Ermittlung der Kennwerte eines Prozesses mit Hilfe selbsteinstellender Systeme

Complete English translation of the original German-language document (30 pages).

AEG Telefunken — Data Processing: Analog Computers and Hybrid Systems

Roland Konkart

Determination of the Characteristic Parameters of a Process with the Aid of Self-Adjusting Systems

The Concept of Self-Adjustment

Process Identification by Four Different Methods

2.1. Gauss-Seidel Iteration Method

2.2. Gradient Method

 2.2.1. Iterative Gradient Method

 2.2.2. Continuous Gradient Method

     2.2.2.1. Process Identification with the Aid of a Series Model

     2.2.2.2. Process Identification with the Aid of a Parallel Model

Example: Identification of a Second-Order System

3.1. Identification with the Aid of the Gauss-Seidel Iteration Method

3.2. Identification with the Aid of the Iterative Gradient Method

3.3. Identification with the Aid of the Continuous Gradient Method
Comparison of the Methods
Parameter Trajectories during the Identification of a Second-Order System

References

1. The Concept of Self-Adjustment

A dynamic system is called self-adjusting when it controls itself — using its own behaviour — toward a prescribed behaviour. The goal of self-adjustment (adaptation) is to achieve the prescribed behaviour exactly or in some sense optimally. A system is considered optimal when a previously chosen objective function Z, expressed as a function of the adjustable parameters α (with α = α₁ … αₙ) of the system, attains an extremum:

Z(α) = Extremum!

2. Process Identification by Four Different Methods

The following shows how the characteristic parameters a (with a = a₁ … aₙ) of a process can be determined with the aid of models using four different methods. The problem can be formulated as follows:

In the differential equation describing the process,

aₙ y⁽ⁿ⁾ + aₙ₋₁ y⁽ⁿ⁻¹⁾ + … + a₁ y’ + a₀ y = x(t)

or equivalently in the transfer function,

P = Y/X = 1 / (aₙpⁿ + … + a₁p + a₀)

the coefficients a are to be determined from the measured signals x and y. To do so, a model whose structure corresponds to that of the process is matched using the transfer function

M = Yₘ/X = 1 / (αₙpⁿ + … + α₁p + α₀)

until, at the matched condition, α = a.

For the matching of the model, an objective function — which may be, for example, the mean-square value of the deviation e — is minimised (Fig. 1):

Z(α, t) = ∫₀^∞ e²(α, t) dt = Min!

Fig. 1: Basic system arrangement for the determination of process parameters with the aid of a parallel model

The block diagram shows input x feeding both the Process P(a) (output Y) and the Model M(α) (output Yₘ). The difference e = Y − Yₘ is squared (e²), then integrated (1/p) to yield Z.

2.1. Gauss-Seidel Iteration Method

The Gauss-Seidel iteration method operates during process identification with a system arrangement as in Fig. 1. From the adjustable model parameters α, only one parameter αᵢ at a time is varied by the amount Δ under repeated deterministic excitation, until

Z(α) = ∫₀^∞ e²(α, t) dt, αⱼ = const.; j ≠ i

is minimal. The condition is therefore:

Z{α, αᵢ + m·Δ} < Z{α, αᵢ + (m−1)·Δ}

where m = number of variation steps. The objective function for each parameter αᵢ is thus iteratively brought to a relative minimum. With each iteration step one obtains a successively better approximation to αᵢ.

2.2. Gradient Method

In the gradient method the adjustable parameters α of a system are adjusted so that the objective function

Z = Z(α)

is minimised along the gradient of Z with respect to α.

A necessary condition for achieving the minimum is that the gradient of the objective function vanishes at the optimal parameters:

grad Z(α) = 0

For a single parameter the derivative becomes:

∂Z/∂αᵢ = (∂/∂αᵢ) Z(αᵢ)

With Z = ∫₀^∞ e² dt, this yields:

∂Z/∂αᵢ = 2 ∫₀^∞ e · (∂e/∂αᵢ) dt

The parameter α is now adjusted by an amount Δα proportional to the gradient:

Δα = −h · grad Z

For a single parameter this becomes:

Δαᵢ = −h · ∂Z/∂αᵢ

It follows that:

