DETERMINATION OF THE QUANTIZATION INTERVAL OF THE CONTROL LAW IN A PROGRAM CONTROL SYSTEM

V. Z. Yampolsky, A. I. Zaitsev

In complex automation systems, continuous program-control systems are widely used whose inputs receive discrete signals. Discrete input signals arise if the control law is formed with the aid of an electronic digital computer, is received from a discrete communication channel, or is implemented with the aid of a specialized digital device.

In the present work, a system of program control of the speed of an electric motor with rigid negative feedback in speed is considered, in which the control law is implemented with the aid of a digital device (Fig. 1a). The digital device consists of a digital automaton, DA, and a code-to-voltage converter, CV. The sequence of binary numbers formed by the digital automaton is converted by means of the converter into a step signal, which is then applied to the input of a continuous follow-up system with transfer function \(W(p)\) in the open-loop state.

The transition from the actual circuit to the computational one is carried out by replacing the digital device with a switch and a zero-order holding element \(H\). Two variants of the computational circuit for first- and second-order follow-up systems are shown in Fig. 1b and c, respectively.

In such a system, the magnitude of the quantization interval of the control law in time (the discreteness interval \(T\)) substantially affects the amount of electronic equipment in the digital device. The smaller the quantization interval, the more accurate the approximation of the control law, but the more elements are required to construct the digital device.

Theoretically, the approximation error vanishes at \(T = 0\); however, in this case the digital device must reproduce an infinite number of levels of the continuous signal, which drives to infinity the number of elements needed for its construction.

In practice, however, there is no need to reproduce signals with zero error.

Every real automatic-control system that receives continuous information has a number of limitations,

not allowing it to distinguish many subtle properties of the signals carrying information.

The limited sensitivity and resolving power of the system leads to the fact that sufficiently close signal values are perceived by it as a single value.

The limited passband of the system does not allow it to distinguish very close instants of time; therefore one may use discrete time, splitting real continuous time into such

Figure 1.

a. Block diagram with a dashed control unit containing blocks \(IA\) and \(D/n\); signal \(u\) enters a summing junction, then block \(W(p)\), output \(n\), and feedback through \(\gamma\).

b. Block diagram: input \(G(t)\), switch, block \(H_0\), summing junction, block
\[ \frac{-k}{T_M p+1}, \]
output \(X_T(t)\), and feedback through \(\gamma\).

c. Block diagram: input \(G(t)\), switch, block \(H_0\), summing junction, block
\[ \frac{k}{T_e T_M p^2+T_M p+1}, \]
output \(X_P(t)\), and feedback through \(\gamma\).

small intervals, so that they are perceived by the system as a single instant of time.

The equivalence of a continuous and a discrete signal from the standpoint of the information contained in them is established in information theory by V. A. Kotelnikov’s theorem [1].

According to this theorem, any continuous signal \(g(t)\) whose spectrum is zero for \(\omega > \omega_c = 2\pi f_c\) can be replaced by an equivalent discrete signal with discreteness interval

\[ T < \frac{\pi}{\omega_c} = \frac{1}{2f_c}. \tag{1} \]

The difficulty of applying V. A. Kotelnikov’s theorem to the choice of the quantization interval of the control law lies in the fact that continuous automatic-control systems have spectra that are unbounded in frequency.

In practice, on energy grounds, it is possible to replace the unbounded frequency spectrum of an automatic-control system by a bounded one, since the area bounded by the spectral curve corresponds to the energy

Fig. 2.

Visible labels in the figure: \(G\), \(t\), \(t_1\), \(t_2\), \(t_3\), \(t_4\), \(A_1 t\), \(A_4(t-t_3)\), \(A_5(t-t_4)\), \(A_2(t-t_1)\), \(A_3(t-t_2)\).

Fig. 3.

of the signal, which must be finite. Moreover, infinitely high frequencies are impossible in practice, just as absolutely inertia-free bodies and processes are impossible.

However, limiting the spectrum does not remedy the situation, since it does not appear possible, by methods accessible to engineering practice, to estimate the error of the processes in the system due to neglecting the high-frequency part of the spectrum and, consequently, to justify the choice of one or another value of the sampling interval.

In the present work an attempt is made, on the basis of an analysis of transient processes in the system, to obtain analytical relations that make it possible to determine the quantization interval of the control law from known system parameters and the permissible error in the transient process.

Of particular interest is the problem of determining the largest value of the quantization interval for which the maximum error of the transient process in the system would not exceed the prescribed value \(\delta_z\).

