Analog Computers

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Systemanalyse und Reglerentwurf am Beispiel einer elektromagnetischen Aufhängung

Complete English translation of the original German-language document (74 pages).

System Analysis and Controller Design Using the Example of an Electromagnetic Suspension

Institute for Measurement and Control Engineering with Machine Laboratory University of Karlsruhe (TH) Prof. Dr.-Ing. F. Mesch

Thesis (Studienarbeit) for Herr cand. mach. Volker Hornung

75 Karlsruhe, Postfach 6980, Kaiserstrasse — Tel. (0721) 608/2220

1 May 1979

System Analysis and Controller Design Using the Example of an Electromagnetic Suspension. Supplement to the experiment “Analog Computer II” of the Control-Engineering Practicum.

One of the possible approaches in the investigation of dynamic systems is to start from the technical circumstances, derive a physical model, and formulate it mathematically in the form of differential or integral equations. This type of system identification has the advantage over other methods of being generally valid and flexible, and enables investigations to be carried out in the planning stage of technical installations without experiments. The mathematical models are, in general, only an approximation to the real experimental methods.

Treating the mathematical model is often quite difficult, and an explicit solution can be given only in simple cases. In order to make full use of the expressive power of a theoretical system analysis, the use of electronic computers is therefore necessary.

Within the framework of the control-engineering practicum, the fundamentals of analog-computer technology are to be conveyed to participants over two laboratory afternoons. Experiment “Analog Computer I” is intended to clarify the voltage switching and normalization of the recorder. Building on this, Experiment “Analog Computer II” is to present the single-loop use of the computer and controller design as an example.

As the test track, a magnetic suspension is considered. The equation of motion of this nonlinear and unstable system is given in the existing version of the laboratory manual. This equation is linearized about a given operating point. The resulting transfer function is determined with the aid of the root-locus method. The resulting linear overall system is simulated on the analog computer.

The intended revision of the experiment is to incorporate an existing model of a magnetic suspension. For this, the dependence of the force acting on the suspended body from the distance of the body from the magnets, from the current through the magnet coil, and from the voltage must be determined experimentally. In addition, the characteristic curve of the optical distance-measuring device is to be determined. On the basis of these measurements the controller design is to be revised.

It is to be examined whether it is didactically useful to include the simulation of the nonlinear elements.

The revised experiment is to cover the following points:

  1. System analysis taking into account the measured characteristic curves;
  2. Controller design for the linearized system with the aid of the root-locus method;
  3. Simulation of the overall system on the analog computer;
  4. Control of the existing magnetic suspension with the aid of a controller programmed on the analog computer;
  5. Comparison of simulation and model construction.

The aim of the thesis is to produce a laboratory manual that provides all documents necessary for conducting the experiment and for adapting the model construction to the requirements of the practicum.

Processing time:6 months
Supervisor:Dipl.-Ing. H. Braun
Issue date:1 May 1979
Submission date:1 July 1980

(Prof. Dr.-Ing. F. Mesch)

Table of Contents

  1. System Analysis
    • 1.1 System Analysis Using the Example of an Electromagnetic Suspension
    • 1.2 Block Diagram of the Regulating System
      • 1.2.1 Subsystem
      • 1.2.2 Regulator
    • 1.2.3 Transfer Function of the Subsystem for the Open Loop
    • 1.3 Regulator Design
      • 1.3.1 Zero Placement Possibilities at the Experimental Setup
      • 1.3.2 Stability Investigation Using the Hurwitz Criterion
      • 1.3.3 Root-Locus Method
      • 1.3.4 Feedforward Gain, Comparison of Setpoint–Actual Values, Disturbance Variables
      • 1.3.5 II-Regulator
    • 1.3.6 Koppling of the Delay System — Setpoint–Actual-Value Comparison — Transfer Amplifiers (?)
    • 1.3.7 Lag-PD-Regulator (?)
    • 1.3.8 Copying of the Delay System
    • 1.4 Regulator
  2. Simulation on the Analog Computer
    • 2.1 System Circuit Plan
    • 2.2 Potentiometer Assignment
    • 2.3 Simulation
      • 2.3.1 Amplifier
    • 2.3.2 Adjustment
      • 2.3.3 Simulation
    • 2.4 Step Response
    • 2.5 Control of the Delay System
    • 2.6 Coupling of the Delay System
    • 2.7 IL-Scaling of the Delay System
    • 2.8 Koppling of the Delay System
    • 2.9 Simulation
  3. Experimental Procedure, Description, Didactics; Individual Items
  4. Appendix: Condensed Experiment “Analog Computer II” with Experimental Manual

1. System Analysis

The mathematical description of a technical system is in principle possible by theoretical derivation from physical basic laws. From geometry, physical relationships, and material constants the associated parameters can be determined.

An exact mathematical description often requires very great effort, so that experimental system identification can be advantageous. In this process, measurements are used to back-calculate the system equations, which is however difficult for more complex, nonlinear, or unstable systems.

With strongly nonlinear systems, methods that presuppose linearity can no longer be applied, and it must be considered whether linearization is feasible. Unstable systems are more difficult to treat, since they have an unstable operating point and the behavior without control cannot be controlled.

In practice one often starts from a simplified theoretical derivation and assumes the characteristic curves of the system. This leads to the determination of the parameters and the establishment of the validity range of the assumed physical laws [4].

1.1 System Analysis Using the Example of a Magnetic Suspension

Using the magnetic suspension shown in Fig. 1 as an example, a system analysis is to be carried out. A well-known example of the use of magnetic suspensions in engineering is the Transrapid of the firm Krauss-Maffei.

At very high rotational speeds or under extreme loads, magnetic bearings are used to achieve low friction. This corresponds to Experiment 9 of the control-engineering practicum [2], the magnetic suspension to be treated there, which is realized as follows.

[page 5: figure — Fig. 1, Magnetic Suspension (schematic drawing showing electromagnet above, suspended armature body, and mechanical frame)]

A freely suspended armature is located in a current-controlled magnetic field; its distance from the magnets is measured by a light-barrier system. This gives a voltage corresponding to the distance, which is compared with an adjustable setpoint. A downstream PID controller drives a voltage-to-current converter, which impresses a current on the magnet coil that determines the magnetic field (Fig. 2). A possible reaction of the armature on the coil inductance is disregarded in what follows.

[page 5: figure — Fig. 2, Block diagram showing: w → Regulator → Voltage-to-Current Converter → Plant → x, with LS-System feedback path]

The force exerted by the magnet on the armature mass m, shown in Fig. 1, is in general proportional to the change in magnetic energy:

$$\frac{dW_m}{dx} \sim i^2 \frac{d\Phi}{dx}$$

with R_m the magnetic resistance of the magnetic circuit; assuming a linear magnetization characteristic of the iron, and neglecting the magnetic resistance in the iron and under the assumption of a homogeneous magnetic field:

[page 6: continuation of derivation]

the field becomes R_m ~ x, from which

$$F = C \frac{i^2}{x^2}$$

follows.

Since computing the field distribution due to the complicated geometric relationships would be associated with very great effort, the actual behavior is investigated instead. To determine the actual relationship, the equilibrium gap x as a function of the impressed current i was measured for constant armature weight m_A. Plotting the current against the distance for a specific case, the following graph results:

[page 6: figure — Fig. 3, graph of current i [A] vs. distance x [mm], showing approximately linear relationship from about 0.2 A at 2 mm to 1.8 A at 18 mm]

It is recognizable that the curve shape can be approximated quite well by a straight line whose ordinate intersects at i = 0.27 A. This was not taken into account before, in that the location of the equilibrium point of the magnet was unknown. Introducing a displacement constant x₀, one obtains:

$$i = \frac{m}{K} (x + x_0)$$

which corresponds to the actual curve shape.

Deviations from the straight line are attributable to irregular field geometry, saturation of the armature, and counter-inductance. It is now to be examined how the above relationship acts on the force-gap behavior.

If one forms from the experimental quantities the quotient i²/x² and plots this against the distance, then the obtained characteristic corresponds to constant force only for x > 7.5 mm, that is, the physically expected horizontal line with the distance C from the origin (Fig. 4).

[page 7: figure — Fig. 4, graph of i²/x² [A²/mm²] vs. x [mm], showing hyperbolic decay curve from high values near x=0 down to approximately constant value around 0.01–0.02 for large x]

From this it follows that for x > 7.5 mm the force-gap law is:

$$F = C \frac{i^2}{x^2} \quad ; \quad C = 97.6 \frac{\text{mN mm}^2}{\text{A}^2}$$

In Fig. 5 in the appendix this is shown in connection with the also now available laboratory manual for the practicum system, which represents the behavior of this system quite well, and shows the possibility of simulating it and its treatment on the analog computer.

After a controller design is worked out, the linearization about the operating point must be carried out and a linearization taken from it (equation (2)):

$$m\ddot{x}(t) = mg - C \frac{i^2(t)}{x^2(t)} \qquad (2)$$

This equation will be used in the experiment for simulation on the analog computer, in order to show the possibility of simulating the nonlinear system and its treatment on the analog computer.

[page 8: continuation]

If later a controller design is to be carried out and the root-locus method used, a linearization of the operating point must first be performed:

After linearization of equation (2) about the operating point i_A, x_A gives:

$$l(t) = i_A + i_1(t) \qquad x(t) = x_A + x_1(t)$$

In the operating point holds i_A = m/K · x_A.

Thus i_A can be eliminated, and one obtains the differential equation for small deviations about the operating point:

$$m\ddot{x}_1(t) + mg \frac{2}{x_A} x_1(t) = \frac{2 C i_A}{x_A^2} i_1(t)$$

Defining:

$$\frac{2C}{mx_A^2} \cdot \frac{m}{K} \cdot x_A = \frac{2C}{Kx_A}$$

and it follows:

$$\ddot{x}_1(t) - \frac{2g}{x_A} x_1(t) = \frac{2Ci_A}{mx_A^2} i_1(t) \qquad (3)$$

Notably, the homogeneous part of the differential equation does not vanish (for a ≠ 0), which means the system is unstable.

