Optimum amount

The optimum amount is a term from control theory , more precisely a control-related optimization criterion in the frequency domain . A regulation is generally then called optimal if the controlled variable to the value of the reference variable can follow with minimal delay. When optimizing using the optimum amount, the settling time of a control system is optimized.

Another optimization criterion in the frequency range is the symmetrical optimum .

motivation

The possible courses of the amplitude response of a closed control loop in the management behavior. The parameters are: - resonant frequency , - crossover frequency , - bandwidth , M _max - value of the frequency response at the resonance point. ${\displaystyle \omega _{R}}$ ${\displaystyle \omega _{d}}$ ${\displaystyle \omega _{B}}$

A short rise time or rise time as with the step response requires a large bandwidth of the closed control loop. There is a direct relationship between the rise time T _An and the bandwidth of the lead transition function. Mathematically speaking, the relation is as follows: ${\displaystyle \omega _{B}}$

${\displaystyle T_{\text{An}}={\frac {\pi }{\omega _{B}}}={\frac {\pi }{2\pi \cdot f_{B}}}={\frac {1}{2\cdot f_{B}}}}$

For good guidance behavior, the ratio of the output to the input amplitude (amplitude frequency response ) is optimized for the absolute value . In the ideal case, the magnitude of the frequency response F for all circular frequencies ω is:

${\displaystyle |F_{w}(\mathrm {j} \omega )|=1}$.

It is also said that F _w (jω) clings to one. In real control loops, however, there are always delays , which is why this ideal absolute value can only be reached in approximation and the amplitude-frequency response is reduced at higher frequencies. The optimization method of the optimum amount attempts over the widest possible frequency range from ω to hold the value 1 to the amount of the frequency response on or near, to be the parameters of the controller from the time constant of the controlled system is calculated.

Requirements for applying the optimum amount

Block diagram of a simple standard control loop , consisting of the controlled system, the controller and a negative feedback of the controlled variable y (also the actual value). The controlled variable y is compared with the reference variable (setpoint) w . The control deviation e = w - y is fed to the controller, which uses it to generate a manipulated variable u according to the desired dynamics of the control loop . The stability of the system depends, among other things, on the poles of the route . The explanation is shown opposite.

In order to apply the optimum amount, certain prerequisites have to be met. If the requirements are not met, undefined states can arise. For example, the manipulated variable limit could be exceeded. One consequence could be that the system causes an unexpected reaction, such as the destruction of electrical components. Thus, it is assumed that, in the controller design the measuring device to the control path is counted, and thus a standard control circuit is present. The parameters of the controlled system, time constants and gain factor must already be known. Furthermore, in the derivation and in the setting rules for the optimum amount , it is assumed that the controlled system is a combination of delay systems. With non-real poles of the line there is a risk of stability difficulties . If there are complex conjugate poles in the controlled system, they should be sufficiently damped. However, the use of purely real poles would be preferable. A non-oscillatory control system with compensation is therefore a prerequisite. When using dominant time constants, i. H. one or two large time constants compared to the equivalent time constant T _E , particularly useful results are to be expected. If the requirements are met, good management behavior is guaranteed.

Math background

The method was derived for a second order control loop. In addition, a distinction is made between two types of derivation: application of the optimum absolute value for first-order controlled systems and application of the optimum absolute value for higher-order controlled systems. There are auxiliary sets to simplify the controlled system; these are used for systems of higher order and dead time elements.

Theorem of the sum of all small time constants

Is a controlled system in front of higher order, which has the following form: , then an equivalent time constant T _E , which is composed of the sum of all small time constant can be formed. A distinction is made here as to whether there are one or two dominant time constants. This is also known as the theorem of the sum of all small time constants . ${\displaystyle G_{s}(s)={\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{2}\cdot s)...(1+T_{n}\cdot s)}}}$

In the case of a dominant time constant: ${\displaystyle \quad \quad \quad T_{E}=\sum _{i=2}^{n}T_{i},\quad T_{1}\gg T_{E},\quad G_{s}(s)={\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{E}\cdot s)}}}$

In the case of two dominant time constants: ${\displaystyle \quad T_{E}=\sum _{i=3}^{n}T_{i},\quad T_{1}>T_{2}\gg T_{E},\quad G_{s}(s)={\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{2}\cdot s)\cdot (1+T_{E}\cdot s)}}}$

Simplification of dead time elements

If a dead time T _{t is} significantly smaller than the time constant T _{1 of} a delay system, the same would apply to an I element and its integration time T _I , then this can be replaced as a PT ₁ element . This consideration is based on the open control loop G ₀ (s). "The series expansion of the exponential function for the dead time element is aborted after the first term:"

