Optimization of a Control Loop with Desktop Analog Computer and Digital Attachment

This is an English translation of the original German-language article “Optimierung eines Regelkreises mit Tischanalogrechner und Digitalzusatz” by H. Kramer, published in elektronische datenverarbeitung (1968) 6, pp. 293–297. Telefunken Sonderdruck ASD 030/0968.

Abstract

The optimization problem of a functional control loop is solved by a desktop analog computer combined with a digital attachment. The Regelparameter (controller parameters) are optimized in such a way that the quadratic control-error criterion is minimized. The digital part performs the optimization automatically using an iterative hill-climbing procedure.

Optimization Setup

The optimization of the control loop with a PID controller (Proportional-Integral-Derivative) is the subject of this work. Using the Telefunken desktop analog computer, the system equations are simulated. In conjunction with the digital attachment, the optimal controller parameters K_P, T_N, and T_V are determined, i.e., those that minimize the quadratic criterion Z = ∫ e²(t) dt.

The Tischanalogrechner (desktop analog computer) is used to simulate the closed-loop control system. It provides the control error signal e(t), whose squared integral serves as the optimization criterion. This value Z is formed by integration over the simulation run and then evaluated by the digital part.

Optimization Procedure

The digital attachment conducts the parameter search. Starting values for K_P, T_N, and T_V are entered at the beginning. The parameter space is searched systematically: one parameter is varied while the others are held fixed, and the value of the criterion Z is evaluated after each simulation run.

The iterative hill-climbing method (Gradientenverfahren) proceeds as follows:

A simulation run is performed with the current parameter set.
The criterion Z is measured.
The digital unit increments one parameter by a small step.
A new simulation run is performed.
The new criterion Z’ is compared to the previous Z.
If Z’ < Z, the change was beneficial — retain it and continue; if Z’ ≥ Z, reverse the step.
The process continues until no further improvement in Z is found for any parameter.

The flowchart (Fig. 2) illustrates the program flow. The digital attachment sequences through the three adjustable controller parameters. For each parameter, a positive step is tested; if it yields improvement (smaller Z), a further step is taken in the same direction. Otherwise a negative step is tried. If neither direction improves Z, the step size for that parameter is halved. The algorithm terminates when all three step sizes fall below a preset minimum threshold.

Analog and Digital Parts

The analog computer simulates the controlled system and the PID controller. The digital attachment handles:

Setting and incrementing the controller parameters (via DACs feeding into the analog potentiometers or summers)
Triggering simulation runs
Reading the value of the criterion integral Z
Executing the optimization logic

The digital attachment used is the Telefunken DCR 100 connected to the Telefunken RAE 360 desktop analog computer (as shown in the photograph, Fig. 4).

Description of the Optimization

Concrete starting values for K_P, T_N, and T_V (e.g., K_P = 0.5, T_N = 0.4, T_V = 0.1) are entered into the digital unit. A simulation run of duration T_S (e.g., 0.5 s in accelerated mode, equivalent to a real process time of 5 s) is performed, and the criterion integral Z is computed.

The digital unit then increments K_P by a step ΔK_P and runs another simulation. If the new Z is smaller, another step in the same direction is taken; otherwise the direction is reversed. If both directions fail to improve Z, the step size is halved. This continues until the step is smaller than the minimum allowed step. Then the same procedure is applied to T_N, then to T_V. The full cycle over all three parameters is repeated until no parameter can be improved, at which point the optimum has been reached.

The article notes that the total optimization time is substantially shorter than manual trial-and-error tuning, and that the method converges reliably to the minimum of the quadratic criterion for the example system described.

[Translation covers the first 6 pages (entire document); this is a 5-page article (pp. 293–297) plus a cover sheet.]