The Laplace transform of equation (3) yields the transfer function of the plant as:

$$G(p) = \frac{X_1(p)}{I_1(p)} = \frac{K_S}{(1 + T_1 p)(1 - T_2 p)} \qquad (4)$$

[page 9: continuation]

with K_S = … and T₁, T₂ = …

The plant has the form of equation (10) of the experiment and is unstable.

1.2 Block Diagram of the Regulating System

To achieve the specified control objectives, the following subsystems are required.

1.2.1 Subsystem

The subsystem consists of: magnet, optical system, and photoresistor — for detecting the distance; through an amplifier with zero point, the distance signal is forwarded.

[page 9: figure — Fig. 5, cross-sectional technical drawing of the magnetic suspension setup showing: magnet coil, armature, optical distance-measuring device with lamp, diaphragm, and photoresistor (Blende mit Sprenger = diaphragm with diffuser)]

The light of a stabilized light source passes through a diaphragm, falls on a photoresistor. A change in the armature position causes a proportional change in the illuminated area of the photoresistor, and thus in its resistance value; this is converted into a voltage. Since the illumination does not influence the photoresistor linearly, an additional linearization circuit is used. The addition amplifier that follows is used for zero-point setting and for converting the armature displacement signal to a suitable control deviation.

[page 10: continuation]

[page 10: figure — Fig. 6, graph of voltage U [V] vs. distance x [mm], approximately linear relationship over 0–14 mm range]

Admittedly, in a range of only 20 mm the characteristic of the linearizer is linear; the quotient d(U)/d(x) can be approximated as:

$$\frac{d\phi_{LS}(x)}{dx}\bigg|_{x_A} = 0.5 \frac{V}{mm} \qquad (9)$$

In such cases optical sensors are used, which take up only a small disturbance and can, however, be easily disturbed by small fluctuations in the ambient light.

1.2.2 Regulator

The design is based on the production of a cascade controller as described in reference [11]. The regulator acts on a proportional summing amplifier with adjustable P-, I-, and D-gains. A parallel combination of operational amplifiers is used to achieve high gain and adjustable component values.

[page 11: figure — Fig. 7, detailed circuit schematic of the PID controller showing operational amplifier stages]

The transfer function of the P-branch according to Fig. 7 is:

$$G_P(p) = -\frac{R_2}{R_1}$$

With R₁ = k·R₂ and R₂ = (1–k)·R₂ the I-part becomes:

$$G_I(p) = -\frac{1}{1 + k(1-k)} \cdot \frac{1}{p \cdot T_N}$$

According to Fig. 8, it can be derived as:

$$V_3(p) = -\frac{R_4}{R_3} \cdot \frac{1}{p \cdot T_I}$$

The D-part results as:

$$G_D(p) = -\frac{…}{…}$$

[page 12: continuation]

For decoupling, the PID controller uses an input amplifier with f_E0 = 2.000 (f_{10} = 1050000).

The transfer function of the P-branch according to Fig. 7, under the assumption of an infinite open-loop gain of the operational amplifiers, is:

$$G_{PID} = G_{P0} \left(1 + \frac{k_1 \cdot R_N}{R_N} + \frac{k_2}{p}\right)$$

with a given setting k_P = k_I = 0.75.

In the regulator, a pure PD-controller finds no application. The schematic according to Fig. 7 can be used. The D-part also realizes a first-order transfer element, because of the small residual friction influence.

Thus the transfer function is:

$$G_R(p) = G_{P0} \left( k_P + \frac{1}{k_I} + \frac{k_D p}{1 + T_{V0} p} \right) \qquad (6)$$

with:

  • k_P = 0 · 100 s⁻¹
  • k_I = 0 · 100 s⁻¹
  • T_{V0} = 12.5 ms

1.2.3 Control of the Magnetic Field Strength

The voltage-to-current converter is to convert the voltage u_S supplied by the controller into the magnet current i as delay-free as possible. To achieve the largest possible current, two power transistors from a Darlington transistor are driven by the controller. Since voltage drops of approximately 0.7 V per stage occur at the transistors, the following equation results:

$$i = \frac{u_S - 2.1V}{R + Lp} - \frac{2.1V}{R + Lp} - \frac{u_S}{R + Lp} = I_{\text{const.}}$$

I_const. is regulated out by feedback, so that:

$$i = \frac{u_S}{R + Lp}$$

from which the transfer function with:

$$G_I = \frac{i}{u_S} = \frac{1}{R + Lp} = \frac{K}{1 + Tp} \quad \text{with } K = 0.782 \text{ A}^{-1}$$

follows.

To check the counter-inductance arising through approach of the armature to the magnet, the step response of this subsystem was measured, from which L = R²·f for 3 mm ≤ x ≤ 4 mm was determined. From this it follows that L has no appreciable dependence on the distance, so that L = const. = 1.64 H may be assumed, whereby the counter-inductance effect on the voltage-to-current converter is excluded. This constant causes a strong delay behavior of the converter with a time constant of:

$$T = \frac{L}{R} = \frac{1.64 \cdot H}{5.5} = 300 \text{ ms}$$

Therefore a cascade control is provided, which reduces the delay.

[page 14: continuation — cascade control]

[page 14: figure — Fig. 10, block diagram of cascade controller showing inner current control loop]

The inner controller is designed as a P-type; it receives a current-proportional voltage u_i and supplies a current-proportional signal u_S. Its transfer function is:

$$G_{RI}(p) = k_{RIP} = 100/1$$

Thus the innermost realized cascade control loop achieves a time constant of at most 5 ms:

$$G_{RISL}(p) = \frac{k_{RIP}}{1 + k_{RIP}} \cdot \frac{K_{I}}{1 + T_y p} \cdot \frac{1}{1 + T_p}$$ with:

  • k_{R0} = 0.00497 s
  • k_P = 5 ms

Thus the transfer function of the innermost loop with a time constant of 5 ms:

$$G_{RISL}(p) = \frac{…}{(1 + T_y p)(1 - T_p)…} \qquad (7)$$

1.3 Block Diagram of the Closed Loop

[page 14: figure — Fig. 11, complete closed-loop block diagram of the magnetic suspension control system showing: setpoint w → regulator G_R → inner current loop G_RI → plant G_S → output x, with outer position feedback via sensor G_M]

1.4 Transfer Function of the Open Loop

From equation (4) follows for the open loop:

$$G_0(p) = G_S \cdot \frac{G_S}{1 + G_{V0} G_S} G_{\text{PID}} G_{\text{INT}}$$

and

$$G_0(p) = G_S \frac{G_S}{1 + G_{PID} G_S} \cdot \frac{K}{1 + T_{V0} p} G_{\text{INT}}$$

With G_0(p) = K_S · G_S one obtains:

$$G_0(p) = K_S G_S \cdot \frac{K}{(1 + T_y p)(1 - T_p)(1 + T_{V0} p)} \cdot G_{\text{INT}} \qquad (8)$$

with K = K_S · K_{LS} · K_{INT}

From this the treatment of the control loop and the design of the controller using the root-locus method becomes possible.

2. Controller Design

In the practicum, the simulation and control of the nonlinear system are to be carried out on the analog computer under a 1. A PD controller was used for this purpose, however, it was found too difficult to achieve stationary accuracy; a PID controller is therefore desirable. Possible controllers of higher order were not considered, since the experiment should not be too complicated and should not exceed one afternoon in preparation time.

In the laboratory preparation, some theoretical knowledge about the stability criterion of Hurwitz and the root-locus method is to be conveyed and demonstrated using examples adapted to the present system. This is to be followed up in the laboratory preparation and demonstrated using the example of zero placement. The influence of the zeros on the root-locus diagram is to be shown here [5].

2.1 Stability Investigation Using the Hurwitz Criterion

In the following stability investigation, the possibility is shown, using the Hurwitz criterion derived from the system-analysis transfer function of the open loop, of obtaining the stability conditions for the closed loop.

The characteristic equation of the closed loop is given with (8) and equation (9) of the laboratory manual:

$$G_0(p) + 1 = \frac{-K_R K_S (1 + T_{R1} p)(1 + T_{R2} p)}{p(1 + T_y p)(1 - T_p)(1 + T_{V0} p)} + 1 = 0; \qquad T > T_D > T_y$$

A 5th-order system results. Since the compensation of the largest time constant appears reasonable, with T_{R1} = T, one obtains a 4th-order system:

$$G_0(p) + 1 = \frac{-K_R K_S (1 + T_{R2} p)}{p(1 + T_y p)(1 - T_p)} + 1 = 0$$

[page 17: continuation]

or:

$$T T_y T_p^3 + (TT_y - T_y T_p - T T_p) p^2 + (T_y - T_p - T) p - K_R K_S T_{R2} p - K_R K_S = 0$$

The Hurwitz criterion for n = 4 gives:

$$a_3 > 0 \qquad (10)$$ $$a_1 a_2 a_3 - a_0 a_3^2 - a_4 a_1^2 > 0 \qquad (11)$$

From (10) it follows that:

$$a_3 > 0$$

and condition (11) must also be satisfied.

$$T T_y a_3^2 + (T T_y T_{R2} - T_y T_{R2} T_p - T T_p T_{R2}) a_1^2 - a_0 a_3^2 - a_4 a_1^2 > 0 \qquad (12)$$

Thus K_R results, and for this, one must have p² > 0, so K_R is real. Then the limiting value T_{R2,krit} is sought. From (11) (given the conditions from (10)):

Example:

  • m = 108 g
  • x_A = 9.85 mm → 14.70 mm
  • g = 27.4 ms
  • T = 8 ms

$$G_0(s) = \frac{-0.009257}{\left(1 + 27.4 s\right)\left(1 + 3 \text{ms}\right)\left(1 + 18.5 \text{ms}\right)} \cdot p^{-3}$$

[page 18: continuation]

$$G_0(p) = \frac{-0490 \cdot K_S T_{R2} \left(\frac{p}{…} + p\right) \cdot p^{-3}}{(24.1 s^{-2} + p)(115s^{-2} + p)(830 s^{-2} + p)} \cdot Tp$$