Z(α + Δα) ≈ Z − h · Σᵢ₌₁ⁿ (∂Z/∂αᵢ)

and therefore:

Z(α + Δα) ≤ Z(α)

If the derivatives ∂Z/∂αᵢ are not simulatable or not accessible, then an increasingly optimal α can nevertheless be determined iteratively. If, however, the derivatives ∂Z/∂αᵢ are accessible, then αᵢ can be adjusted continuously by ∂αᵢ so that the optimal parameters emerge as the sum of all adjustments:

αᵢ := αᵢ − h · ∂Z/∂αᵢ = αᵢ − 2h · ∫₀^∞ e · (∂e/∂αᵢ) dt

2.2.1. Iterative Gradient Method

Just as the Gauss-Seidel iteration method also works with the iterative gradient method, using a system arrangement as in Fig. 1. The gradient of Z with respect to α must be determined iteratively using deterministic test signals. Each parameter αᵢ is adjusted by a fixed step s; the effect of this adjustment on the objective function is evaluated. The step correction to approximate the necessary derivative for the parameter is:

s · αᵢ := −h · ∂Z/∂αᵢ

2.2.2. Continuous Gradient Method

With the continuous gradient method, deterministic test signals can be dispensed with, since here the gradient of the objective function with respect to the adjustable parameters is continuously available and does not have to be formed iteratively. The derivative

∂Z/∂αᵢ = 2 ∫₀^∞ e · (∂e/∂αᵢ) dt

must now be generated for each adjustable parameter. For this purpose, two different system configurations are formulated, which are described in the two following sections.

2.2.2.1. Process Identification with the Aid of a Series Model

In this identification procedure the process with transfer function

P = Y/X = 1 / (aₙpⁿ + … + a₁p + a₀)

is placed in series with a model having transfer function

Mₛ = Yₛ/Y = (αₙpⁿ + … + α₁p + α₀) / (bₙpⁿ + … + b₁p + b₀)

and this combined system is compared with a parallel model having transfer function

Mₚ = Yₘ/X = 1 / (bₙpⁿ + … + b₁p + b₀)

(Fig. 2).

The adjustable parameters of the system are adjusted by the gradient method so that the objective function

Z = ∫₀^∞ e² dt

is minimised. At the matched condition, the numerator of the model compensates the denominator of the process. It then holds that α = a.

Fig. 2: Principle of process identification with the aid of a series model

The block diagram shows: input x → Process P(a) → output y → Series model Mₛ(α) → output Yₛ. Separately, Mₚ produces Yₘ. The difference Yₛ − Yₘ = e is formed.

The derivatives required for automatic execution of this matching,

∂Z/∂αᵢ = 2 ∫₀^∞ (∂e/∂αᵢ) · e dt

can be taken directly from the model without additional computational effort (Fig. 3). With

e = Yₛ − Yₘ = X · (P · Mₛ(α) − Mₚ)

it follows that:

∂e/∂αᵢ = ∂Yₛ/∂αᵢ = X · P · (∂/∂αᵢ) Mₛ(α)

Fig. 3: Computing circuit for the series model with the derivatives ∂Yₛ/∂αᵢ

The circuit shows the process output y passed through a chain of integrators (with feedback coefficients b₀, b₁, …, bₙ₋₁), with tap points providing the partial derivatives ∂Yₛ/∂αₙ, ∂Yₛ/∂αₙ₋₁, …, ∂Yₛ/∂α₀; these are multiplied by the respective parameters αₙ, αₙ₋₁, …, α₀ and summed to give Yₛ.

After substituting the transfer function this becomes:

∂Yₛ/∂αᵢ = [1 / (aₙpⁿ + … + a₁p + a₀)] · [pⁱ / (bₙpⁿ + … + b₁p + b₀)]

For each parameter one then obtains:

αᵢ := αᵢ − 2hᵢ ∫₀^∞ (∂Yₛ/∂αᵢ) dt

This equation is represented as a computing circuit in Fig. 4.

Fig. 4: Computing circuit for the determination of the parameters

The circuit shows, for each parameter αₙ, αₙ₋₁, …, α₀: the error signal e multiplied by the corresponding derivative ∂Yₛ/∂αᵢ, scaled by 2hₙ (or 2hₙ₋₁, …, 2h₀), and fed through an integrator to produce the updated parameter value.