${\displaystyle G_{0}(s)={\frac {K_{R}\cdot K_{S}\cdot e^{-T_{t}\cdot s}}{1+T_{1}\cdot s}}={\frac {K_{R}\cdot K_{S}}{(1+T_{1}\cdot s)\cdot (1+{\frac {T_{t}\cdot s}{1!}}+{\frac {(T_{t}\cdot s)^{2}}{2!}}+...)}}={\frac {K_{R}\cdot K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{t}\cdot s)}}}$if applies. ${\displaystyle T_{1}\gg T_{t}}$

Application of the optimum amount for controlled systems Ι. order

A control loop of the 2nd order is shown with an I controller and a PT ₁ control system . The input signal w (j ) is switched on ${\displaystyle \omega }$ and the output signal x (j ) is ${\displaystyle \omega }$ called up at the output .

In the following, the derivation for the application of the optimum amount for controlled systems of the first order is described; this is based on the facts of the Motivation section . A second-order control loop, composed of an I controller and a PT ₁ control system, has the following frequency response function as an open control loop :

${\displaystyle F_{\text{RS}}(j\omega )={\frac {K_{S}}{T_{I}\cdot j\omega \cdot (1+T_{E}\cdot j\omega )}}}$, with . ${\displaystyle {\frac {1}{T_{I}}}={\frac {K_{R}}{T_{N}}}}$

Now the closed control loop is formed by means of the open control loop , which has the following frequency response function:

${\displaystyle F(j\omega )={\frac {F_{\text{RS}}(j\omega )}{1+F_{\text{RS}}(j\omega )}}={\frac {K_{S}}{T_{I}\cdot j\omega \cdot (1+T_{E}\cdot j\omega )+K_{S}}}={\frac {K_{S}}{K_{S}-T_{I}\cdot T_{E}\cdot \omega ^{2}+T_{I}\cdot j\omega }}}$

Since the amount should be equal to 1 for as large a range as possible, the following applies (the square of the amount is used to remove the root in the denominator):

${\displaystyle |F(j\omega )|={\frac {K_{S}}{\sqrt {(K_{S}-T_{I}\cdot T_{E}\cdot \omega ^{2})^{2}+T_{I}^{2}\cdot \omega ^{2}}}}{\stackrel {\mathrm {!} }{\approx }}1}$

${\displaystyle |F(j\omega )|^{2}={\frac {K_{S}^{2}}{(K_{S}-T_{I}\cdot T_{E}\cdot \omega ^{2})^{2}+T_{I}^{2}\cdot \omega ^{2}}}={\frac {K_{S}^{2}}{K_{S}^{2}+(T_{I}^{2}-2\cdot K_{S}\cdot T_{I}\cdot T_{E})\cdot \omega ^{2}+T_{I}^{2}\cdot T_{E}^{2}\cdot \omega ^{4}}}{\stackrel {\mathrm {!} }{\approx }}1}$

So that the approximation is valid for a large range, as many coefficients of the numerator and denominator polynomial as possible must be the same. So this equation follows:

${\displaystyle |F(j\omega )|^{2}={\frac {\quad K_{S}^{2}+\quad \quad \quad \quad \quad \quad \quad \quad \quad 0\cdot \omega ^{2}+\quad \quad \quad 0\cdot \omega ^{4}}{K_{S}^{2}+(T_{I}^{2}-2\cdot K_{S}\cdot T_{I}\cdot T_{E})\cdot \omega ^{2}+T_{I}^{2}\cdot T_{E}^{2}\cdot \omega ^{4}}}{\stackrel {\mathrm {!} }{\approx }}1}$

A realization table for the coefficient comparison can be derived from the equation :

Numerator polynomial	Denominator polynomial	realization
${\displaystyle K_{S}^{2}\cdot \omega ^{0}}$	${\displaystyle K_{S}^{2}\cdot \omega ^{0}}$	is satisfied
${\displaystyle 0\cdot \omega ^{2}}$	${\displaystyle (T_{I}^{2}-2\cdot K_{S}\cdot T_{I}\cdot T_{E})\cdot \omega ^{2}}$	realizable
${\displaystyle 0\cdot \omega ^{4}}$	${\displaystyle T_{I}^{2}\cdot T_{E}^{2}\cdot \omega ^{4}}$	not feasible

This results in the following optimization equation:

${\displaystyle (T_{I}^{2}-2\cdot K_{S}\cdot T_{I}\cdot T_{E})\cdot \omega ^{2}=0}$