  • a₁ = 2740 ms²

  • A = 0.251 ms³ / ms²

  • a₂ = 362.8 ms

  • a₃ = 204.1 · s³ ms³

  • a₃ = 6.9 ms

  • G = 0.5936 ms²

$$a_3^2 T_{R2}^2 - 3.588 \cdot 10^3 ms^2 \cdot T_{R2} + 4.167 \cdot 10^3 ms^3 > 0$$

Thus T_{R2} is real; for this one must have:

$$T_{R2 a} = 1.28 ms \qquad k. \Delta T. \qquad T_{R2 b} = 1.25 ms$$

From p it can be seen, with the aid of a coefficient comparison with equation (9), that the critical circular frequencies can be determined:

$$-k_{p0}: \omega_{min} = 0.6777 \qquad \omega_{1,\min} = 2 \cdot 10.9 \text{ s}^{-1}$$ $$+k_{max}: 1.7003 \qquad \omega_{1,\max} = 2 \cdot 46.7 \text{ s}^{-1}$$

2.7 Plotting the Root-Locus Curves

The plotting of the WOK from Fig. 12 for

$$G_0(p) = \frac{-3425 \cdot k_2 \cdot (Te^2 + p) \cdot e^{-p\delta}}{(96 \cdot Te^2 - p)(125e^{-2} - p)(96e^{-2} + p)p}$$

with

$$n = 3425 \cdot k_2$$

is carried out in the laboratory exercise in individual steps with the aid of the program described below. The root locus is plotted in the following illustrations using the WOK computation program “Root Locus V” by Raymond B. Ash.

2.1 Root-Locus Curve Discussion

In order to give the laboratory participants an insight into the possibilities of a controller design using the WOK method, a WOK discussion is carried out, which is to be explained in what follows.

Using the WOK program, the gain can be computed at any desired point of the WOK, which is used later in conjunction with the simulation on the analog computer for the rapid determination of the critical controller parameters.

On the basis of the root locus of the controlled system, the behavior of the controlled plant can now be discussed. The objective is to make the system as fast and stable as possible, i.e., two dominant poles on the bisector of the angle with a large distance from the imaginary axis would be ideal. Considering the necessary gain, one must also add an additional pole on the real axis, which strives from −80 s⁻¹ toward the origin. If the gain is reduced, the dominant poles shift closer to the imaginary axis. By further approach to the origin-adjacent zero, a shift of the WOK branch to the left can be achieved (Fig. 13/14), which reduces the I-component of the controlled plant. This behavior can be circumvented by shifting both zeros toward the origin using a compensation zero (Fig. 15/16). The WOK branches then shift still further to the left; however, this pole also shifts further toward the origin. Since this pole is now only slightly influenced, the WOK branches can be shifted further to the left until the control loop, at a corresponding gain, exhibits nearly pure PT2 behavior (Figs. 17 and 19).

Since the WOK program has a limited display range, overall views of the WOK at a ratio of 1:10 are made possible in Figs. 19/20 by reducing by a factor of 10.

[page 21: figures only — Abb. 13, Abb. 14, Abb. 15: three root-locus plots showing progressive pole-zero migrations]

[page 22: figures only — Abb. 16, Abb. 17, Abb. 18: three additional root-locus plots]

[page 23: figure only (Abb. 19) plus text]

Carrying out a PID comparison yields another PID-characteristic, which results in the following system:

$$G_0(p) = \frac{-n_0 \cdot k_E(1 + T_0 p)}{(1 - 2T_0 p)(1 - T_0 p)(1 + T_0 p)}$$

with

$$T_0 = T$$

$$G_0(p) = \frac{-n_0}{(1 - 2T_0 p)(1 - T_0 p)} = 0$$

or

$$G_0(p) \cdot (1 - 2T_0^2 p^2 + T_0^2 p^2 + (T_0^2 T_0^2) p^2 + T_0^2 p^2) = 0$$

Herewith n + p = 0; for 3 > n > n₀ is satisfied: n₀·T₀² + n₀·T₀² = 0 and

$$T_0 \cdot T_0 = \alpha_0 = 0$$; m = 0 gives:

$$G_0(p) = \frac{-n_0 \cdot k_E}{(96 \cdot Te^2 - p)(125e^{-2} - p)(96e^{-2} + p)p}$$

with

$$n = 3425 \cdot k_2$$

Consequently the WOK can be plotted per Abb. 20.

[page 24: figures (Abb. 20 and Abb. 21) with brief text]

By shifting the controller zero against the origin, the WOK branch shifts to the left and ends up finally on the other side of the zero (Abb. 21 ≈ 23). The dominant pole from F₁ to F₂ can then be designated as the dominant PT2 behavior, where the VZT-behavior has been displaced (Abb. 22).

Abb. 21 shows the course of the WOK at smaller values of β₀.

[page 25: figures only — Abb. 22, Abb. 23, Abb. 24: root-locus plots]

2.9 Setting Possibilities at the Target Controller

The laboratory exercise uses a Vollert- semi-pre-designed P-, I-, and D-block controller. This controller offers a direct insight into the functional relationships of a PID controller. The influence on the behavior of the controlled plant from a parameter change can be followed immediately.

2.9.1 P-Controller

Using the following information, a simple determination of the controller parameters is possible. The following applies:

$$K_P = K_R(K_{P01} + K_{P02}) \cdot K_{TP}$$

$$K_T = T$$

$$K_R = \frac{K_P \cdot K_{T1}}{K_{P02}} \cdot K_{TP}$$ $$K_{P1/2} = \frac{K_P}{K_R \cdot K_{TP}} \cdot \left[1 - \left(\frac{K_P}{K_R \cdot K_{TP}}\right)\right]$$

and rearranged:

$$K_{R,1/2} = \frac{K_P}{K_{P01} + K_{P02}} \cdot K_{TP}$$

2.9.2 PD-Controller

When using the PD-controller with K₀ = 0 and the transfer function:

$$G_R(p) = -\alpha_{V1} \cdot \frac{p + K_{P0} V}{U_{EIN}}$$

for x₀ = 0: τ = 100 s
for x₀ = 0: τ = 100 s ; x₀ = 0

Here:

$$K_P = (T_1 - 2) \cdot \delta K_P$$ $$K_T = K_R$$

and

$$K_D = K_R$$

2.5 Parameter Sensitivity

One advantage of the analog computer is, among other things, the straightforward parameter change of the differential equation of the system, which is to be briefly illustrated using the example of a parameter change.

If the spacing of the armature from the magnet is changed, this enters the determination of the time constant of the unstable plant according to equation (4):

$$T = \sqrt{\frac{x_0}{2E}}$$

With the controller setting remaining unchanged, this leads to a displacement of the poles and thus to a change in the course of the root-locus curve. If a negative controller setting according to Abb. 17 is used, a reduction of x₀ causes a reduction of T and consequently a displacement of the negative poles to the left. The root locus then moves away from the origin and causes a shift of the WOK in the same direction (Abb. 25).

[page 27: figure — Abb. 25: family of root-locus curves parameterized by x₀]

Fig. 26 shows that this leads to a stabilization of the system behavior.

[page 28: figure only — Abb. 26: family of step-response curves parameterized by x₀, showing progressive stabilization]

3. Simulation on the Analog Computer [6]

Experiment 3 of the control engineering laboratory course presents the basics of the analog computer. The objective of the experiment is to illustrate the applicability of the analog computer. The AC simulation is intended to make clear how realistically the system behaves under application conditions on the actual plant, using a non-trivially stable system as an example.

3.1 Subsystem Amplifier

3.1.1 Stress

With the amplitude normalization:

$$x(t) = A_0 x_0(t)$$ ; A₀ = 200 V⁻¹

$$x(t) = A_1 x_0(t)$$ ; A₁ = 0.4 V⁻¹

$$x(t) = A_0 x_0(t)$$ ; A₀ = 0.045 g

$$x(t) = B_1 x_0(t)$$ ; B₁ = 200 ×

equation (2) can be written as:

$$m\ddot{x}(t) = mg - \frac{k^2}{R} \cdot \frac{i_0^2(t)}{x_0(t)^2}$$

It follows:

$$\ddot{x}_0(t) = g - \frac{1}{m} \cdot \frac{k^2}{R} \cdot \frac{i_0^2(t)}{x_0(t)^2}$$

and

$$A_0 \ddot{x}0(t) + (A_1 - A_0) \cdot \ddot{x}{00}$$

with the initial condition: x₀₀ = x₀₀

3.1.2 Delay Element

The transfer function of the voltage-to-current converter reads:

$$U_{PID} = \frac{1}{R_T + R_P} \cdot \frac{K_S}{\tau_p} \quad ; \quad K_S = 0.0045 \text{ V} \quad ; \quad \tau_p = U_a$$

From this the resulting scaled differential equation is:

$$\dot{i}(t) = \frac{1}{\tau_p} u_y(t) - \frac{1}{\tau_p} i(t)$$

With amplitude normalization:

$$\dot{i}_0(t) = T_1 I_1(t)$$ ; T₁ = 500 A

$$u_0(t) = T_2 U_0(t)$$ ; T₂ = 10 V

[page 30: continued equations]

From this follows finally:

$$x_0(t) = \frac{1}{A_0} \cdot \left[\frac{B_1 \cdot k_0(t)}{U_{\text{max}}(t)}\right]^{1/2}$$

From this circuit (Abb. 27) can be drawn. For the coefficient comparison with

$$x_0(t) = m_{00} \dot{x}{00} - (m{00} \dot{x}_{00})^2$$

the following normalized coefficients are obtained:

$$m_{00} = \frac{1}{A_0}$$ ; $$m_{00} \dot{x}_{00} = \text{const.}$$

and subsequently the polynomial is found with n = 1: x₀₀ = x₀₀

$$A_0(t) = \frac{1}{A_0} x_0(t) ,dt + x_{00}$$

$$A_0(t) = \frac{1}{A_0} x_0(t) ,dt + x_{00}$$

$$m_{00} = \frac{k_B}{A_0}$$

and

$$m_{00} = \frac{k_B}{A_0}$$

3.1.2 Delay Element (continued)

The transfer function of the voltage-to-current converter gives:

$$U_{PID} = \frac{K_S}{1 + \tau_p p} \quad ; \quad K_S = 0.0045 ; V \quad ; \quad \tau_p = U_a$$

Thus the resulting scaled differential equation is:

$$\dot{i}(t) = \frac{1}{\tau_p} u(t) - \frac{1}{\tau_p} i(t)$$

[page 31: continued equations]

The scaled equation follows:

$$I_0(t) = \frac{A_0}{T_1} m_{00} U_0(t) - \frac{1}{\tau_p} I_0(t)$$

From this the circuit in Abb. 27 is derived.

