Analog Computers

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Simulation and Investigation of Sampled-Data Systems on an Electric Analog Computer

This document translates the original German-language article (8 pages).

Author: G. Schneider
Institution: Institut für Automation der AEG (AEG Institute for Automation), Frankfurt (Main)
Published in: Elektronische Rechenanlagen, Vol. 2 (1960), Issue 1, pp. 31–37
Manuscript received: 16 December 1959
Reprint (Sonderdruck) No.: AH 507

Abstract

In order to be able to investigate sampled-data systems on the analog computer it is necessary to synthesize, besides the actual sampling units, also those compensating elements whose transfer function is described by a linear difference equation (digital computer in the control loop). In this paper a new method for the realisation of these units is shown and applied to an example, including the finding of appropriate compensating elements. Finally, the general solution of difference equations by using an analog computer is shown.

Introduction

Sampled-data systems form a distinct class of systems for automatic control and regulation, characterised by the presence of one or more sampling elements. The transfer behavior of a sampling element can be described by:

a(t) = e((n + q)Ts), for (n + q)Ts ≤ t < (n + 1 + q)Ts; n integer, 0 ≤ q < 1

The sampling element stores the value its input quantity takes at the instants Ts, (1 + q)Ts, (2 + q)Ts, … for the duration of one sampling period Ts. The value of q gives the position of the first sampling relative to the chosen time origin.

A sampled-data system is therefore one in which measurement of the control deviation or of another system variable does not occur continuously but only at regular time intervals. Many systems that contain digital measuring devices, or systems in which the measured values of several quantities are transmitted sequentially through a single transmission channel, have this characteristic.

There are fundamentally two possibilities for influencing the transfer behavior of sampled-data systems:

  1. Converting the staircase waveform of the actuating variable into a manipulated variable that is no longer a step function — this is the case when a compensating element is inserted whose transfer behavior is described by a differential equation.

  2. Converting the staircase function into another, more suitable staircase function (Fig. 5) — this occurs when a compensating element is inserted whose behavior is described by a difference equation.

Simulation of Sampling Elements

For the simulation and investigation of sampled-data systems on an electric analog computer, a method is required for simulating both sampling elements and difference-equation elements, since these are the elements that can appear in sampled-data systems in addition to those of normal “continuous” systems (whose simulation is assumed known).

A simple method for constructing sampling elements is shown in Fig. 6, using only components available in any electric analog computer — with the possible exception of a relay element, whose operation is explained in Fig. 7. In the circuit of Fig. 6 the relay closes for the interval:

(n + q − ε/2)Ts ≤ t ≤ (n + q + ε/2)Ts

where ε is chosen small enough that the input quantity e does not change substantially during this interval. During this period the circuit behaves as a first-order lag element with unity gain and time constant Tn = (1 − α)RC. Provided (1 − α)RC ≪ εTs (achievable by appropriate choice of capacitor), by the end of the relay closure time the operational amplifier output has substantially reached the value of the input and then remains constant, since the capacitor cannot discharge. Oscillograms are shown in Figs. 9 and 10 (the sawtooth voltage used for triggering is generated as shown in Fig. 10).

Simulation of Difference-Equation Elements

For simulating difference-equation elements one can confine attention to those whose transfer behavior is described by a linear difference equation with constant coefficients (per Fig. 5):

y(t) + c₁y(t − T) + … + cₖy(t − kT) = d₀a(t) + d₁a(t − T) + … + dₖa(t − kT)

with the additional assumption that T and the step width Ts of the input function a stand in a rational relationship:

T = Ts/m, m > 0, integer

It suffices to consider the case m = 1.

The transfer function of this difference-equation element is:

Y(p)/A(p) = (d₀ + d₁e^{−Tp} + … + dₖe^{−kTp}) / (1 + c₁e^{−Tp} + … + cₖe^{−kTp})

A system consisting essentially of dead-time elements that also possesses this transfer function is shown in Fig. 11 (the “delay-line synthesizer” principle). Since the input function a is a staircase function with step width Ts and all system variables vanish for t < 0, each dead-time element can be replaced by a cascade of two sampling elements (Figs. 13 and 14): the first samples x(t) at instants T/2, 3T/2, 5T/2, … (q = 1/2), producing the staircase function x(t − T/2); the second samples that result at 0, T, 2T, … (q = 0), producing x(t − T). The complete simulation of a difference-equation element then proceeds according to the scheme in Fig. 15, requiring at most 2k sampling elements built as in Fig. 6.

Simulation and Investigation of a Specific Sampled-Data System

To illustrate these methods, a non-trivial example of a sampled-data system is treated in detail. The structural diagram and numerical constants are given in Fig. 16.

Finding Suitable Compensating Elements — Reference (Command) Behavior

First, only the reference behavior is considered (disturbance z(t) = 0). The requirement is that after a step change in the reference variable r, the controlled variable c returns to exact agreement with r after as few sampling instants as possible (finite settling time, Fig. 17). This requirement — not achievable for continuous systems — can be met for sampled-data systems.