2.2.2.2. Process Identification with the Aid of a Parallel Model

For the identification of a process with transfer function

P = Y/X = 1 / (aₙpⁿ + … + a₁p + a₀)

a parallel model (Fig. 1) with transfer function

M = Yₘ/X = 1 / (αₙpⁿ + … + α₁p + α₀)

is matched by the gradient method so that the objective function

Z = ∫₀^∞ e² dt

is minimised. The derivatives required for the matching are not directly available from this model but must first be generated (Fig. 5).

Fig. 5: Computing circuit of the parallel model and the formation of the derivatives ∂Yₘ/∂αᵢ

The circuit shows: on the left, the Model M(α) — input x through a chain of integrators with feedback coefficients α₀, α₁, …, αₙ — producing Yₘ. On the right, a duplicate circuit (the derivative generator) with the same feedback structure produces the outputs ∂Yₘ/∂αₙ, ∂Yₘ/∂αₙ₋₁, …, ∂Yₘ/∂α₀.

It holds that:

∂Z/∂αᵢ = 2 ∫₀^∞ (∂e/∂αᵢ) · e dt

with e = Y − Yₘ = X(P − M), so that:

∂e/∂αᵢ = −(∂Yₘ/∂αᵢ) = −X · (∂/∂αᵢ) M(α)

and it follows that:

∂Yₘ/∂αᵢ = pⁱ / (αₙpⁿ + … + α₁p + α₀)²

With these derivatives the parameters are again determined by:

αᵢ := αᵢ − 2hᵢ · ∫₀^∞ e · (∂Yₘ/∂αᵢ) dt

as in Section 2.2.2.1, using the computing circuit of Fig. 4.

3. Example: Identification of the Parameters of a Second-Order System

Using the example of the identification of a process with transfer function

P = Y/X = 1 / (p² + a₁p + 1 + a₀) = 1 / (p² + 0.5p + 1 + 0.5)

it is now shown how the methods described in Chapter 2 are to be realised on the hybrid analog computer. Fig. 6 shows the computing circuit and Fig. 7 shows the step-response of the process to be identified.

Fig. 6: Computing circuit of the process to be identified

The circuit shows input x fed through two integrators with parameter feedback coefficients a₁ and a₀ (and summing) to produce output y.

Fig. 7: Step response of the process to be identified

The plot shows the output Y versus time t: an underdamped second-order response — an initial overshoot followed by damped oscillations converging to steady state.

3.1. Identification with the Aid of the Gauss-Seidel Iteration Method

When determining the parameters a₀ and a₁ with the aid of the Gauss-Seidel iteration method described in Section 2.1, the following procedure (Fig. 8) results:

The parameter α₁ is varied around the value 1 in steps; the effect of this variation on the objective function is evaluated. A negative ΔZ means that the variation of the parameters is proceeding in the right direction. The variation must then proceed in another direction, to seek a better approximation for the desired parameter a₁. This can have two causes:

The variation of α₁ around the value 1 was in the wrong direction. In this case a negative ΔZ arises already at the first approximation step. The variation must then be performed in another direction; the parameter approximation to a₁ can be successively improved in this way.
The parameter a₀ can no longer be more closely approximated by the parameter a₁ through successive variation about the value 1. The last variation must then be reversed and the next parameter a₁+₁ is varied instead.

Whichever of the two possible causes has produced ΔZ < 0 is indicated by a counter that counts, for each parameter, how many times in succession the parameter variation gives ΔZ < 0. When the count exceeds m > 2, a switchover to the next parameter is triggered. A flip-flop Z (Fig. 10) permits the decision m > 2 to be detected, and thereby makes it possible to switch over to the next parameter.

Figs. 8 and 9 show the basic mode of operation of the method; Fig. 10 shows the computing circuit that was realised on a hybrid analog computer (RA 770 or RA 800 H).

Fig. 8: Flow diagram for the determination of the parameters a₀ and a₁ of a second-order system using the Gauss-Seidel iteration method

The flow chart begins at Start with initial values α₀ := 0, α₁ := 0, m := 1. Two parallel branches vary α₀ and α₁ respectively: each tests ΔZ > 0? (yes: keep going; no: check m > 2? — yes: adjust opposite parameter and reset m; no: reverse step and increment m) in a cyclic manner, alternating between the two parameters.