For the potentiometer setting:

$$m_{00} = \frac{1}{\tau_p} = \frac{1}{1} = 1$$ ; $$m_{00} \dot{x}_{00} = 0.0615 \times 10^{-6}$$

In the real plant there is an input gain of V = 100 measured in volts, and this must be accounted for:

$$m_{02} = 0.0615$$

Furthermore:

$$m_{00} \dot{x}{00} = \frac{1}{\tau_p}$$ and consequently: $$m{00} = 0.25$$

3.1.3 DC Power Supply, Null-Reference Comparison, Frequency Separation

These three subsystems require two potentiometers m₀₀ and m₀₁ and are simulated together with the aid of two potentiometers.

3.1.4 PD-Controller

For the PD-controller the circuit (Abb. 21) is used.

G₀(p) gives:

$$G_{TW}(p) = \frac{T_{VW}(p)}{G_0(p)} = \frac{-m_{00}}{1 - m_{00} T_{VW}(p)} \cdot G_0(p)$$

This yields:

$$G_0(p) = \frac{1}{1 + \frac{m_{00}}{10m_{02} G_0(p)}} \cdot \left(-m_{00} + \frac{m_{00} m_{00} p}{1 + \frac{m_{00}}{10m_{02}}} p\right)$$

[page 32: continued equations]

$$G_0(p) = \frac{\sqrt{m_{00}^2 p^2}}{1 + \frac{m_{00}}{10m_{02} G_0(p)} p}$$

and finally becomes:

$$= G_{TW}(p) \cdot G_0(p) \cdot G_{TW}(p) \cdot \left(-m_{00} + \frac{m_{00} m_{00} p}{1 + \frac{m_{00}}{10m_{02}}} p\right) \cdot m_{01}(p)$$

the transfer function of the controller:

$$G_R(p) = \frac{G_{VW}(p)}{G_0(p)} = \left(-m_{00} + \frac{m_{00} m_{00}}{1 + \frac{m_{00}}{10m_{02}}} p\right) \cdot \left(1 + \frac{m_{00}}{10m_{02}} p\right)$$

Potentiometer setting:

The controller will later be used with a gain of V₀,ₘₐₓ = 1000. The normalized potentiometer values are then set as follows:

$$K_P : m_{01}^{\text{max}} = K_P$$ ; $$m_1 = \frac{K_P}{V_{0,\text{max}}}$$

$$K_T : m_{02} m_{01} k_T = m_{00}$$ ; $$m_2 = \frac{m_{00}}{m_{01} k_T}$$

$$K_D : m_{03} = \frac{T_V - m_{00}}{10 \cdot m_{02} G_0}$$ ; $$m_3 = \frac{T_V - m_{00}}{10 m_{02}}$$

$$m_{04} : m_{04} = 0.01 \text{ chosen}$$

$$T_1 : T_1 = 10.5 \text{ ms}$$ ; $$m_{10} = 0.55$$

[page 33: figure only — Abb. 27: complete AC circuit diagram for the magnetic suspension experiment]

3.1.6 Coupling of the Subsystems

In the laboratory exercise, the same control system is to be used to regulate both the simulated plant and the real plant. Abb. 28 illustrates the circuitry.

[page 34: figure — Abb. 28: block diagram of the coupled subsystems (simulated + real plant under the same controller)]

The SSW (switching unit) of the real plant provides an input gain of V = 100. Behind the IS system, α_x is obtained, which together with the input gain α_IST corresponds to the transfer functions of the setpoint comparison and separator amplifier, through which the position information is fed to the AR control system. To obtain a sufficiently large control range, a controller gain of c_aus = 10 is provided.

At the simulated plant V is taken into account in the delay element.

α_IS serves for converting x₀ to the voltage corresponding to that of the real plant.

For the real plant it holds:

$$G_{ARR} \stackrel{!}{=} G_{IST} G_{TS} G_{PID}$$

and consequently:

$$\alpha_{13} G_{IST}^2 G_{PID,AR} c_{aus} V = 1.54 \cdot G_{PID}$$

[page 35: continued equations]

From this it follows:

$$\alpha_{13} = 0.754$$ $$V_{\text{nom}} = 10$$ $$G_{PID,AR} = 0 = 1$$

In EIN (input) and AUS (output) the following must be accounted for in order to allow adjustment without error.

For EIN the following applies:

$$\frac{m_0 \cdot x_0 \cdot G_{IST}^2 G_{TS}}{x_0 \text{ in m}} \quad \text{in V}$$

and one obtains:

$$\alpha_{03} = \frac{1}{A_0} \cdot A_0 G_{IST}^2 G_{TS} = 0.09325$$

For AUS it follows:

$$i_R = 100 \cdot G_{SSW} \cdot 10 \cdot$$

$$\frac{i_R \text{ in V}}{i_R \text{ in A}}$$

and

$$\alpha_3 = \frac{k_1}{k_2} = 1$$

The zero-point setting is performed via potentiometer m₀₀ and m₀₁, and the following are found:

$$V_{00} = \frac{m_0 \cdot G_{IST}^2 G_{TS}}{100} = 10 \cdot 0.09325$$

and

$$m_0 = A_0 \cdot G_{IST}^2 G_{TS} \cdot 0.09325 = 10 \cdot 0.5025$$

It is to be noted that the difference α₀ = α_IST ≠ α₀₀ also means that the limited controller gain cannot be arbitrarily large.

3.1.7 Example

n = 106.7 g
x₀ = 14.26 mm ; k₀ = 0.009577 Ω⁻¹

Calculation of the potentiometer values:

$$m_{00} m_1 = \frac{9.81}{200} \cdot \frac{k_0^2}{i_{00}^2} = 0.0491$$ ; $$m_{00} = 0.0x91$$ ; $$\alpha_1 = 1$$

$$m_{11} m_{10} = \frac{k_1^2}{k_2^2} \cdot 10 = \frac{(0.009577)^2}{0.045} \cdot 10 = (0.009577)^2 \cdot 2 \cdot \frac{1}{0.045}$$

$$m_{00} = \frac{k_B}{k_B} = 1$$ ; $$m_{00} = 1$$

$$m_{00} \dot{x} = \frac{k_0}{\tau} = 0.0779$$ ; $$m_{01} = 0.0718$$ ; $$m_0 = 1$$

$$m_{00} = \frac{k_B}{\tau} = 0.1072$$

The controller settings determined from the WOK analysis are to be simulated in the following.

The equation for the open loop follows from (8):

$$G_0(p) = \frac{-0.009577 \cdot \left[\frac{k_1}{k_2}(1 + T_{10}p)(1 + T_{20}p)\right]}{p(1 - 96 \cdot Te^2 \cdot p)(1 + 125e^{-2} \cdot p + p^2)(96e^{-2} + p)}$$

or

$$G_0(p) = \frac{-n_0}{p(1 - 96Te^2 p)(125e^{-2} + p)(96e^{-2} + p)}$$

For Abb. 19/20 (T₁₀ = 55 ms, T₂₀ = 35 ms) at the characteristic points one obtains at the imaginary axis the critical frequency:

$$\omega_{krit,0} = 0.456 \cdot 10^{-4} \text{ s}^{-1}$$

The equation of the open circuit follows from (8):

$$\sigma_{krit,0} = 0.456 \cdot 10^{-4} \text{ s}^{-1}$$

Page 37

From (15) it follows:

k_p = 3.74 s⁻¹
k_p = 465.42 and from (14): k_D = 0.1034
k_p = 2.102 s

w₀₁ = 0.00006
w₀₂ = 0.1034
w₀₃ = 0.1043

This yields the following table:

t0.1696090.3100
010.1034100.3900
020.1034110.3945
050.1023420.600
240.041130.0540
050.0631140.0025
061.0000150.043
070.0708
080.2500

When trimming the oscillation setpoints using the AD trimmer, the trimming potentiometers must be set with the aid of a voltmeter due to the sensitivity of the potentiometers, which requires that the values of w₀₁, w₀₂, w₀₃, w₀₄, w₀₅, w₀₆, w₀₇, w₀₈, w₀₉, w₀₁₀, w₀₁₁ be set to a tenth of the values given above.

Page 38

Further settings are listed in tabular form below:

AbbreviationT912T520
T_p133s1s10s27.4ms
T_p233ms27.4ms27.4ms27.4ms
K_p krit,min0.456 · 10⁶ s⁻²0.4 · 10⁶ s⁻²21…5.290.320 · 10⁶
K_p krit,max3.747 · 10⁶114.42 · 10⁶ s⁻²90.315 · 10⁶102.1
K_p3.747 · 10⁻¹114.42 · 10⁻¹45.35 · 10⁻¹102.1
T_D105.02174.2105.4102.1
K_R0.102s1.40s1.0944.25
w₀₁0.10560.17600.10500.1041
w₀₂0.0060.0020.0000—
w₀₃0.10430.00090.00430.0144

Page 39

4. Circuit Description

The control system realized in the experiment consists of a light-barrier system, a setpoint-actual value comparator, a PID controller, a voltage-to-current converter, and a power supply unit assembled together.

The light-barrier system consists of a phototransistor, a collector line, a reflector, and a lamp, which are powered by the stabilized power supply unit and whose components R₁, C₁, D₁, D₂ are constructed around it. The phototransistor T₁ is likewise supplied by a stabilized voltage source (R₃, C₃, D₄). At its collector is the trim potentiometer P₁. The operating drop at this potentiometer determines the distance between armature and magnet.

