USE OF A DISCRETE CONTROL DEVICE TO OBTAIN A PRESCRIBED TRANSIENT PROCESS

V. Z. Yampolsky, A. I. Zaitsev

The basic requirement imposed on the electric-drive systems of high-speed passenger elevators is to ensure high productivity and comfort.

Comfort, or the measure of smoothness of elevator operation, is the rate of increase of acceleration (deceleration) in transient conditions during starting and stopping. In elevator engineering this quantity is called jerk and is mathematically expressed as the time derivative of acceleration (deceleration):

\[ \rho=\frac{da}{dt}. \]

An acceleration during starting or deceleration of \(2\ \text{m/sec}^2\) does not cause unpleasant sensations for the passenger if the jerk does not exceed \(3.5\ \text{m/sec}^3\) [1]. Combining comfort with high productivity is very difficult, since the speed diagram of the car’s motion must take into account, on the one hand, the requirements of comfort, which limit the magnitude of acceleration and jerk, and, on the other hand, satisfy the requirement of maximum productivity, which dictates the need to increase those same parameters.

The optimal speed diagram is shown in Fig. 1, where the combination of the requirements of comfort and productivity is achieved by changing the speed of motion on sections 0—1 and 2—3 according to a parabolic law. A modern elevator electric drive using the G–D system with a closed automatic speed-control system is practically incapable of accurately carrying out the optimal motion diagram in transient conditions.

Obviously, complete correspondence of the diagram to the actual process in the system can be achieved by introducing into the drive system an automatic control system for starting and braking that ensures a change in speed according to the prescribed time law.

This article proposes a new method for obtaining the desired transient process of starting and braking, based on the use of a discrete control device of the sequential type. A method is given for the optimal synthesis of a discrete dev-

Fig. 1. Optimal velocity graph of elevator motion.

Fig. 2. Schematic diagram of the ACS.

…properties of the control device. Among regulated electric drives for high-speed passenger elevators, the G–D system has found the widest application. To implement a controlled transient process, a G–D system with rigid negative speed feedback is proposed, supplemented by a discrete control device (DCD), Fig. 2.

The discrete control device is intended for the automatic formation and time sweep of a signal at the input of the EMU proportional to the prescribed speed during starting and braking.

Fig. 3. Graph of the control law.

Consequently, finding the control law reduces to constructing the dependence \(i_{\mathrm{vzmu}} = f(t)\). This dependence can be found by means of the method of A. V. Basharin [2]. The graphical method developed by him makes it possible to synthesize the control law as a function of distance, speed, or time for a given optimal speed-

schedule in complex electric-drive systems with feedback, taking account of nonlinear relationships.

In Fig. 3a is shown the result of constructing the control law as a function of time \(i_{\text{exc}}=f(t)\) during the starting of an elevator installation.

Synthesis of a Discrete Control Device

The discrete control device is implemented by means of sequential logic circuits. The operation of sequential circuits is described by a degenerate recurrent Boolean function of the first kind (abbreviated DRBF-1).

A recurrent Boolean function of the first kind is a function defined for any \(t\) on sets of the type

\[ <X_{1t}=X_{2t},\ldots,X_{nt},Y_{t-1},Y_{t-2},\ldots,Y_{t-k}> \]

and taking as its possible values either 0 or 1.

In these sets:

\(X_{1t}, X_{2t},\ldots, X_{nt}\) are binary arguments whose values are taken at the time instant \(t\).

\(Y_{t-1}, Y_{t-2},\ldots, Y_{t-k}\) are the values of the function \(Y_t\) at the instants \(t-1, t-2,\ldots,t-k\).

In contrast to an RBF-1, the values of a DRBF-1 do not depend on the arguments \(X_{it}(i=1,2,\ldots,n)\).

Physically, in a circuit corresponding to a degenerate recurrent Boolean function of the first kind there are no active inputs \((X_{1t}, X_{2t},\ldots,X_{nt})\), and there are \(K\) feedback lines (Fig. 4).

Fig. 4.