${\displaystyle T_{I}=2\cdot K_{S}\cdot T_{E}}$, it applies . ${\displaystyle K_{I}={\frac {1}{T_{I}}}={\frac {1}{2\cdot K_{S}\cdot T_{E}}}}$

If one now uses the integration constant T _I in the frequency response function of the closed control loop, the result is:

${\displaystyle F(j\omega )={\frac {K_{S}}{K_{S}+T_{I}\cdot j\omega +T_{I}\cdot T_{E}\cdot (j\omega )^{2}}}={\frac {K_{S}}{K_{S}+2\cdot K_{S}\cdot T_{E}\cdot j\omega +2\cdot K_{S}\cdot T_{E}^{2}\cdot (j\omega )^{2}}}={\frac {1}{1+2\cdot T_{E}\cdot j\omega +2\cdot T_{E}^{2}\cdot (j\omega )^{2}}}}$

If you now go into the Laplace range and set up the transfer function of the closed control loop G (s), you can compare this with a standardized PT ₂ element .

${\displaystyle G(s)={\frac {1}{1+2\cdot T_{E}\cdot s+2\cdot T_{E}^{2}\cdot s^{2}}}{\stackrel {\mathrm {!} }{=}}{\frac {1}{1+2\cdot D\cdot T_{0}\cdot s+T_{0}^{2}\cdot s^{2}}}}$

The damping D and the characteristic angular frequency can be determined by comparing coefficients . ${\displaystyle \omega _{0}}$

${\displaystyle 2\cdot T_{E}=2\cdot D\cdot T_{0},\quad 2\cdot T_{E}^{2}=T_{0}^{2}}$, it applies . ${\displaystyle \omega _{0}={\frac {1}{T_{0}}}}$

${\displaystyle T_{0}={\sqrt {2}}\cdot T_{E},\quad \omega _{0}={\frac {1}{{\sqrt {2}}\cdot T_{E}}}}$

${\displaystyle D={\frac {T_{E}}{T_{0}}}={\frac {1}{\sqrt {2}}}}$, thus a fixed damping applies to all frequencies .

Use of the optimum amount for higher-order controlled systems

The derivation of the application of the optimum absolute value for higher-order controlled systems is based on the facts of the Motivation section on the one hand, and the optimal setting determined for T _I from the section Application of the optimum absolute value for the Ιth order controlled systems . In addition, the derivation is divided into the compensation of one large time constant and the compensation of two large time constants .

Compensation of a large time constant

Is a time constant of the controlled system significantly larger than the others, the reset time T can _N of a PI controller are used to the big time constant T ₁ to compensate. The advantage of this compensation is that the speed of the control is improved and the mathematical calculation is simplified. The mathematical calculation is simplified because the compensation is like a reduction . With the set of the sum of all small time constant , the small time constant of the controlled system are the equivalent time constant T _E summarized.

Provided is a PI controller and a PT ₂ -distance or PT _n -distance. With a PT ₂ system, the following transfer function is formed for the open control loop:

${\displaystyle G_{\text{RS}}(s)=G_{R}(s)\cdot G_{S}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)}{T_{N}\cdot s}}\cdot {\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{E}\cdot s)}}}$, it applies . ${\displaystyle T_{1}\gg T_{E}}$

If the reset time T _{N is} now chosen to be equal to the large delay time constant T ₁ , the result is a reduction. This creates the optimization rule . ${\displaystyle T_{N}=T_{1}}$

${\displaystyle G_{\text{RS}}(s)=G_{R}(s)\cdot G_{S}(s)={\frac {K_{R}}{T_{N}\cdot s}}\cdot {\frac {K_{S}}{(1+T_{E}\cdot s)}}}$

On the basis of the portion of application of the optimum amount for control systems Ι.Ordnung follows using the optimum setting for T _I , the optimization rule for K _R .

${\displaystyle T_{I}=2\cdot K_{S}\cdot T_{E}={\frac {T_{N}}{K_{R}}},\quad K_{R}={\frac {T_{N}}{2\cdot K_{S}\cdot T_{E}}}}$

Compensation of two large time constants

To compensate for two large time constants, the small time constants are combined to form the equivalent time constant T _E , as in the section Application of the optimum absolute value for higher-order controlled systems. Further, a PID controller used to using the reset time T _N and the derivative-action time T _V , the two large time constants of the regulator control loop to compensate. The advantage of this compensation is that the speed of the control is improved and the mathematical calculation is simplified. The mathematical calculation is simplified because the compensation is like a reduction .