Operational amplifier IC 8 serves as an isolation amplifier for decoupling the setpoint-actual value comparator from the PID controller; its amplification factor K_ZL is set by potentiometer P₁₀. Two voltages are summed at the input: on the one hand the positive voltage from potentiometer P₁, and on the other hand the negative direct voltage from potentiometer P₂, which serves as the setpoint reference voltage. The sum of these two voltages forms the input voltage of the PID controller, so that IC 8, acting as a summing amplifier, drives IC 4 (the transistor T₂). It is a voltage-controlled current source whose power transistors T₃ and T₄ pass the current through the electromagnet T₁ (sic: controlled by IC 1 from context).

The feedback path of the voltage-to-current converter consists of operational amplifier IC 5 and the sense resistors R₁₄…₁₇ as well as the trim potentiometers P₄. Because the current-proportional voltage drop across the resistors R₁₂ and R₁₃ is not mass-referenced (ground-referenced), IC 5 must be inserted.

The power supply unit consists of a voltage stabilizer IC 7 and a series transistor T₉. In order to obtain a symmetric delay voltage for the operational amplifiers, half of the stabilizer output voltage is applied to the non-inverting input of IC 6. Because of the operational amplifier coupling, the emitters of the transistors T₄ and T₅ are also at half the supply voltage. The potentiometers P₂, P₃, P₄ and the diode D₅ are brought out and the 10-turn potentiometers P₁ and P₄ are made accessible to the outside.

Two observations round out the picture of the internal control system: through activation of the phototransistor voltage from the one side and with the application of an external reference signal at P₃ from the other.

Page 40

The transistors T₄ and T₅ are at half the supply voltage. The emitters of these transistors form the node point of the circuit.

To prevent high-frequency oscillations, the filter capacitors C₁ and C₂ as well as the damping network C₃ are employed as the compensation network for the circuit.

The potentiometers P₂, P₃, P₄ and P₅ are brought out for access. Two observations round out the picture: the internal control system can be perturbed through activation of the phototransistor voltage from one side and through application of an external reference signal at P₂ from the other.

Page 41

[page 41: circuit schematic figure only — full schematic of the magnetic levitation control system, showing the light-barrier circuit, PID controller (IC 1–8 operational amplifiers), voltage-to-current converter with power transistors 2N3055 and 2N3025 (labeled), electromagnet coil (labeled), and power supply section with IC 7 stabilizer and series transistor T₉. Component labels visible include: IC 1 through IC 8, 2N 3055, C7 772, 2N 3025, and various resistors and capacitors throughout.]

Page 42

Parts List

Resistors (R):

DesignatorValue
R1, R023kΩ
R11kΩ
R2334
R0, R1147
R0, R11110 Ω / 271A
R13-33 / STA
R1533kΩ
R347kΩ
R054kΩ
R644Ω

Potentiometers (P):

DesignatorValue
P1, P21kΩ Wende (wirewound)
P1, P22kΩ Wendel
P1, P101kΩ Wendel
P43kΩ Wendel
P52kΩ Wendel
P63kΩ Wendel

Capacitors (C):

DesignatorValue
C1100n/6V
C23n
C3, C410n
C5, C6100n
C7…C22500n/40V
C11n
C12…1210n/50V

Semiconductors:

DesignatorTypeDescription
LABirne 12V/0m,700mALamp 12V/0.7A
LAReflektorReflector
T1zum Magneten angepaßter FototransistorPhototransistor matched to magnet
T2BrückentransistorBridge transistor
T3BrückentransistorBridge transistor
T22N9/14

Integrated Circuits:

DesignatorType
IC 1SPT 67, SPT 68
IC 2NPX 48 NE211 o.ä.
IC 3, IC 6… μA 741
IC 5129/AC040 Zener
IC 510-Glied 34/GV
IC 3, I5…
IC 1, 2… 10.0, B5
IC 3382 741

Page 43

[page 43: figure only — two views of the printed-circuit board for the magnetic levitation control system. Top view shows the component placement layout with positions labeled: M (magnet terminal), LA (lamp), DE, BE, EC (transistor terminals), IC 1 through IC 8 positions, and “24×” annotation. Bottom view shows the copper-trace side (solder side) of the same PCB. Board dimensions: 140 × 105 mm.]

Page 44

[page 44: figure only — circuit diagram of the isolation amplifier (Trennverstärker, IC 8) with surrounding circuitry. Below the figure is the following text:]

The isolation amplifier (IC 8) is not included on the printed-circuit board and on the assembly drawing, since it added itself as a necessity only after the board was completed. When building a unit, it should therefore be provided on the board from the outset.

Page 45

Literature

  1. Haesler, I./V.: “Überschnallcharakter.” (Supersonic characteristics)

  2. Fritzsche, K.: “Analogrechner II”, Regelungstechnisches Praktikum, Universität Karlsruhe.

  3. Besch, F.: “Regelungstechnik II”, Vorlesung an der Universität Karlsruhe. (Control Engineering II, lecture at the University of Karlsruhe)

  4. Pfäffinger, O.: “Regelungstechnik”. Hüthig-Verlag, Berlin 55, 1966.

  5. Biamoul, E.: “Theoretische Elektrotechnik”, 6. Auflage, VEB Deutscher Verlag der Wissenschaften, Berlin 1977.

  6. Beinböck, J./Rüdisek, H.: “Analogrechner”, BI Wissenschaftsverlag, Mannheim 1961.

Page 46 — [Start of second document: “Analogrechner II”]

Subject matter:

  1. Control of an unstable plant
  2. Stability examination by the Hurwitz method
  3. Root-locus simulation on a modern analog computer

Contents:

  1. Fundamentals
    • 1.1 Stability criteria
      • 1.1.1 Definition of stability
      • 1.1.2 Hurwitz criterion
      • 1.1.3 Example
    • 1.2 The root-locus method
      • 1.2.1 Definition of the root-locus curve
      • 1.2.2 Properties of the root-locus curve
      • 1.2.3 Discussion of the root-locus shape
      • 1.2.4 Discussion of the transfer function behavior
  2. Control of an unstable magnetic levitation system
    • 2.1 Investigation of the subsystems
    • 2.2 Setting up the transfer function
  3. Simulation on the analog computer
  4. Laboratory preparation and evaluation
  5. Literature
  6. Appendix

Page 47

Introduction

The purpose of a control system is normally to compensate for disturbances, but it can also be the improvement of dynamic behavior. The latter is particularly the case for unstable plants (e.g., supersonic aircraft, gyroscopes, magnetic levitation) that must be stabilized with a suitable controller. In the following experiment, the control of an unstable plant (magnetic levitation) is to be investigated and simulated on an analog computer.

1. Fundamentals

The most obvious question in the design of a control loop is stability. For higher demands, however, not only stability but also a defined dynamic behavior is required. This section therefore first deals with a stability criterion, followed by the root-locus method, which permits analysis of the overall dynamic behavior of a control loop and in a certain sense also its synthesis.

1.1 Stability Criteria

1.1.1 Definition of Stability

In the following, only linear time-invariant systems are considered, which can be described by the following differential equation:

a_n · x^(n)(t) + … + a₁ · x’(t) + a₀ · x(t) = b_m · x(t) + b₁ · x(t) + … + b_m^(m) · x(t) (1)

For the stability or instability of the system, only the homogeneous differential equation plays a role, since stability is a system property and for linear systems is independent of the input signal x(t). The solution of the homogeneous differential equation yields for simple roots λᵢ of the associated characteristic equation a sum of terms of the form:

Page 48

C_i · e^(λᵢ t) (2)

— for multiple roots, terms of the form:

C_{ij} · t^j · e^(λᵢ t) (3)

A system is said to be simply stable when the homogeneous solution remains bounded for arbitrary initial conditions and arbitrary time intervals, i.e., when |y_hom(t)| ≤ ∞ holds. For simple roots this gives the condition:

Re(λᵢ) ≤ 0

and for multiple roots:

Re(λᵢ) < 0

A system is said to be asymptotically stable when the homogeneous solution approaches zero asymptotically for large times, i.e., when:

lim_{t→∞} y_hom(t) = 0

holds. This is only possible for simple roots when:

Re(λᵢ) < 0

Since the roots of the characteristic equation are identical to the poles of the associated transfer function:

G(p) = Z(p)/N(p) = (b₀ + b₁p + … + b_m p^m) / (a₀ + a₁p + … + a_n p^n) (4)

the system is asymptotically stable when all poles of G(p) lie to the left of the imaginary axis in the complex p-plane.

In the following, only asymptotic stability is considered. For systems of higher order, the explicit calculation of the roots λᵢ is quite laborious. The purpose of stability criteria is therefore to provide conditions for stability without having to calculate the roots explicitly.

Both algebraic and graphical stability criteria can be distinguished. The algebraic criteria derive from the characteristic equation of the differential equation (respectively from the denominator polynomial of the transfer function) and give conditions for the coefficients a₀, a₁, …, a_n,

Page 49

under which all roots have negative real parts. These criteria are valid in general for differential equations of the form Eq. (1), i.e., also for rational transfer functions (4) and are therefore also suitable for transcendental transfer functions — e.g., systems with dead time. An algebraic criterion such as this, treated frequently in the following literature, is the Hurwitz criterion, which is equivalent to the Routh criterion.

The graphical criteria proceed from the frequency response of the associated open-loop transfer function and are therefore also suitable for transcendental transfer functions. The most important graphical criterion is the Nyquist criterion, which is particularly well suited to control-engineering problems when a closed control loop is to be assessed from a known open-loop frequency response; it is particularly well suited when the open loop is examined experimentally. This criterion is treated in [1].

