Der Analogrechner als Lehr- und Übungsgerät in der Regelungstechnik

Complete English translation of the original German-language document (20 pages).

Operating and Demonstration Instructions

The Analog Computer as a Teaching and Laboratory Instrument for Control Engineering

(Based on a lecture by Dipl.-Ing. G. Jarbe, Siesti, Potentiometer-Werk, Waldburg)

A. General

Two possibilities for simulating dynamic processes in control engineering systems will be demonstrated. First-order and second-order closed-loop control systems will be demonstrated.

I. Introduction

Three problem areas are addressed:

Simulation of systems with the aid of the analog computer — an entirely new field in the sector of electronics: operational amplifiers in integrated circuit technology.
Deployment of the analog computer in the field of control engineering, since it allows the influence of individual system parameters to be investigated very easily and quickly.
Second-order closed-loop control systems will be demonstrated.

The analog computer is particularly well suited to the simulation of dynamic processes in control engineering systems, because it can represent physical systems as regulatory circuits in a very simple way. The operational amplifier, which is the key active element of the analog computer technology, has found its latest form of development in the electronics sector — operational amplifiers in integrated circuit technology. The analog computer is used operationally and has long since been developed to a level of technical maturity that allows its smooth integration within a system through the use of electronic components.

For the operation of analog computers, fundamental concepts are indispensable in the digital age in order to understand dynamic processes in an illustrative and physically correct manner.

II. Simulation of Dynamic Processes Using Analog Models

An analog computer can be used to simulate:

analogous mechanical systems
dynamic systems

An analog model is sought when physical systems are to be studied, or when electrical oscillations and electronic oscillations in their frequency bands influence the operating behavior of the plant, as control systems. In the case of irregular operation, control systems can trigger simple control responses. Second-order closed-loop control systems can be demonstrated easily.

The analog computer is built essentially using operational amplifiers, which form its key active element. Simulations of the following types are possible with an analog computer:

Analogy:

Springs
Velocity
Dynamics

Bild 1 (Figure 1) illustrates the block diagram of a simple mechanical force system.

An actuation (input) excites a system. The system itself has a certain state (position) and produces a reaction.

Bild 2 (Figure 2) illustrates the general system representation:

Loading → Vehicle → Position → System reaction

In springs and masses, as well as oscillatory (swinging) properties, the system consists of a first step that is generally controlled by a second step. The system behavior is developed through the study of system outputs. The system response is developed through a differential equation.

Bild 2 shows the general system behavior in overview: Input → System → Reaction (Systemverhalten / system behavior).

By feeding an input signal into a system, energy is stored in the system (kinetic, potential, or in the form of electrical stored energy). The system response is developed through the study of system outputs, which can be measured electrically or pneumatically. The fundamental behavior is established through a differential equation that describes the dynamic behavior of the system.

(1) Mechanical System

The mechanical system shown in Bild 3 consists of a spring, a mass, and a damper. The excitation is an applied force z. The mechanical system obeys Newton’s law, which leads to a linear differential equation:

Figure 3: Mechanical oscillatory system

A · x”(t) + C · x’(t) + B · x(t) = M · z(t) … (1)

A · x”(t) = C · x’(t) + B · x(t) = M · z(t) … (2)

where the coefficients are:

A = C · x: damping constant (times spring constant)
R = M · z

with the abbreviations denoting damping, spring constant, and mass. For the electrical system: inductance, resistance, and capacitance.

The differential equation of the mechanical system, after rearrangement in Eq. (2) above, takes the form of a general differential equation:

x” + (s₁/s₂) · x’ + (s₀/s₂) · x = (M/s₂) · z(t) … (6, general form)

Here z denotes the excitation, x denotes the first derivative of the variable with respect to time, and so on. For the mechanical system the variable x denotes position; for the electrical system it denotes current. The differential equation of the electrical system is analogous in form. The coefficients depend on the physical quantities — for the mechanical system: damping, spring constant, and mass; for the electrical system: inductance, capacitance, and resistance.