In this formulation, the problem stated may be regarded as the problem of determining the optimal quantization interval,

providing the specified accuracy of the transient process in the system with the smallest hardware capacity of the digital device.

The solution of the problem posed is carried out by comparing the response of a continuous tracking system to a continuous input action and to a discrete one obtained from the continuous action by a stepwise approximation with discreteness interval \(T\).

Taking into account the considerable complexity of the problem, we shall confine ourselves to considering linear control systems, which, despite all the rigor of the mathematical formulations, leads to results that are approximate for practical use.

In addition, an arbitrary input action \(G_{(t)}\) is replaced by a series of linearly increasing actions shifted in time, as shown in Fig. 2. This assumption makes it possible to reduce the problem of determining the quantization interval of an arbitrary input action to the problem of determining it for a typical linear action.

Figure 3 shows transient processes in a first-order tracking system (Fig. 1b) for different values of the discreteness interval \(T\), with the following values of the system parameters:

\[ K = 250, \qquad T = 0.267, \qquad \gamma = 0.04, \qquad A = 120 \]

As follows from the curves shown, the maximum error in reproducing the discrete linear input signal relative to the corresponding continuous one arises within each discreteness interval \(T\). Since the error arises due to the replacement, in each interval, of the continuous linear input signal by a stepwise one (Fig. 4a), it can be estimated as the difference between the responses of the system to these signals.

Fig. 4.

The transient process in a first-order tracking system for a linear input signal \(G_{(t)} = At\) is described by the equation:

\[ X_{1(t)\ell} = \frac{Ak}{T_m a}\cdot \frac{1}{a}\left( at + e^{-at} - 1 \right), \tag{2} \]

where

\[ a=\frac{1+\gamma k}{T_{\mathrm{m}}}. \]

For a step input signal, the equation of the transient process (the original function of the output quantity) of the system has the form:

\[ X_{i(t)\mathrm{c}}=\frac{AkT}{T_{\mathrm{m}}a}\left(1-e^{-at}\right), \tag{3} \]

where \(AT\) is the amplitude of the step.

Equations (2) and (3) are obtained from correspondence tables [2], from the corresponding images of the output quantity in the sense of the inverse Laplace transform.

The error of the transient process on each interval is determined by the equation:

\[ \delta=X_{i(t)\mathrm{c}}-X_{i(t)\mathrm{л}} \tag{4} \]

or, after substituting into it the values of \(X_{i(t)\mathrm{л}}\) and \(X_{i(t)\mathrm{c}}\) from equations (2) and (3):

\[ \delta=\frac{Ak}{T_{\mathrm{m}}a}\left[T-\frac{1}{a}e^{-at}(Ta+1)-t+\frac{1}{a}\right]. \tag{5} \]

The optimum is the largest value of \(T\) for which the specified process error \(\delta_{\mathrm{з}}\) coincides with the maximum error in the system \((\delta_{\mathrm{з}}=\delta_{\mathrm{м}})\).

To find the maximum error, we subject equation (5) to investigation for a maximum in the interval \(0\le t\le T\):

\[ \delta'_{(t)}=\frac{Ak}{T_{\mathrm{m}}a}\left[e^{-at}(Ta+1)-1\right]=0, \tag{6} \]

whence we find the time at which the maximum error occurs,

\[ t_{\mathrm{м}}=-\frac{\ln\left(\dfrac{1}{Ta+1}\right)}{a}, \tag{7} \]

Substituting the value \(t_{\mathrm{м}}\) into equation (5), we obtain the equation of the maximum error:

\[ \delta_{\mathrm{м}}=-\frac{Ak}{T_{\mathrm{м}}\cdot a^{2}}\left[Ta+\ln\left(\frac{1}{Ta+1}\right)\right]. \tag{8} \]

It is not difficult to show that, as an estimate for the maximum error of the process in the system, one may take a quantity equal to half the value of \(\delta_{\mathrm{м}}\) determined from equation (8).

Indeed, in the derivations given above it was assumed that quantization of the control law is performed with advance, as shown in Fig. 4, a. For the same case, the graph of the transient process is plotted in Fig. 3.

However, if the method of quantization is changed and it is carried out as shown in Fig. 4 b, we arrive at a significant reduction in the magnitude of the maximum error (approximately by a factor of two).

The latter is readily seen from Fig. 5, which presents the graph of the transient process in the first-order system (Fig. 3),

Fig. 5.

reconstructed in accordance with the modified method of quantization of the input signal.