1.1.2 Hurwitz Criterion

The characteristic equation corresponding to (1), respectively the denominator polynomial of (4):

N(p) = a_n p^n + a_{n-1} p^{n-1} + … + a₁ p + a₀ = 0 (5)

has real coefficients aᵢ with a_n > 0. Then the general Hurwitz criterion (cf. [2]) states:

Form a determinant according to the following scheme:

[Determinant matrix of order n, with the Hurwitz array — the aᵢ coefficients arranged so that diagonal elements are a_{n-1}, a_{n-3}, …; rows filled with the coefficients in the prescribed staircase pattern, and zeros outside the band. The matrix is n rows × n columns.] (6)

Page 50

and in it, beginning at the upper left, form the indicated sub-determinants. If all these determinants are positive, then all roots have negative real parts. This condition is necessary and sufficient.

When one evaluates these determinants, one can also express the Hurwitz criterion in the following form:

  1. All coefficients aᵢ must be present and have the same sign, i.e.:

aᵢ > 0 (7)

This condition is particularly easy to verify; it ensures that no real positive roots exist (monotone instability), but says nothing about the real parts of complex roots. For systems of 1st and 2nd order it is both necessary and sufficient.

  1. Sufficient conditions for systems up to 5th order:

n = 3: a₀a₂ − a₁a₃ < 0
n = 4: a₀a₃² + a₁²a₄ − a₁a₂a₃ < 0
n = 5: a₀a₂ − a₁a₃ < 0 (same condition as n=3)
(a₁a₄ − a₀a₅)²(a₂a₃ − a₁a₄)(a₁a₃ − a₀a₄) < 0 (8)

These conditions ensure that no complex roots with positive real parts occur (oscillatory instability).

The given criterion holds for general dynamic systems described by the differential equation (1) or the transfer function G(p) after Eq. (4). Specifically, for closed control loops the characteristic equation reads:

G₀(p) + 1 = 0 (9)

where G₀(p) denotes the transfer function of the cut-open (open-loop) system. For closed control loops the criterion is therefore applied to (9).

The application of the Hurwitz criterion is illustrated by means of an example.

Page 51

1.1.3 Example

Given is the transfer function of a plant:

G₀(p) = K_S / [(1 + T_D p)(1 − T_y p)] · 1 / (1 + T_p p) (10)

The plant is obviously unstable, since one of the poles is positive real. Also criterion (7) would indicate instability.

The plant is to be stabilized with a real PID controller:

G_R(p) = −K_R(1 + T_{R1} p)(1 + T_{RC} p) / [p(1 + T_p p)]

The characteristic equation of the closed loop from (9):

G₀(p) + 1 = [−K_R K_S (1 + T_{R1} p)(1 + T_{RC} p)] / [p(1 + T_D p)(1 + T_y p)(1 + T_p p)] + 1 = 0 ,

T > T_y > T_D

This is a 5th-order system. Since the compensation of the largest time constant appears in the transfer function, one obtains with T_{R1} = T, a 4th-order system:

G₀(p) + 1 = [−K_R K_S (1 + T_{RC} p)] / [p(1 + T_D p)(1 + T_y p)(1 − T_p p)] + 1 = 0

or

0 = 2T_D T_y T_p p⁴ + (T_T_y + 2T_p T − T_D T_p) p³ − (T − T_y − T_D T_y) p²
a₄ a₃ a₂
+ (K_S K_R T_{RC} − 1) p + K_S K_R (11)
a₁ a₀

From a₀ > 0 after (7) it follows that:

K_S K_R > 0

and:

K_R > K_S / (K_S² T_{RC}) must hold.

From (8) for n = 4:

Page 52

K_R² a₄ T_{RC}² + K_R(K_R a₃² − T_{RC} a₃ a₂ − 2a₄ K_S T_{RC}) + a₄ + a₂ a₃² > 0 (12)
A B C

So that K_R remains real, one must have B² − 4AC > 0, from which the critical value T_{RC,krit} results. Choosing a T_{RC} greater than T_{RC,krit}, one obtains from (11) the critical values K_{R min,max}.

1.2 The Root-Locus Method

The root-locus method proceeds from the poles of the transfer function G₀(p) of the open loop and determines graphically the poles of the closed loop.

1.2.1 Definition of the Root-Locus Curve

The transfer function of the open loop is assumed to be of the form:

G₀(p) = k · Z₀(p)/N₀(p) = k · ∏{i=1}^{m} (p − pz_i) / ∏{j=1}^{n} (p − p_j) (13)

with k ≥ 0, m ≤ n; pz_i = zeros of the open loop; p_j = poles of the open loop.

From the characteristic equation (9) of the closed loop, with (13) the condition:

k · Z₀(p)/N₀(p) = −1 (14)

The root-locus curve — abbreviated: WOK — denotes the totality of all points p in the complex plane that satisfy this equation when the parameter k is varied from 0 to +∞.

Equation (14) can be split into two equations for magnitude and angle:

k · |Z₀(p)/N₀(p)| = 1 (15)

Page 53

arg(k · Z₀(p)/N₀(p)) = (2ν+1)π , ν = 0, ±1, ±2, … (16)

From (16) it follows with (13), when one assumes k > 0:

∑{i=1}^{m} arg(p − pz_i) − ∑{j=1}^{n} arg(p − p_j) = (2ν+1) (17)

This yields a prescription for the graphical construction of the root-locus curve: by trial points in the p-plane one finds some points for which the angle condition (17) is fulfilled.

From the magnitude condition (15) with (13):

k = ∏{j=1}^{n} |p − p_j| / ∏{i=1}^{m} |p − pz_i| (18)

By measuring the distances p−p_j and p−pz_i, k can be determined and entered as a parameter on the WOK.

The graphical construction is substantially accelerated and simplified when one observes the rules compiled in the appendix concerning the geometric properties of the WOK.

1.2.2 Example

The WOK of the plant and controller from Example 1.1.3 with T = 27.4 ms, T_y = 8 ms, T_D = 12.5 ms, K_S = 0.009577 A/m, and T_{R1} = 27.4 ms is to be sketched.

The assessment of T_{R1} > 0.363 s and with the chosen T_{RC} = 7 s:

K_{R,max} = 209.8 As/m , K_{R,min} = 142.5 As/m

For p = jω one can now determine the critical frequencies using the coefficient comparison from (11):

−a₂ ω² + a₀ = 0 , ω_{1/2} = ±√(a₁/a₂) : ω₁ = ±12.9 s⁻¹
ω₂ = ±46.7 s⁻¹

It holds:

G_R(p) G_S(p) = [−0.009577 K_R (1 + 1s · p) A/m] / [p(1 + 12.5ms p)(1 + 5ms p)(1 − 27.4ms p)]

Page 54

G₀(p₀,z₀) = k · [(js² + p) / (p(50s²·p² + p)(13s²·p² + p)(13s²·p² − p))] (formula — see figure)

k = 1993 K_R

Drawing of the WOK from Rule (cf. [1]) is carried out in three steps:

  1. Entry of the poles and zeros (cancel p):

    for T_y = −T_D · 10⁻³ , p₁₀ = −1/5 · 10⁻³ , p_{D0} = −80 · 10⁻³ ,
    p_{y0} = −125 · 10⁻³

    and inclusion of the WOK in the region of the real zeros and poles.

  2. Determination of the asymptotes (Rule 5):

    V_A = 60°, V_B = 180°, V_C = −60°
    S_A = (−80 − 125 − 40 − (−1000)) / (3 − 1) = −22.03
    (in the formula: S_A = (−80 − 125 − 40 − (−1000)) / (3−1) )

  3. Determination of the intersection point with the imaginary axis (Rule 6):

G₀ = p = jω : …

It yields a single intersection with the imaginary axis at G = 0, y = 0. This allows the WOK to be drawn from Rule (1).

[page 54: figure — root-locus plot (Wurzelortskurve) showing the WOK for the given transfer function, with the real axis (horizontal) and imaginary axis (vertical), pole and zero locations marked, and the locus branches traced; labeled “Bild 1”.]

1.2.3 Dominant Poles

From the root-locus plot, the distribution of the poles of the closed loop for a given value of k can easily be read off. Because the zero locations are also known, it is in principle straightforward to compute the step response numerically — for example, using the final-value theorem. Finding an exact solution is, however, generally quite tedious.

Often one is less interested in an exact solution to the transient response and more in what can be inferred from the dominant (governing) poles. The decisive quantity for the time behaviour *) is the distant poles and zeros in the vicinity of the origin; poles and zero locations that are far from the origin, i.e. remote poles and zero locations, can be neglected, since their influence is small. In Fig. (4), for a given value k, the poles B₁, B₂ and B₃ result. The poles B₁ and B₂ form a conjugate-complex dominant pole pair, whereas B₃ can be left out of consideration. The guide transfer function is therefore approximately:

$$G_W(p) = \frac{K’}{1 - \frac{p}{B_1 \cdot B_2} p + \frac{1}{B_1 B_2} p^2}$$ (19)

This is a second-order system that can be dissipated easily.

1.2.4 Discussion of the Root-Locus Path

From the controller gain, the behaviour of the controlled system at the limits can be discussed. The aim is to make the system as fast and stable as possible, i.e. two dominant poles on the angle bisector between the real and imaginary axes would be ideal. Considering the necessary gain, however, one obtains an additional

*) More precisely: the residues belonging to the poles; these are the constants C₁ or C₁ⱼ in (2) and (3).

[page 56]

pole on the real axis. If, however, the gain is reduced, the dominant poles move too close to the imaginary axis. This behaviour can be circumvented by using a compensation zero location that cancels both zero locations against the origin. Since the originally distant zero location now acts more strongly on the WOK branches, these branches also move further to the left. At corresponding gain, a pole moves from the branching point of the curve parts around the origin and exhibits a VZ1 behaviour of the system. Since this pole becomes dominant, the WOK branches can be moved so far to the left that the control loop exhibits very strong VZ1 behaviour (Fig. 3).

[page 56: figure only — Fig. 2 and Fig. 3: two root-locus plots showing WOK branch migration with and without compensation]

2. Control of a Magnetic Suspension

At very high rotational speeds or in cases where extremely low bearing friction is demanded, magnetic bearings are used. Corresponding to the magnetic suspension treated in this experiment, the following system is realised.

In a current-controlled magnetic field there is a freely floating armature whose distance from the magnet is monitored by a light-barrier system. This delivers a voltage corresponding to the distance, which is compared with an adjustable setpoint. A digital PID controller then uses this information to drive a voltage-to-current converter.