The following transfer function results for the open control loop when using a PID controller and a PT ₃ control system:

${\displaystyle G_{\text{RS}}(s)=G_{R}(s)\cdot G_{S}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)\cdot (1+T_{V}\cdot s)}{T_{N}\cdot s}}\cdot {\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{2}\cdot s)\cdot (1+T_{E}\cdot s)}}}$, it applies . ${\displaystyle T_{1}>T_{2}\gg T_{E}}$

The following controller setting results from the compensation of two large time constants:

${\displaystyle T_{N}=T_{1},\quad T_{V}=T_{2},\quad }$for . ${\displaystyle T_{1}>T_{2}}$

After the compensation, the same transfer function results as in the section Applying the optimum absolute value for higher-order controlled systems :

${\displaystyle G_{\text{RS}}(s)=G_{R}(s)\cdot G_{S}(s)={\frac {K_{R}}{T_{N}\cdot s}}\cdot {\frac {K_{S}}{(1+T_{E}\cdot s)}}}$

The optimization rule for the controller gain factor K _R is therefore the same as when using a PI controller and a PT _{2 system} :

${\displaystyle K_{R}={\frac {T_{N}}{2\cdot K_{S}\cdot T_{E}}}}$

Setting rules for the optimum amount

The following table shows the route and control structures , including the specification of the transfer behavior shown. In addition, the respective setting rules for the optimum amount have been added. The setting rules only apply if the requirements are met.

Controlled system		Regulator
Type	Transfer function	Type	Transfer function
PT 1	${\displaystyle G_{S}(s)={\frac {K_{S}}{1+T_{1}\cdot s}}}$ ${\displaystyle T_{E}=T_{1}}$	I.	${\displaystyle G_{R}(s)={\frac {K_{I}}{s}},\quad G_{R}(s)={\frac {1}{T_{I}\cdot s}}}$ ${\displaystyle K_{I}={\frac {1}{2\cdot K_{S}\cdot T_{E}}},\quad T_{I}=2\cdot K_{S}\cdot T_{E}}$
PT 2	${\displaystyle G_{S}(s)={\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{2}\cdot s)}}}$ ${\displaystyle T_{1}>T_{2},\quad T_{E}=T_{2}}$	PI	${\displaystyle G_{R}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)}{T_{N}\cdot s}}}$ ${\displaystyle T_{N}=T_{1},\quad K_{R}={\frac {T_{N}}{2\cdot K_{S}\cdot T_{E}}}}$
PT n	${\displaystyle G_{S}(s)={\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{E}\cdot s)}}}$ ${\displaystyle T_{1}\gg T_{E},\quad T_{E}=\sum _{i=2}^{n}T_{i}}$	PI	${\displaystyle G_{R}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)}{T_{N}\cdot s}}}$ ${\displaystyle T_{N}=T_{1},\quad K_{R}={\frac {T_{N}}{2\cdot K_{S}\cdot T_{E}}}}$
PT n	${\displaystyle G_{S}(s)={\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{2}\cdot s)\cdot (1+T_{E}\cdot s)}}}$ ${\displaystyle T_{1}>T_{2}\gg T_{E},\quad T_{E}=\sum _{i=3}^{n}T_{i}}$	PID	${\displaystyle G_{R}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)\cdot (1+T_{V}\cdot s)}{T_{N}\cdot s}}}$ ${\displaystyle T_{N}=T_{1},\quad T_{V}=T_{2},\quad K_{R}={\frac {T_{N}}{2\cdot K_{S}\cdot T_{E}}}}$

Optimal amount in comparison

Comparison of the optimum amount with an empirical setting rule , T-sum rule , and a further optimization criterion from the frequency range, symmetrical optimum . Mapping of the management transition function for a target jump w ₀ = 1.

Comparison of the optimum amount with an empirical setting rule , T-sum rule , and a further optimization criterion from the frequency range, symmetrical optimum . Mapping of the disturbance transition function for a target jump z ₀ = 1 at the line entrance .

To illustrate the strengths of the optimum amount, a comparison with several methods is presented below. The regulation of a PT is ₃ - route with a PI controller in focus. For comparison, an empirical setting rule from the time domain and another optimization criterion from the frequency domain are used. Fast control according to the T-sum rule is taken from the spectrum of empirical setting rules . The symmetrical optimum is selected from the optimization criteria in the frequency range .