The differential equation is solved simultaneously analogously in this way. The interesting aspect of it is the particular forced response component.

In an analog computer, it is possible to specify which element is to be handled; the initial conditions on the system can be entered as initial values. This eventually leads to the representation of the system response in the form demanded.

Bild 5 shows the complete circuit for solving a first-order differential equation, though the right-hand side of the differential equation must be kept in mind.

Bild 5: Circuit for solving a first-order differential equation (1st order)

(2) Electrical System

The electrical series circuit shown in Bild 4 contains a coil, a capacitor, and a resistance. The current represents the excitation. The system obeys the following differential equation:

A · x”(t) + R · x’(t) + (1/C) · x(t) = (1/L) · z(t) … (5)

Figure 4: [Circuit diagram with components R, L, C in series]

Both the mechanical and the electrical system behave in the same way under corresponding excitation, and the differential equation of the mechanical system is analogous in form. The two systems are analogous and satisfy a differential equation of the general form:

s₂ · x”(t) + s₁ · x’(t) + s₀ · x(t) = z(t) … (6)

where z denotes the excitation, s₀, s₁, s₂ are constant coefficients. The differential equation of the mechanical system has the position x as the dependent variable, and the coefficients k₀ = B (spring), k₁ = C (damping), k₂ = A (mass). The differential equation of the electrical system has the charge x as dependent variable and the coefficients k₀ = 1/C (capacitance), k₁ = R (resistance), k₂ = L (inductance).

The differentiation of the mechanical system gives Eq. (2). From the transformed form, when the system is described by a differential equation with a continuous Heaviside excitation function, z = 1, the particular solution is obtained.

(3) Simulation of Oscillatory Systems

The described oscillation equation of 2nd order is rearranged and solved numerically by the analog computer. This means the time derivative is determined, i.e., the time solution is obtained.

From this follows:

x = z - 0.3 · x’ - 0.8 · x … (7)

Then follows:

x = x - 0.3 · x - 0.8 · x … (8)

The variable x of the highest order is obtained as a sum, which in turn contains the derivatives of x and the excitation function x. With one integrator, the variable of highest order yields the next-lower variable. From the 2nd integrator the following variable (derivative) is obtained. The result is the computation of the particular time characteristic.

An interesting aspect of it is the particular forced response. A sign inversion occurs at each integration step. This is of importance at the operation of the operational amplifier.

Bild 5 shows the complete circuit for solving a 1st-order differential equation; the right-hand side of the equation must be considered, with which the problem can be described by differentiating in the appropriate way, with pre-inversion of the sign.

Bild 5: Circuit for solving a first-order differential equation (1st order)

From the following equations, the damping ratio and the undamped natural frequency are obtained for the system under consideration:

Damping ratio:

D = (1/2) · √(S₁ / (S₀ · S₂)) … (9)

Natural frequency:

ω₀ = 2π · f₀ = √(S₀ / S₂) … (10)

The integrators are connected according to Bild 6. A time transformation thereby results, which causes the oscillatory process to run 1 : 2.2 times more slowly than the differential equation would correspond to.

Bild 6: Integrator

(Circuit: three 220 kΩ input resistors, 10 µF feedback capacitor; scaling amplifier with gain 1/2.2)

Exercise Tasks

Exercise 1: At the output of the 1st integrator, the time course of z is observed. The system begins with an initial condition of a weakly damped oscillation. (Figure is shown on page 7.)

Exercise 2: The parameter s₀ is varied, thereby changing the damping ratio of the system. If s₀ is reduced, the natural frequency is lowered. (Figure is shown on page 7.)

Exercise 3: The parameter s₁ is varied. If s₁ is increased, the damping ratio increases, giving greater damping. (Figure is shown on page 7.)