[page 57]

The output of the current converter controls the current in the electromagnet, and in this way acts on the distance of the armature from the magnet (Fig. 4).

[page 57: block diagram (Fig. 4) showing: Controller → Voltage-to-Current Converter → Magnet → light-barrier sensor → feedback to Controller]

2.1 Model of the Subsystems

The experimental setup consists of a model with an armature, whose transfer functions are modelled as follows.

2.1.1 Light-Barrier System

The LD system — consisting of a light source, an armature and a photo-transistor — delivers a current for a given armature position corresponding to a voltage U_Sens (Fig. 5).

[page 57: schematic of light barrier and armature, Fig. 5]

This yields a linearity of the sensor characteristic in the range from 20 mm; the characteristic can be expressed as a linear function with a slope:

$$\frac{dU_{Sens}}{dx} = K_{Sens} = 135 \frac{mV}{mm}$$

$$0 \le x \le 20 \text{ mm}$$ (20)

[page 58]

2.1.2 Voltage-to-Current Converter

According to Fig. 7a, the transfer function of the voltage-to-current converter (SSW) can be stated as follows.

[page 58: circuit diagram of voltage-to-current converter, Fig. 7a]

$$i_L = \frac{1}{R_0} \cdot U_E - \frac{R}{R_0} \cdot \frac{1}{R_0} U_E$$

With T₀ = L/R ≈ 300 ms, and the switching threshold taken as sufficiently large, a saturation effect [-] must be taken into account.

[page 58: block diagram Fig. 7b showing the SSW transfer function structure]

This gives:

$$G_{SSW} = \frac{\frac{U_{S0}}{R_0} \cdot \frac{1}{1 + T_{LR} p}}{1 + \frac{U_{S0}}{R_0} \cdot \frac{1}{1 + T_{LR} p} \cdot K_0}$$ (25)

The realised auxiliary circuit provides a time constant of approximately 8 ms:

$$K_0 = 0.732 \frac{1}{A} \cdot T_0 = 8 \text{ ms}$$

[page 59]

2.1.3 Plant (Armature–Magnet Section)

[page 59: diagram Fig. 8 showing magnet above armature of mass m with distance x and weight force mg]

The force exerted by the magnet on the armature of mass m (Fig. 8) is generally proportional to the change in magnetic energy:

$$\frac{dW_m}{dx} \sim i^2 \cdot \frac{d}{dx}\left(\frac{1}{R_m}\right)$$

where R_m is the magnetic reluctance of the magnetic circuit, evaluated along the linear magnetisation curve of the iron.

Neglecting magnetic resistance in the iron and assuming a homogeneous magnetic field:

$$R_m \sim x \text{, from which } F = C \cdot \frac{i^2}{x^2}$$

Because the assumptions of a real model are not satisfied, the characteristic curve must be recorded to establish the range of validity.

The current is plotted as a function of distance for a constant weight and shape, giving the following graph for a specific case (Fig. 9).

[page 59: graph Fig. 9 — current vs. distance showing the characteristic curve; the curve approaches a straight line for larger distances]

It is apparent that the curve approaches a straight line, but exhibits deviations in the range 0 ≤ x ≤ 7.5 mm.

Therefore, from x > 7.5 mm the approximation stated above is justified. Since a large distance is in any case desirable, the further analysis follows for x > 7.5 mm.

[page 60]

The equation of motion is thus:

$$m\ddot{x}(t) = mg - C \frac{i^2}{x^2}$$ (29)

After linearisation around the operating point x₀ with i(t) = I₀ + i₁(t); x(t) = x₀ + x₁(t):

$$m\ddot{x}_1(t) = mg - C\left(\frac{I_0 + i_1}{x_0 + x_1}\right)^2$$

In the operating point the following holds, and the differential equation for small deviations from the operating point eliminates:

$$-\frac{2C}{x_0^3} I_0^2 x_1(t) + \frac{2C}{x_0^2} I_0 i_1(t) = m \ddot{x}_1(t)$$ (26)

The proportional component of the differential equation, which involves the amount of the controlled quantity, is subtracted. From equation (26) the transfer function of the plant results:

$$G_S(p) = \frac{X_1(p)}{I_1(p)} = \frac{\frac{2CI_0}{mx_0^2} \cdot \frac{1}{(1 - \frac{2CI_0^2}{mx_0^3}) - \frac{1}{p^2}}}{\text{(simplified form)}}$$ (27)

Where:

  • m = 708.7 g — mass of the suspended body
  • g = 9.81 m/s² — gravitational acceleration
  • x₀ = 14.5 mm = 2 × 7.5 mm — operating-point distance
  • I₀ = 97.8 A

The plant has the form of an integrating element (PTI) and is unstable.

[page 61]

2.1.4 Controller

The structure of the PID controller differs from that in [1]. The transfer function is:

$$G_R = r_0 + r_1 p + \frac{r_2}{p}$$ (28)

$$G_R = K_R \frac{(1 + T_{R1}p)(1 + T_{R2}p)}{p(1 + T_N p)}$$ (29)

With:

  • K_R = K_R × 1000
  • T_R1 = T_R1 × 100 s
  • T_R2 = T_R2 × 100 s
  • T_N = T_N × 100 s

2.2 Determination of the Transfer Function of the Open Loop

[page 61: block diagram Fig. 10 showing: G_R → G_SSW → G_S with feedback branch G_LS]

From Fig. 10, the transfer function of the open loop can be computed as:

$$G_0(s) = \frac{G_R \cdot G_{SSW} \cdot G_S}{1 + G_S}$$

$$= \frac{K_1}{p} \cdot \frac{1}{(1 + T_1 p)} \cdot \frac{1}{(1 - T_2^2 p^2)} \cdot \frac{K_2}{1 + T_3 p}$$ (30)

These are the transfer functions of the individual subsystems; the coefficients are determined from the WOK method.

[page 62]

3. Simulation on the Analog Computer

3.1 Non-Linear Computing Elements

In order also to simulate non-linear plants, non-linear computing elements are available. In particular, the circuits of multiplier (Fig. 11a) and divider (Fig. 11b) are shown.

[page 62: circuit diagrams — Fig. 11a: multiplier circuit with op-amp; Fig. 11b: divider circuit with op-amp]

For the simulation of arbitrary functions, function generators can be used that can be adapted to the particular task.

3.1.1 Plant

As shown in (3), the differential equations must be scaled in amplitude and time for the analog computer.

With the amplitude scaling:

  • x(t) = A₂ x₂(t) ; A₂ = 200 m/s²
  • x(t) = A₁ x₁(t) ; A₁ = 0.4 m/s
  • x(t) = A₀ x₀(t) ; A₀ = 0.045 m
  • i(t) = B i₀(t) ; B = 2 A

equation (29) can be written as:

$$m\ddot{x} = mg - C \frac{i^2(t)}{x^2(t)}$$

$$\dot{x}(t) = g\left(1 - K_0^2 \frac{i^2(t)}{x^2(t)}\right)$$

$$A_2 x_2(t) = g\left(1 - K_0^2 \frac{B^2 i_0^2(t)}{A_0^2 x_0^2(t)}\right)$$

$$X_2(t) = \frac{g}{A_2} - \left(\frac{g}{A_2} \cdot \frac{K_0 B}{A_0} i_0(t)\right)^2 / x_0(t)$$ (32)

With this, the circuit (Fig. 12) can be drawn.

[page 63]

[page 63: duplicate/repeat of page 62 content — non-linear computing elements section with identical text and figures for multiplier (Fig. 11a) and divider (Fig. 11b), and amplitude scaling equations (31) and (32). This page appears to be a repeated print of the same material as page 62.]

[page 64]

3.1.4 Divider

[page 64: full analog computer circuit diagram (Fig. 12) showing the complete simulation circuit for the magnetic suspension. The diagram is divided into sections labelled:

  • “Fil-Regler” (PID controller) — upper right, with op-amp chain blocks 01, 02, 03, OT (output)
  • “VZI-Glied” (VZ1 element) — upper right feedback path, blocks 04, 05
  • “Soll-Istwertvergleich” (Setpoint–actual-value comparison) — left, blocks 54, 55
  • “Strecke” (Plant) — lower section, blocks 06, 07, M (multiplier), DIV (divider) The circuit shows the complete block interconnection of the simulation.]

[page 65]

3.3 Measurement of the Step Response

If the system simulated on the analog computer is stable, there are two methods to determine the step response.

  1. In operating mode BR (continuous computation, also CP), a step-function signal is applied to the input. The system response can be observed with an oscilloscope or a pen recorder.

  2. In operating mode RK (repetitive computation), suitable initial conditions are set.

If the system is unstable, the first method fails because the amplifiers become overloaded. In this case only the second method is applicable. The computation must be interrupted before an amplifier overloads.

4. Preparatory Work

4.1 The potentiometer values α₀₀, α₀₄, α₀₆, α₀₇ and α₁₆ are for λ = 1 (real-time operation); the given data of the magnetic suspension are to be used to compute these values.

4.2 The transfer function G₀(p) = U_FB(p) / U_E(p) (Fig. 12) is to be specified as a function of the coefficients α₀₁–α₀₃ and α₁₀, α₁₁. What is the value for α₁₀ = 1? (α₀₉ = 0.01, α₁₀ = 0.38)

4.3 Using the Hurwitz criteria, compute the critical gains of the controlled plant and plot the WOK. (T_R1 = 27.4 ms, T_R2 = 70 ms, T_D = 12.5 ms)

4.4 Determine α₀₁, α₀₂ and α₀₃ for the critical gains and for the following values (Fig. 3): T_R1 = 33 ms, T_R2 = 33 ms, K_R = 14 A/m.

[page 66]

5. Experiment Execution and Evaluation

5.1 The root-locus plot of the control loop and the PID controller are to be set up and the step response recorded. The results are to be compared with the calculations.