Representing the desired staircase actuating variable y as a superposition of step functions (Eq. 9) and using the plant step response (Eq. 10), the required coefficients are determined by eliminating the time-dependent terms. The solution yields:

y(0) = 16.70, y(Ts) = −8.18, y(2Ts) = y(3Ts) = … = 1

Working out the input staircase function a(t) of the compensating element (which is the error signal e(t) at the sampling instants) gives:

a(0) = −1, a(Ts) = 0, a(2Ts) = 0.450, a(3Ts) = a(4Ts) = … = 0

The resulting difference equation for the compensating element is (Eqs. 25–26):

y(t) − 0.550y(t − 2Ts) − 0.450y(t − 3Ts) = 16.3a(t) − 24.8a(t − Ts) + 9.2a(t − 2Ts)

The connection diagram of the complete system including the simulated difference-equation element is shown in Fig. 21. The reference behavior of the model system is shown by oscillograms in Fig. 22a; close agreement with the theoretically found values was obtained.

Disturbance Behavior

The disturbance behavior with compensating element (26) is shown in Fig. 23a and must be characterized as unsatisfactory. By choosing a different difference-equation element to optimize primarily the disturbance behavior (Eq. 35), the disturbance behavior is improved (Fig. 23b) but the reference behavior becomes very unsatisfactory (Fig. 22b).

A compensating element that produces finite settling for both reference and disturbance step responses is given by Eq. (36):

y(t) + 1.49y(t − Ts) − 1.04y(t − 2Ts) − 1.45y(t − 3Ts) = 82.4a(t) − 95.6a(t − Ts) + 29.6a(t − 2Ts)

This still shows excessive overshoot in the reference response (Figs. 22c, 23c). An excellent reference and disturbance behavior can however be obtained by feeding the reference variable through a second compensating element (Fig. 24):

ȳ(t) − 1.16ȳ(t − Ts) + 0.359ȳ(t − 2Ts) = 0.204r(t)

The resulting responses are shown in Figs. 22d, 23d, and the final trajectory V(t) for a reference step is shown in Fig. 25.

Method for Solving General Difference Equations

For the general difference equation written in the form:

Φ(yₙ, yₙ₊₁, …, yₙ₊ₖ, n) = 0

the solution sequence is represented on the analog computer by a staircase function with freely selectable step width T. Solving for the highest-index term:

yₙ₊ₖ = F(yₙ, yₙ₊₁, …, yₙ₊ₖ₋₁, n)

and using the scheme of Fig. 26 — successive delay by sampling elements, then forming F (which need not be a linear operation) — the sequence is generated. Initial values are set by pre-charging the capacitors of the sampling elements to y₀ through yₖ₋₁.

Conclusion

The author believes to have shown that the simulation and investigation of sampled-data systems can be carried out on an electric analog computer using normal components, and that the simulation of difference-equation elements for use as compensating elements presents no difficulty. Such elements in a real sampled-data system (with digital measurement of the control deviation) are usually implemented as small digital computers; however, the analog simulation method suggests using such “analog” difference-equation elements also for correcting real sampled-data systems. For systems with continuous measurement of the control deviation — especially dead-time systems — one should also consider whether inserting a sampling element to create an “artificial” sampled-data system and then using analog difference-equation elements might yield better transfer characteristics than conventional continuous-control methods.

References

  1. W. Hurewicz, Filters and Servo Systems with Pulsed Data, in James/Nichols/Phillips, Theory of Servomechanisms, Chap. 5. McGraw-Hill, New York, 1947.
  2. J. M. L. Janssen, Discontinuous Low-Frequency Delay Line with Continuously Variable Delay. Nature, Vol. 169, Jan. 1952.
  3. G. A. Philbrick, Bucket-Brigade Time Delay. Palimpsest on the Electronic Analog Art. Philbrick Researches, Inc., 1955.
  4. J. S. Tsypkin, On the Synthesis of Pulsed Automatic Control Systems, in Regelungstechnik: Moderne Theorien und ihre Verwendbarkeit. Oldenbourg, Munich, 1957.
  5. J. R. Ragazzini & G. F. Franklin, Sampled-Data Control Systems. McGraw-Hill, New York, 1958.
  6. O. I. Elgerd, An Analog Computer Study of the Transient Behavior and Stability Characteristics of Serial-Type Digital Data Systems. Trans. AIEE, Pt. 1, July 1958.
  7. H. Chestnut, A. Dabul & D. Leiby, Analog Computer Study of Sampled Data Systems. Trans. AIEE, Pt. 2, Jan. 1959.
  8. F. J. Mullin, The Stability and Compensation of Saturating Sampled-Data Systems. Trans. AIEE, Pt. 2, July 1959.