Fig. 9: Block diagram for the determination of the parameters of a second-order system using the Gauss-Seidel iteration method

The block diagram shows: input x → Process (1/(p² + a₁p + 1 + a₀)) → y; and Model (1/(p² + α₁p + 1 + α₀)) → Yₘ. The difference e = y − Yₘ is passed through the time-function formation block (Z = ∫e² dt) to produce Z(α); the comparison block produces ΔZ = Z(αᵢ) − Z(αᵢ + Δ). The control logic (Steuer-Logik) receives ΔZ and the timer clock (Zeitgeber Takt) and drives: a sign-of-Δ block (Vorz. von Δ), a Δ memory, a switchover block (Umschaltung von α₀ ↔ α₁), and a memory for α₀ and α₁ (Speicher für α₀ und α₁), with separate Δ increments for α₀ and α₁.

Control conditions for the individual operations:

Sign reversal of Δ — the sign reverses when ΔZ < 0.
Switchover from α₀ to α₁ or vice versa — switchover occurs when ΔZ < 0 and the number of variations by Δ exceeds 2. At the switchover both parameters are adjusted.

Fig. 10: Computing circuit for the determination of the parameters of a second-order system using the Gauss-Seidel iteration method

The detailed circuit diagram shows: Process block (1/(p² + a₁p + 1 + a₀)) producing y, and Model M with parameters α₁ and α₀. The error signal e = y − Yₘ is squared (×, ×), multiplied by a scaling factor p, and integrated by ZG1 to form Z(α) and Z(α + Δ). A comparison block computes Z(α + Δ) − Z(α) = ΔZ. A sign-detection stage (Vorz. von Δ) drives the Δ-sign logic (+1/−1). A switchover stage (Umschalten α₀ ↔ α₁) driven by ZG2 selects which parameter is currently being varied. The Zeitgeber (timer clock) controls the sequence: Pause → |Δt| → Rechnen (compute) → Halt, cycling with ZG1, ZG2, ZG3.

Timing diagram of the timer (Zeitdiagramm der Zeitgeber):

Signal	Pause-Taste gedrückt	Δt	Pause	Rechnen	Halt
P	1 / 0
ZG1	0
ZG2	0
ZG3	0

Z checks whether the count of parameter variations m satisfies: m > 2 → parameter switchover α₀ ↔ α₁; m < 2 → remain on parameter α₀.

3.2. Identification with the Aid of the Iterative Gradient Method

When determining the parameters a₀ and a₁ of a second-order system with the aid of the iterative gradient method described in Section 2.2.1, the control logic required for automatic operation must perform the following steps (Fig. 11):

The initial value of the objective function Z(α) must be stored so that the effect of the variation of the parameters α₀ and α₁ by the step s on the objective function can be established. In doing so, Z(α) (the initial value) is compared with Z(α, αᵢ + s) (with i = 0, 1).
From the difference Z(α) − Z(α, αᵢ + s), a value Δαᵢ must be formed and stored:

Δαᵢ = h · {Z(α) − Z(α, αᵢ + s)} := −h · ΔZ
After performing both test steps for the parameters α₀ and α₁, these must be adjusted by the values Δα₀ and Δα₁ respectively:

α₀ := α₀ + Δα₀ α₁ := α₁ + Δα₁

With these values a new initial value for the objective function is obtained:

Z(α) := Z(α + Δα)
This procedure must be repeated until Z(α) = Min has been reached.

Figs. 11 and 12 show the mode of operation of this method. Fig. 13 shows the computing circuit that was realised on a hybrid analog computer (RA 770 or RA 800 H).

Fig. 11: Flow diagram for the determination of the parameters a₀ and a₁ of a second-order system using the iterative gradient method

The flow diagram begins at Start with: α₀ := initial value, α₁ := initial value, Δα₀ = 0, Δα₁ = 0, i = 0.

