action in the form of a unit step in the load torque, without knowing the roots of the characteristic equation of the closed pulse system. The latter is especially important, since determining the values of the roots of the characteristic equation of a closed pulse system presents a known difficulty in systems of the type under consideration.

Fig. 3. Influence of the system parameters on the duration of transient processes.

In Fig. 3 the dependences are presented of the relative transient-process time on the gain coefficient of the open-loop system \(K_0\) for various values of

\[ \psi=\frac{T_{\mathrm{m}}}{T_{\mathrm{dm}}} \]

and \(\gamma\).

It follows from Fig. 3 that a decrease in the gain of the open-loop system \(K_0\) leads to a reduction in the duration of the transient process; in this case \(n\) becomes increasingly less dependent on changes in \(K_0\).

An increase in

\[ \psi=\frac{T_{\mathrm{m}}}{T_{\mathrm{dm}}} \]

increases the speed of response of the system for the same

\[ \frac{k_{\mathrm{ogr}}}{k_0}. \]

With an increase in the relative duration of the control pulses \(\gamma\) in the nominal regime, with

\[ \frac{k_{\mathrm{gr}}}{k_e}=\mathrm{const} \]

and \(\psi=\mathrm{const}\), the relative transient-process time increases.

The investigations carried out on the influence of the parameters of pulse systems on accuracy and speed of response make it possible to optimize systems with respect to each of these quality characteristics of regulation. When the task is to optimize these contradictory characteristics simultaneously, it is necessary to find a compromise solution, i.e., at some sufficient speed of response, to strive for the minimum steady-state error of the pulse system.

Analysis of expressions (8), (11), (15) and of the dependences \(\Delta\omega_{\mathrm{st}}=f(k_0)\) and

\[ \frac{t_p}{T}=f(k_0) \]

makes it possible to conclude that the most optimal parameter values, from the standpoint of the minimum steady-state error and transient-process time, lie within the limits:

\[ \frac{T_{\mathrm{m}}}{T_{\mathrm{dm}}}=7.7\div 10;\quad \gamma=0.7\div 0.9;\quad k_0=(0.7\div 0.8)\,k_{\mathrm{ogr}}. \]

The experimentally obtained values of the steady-state error and of the transient-process time of the rotational-speed control system for a DPR-type DC micromotor (Figs. 2, 3) confirm, with accuracy sufficient for practice, the correctness of the theoretical investigations carried out.

REFERENCES

1. I. V. Bulin-Sokolov, S. M. Mironov. Stability of a discrete system for stabilizing the rotational speed of a DC micromotor. Proceedings of the seminar “Low-Power Electric Motors in Electric-Drive Systems,” held in Leningrad, November 25–27, 1968. Energiya Publishing House, 1969.

2. Ya. Z. Tsypkin. Theory of Linear Pulse Systems. Fizmatgiz, 1963.