The system of proper functions of sequential (nonprimitive) circuits has the form

\[ \left\{ \begin{aligned} Y_{1(t+1)}&=\varphi_1(Y_{1t},Y_{2t},\ldots,Y_{st});\\ Y_{2(t+1)}&=\varphi_2(Y_{1t},Y_{2t},\ldots,Y_{st});\\ &\cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\\ Y_{s(t+1)}&=\varphi_s(Y_{1t},Y_{2t},\ldots,Y_{st}); \end{aligned} \right. \tag{1} \]

with initial conditions

\[ \left\{ \begin{aligned} Y_{10}&=\alpha_1;\\ Y_{20}&=\alpha_2;\\ &\ldots\quad \ldots;\qquad \alpha_i=0,1;\quad i=1,2,\ldots,s.\\ Y_{s0}&=\alpha_s; \end{aligned} \right. \]

The behavior of sequential functions was investigated from various

points of view in the work of a number of authors [3, 4, 5]. The problem of synthesizing a discrete control device consists in constructing a certain real circuit on the basis of a physical description of its operation.

The physical description of the operation of the DCD is the control law \(i_{\text{exc}} = f(t)\) (Fig. 3), which, by means of quantization in time and in level, is represented in the form of a sequence of numbers. These numbers are written in Fig. 3, b, as decimal digits, and in Fig. 3, c—as binary digits. The state of the elements of the sequential circuit at various instants of time is shown in Fig. 3, d. The digital notation of the control law makes it possible to depict the operation of the discrete control device by the timing diagram shown in Fig. 5.

In the timing diagram, circles denote all possible states of the circuit at discrete instants of time. The shaded circle corresponds to the initial state of the circuit. The arrows show transitions from the preceding state of the circuit to the following one.

Fig. 5. Timing diagram of the DCD.

For the synthesis of the DCD we compile a table which, from given values \(Y_{1t}, Y_{2t}, \ldots, Y_{st}\), permits finding the quantities \(Y_{1(t+1)}, Y_{2(t+1)}, \ldots, Y_{s(t+1)}\). The left \(S\) columns of Table 1 contain \(2^S\) rows, which exhaust all possible sets of \(S\) binary arguments.

The columns \(Y_{1(t+1)}, Y_{2(t+1)}, \ldots, Y_{s(t+1)}\) are filled with rows in accordance with the timing diagram of the device being synthesized.

Applying known methods of the algebra of logic, one can carry out the synthesis of a functional circuit whose operation is specified by Table 1.

However, synthesis is not a unique process; therefore, many functional circuits may be obtained, differing both

Table 1

in complexity, as well as in the amount of required equipment, possessing one and the same proper function.

Of greatest interest are circuits that require a minimum amount of equipment for their implementation.

The synthesis of functional circuits under this additional condition is called optimal. To obtain an optimal version of the functional circuit of the control device, we shall use the method of minimal normal forms [6].

Let us write the DNF of the functions for the example under consideration:

\[ \tag{2} \left\{ \begin{aligned} Y_{1(t+1)}={}& \overline{Y}_{1t}\cdot Y_{2t}\cdot \overline{Y}_{3t}\cdot \overline{Y}_{4t} +Y_{1t}\cdot \overline{Y}_{2t}\cdot \overline{Y}_{3t}\cdot \overline{Y}_{4t}+{}\\ &+Y_{1t}\cdot \overline{Y}_{2t}\cdot Y_{3t}\cdot \overline{Y}_{4t} +Y_{1t}\cdot Y_{2t}\cdot \overline{Y}_{3t}\cdot \overline{Y}_{4t};\\ Y_{2(t+1)}={}& Y_{1t}\cdot \overline{Y}_{2t}\cdot Y_{3t}\cdot Y_{4t} +Y_{1t}\cdot Y_{2t}\cdot Y_{3t}\cdot Y_{4t};\\ Y_{3(t+1)}={}& Y_{1t}\cdot Y_{2t}\cdot \overline{Y}_{3t}\cdot \overline{Y}_{4t} +Y_{1t}\cdot Y_{2t}\cdot \overline{Y}_{3t}\cdot Y_{4t};\\ Y_{4(t+1)}={}& Y_{1t}\cdot Y_{2t}\cdot \overline{Y}_{3t}\cdot \overline{Y}_{4t}. \end{aligned} \right. \]