The main loop branches on i = 0, 1, 2:

i = 0: α₀ := α₀ + Δα₀; α₁ := α₁ + Δα₁ → compute Z(α) := ∫e² dt → set i := 1
i = 1: α₀ := α₀ + Δ → compute ΔZ(α)₀ := Z(α) − Z(α₀ + Δ, α₁) → compute Δα₀ := h·Z(α)₀ → α₀ := α₀ − Δ → set i := 2
i = 2: α₁ := α₁ + Δ → compute ΔZ(α)₁ := Z(α) − Z(α₀, α₁ + Δ) → compute Δα₁ := h·Z(α)₁ → α₁ := α₁ − Δ → set i := 0

The loop repeats until convergence.

3.3. Identification with the Aid of the Continuous Gradient Method

[page 18: figure only — Fig. 16 continues on subsequent pages]

Fig. 16 (page 18): Flow diagram for the determination of the parameters a₀ and a₁ of a second-order system using the iterative gradient method

[Duplicate/continuation of Fig. 11 flow diagram at higher detail, showing the same three-branch structure i = 0, 1, 2 with the same update rules, confirmed as belonging to Section 3.2 from caption.]

[page 19: Figure 12 — block diagram and procedure description]

The adjustment of the model parameters is carried out according to the following scheme:

Advance α₀ by Δ
Store Δα₀ = −h (∂Z/∂α₀) = −h { Z(α₀, α₁) − Z(α₀ + Δ, α₁) }
Reset α₀ by Δ and advance α₁ by Δ
Store Δα₁ = −h { Z(α₀, α₁) − Z(α₀, α₁ + Δ) }
Reset α₁ by Δ, and advance α₀ by Δα₀ and α₁ by Δα₁
Repeat steps 1–5 until Z = minimum.

Figure 12: Block diagram for determining the parameters of a 2nd-order system by the iterative gradient method.

The block diagram shows the following elements:

Process (transfer function 1 / (p² + a₁p + 1 + a₀)) receiving input x and producing output y
Cost-function formation block computing Z from the error signal e = y − y_m, producing ΔZ output
Model (transfer function 1 / (p² + α₁p + 1 + α₀)) receiving input x and producing output y_m
Storage for α₀ and α₁, receiving increments Δα₀ (or Δ) and Δα₁ (or Δ), feeding back α₀ and α₁ to the model
Storage block for values Δαᵢ = −h (∂Z/∂αᵢ)
Sign-of-Δ block
Test step Δ
Control logic
Clock / timing pulse (Zeitgeber Takt)

[page 20: Figures 13 — computation circuit for iterative gradient method]

Figure 13: Computation circuit for determining the parameters of a 2nd-order system by the iterative gradient method.

The upper portion of the figure shows the complete analog computation circuit. The circuit comprises:

Process block (transfer function 1 / (p² + a₁p + 1 + a₀)) with input x and output Y
Model M (a) implemented with summers and integrators, receiving parameters α₁ and α₀, producing output y_m
Error-signal path computing e = Y − y_m
Multiplier and squaring stages producing z² and then Z (via integrator/summer path), with ΔZ output driving Δα₁
Test-step magnitude (Testschrittgröße) block
Sign / VZ (Vorzeichen) block with ZG3

The lower-left detail shows the parameter counter i sub-circuit:

ZG1 gate feeding the counter, with outputs f_s (set) and f_p (reset), and ZG1 feedback

Below is the timing diagram of the clock (Zeitdiagramm des Zeitgebers) showing four operating phases:

Pause key pressed (Pausentaste gedrückt)
Pause
Compute (Rechnen)
Halt

Signal waveforms for P, ZG1, ZG2, and ZG3 are shown across these phases.

3.3 Identification with the Aid of Continuous Gradient Methods

As described in Section 2.2.2, continuous gradient methods for identifying the parameters of a system require the derivatives ∂y_m/∂αᵢ of the model output with respect to the adjusted system parameters, which must be available continuously — not formed iteratively — for the comparison process.

With these derivatives, the equation

α̇ᵢ = −2hᵢ ∫ (∂y_m/∂αᵢ) e dt (with i = 0, 1)

can be modelled directly. In a control loop, the model parameter αᵢ then automatically adjusts to the identified system parameter. For the 2nd-order system being identified, two parameters α₀ and α₁ must therefore be modelled with two control loops.

The adjustment speed depends in these methods on the proportionality factor hᵢ.