During the demonstration of Exercise 1, an inverting amplifier is inserted; this does not alter the result. The phase voltage at the output of this amplifier changes, which causes the oscillation to begin at the opposite sign. The simulation of an analog model for a mechanical oscillatory system has been achieved. An analog model has been found that can be described by the differential equation for spring, mass, and damping — thereby allowing these physical quantities to be reproduced.

Bild 7: Oscillatory process for Exercise 1 (P-system, 2nd order; S₀ = 0.8, S₁ = 0.1, S₂ = 1)

Bild 8: Oscillatory process for Exercise 2 (S₀ = 0.8, S₁ = 0.3, S₂ = 1)

Bild 9: Oscillatory process for Exercise 3 (S₀ = 0.6, S₁ = 0.3, S₂ = 1)

Bild 10: Oscillatory process for Exercise 4

[page 9: figure only — phase-plane trajectory of a 2nd-order oscillatory system]

(4) Establishing a Control Loop

The previously discussed weakly damped second-order P-system can now be used as a controlled plant (Regelstrecke) on which various controller types can be tested. As controller types: P-controller, PD-controller, PI-controller, and PID-controller are considered. The controller is preceded by a differential amplifier so that the control deviation (error signal) can be formed.

The controller gain (set-point value) of the controller is y; it is compared with the actual value x₀ of the plant, and the control deviation forms the basis for the controller output.

Bild 11 shows the control loop and the P-controller circuit.

Bild 11: Control loop and P-control device

(Block diagram: set-point W input, summing junction producing error y = W − x, P-controller block, plant block (Regelstrecke), output x, feedback loop back to summing junction)

Exercise 5:

On the path of the plant, a voltage step occurs whose height is equal to the range of the controller (set-point value). The plant begins to oscillate with a step response. The closed-loop gain K_p = 0 is set at the potentiometer. This means the controller does not yet act.

In Teilbild (a) (sub-figure a), the proportional gain of the controller is K_p = 0 and the loop is still open. The set-point step is applied, and the plant oscillates.

The gain is then set to K_p = 3. The controller is now strongly excited and the oscillation is well damped.

Further increasing K_p increases the damping of the oscillation still more. At the same time, however, a steady-state error remains; the controlled variable x does not fully reach the set-point W.

Bild 12 is located on page 13.

A PD-controller equation is derived from a differential network. The controller circuit shown in Bild 13 is an extension of the P-controller system consisting of a differentiating network whose output is further processed. The control of the P-system consisting of the plant is described in Bild 13; the D-portion (derivative) is set by a potentiometer that can be adjusted in the range 0 to 1.

Bild 13: PD differential network

(Circuit: op-amp differentiator stage combined with proportional amplifier)

The behavior of the controller is particularly good when the PD action is also active.

Bild 12: Control loop with P-controller

[page 13: figure only — three oscillatory step responses (a), (b), (c) for K_p = 0, K_p = 3, and K_p = 6.2 respectively]

Bild 14: PD control device

[Block diagram: dashed-border box labeled “PD-Regeleinrichtung nach Bild 13” (PD control device per Fig. 13) receiving −y; dashed-border box labeled “Regelstrecke nach Bild 5” (controlled plant per Fig. 5) receiving disturbance Z; output −x_W; inverting amplifier (gain −1); terminals x and W]

Exercise 6:

Bild 15 shows the result achieved with a PD-controller compared with the effect of a pure P-controller set to K_p = 3. Noteworthy is the damping of the transient oscillation. The differentiating action of the controller can be set on a potentiometer. Sub-figure (b) shows, compared with sub-figure (a), a stronger damping of the controlled variable.

(Bild 15 is located on page 14)

Bild 15: Control loop with PD-controller

[page 14: figure only — two step response curves (a) and (b): (a) shows underdamped oscillation (P-controller), (b) shows strongly damped, nearly monotonic convergence to set-point (PD-controller)]

Exercise 7:

A PI-controller can be realized by the parallel connection of an integrator to a P-controller. The circuit is shown in Bild 16.