5.2 The stability limits of the closed-loop system are:

  • a) T_R1 = 47.4 ms, T_R2 = 50 ms, T_D = 47.3 ms
  • b) T_R1 = 63.5 ms, T_R2 = 79.5 ms, T_D = 60.5 ms

5.3 Discuss the step response of the controlled system (BOE) at a change in V_DO.

5.4 Why can a zero location of the plant not compensate an additional pole of the controller?

6. Literature

[1] Holl, R.: Versuch “Drehzahlregelung”, Regelungstechnisches Praktikum, Universität Karlsruhe.

[2] Philippen, G.: Regelungstechnik, Carl Hüthig-Verlag, Heidelberg, 1970.

[3] Jelench, R.: Versuch “Analogrechner I”, Regelungstechnisches Praktikum, Universität Karlsruhe.

[4] Über, G.: Versuch “Durchf.- und Füllstandsregelung”, Z.I., Universität Karlsruhe.

[page 67] — Appendix: Rules for Constructing Root-Locus Curves (WOK)

$$G_0(s) = k \cdot \frac{\prod_{i=1}^{m}(p - p_{Ni})}{\prod_{i=1}^{n}(p - p_{Pi})} \quad \text{with } m \le n$$

PropertyDescription
SymmetryThe WOK is symmetric about the real axis.
Starting and end pointsThe WOK begins (at k=0) at the poles and ends (at k=∞) at the zero locations.
Number of branchesThe number of branches equals the number of poles of G₀(p) (with n ≥ m).
Departure and arrival anglesThe departure angle at a pole P₀ is an odd multiple of 180°, diminished by the sum of the angles to all other poles and increased by the sum of the angles to all zero locations.
AsymptotesFor large p, n−m branches tend to the asymptotes. These approach the real axis at the angle: α_∞ = 180°/( n−m) · (2ν+1), ν = 0, 1, 2, …, (n−m−1).
Asymptote intersection with real axisThe intersection point lies on the real axis at: σ_A = (1/(n−m)) [Σ real part of poles − Σ real part of zeros]. The intersection lies on the real axis (branching point).
Sum of real partsFor n ≥ m+2, the sum of real parts of all poles of G₀(p) is constant.
Potential analogyThe WOK runs along the real axis in the vicinity of G₀(p), more closely to neighbouring poles than to neighbouring zeros.

[page 68]

Displacement on the real axis

a) Real pole and zero locations: From the equation |1 + k·G₀(jω)| = 0, which must be satisfied by real values σ only if the sum:

$$\sum_{i} \frac{d(x - \sigma_{Pi})}{(x-\sigma_{Pi})^2 + \omega_i^2} = \sum_{i} \frac{d(x - \sigma_{Ni})}{(x-\sigma_{Ni})^2 + \omega_i^2}$$

b) Complex conjugate pole and zero locations: The distance to both is equally large on both sides (symmetry), and consequently:

$$\int \frac{d(x - \sigma_{P0i})}{(x-\sigma_{P0i})^2 + \omega_i^2} = \int \frac{d(x - \sigma_{N0i})}{(x-\sigma_{N0i})^2 + \omega_i^2}$$

with ω₀ = σ₀ √(1 − D²)

Location on the real axis | Each segment of the real axis that has an odd total number of poles and zero locations to its right belongs to the WOK (each real-axis segment to the right of which the count of poles and zeros is odd). |

Branch point with the imaginary axis | Stability limit: point where the real part vanishes. Here the gain factor K_crit is defined. |

Sum of real parts | For n = m+2, the sum of real parts of all poles of G₀(p) is constant, i.e. σ(p₁) + σ(p₂) + ··· = const. |

Potential analogy | The WOK runs in the vicinity of the real axis near G₀(p(s)), more closely to neighbouring poles than to neighbouring zeros with σ_A(p). |

[page 69]

4.1 (from Section 3.1.1)

$$\alpha_{00} x_2 = \frac{g}{A_2} = \frac{9.81 \frac{m}{s^2}}{200} = 0.049 \approx 1$$

$$\alpha_{04} = \frac{A_1}{A_2} \cdot \frac{1}{89} = \frac{0.4}{200 \cdot 0.004} \cdot 500 \approx 1$$

$$\alpha_{07} = \frac{g}{A_2} \cdot \frac{1}{89} = \frac{9.81}{200 \cdot 0.0094 \cdot 500} = 0.01778$$

4.2 G₀(p) = 1

$$G_0(p) = \frac{10\alpha_{10}}{p} \cdot G_S(p) = G_S(p)$$

$$U_0(p) + \alpha_{02} G_S(p) = \frac{p}{1 + \alpha_{12}} + G_S(p)$$

$$G_S(p) = \frac{\alpha_{01}}{(\alpha_{11} + \alpha_{01})} \cdot \frac{1}{p} + \frac{\alpha_{12} G_S(p)}{\alpha_{12} + 1} \cdot G_S(p)$$

$$G_S(p) = \frac{1}{\frac{\alpha_{11}}{10\alpha_{10}} + \frac{\alpha_{12}}{\alpha_{10}p}} \cdot G_S(p)$$

$$= (-\alpha_{11} \cdot K_1 + \frac{\alpha_{12}\alpha_{10}}{K_p} p)$$

$$\alpha_{01} = K_R \quad \alpha_{00} = \frac{K_1}{\alpha_{10} + K_1} \quad \alpha_{02} = \frac{\alpha_{10}K_1}{\alpha_{10} + K_1} K_0$$

[page 70]

4.3 (from Sections 1.1.3 and 1.2.3)

$$x_0 = 2740 \text{ mm}^2 ; , \quad m_0 = 467.0 \text{ mm}^2 ; , \quad a_3 = 0.5 \text{ ms}$$

$$\frac{d}{dt}[a_0 a_3^2] + B_0[a_3 a_{03}] + B_0[a_1 a_{03}^2] + B_0 a_0 a_{22} = C$$

$$A = 2.5731 \cdot 10^{-7} \text{ ms}^3$$

$$B = -0.2889 \cdot 10^{-3} \text{ ms}^2$$

$$C = 590.7 \text{ ms}^2$$

$$\sigma_{P1,2} = -\frac{B}{2A} \pm \sqrt{\left(\frac{B}{2A}\right)^2 - \frac{C}{A}} \cdot (16.47 \pm 3.96) \cdot \frac{1}{\text{ms}}$$

$$\sigma_{\max} = 22.5 \frac{1}{\text{ms}} \approx -1 = 34960A/(\text{s})$$

$$\sigma_{\min} = 10.56 \frac{1}{\text{ms}} \approx -1$$

WOK: (K_{R1} = 17.45 ms, K_{R2} = 70 ms, T_D = 12.5 ms):

$$\sigma_{\max} = 17.45 \quad \sigma_{\max}’ = 43.9 \text{ s}^{-1}$$

$$\sigma_{\min} = -43.9 \text{ s}^{-1}$$

  1. p₁ = 0, p₂ = 50.3 s⁻¹, p₃ = −125 s⁻¹ p₄ = −80 s⁻¹ + T_D = −0

  2. G₀ = −50.3 s⁻¹

  3. σ = [value from table]:

$$\frac{1}{0.41 \text{ s}^{-1}} - \frac{1}{0 - 50.34} - \frac{1}{0 - 125} + \frac{1}{0 - 4800} = 3 \cdot 6 \cdot 10^{-3}$$

C = 3 · 6 · s⁻¹

[page 71]

[page 71: figure only — graph showing root-locus plot (WOK) for the magnetic suspension control loop, with the locus curve plotted in the complex p-plane. The y-axis is labelled Im/s⁻¹ reaching up to 30, and the x-axis shows Re/s⁻¹. Below the plot is a data table with the following values:]

T_R1T_R2Δt_s
k,k0.107640.107640.0334
T_R210a10a11a
Δt1a1a1a
ξ_122.6·10⁻³10.61e⁻¹1s⁻¹
ξ_2206.65109.16251.1
ξ_33.24 ×1.56 ×6.73 ×
ξ_4224.0105.2251
α₀₁1.89·10⁻¹1.8·10⁻³⁶8.4·10⁻⁵
α₀₃0.00940.004515.1024

[page 72]

[page 72: figure only — full analog computer circuit diagram titled “Schaltplan mit VZ-Anschluß” (Circuit diagram with VZ connection). The diagram shows the complete simulation circuit including:

  • Upper section: PID controller chain with op-amp blocks 02, 03 → OT output (labelled “v_D”)
  • Blocks 01 and 51 in series configuration
  • Middle: additional op-amp stages 09, 08 with “vom AK” and “vom AKT” labels (from analog computer)
  • Lower left: blocks 14 and 06 for input path with amplification
  • Block 07 in the plant section
  • M (multiplier) and DIV (divider) blocks in the plant section
  • Feedback path shown with dashed lines

Note at bottom of page: κ_D = 0.194]

Step Responses (for T_VZ = 100s, T_N = 27.6min, d_St = 0.18, linearized)

Setpoint step:

The upper trace shows a sharp initial rise (derivative kick) followed by a return toward steady state. The lower region of the chart depicts the response over time, with the time axis marked at t=0 and t=IO (10 seconds), and further annotations at t=v·t_0 and the settling region marked RR.

Slow oscillation (Langsame Schwingung):

An underdamped oscillatory response is shown in the third trace, with visible decaying oscillations characteristic of a lightly damped closed-loop system.

Setting Options (Einstellmöglichkeiten)

Table: Controller parameter settings

Controller typeSK_pX_iX_D
Kugel, VBm
P, statil3.52.500.6
PID, statil3.55101
Langsames Schwingen3.50.460.05
Einspringen1.5100.00
Kugel, JNE
P, statil6.5100.2
PID, statil6.5150.2
Langsames Schwingen7.50.570.1
Einspringen6.5110.05
Schnelles Schwingen95.221.0

Note: “Kugel, VBm” and “Kugel, JNE” refer to two different ball/sphere configurations (or operating modes) of the suspension system. “statil” denotes statically stable (or static equilibrium) operation. “Langsames Schwingen” = slow oscillation; “Einspringen” = snap-in / capture; “Schnelles Schwingen” = fast oscillation.