When identifying with the aid of the series model (Section 2.2.2.1 and Figures 14 and 15), hᵢ can be chosen quite large, because a change of parameter αᵢ acts directly (without delay) on the deviation e. The loops in which the parameters αᵢ are compared are therefore theoretically stable for all loop gains hᵢ.

When identifying with the aid of the parallel model (Section 2.2.2.2 and Figures 16 and 17), hᵢ cannot be chosen so large, because a parameter variation does not act directly on the deviation e; rather, it acts with a system-dependent delay (delays of the system being identified and of the model approximation). The loop gains hᵢ for which the loops remain stable depend on the order of the system being identified, on the values of the system parameters a, and on the initial values of the model parameters αᵢ.

[page 22: Figures 14 and 15 — series model block diagram and computation circuit]

Figure 14: Block diagram for determining the parameters of a 2nd-order system by the gradient method using a series model.

The block diagram shows:

Process P = 1 / (p² + a₁p + 1 + a₀) receiving input x and producing output y
Series model M_s = 1 / (b₂p² + b₁p + 1) receiving input y and producing output y_s
Model M_p = 1 / (b₂p² + b₁p + 1) receiving input x and producing output y_m
Error signal e = y_m − y_s
Parameter identification circuit (Parametererkennungsschaltung) receiving e, producing parameter α feedback to the series model

Figure 15: Computation circuit for determining the parameters of a 2nd-order system by the gradient method (series model).

The circuit shows:

Process P = 1 / (p² + a₁p + 1 + a₀) with input x and output y
Model M_p (b) implemented with potentiometers b₁ and b₂ and integrators, producing output y_m
Series model M_s (d) implemented with potentiometers b₂ and b₁ (labelled b₂ and b₁ on the diagram) computing ∂y_s/∂α₁ and ∂y_s/∂α₀, producing output y_s and error e
Multiplier stages: (∂y_s/∂α₀) × e fed through integrator 2h₀ → α₀; (∂y_s/∂α₁) × e fed through integrator 2h₁ → α₁

[page 23: Figures 16 and 17 — parallel model block diagram and computation circuit]

Figure 16: Block diagram for determining the parameters of a 2nd-order system by the gradient method using a parallel model.

The block diagram shows:

Process P = 1 / (p² + a₁p + 1 + a₀) receiving input x and producing output y
Model M = 1 / (p² + α₁p + 1 + α₀) receiving input x and producing output y_m
Error signal e = y − y_m
Parameter identification circuit receiving e and the derivatives ∂y_m/∂αᵢ from a circuit 1 / (p² + α₁p + 1 + α₀), producing parameter α fed back to the model

Figure 17: Computation circuit for determining the parameters of a 2nd-order system by the gradient method (parallel model).

Left half — Model:

Process P = p² + a₁p + 1 + a₀ with input x and output y
Model implemented with summers and integrators, with multiplier inputs α₁ and α₀
Error signal e = y − y_m

Right half — Derivatives ∂y_m/∂α:

The derivative path ∂y_m/∂α₁ passes through summer/integrator chain with multiplier α₁
The derivative path ∂y_m/∂α₀ passes through summer/integrator chain with multiplier α₀
Both derivative outputs multiplied by e and fed through integrators 2h₁ → α₁ and 2h₀ → α₀ respectively

4. Comparison of the Methods

In order to obtain a comparative statement about the four methods, a process of 2nd-order was identified by all methods; the process was excited with a repeating step function. In all methods, the identified process parameters were a₀ and a₁; the process was excited via the identification circuit by the same Anregungssprünge [excitation steps]. The process error in all subsequent excitation steps was kept at approximately 1% of the estimated value (no oscillation of the parameters about the determined value).

The fastest identification was achieved by the process from the continuous gradient method using the series model (parameters a in Section 3, Figures 14 and 15). After approximately 7 excitation steps the identification was complete (parameter traces a, Section 5).

The two iterative identification methods operate very much more slowly. The identification speed here depends on the starting value and on the step size Δ (test step size for the iterative gradient method) or on the step size h (iterative Gauß-Seidel method). The closer the starting value is to the correct value, the faster the identification proceeds. The approximation error for small step sizes or test-step sizes is more accurate; for large step sizes the parameter value is approximated less accurately, but more quickly. At the set operating point, the identification by means of these methods gives good results only after relatively many excitation steps.