We carry out minimization of the DNF, applying Blake’s method [7]:

\[ \tag{3} \left\{ \begin{aligned} Y_{1(t+1)}={}& \overline{Y}_{4t}\bigl[ Y_{1t}\cdot \overline{Y}_{2t}(\overline{Y}_{3t}+Y_{3t}) +Y_{2t}\cdot \overline{Y}_{3t}(\overline{Y}_{1t}+Y_{1t}) \bigr] ={}\\ ={}& \overline{Y}_{4t}\bigl(Y_{1t}\cdot \overline{Y}_{2t} +Y_{2t}\cdot \overline{Y}_{3t}\bigr);\\ Y_{2(t+1)}={}& Y_{1t}\cdot Y_{4t}\bigl(\overline{Y}_{2t}\cdot Y_{3t} +Y_{2t}\cdot \overline{Y}_{3t}\bigr);\\ Y_{3(t+1)}={}& Y_{1t}\cdot Y_{2t}\cdot \overline{Y}_{4t} (\overline{Y}_{2t}+Y_{2t}) =Y_{1t}\cdot Y_{2t}\cdot \overline{Y}_{4t};\\ Y_{4(t+1)}={}& Y_{1t}\cdot Y_{2t}\cdot \overline{Y}_{3t}\cdot \overline{Y}_{4t}. \end{aligned} \right. \]

The results obtained are the minimal form of the proper RBF of the discrete control device.

Figure 6 shows a functional diagram whose proper function is the system of equations (3).

Fig. 6. Functional diagram of the DCD.

The delay lines are implemented on ferrite-transistor cells. The duration of the delay of a ferrite-transistor cell is completely determined by the time interval between two adjacent pulses of the clock generator.

If the clock-pulse generator is made to operate not as a function of time, as was the case in the example given, but as a function of speed or path of motion, then automatic control systems are obtained with starting as a function of the corresponding parameters.

On the curves \(S = f(t)\) and \(V = f(t)\) (Fig. 1), triangles indicate the values of the corresponding parameters upon reaching which the clock-pulse generator must switch the discrete control device.

The accuracy of the numerical approximation of the control law is theoretically unlimited and depends only on the quantization interval. The smaller the quantization interval, the higher the accuracy and the larger the amount of equipment required to implement the discrete control device.

It is necessary to note that, as the number of arguments increases, the process of minimizing the Boolean function of the discrete control device becomes more difficult. In this case, matrix minimization methods should be used.

Conclusions

1. The use of a DCD to obtain a prescribed transient process considerably simplifies the structural control scheme in the \(G\)—\(D\) system and makes it possible to obtain high accuracy in approximating the control law.

2. The presence of controllable delay lines in the feedback circuits of the sequential circuit broadens the capabilities of the control system. It becomes possible to control the transient process as a function of travel, velocity, and time, which is an important condition for the automation of hoisting-and-transport mechanisms.

3. The use of sequential circuits whose operation is described by first-kind Boolean functions for implementing the control device makes it possible to algorithmize the process of optimal synthesis of functional circuits according to the criterion of minimizing the amount of equipment.

References

1. G. K. Korneev, M. G. Korotkov. Passenger and Freight Elevators. Mashgiz, 1958.

2. A. V. Basharin. Calculation of Dynamics and Synthesis of Nonlinear Control Systems. GEI, 1960.

3. E. F. Moore. Speculative experiments with sequential machines. In: Avtomaty [Automata], Moscow, 1956.

4. D. D. Aufenkamp, F. E. Hohn. Analysis of sequential machines. In: Matematika, 3:3, 1958.

5. R. Geleski, D. D. Aufenkamp. On the analysis of sequential machines. In: Matematika, 3:4, 1959.

6. L. A. Pospelov. Arithmetic and Logical Foundations of Discrete-Action Computing Machines. Part 3, MEI Press, 1961.

7. S. V. Yablonskii. Functional constructions in \(K\)-valued logics. Proceedings of the V. A. Steklov Mathematical Institute, vol. 51, 1958.

Received by the editors
in June 1962.