Bild 16: PI control device

(Circuit: inverting input −Y; parallel paths through proportional resistors 200 kΩ and through op-amp integrator stage with 10 µF and 210 kΩ; outputs summed via mixing amplifier; reference input x_W)

The set-point value of the PI-controller is composed of two components that can each be set independently. They are summed by a mixing amplifier. The manipulated variable therefore consists of three components. One component is proportional to the control deviation; the other is proportional to the integral of the control deviation over time. This integral portion makes the steady-state error zero.

As a result, the controlled variable x reaches the set-point W without steady-state error, even for step changes. The PI-controller circuit (block diagram) is shown in Bild 17.

Bild 17: PI control device (block diagram)

[Block diagram: dashed box “PI-Regeleinrichtung nach Bild 16” (PI control device per Fig. 16); plant block (Regelstrecke nach Bild 5); set-point W; disturbance Z; controlled variable x_W; feedback loop]

Bild 18: Control loop with PI-controller

[page 16: figure only — oscillatory step response with label text:]

The control result achieved with a PI-controller is shown in Bild 18. At this setting it shows a poorly damped control transient. However, the controlled variable reaches the set set-point after a certain time.

Exercise 8:

The advantages of the previously considered controllers can be usefully combined in a PID control device per Bild 19. The individual components can each be set independently; they are summed by a mixing amplifier. The manipulated variable is therefore composed of three components: P-component, I-component, D-component.

Bild 19: PID control device

(Circuit: inverting input −Y through gain −1; three parallel paths:

Upper path (D-component): 220 kΩ input, 22 kΩ feedback, with op-amp differentiator
Middle path (P-component): 220 kΩ, combined with I-component through 10 µF integrator and op-amp
Lower path (I-component): 10 MΩ input, op-amp integrator, 22 kΩ and 1 µF Outputs mixed and summed to output −x_W)

In Bild 20, the control result achievable with a PID-controller is shown. The controlled variable reaches the set-point very quickly and without overshoot even before the deviation has become significant.

Bild 20: Control loop with PID-controller

(Step response: fast rise, minimal overshoot, settling to set-point — monotonic or near-monotonic approach)

Summary

By meaningfully combining the three components, the control device can be optimally adapted to the available controlled plant.

In summary, it can be stated that an analog computer offers the following advantages:

It allows the dynamic behavior of a controlled plant to be reproduced in the laboratory.
It thereby provides a good insight into the dynamics of the plant and the influence of individual plant parameters.
Controller devices can be simulated.
Complete control systems can be simulated. This provides the possibility of gaining a better understanding of the operation of a complete plant and its dynamic behavior.
In a simple way — namely by changing potentiometer settings — a control loop can be optimally adjusted; the most favorable controller setting for a given problem is easy to find. Bild 21 shows the control results achieved on one plant with various controller devices once more in a single diagram.

(Bild 21 is on page 19)

Figure 22 — Transient Responses in a Control Loop with 2 Delays

[page 19: figure only]

The figure shows a family of step-response curves (transient responses) for a closed control loop containing two lag elements (Regelstrecke mit 2 Verzögerungen). Each curve corresponds to a different controller type, arranged from left to right in order of decreasing oscillatory behaviour:

P-Regler (proportional controller) — leftmost curve; exhibits the largest overshoot and sustained oscillation.
PI-Regler (proportional-integral controller) — second curve; oscillation is somewhat reduced compared to the P-controller alone.
PID-Regler (proportional-integral-derivative controller, “Ideal-PD-Regler” variant) — third curve; markedly faster damping of oscillations.
PID-Regler mit PID-Regler (optimally tuned PID controller) — rightmost curve; best overall response with the quickest settling and least residual oscillation.

The horizontal axis represents time; the vertical axis represents the controlled variable. The curves illustrate that, for a process with two first-order lags, adding integral and derivative action progressively reduces overshoot and settling time, with the fully tuned PID controller yielding the most favourable transient behaviour.

Document reference: A 1362 – 3.19 (January 1972)

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