At a test-step size Δ = 0.02 for α₀ and Δ = 0.02·10⁻² for α₁, approximately 10% error identification in approximately 10 excitation steps of the process is achieved (parameter traces c, Section 5, Figures 19 and 20). At a test-step size Δ = 0.03 for α₀ and Δ = 0.03·10⁻² for α₁, only ca. 25 excitation steps are required for identification with the same error (parameter traces e — note: labelled as e in the text but corresponding to c in the legend).

The Gauß-Seidel iteration method requires, with a step size Δ = 0.02 for α₀ and Δ = 0.03·10⁻² for α₁, approximately 30 excitation steps to identify the process with approximately 10% error (parameter traces c, Section 5, Figures 18 and 19). With a step size Δ = 0.03 for α₀ and Δ = 0.03·10⁻² for α₁, about 25 excitation steps are required for identification at the same error (parameter traces d, Section 5, Figures 20 and 21).

5. Parameter Traces During Identification of a 2nd-Order System

For an example, the progression of the parameters

α = α(t) and α₀ over α₁

is presented here. The process parameters to be identified are a with the values

a₀ = 0.5 and a₁ = 0.5·10⁻²

The letters on the parameter traces indicate the identification method. The key is as follows:

a = continuous gradient method (series model)
b = continuous gradient method (parallel model)
c = iterative gradient method
d = Gauß-Seidel iteration method

[page 24: Figure 18 — parameter traces α = α(t)]

Figure 18: Parameter traces α = α(t) during identification of a 2nd-order system (for iterative methods: Δ = 0.02 for α₀ and Δ = 0.02·10⁻² for α₁).

Upper plot — α₀ vs. time t/s (0 to 50 s):

Correct value α₀ = 0.5 shown as horizontal reference line
Curve a (continuous gradient, series model): fastest rise, reaching 0.5 within ~5 s, then oscillating above correct value
Curve d (Gauß-Seidel): also fast, converges close to correct value by ~5 s
Curve b (continuous gradient, parallel model): somewhat slower convergence
Curve c (iterative gradient): slowest convergence among the four; approaches 0.5 more gradually

Lower plot — α₁ × 10⁻² vs. time t/s (0 to 50 s):

Correct value α₁ = 0.5·10⁻² shown as horizontal reference line
Curve b (parallel model): rises quickly, oscillates near correct value
Curve a (series model): converges to correct value with small residual oscillation
Curve c (iterative gradient): rises more slowly with step-like increments
Curve d (Gauß-Seidel): slowest initial rise; converges from below

[page 25: Figure 19 — parameter traces α₀ over α₁]

Figure 19: Parameter traces α₀ over α₁ during identification of a 2nd-order system (for iterative methods: step size Δ as in Figure 18).

Phase-plane plot with α₁ (×10⁻²) on the horizontal axis (0.1 to 0.6) and α₀ on the vertical axis (0.1 to 0.6):

Correct value indicated at the point (α₁ = 0.5·10⁻², α₀ = 0.5)
Curve a (continuous gradient, series model): smooth approach path moving toward the correct value, ending with fine oscillations around it
Curve b (continuous gradient, parallel model): broader excursion path, spiralling around the correct value but remaining offset to the right
Curve c (iterative gradient): step-like trajectory converging toward correct value from lower-left region
Curve d (Gauß-Seidel): nearly direct trajectory to the correct value, approaching it most directly

[page 26: Figure 20 — parameter traces α = α(t), larger step size]

Figure 20: Parameter traces α = α(t) during identification of a 2nd-order system (for iterative methods: Δ = 0.03 for α₀ and Δ = 0.03·10⁻² for α₁).