The equation for \(\delta_{\mathrm{m}}\) under this method of quantization is found in an analogous manner and has the form:

\[ \delta_{\mathrm{m}} = \frac{Ak}{t_{\mathrm{m}}a^{2}} \left[ \frac{Ta}{2} + \ln\left( \frac{1}{Ta+e^{-\frac{Ta}{2}}} \right) \right]. \tag{9} \]

where

\[ t_{\mathrm{m}} = - \frac{ \ln\left( \frac{1}{Ta+e^{-\frac{Ta}{2}}} \right) }{a}. \]

The use of this equation is complicated by the fact that, for certain parameter values, the value of the maximum error at the boundary of the sampling interval may exceed the value \(\delta_{\mathrm{m}}\) determined from equation (9).

Fig. 6.

Therefore it is more expedient to use an estimate of the maximum error equal to one half of the value \(\delta_{\mathrm{m}}\) from equation (8), which, for any values of the system parameters, is an upper bound on the error that arises.

Determination of the optimal quantization interval \(T\) of the input signal for given parameters \(T_{\mathrm{m}}, a, A, k\) of the first-order system and the magnitude of the permissible error of the transient process \(\delta_3\) is conveniently carried out with the aid of the graph shown in Fig. 6.

To calculate the graphical dependence \(Ta=f(\chi\delta_3)\), equation (8) is represented in the form:

\[ \chi\delta = Ta + \ln\left(\frac{1}{Ta+1}\right), \tag{10} \]

where

\[ \chi=\frac{2T_{\mathrm{m}}a^2}{Ak}, \qquad \delta_3=\frac{\delta_{\mathrm{m}}}{2}. \]

Determination of the optimal \(T\) from the graph in Fig. 6 is carried out as follows: from the specified value \(\delta_3\) and the system parameters \(T_{\mathrm{m}}, a, A, k\), the value \(\chi\delta_3\) is determined; for this value, the quantity \(Ta\) is found from the curve \(Ta=f(\chi\delta_3)\); dividing the found quantity by \(a\), we obtain the value of the optimal quantization interval \(T\) (in the sense indicated above), which ensures that the transient process in the system proceeds with an error not exceeding \(\delta_3\).

For the following second-order system, after carrying out calculations analogous to the derivation of equation (8), we obtain the following equation for estimating the maximum error:

\[ \delta_{\mathrm{m}}=2\delta_3= \frac{Ak}{T_{\mathrm{e}}T_{\mathrm{m}}a^2\beta} \left\{ Ta+\ln\left[\frac{\beta-\alpha}{\beta(Ta+1)}\right] +\frac{\beta+\alpha}{\beta}-1 \right\}, \tag{11} \]

where

\[ \alpha=\frac{1}{2T_{\mathrm{e}}} -\sqrt{\left(\frac{1}{2T_{\mathrm{e}}}\right)^2 -\left(\frac{1+\gamma k}{T_{\mathrm{m}}T_{\mathrm{e}}}\right)^2}, \]

\[ \beta=\frac{1}{2T_{\mathrm{e}}} +\sqrt{\left(\frac{1}{2T_{\mathrm{e}}}\right)^2 -\left(\frac{1+\gamma k}{T_{\mathrm{m}}T_{\mathrm{e}}}\right)^2}. \]

Analysis of equations (8) and (11) shows that the maximum error is proportional to the rate of increase of the input action \(A\); therefore, when choosing the quantization interval for an arbitrary input action, it is necessary to do so on the segment with the greatest steepness.

Obtaining, by an analogous method, relations for determining the quantization interval of the control law in systems of higher order does not appear possible, owing to the absence of analytical relations for the original function of the output quantity. The latter is connected with the impossibility, in the general case, of factoring the denominator polynomial of the transfer function of a closed tracking system of order higher than the second.

Conclusions:

1. The relations obtained make it possible to estimate the error of the transient process due to the discreteness of the input signal in closed continuous tracking systems of the first and second orders, and to select the optimal quantization interval of the control law for a prescribed value of the permissible error.

2. Relations (8) and (11) may be used to determine the quantization interval of the input signal of local control systems of a complex cybernetic control complex, the control laws in which are implemented by means of a control digital computer, which is equivalent to determining a rational rate of data output by the machine.

REFERENCES

1. Kotelnikov V. A. On the transmission capacity of the ether and of wire in electrical communications. Materials of the radio section for the First All-Union Congress on the Reconstruction of Communications. All-Union Energy Committee, 1933.

2. Kuzin L. T. Calculation and Design of Discrete Control Systems. Moscow, 1962.