Upper plot — α₀ vs. time (0 to ca. 40 s):

Correct value α₀ = 0.5 shown as horizontal reference line
Curve a (series model): rises rapidly to ~0.6, then settles at 0.5 with small oscillation
Curve b (parallel model): rapid initial rise, then converges
Curve c (iterative gradient): step-wise rise reaching 0.5 within ~10 s
Curve d (Gauß-Seidel): rapid step rise to near 0.5 by ~5 s, then fine oscillation

Lower plot — α₁ vs. time (0 to ca. 40 s):

Correct value α₁ = 0.5·10⁻² shown as horizontal reference line
All four curves converge toward the correct value; curves b and a (continuous methods) show fastest initial transient; c and d show step-wise convergence with slightly coarser oscillation at steady state

[page 27: Figure 21 — parameter traces α₀ over α₁, larger step size]

Figure 21: Parameter traces α₀ over α₁ during identification of a 2nd-order system (for iterative methods: step size as in Figure 20).

Phase-plane plot with α₁ (×10⁻²) on the horizontal axis (0.1 to 0.6+) and α₀ on the vertical axis (0.1 to 0.6+):

Correct value indicated at (α₁ ≈ 0.5·10⁻², α₀ ≈ 0.5)
Curve a (continuous gradient, series model): curved approach from lower-right, spiralling around correct value
Curve b (continuous gradient, parallel model): broader sweep, remaining somewhat to the right of correct value
Curve c (iterative gradient): step-wise trajectory from origin region toward correct value
Curve d (Gauß-Seidel): very direct vertical-then-horizontal trajectory converging rapidly onto the correct value

The diagram indicates that with the larger step size, all methods converge more quickly to the vicinity of the correct value, though the final residual spread of the iterative methods (c, d) is somewhat larger than with the finer step size used in Figures 18 and 19.

Bibliography (SCHRIFTTUM)

[1] Kramer, H.: Optimierung eines Regelkreises mit Tischanalogrechner und Digitalzusatz [Optimization of a Control Loop with a Desk Analog Computer and Digital Supplement] Sonderdruck elektronische Datenverarbeitung 1968/6 Pages 293–297

[2] Albrecht, P. and Lotz, H.: Die Verwendung der Hybriden Präzisionsrechenanlage RA 770 zur automatischen Parameteroptimierung nach dem Gradientenverfahren [The Use of the Hybrid Precision Computing System RA 770 for Automatic Parameter Optimization by the Gradient Method] Technische Mitteilungen AEG-TELEFUNKEN 5. Beiheft, Pages 41–43

[3] Maršik, I.: Versuche mit einem selbsteinstellenden Modell zur automatischen Kennwertermittlung von Regelstrecken [Experiments with a Self-Adjusting Model for Automatic Characteristic-Value Determination of Controlled Plants] m r s 9, 1966, Issue 6, Pages 210–213

[4] Rake, H.: Selbsteinstellende Systeme nach dem Gradientenverfahren [Self-Adjusting Systems by the Gradient Method] Regelungstechnik, Issue 5, 1967, Pages 211–217

Ermittlung der Kennwerte eines Prozesses mit Hilfe selbsteinstellender Systeme

AEG Telefunken — Data Processing: Analog Computers and Hybrid Systems

Table of Contents

1. The Concept of Self-Adjustment

2. Process Identification by Four Different Methods

2.1. Gauss-Seidel Iteration Method

2.2. Gradient Method

2.2.1. Iterative Gradient Method

2.2.2. Continuous Gradient Method

2.2.2.1. Process Identification with the Aid of a Series Model

2.2.2.2. Process Identification with the Aid of a Parallel Model

3. Example: Identification of the Parameters of a Second-Order System

3.1. Identification with the Aid of the Gauss-Seidel Iteration Method

3.2. Identification with the Aid of the Iterative Gradient Method

3.3. Identification with the Aid of the Continuous Gradient Method

[page 19: Figure 12 — block diagram and procedure description]

[page 20: Figures 13 — computation circuit for iterative gradient method]

3.3 Identification with the Aid of Continuous Gradient Methods

[page 22: Figures 14 and 15 — series model block diagram and computation circuit]

[page 23: Figures 16 and 17 — parallel model block diagram and computation circuit]

4. Comparison of the Methods

5. Parameter Traces During Identification of a 2nd-Order System

[page 24: Figure 18 — parameter traces α = α(t)]

[page 25: Figure 19 — parameter traces α₀ over α₁]

[page 26: Figure 20 — parameter traces α = α(t), larger step size]

[page 27: Figure 21 — parameter traces α₀ over α₁, larger step size]

Bibliography (SCHRIFTTUM)