The following controlled system is given: ${\displaystyle G_{S}(s)={\frac {K_{S}}{(1+T_{1}\cdot s)\cdot (1+T_{2}\cdot s)\cdot (1+T_{3}\cdot s)}}}$

with the route parameters: ${\displaystyle K_{S}=1{,}78,\quad T_{1}=0{,}73,\quad T_{2}=0{,}031,\quad T_{3}=0{,}021}$

The PI controller should look like this: ${\displaystyle G_{R}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)}{T_{N}\cdot s}}}$

• Setting of the control parameters according to the optimum amount (see table, 3rd line): as is valid.
  ${\displaystyle T_{N}=T_{1}=0{,}73}$
  ${\displaystyle T_{E}=T_{2}+T_{3}=0{,}052}$ ${\displaystyle T_{N}\gg T_{E}}$
  ${\displaystyle K_{R}={\frac {T_{N}}{2\cdot K_{S}\cdot T_{E}}}=3{,}9434}$
  ${\displaystyle G_{R}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)}{T_{N}\cdot s}}={\frac {3{,}94\cdot (1+0{,}73\cdot s)}{0{,}73\cdot s}}}$

• Setting the control parameters according to the symmetrical optimum ( unless ): IT ₁ approximation: ${\displaystyle a=2}$
  ${\displaystyle G_{S}(s)={\frac {K_{S}}{s\cdot (1+T_{1}\cdot s)}}={\frac {1{,}78}{0{,}73\cdot s\cdot (1+0{,}052\cdot s)}}}$
  ${\displaystyle T_{1}=0{,}052,\quad K_{S}={\frac {1{,}78}{0{,}73}}=2{,}44}$
  ${\displaystyle T_{N}=a^{2}\cdot T_{1}=0{,}208}$
  ${\displaystyle K_{R}={\frac {1}{a\cdot K_{S}\cdot T_{1}}}=3{,}9434}$
  ${\displaystyle G_{R}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)}{T_{N}\cdot s}}={\frac {3{,}94\cdot (1+0{,}21\cdot s)}{0{,}21\cdot s}}}$

• Setting of the control parameters according to the T-sum rule (fast control):
  ${\displaystyle K_{S}=1{,}78,\quad T_{\sum }=T_{1}+T_{2}+T_{3}=0{,}782}$
  ${\displaystyle T_{N}=0{,}7\cdot T_{\sum }=0{,}5474}$
  ${\displaystyle K_{R}={\frac {1}{K_{S}}}=0{,}5618}$
  ${\displaystyle G_{R}(s)={\frac {K_{R}\cdot (1+T_{N}\cdot s)}{T_{N}\cdot s}}={\frac {0{,}56\cdot (1+0{,}55\cdot s)}{0{,}55\cdot s}}}$

Based on the leadership transition function can be seen that the optimum amount significantly less on - and settling time which, compared to the empirical setting rule. This is due to the difference in size of the controller gain factor K _R , which is significantly smaller with the T-sum rule. Furthermore, the overshoot of the absolute optimum is significantly lower than that of the symmetrical optimum. The reason is the reset time T _N , although this is less with the symmetrical optimum and therefore leads to a slightly better rise time. However, the price to be paid is a larger overshoot and an increased settling time.

Areas of application

Because of its strengths, the optimum amount is undisputed in practice and is preferably used in the field of electrical control. In addition, the process is also used in drive technology .,

The optimum amount is often used for hiring

• Speed,
• Electricity-,
• Torque and
• Force controls

used. The areas of application extend from

• Main drives of machine tools,
• Feed drives for machine tools and industrial robots
• to elevators.

See also

• Rule of thumb procedure (automation technology)
• Symmetrical optimum

literature

• Slobodan N. Vukosavic: Digital Control of Electrical Drives . Springer-Verlag, New York 2007, ISBN 978-0-387-48598-0 .
• Dierk Schröder: Electric drives - control of drive systems . Springer-Verlag, Berlin, Heidelberg 2015, ISBN 978-3-642-30471-2 .
• Gert-Helge Geitner: Design of digital controllers for electric drives . VDE-Verlag, Berlin, Offenbach 1996, ISBN 3-8007-1847-2 .

Web links

• Controller design based on the optimum amount (PDF; 40 kB)
• BOD - The digital amount optimum

Individual evidence

1. ↑ Ekbert Hering, Heinrich Steinhart: Taschenbuch der Mechatronik . Hanser Verlag, Leipzig 2005, ISBN 978-3-446-22881-8 , pp. 100 .
2. ↑ ^{a b c d e f g h i} Holger Lutz, Wolfgang Wendt: Pocket book of control engineering with MATLAB and Simulink . 11th edition. Verlag Europa-Lehrmittel, Haan-Gruiten 2019, ISBN 978-3-8085-5869-0 , p. 504-515 .
3. ↑ ^{a b c} Gerd Schulz: Control engineering: Fundamentals, analysis and design of control loops, computer-aided methods . Springer, Berlin 1995, ISBN 3-540-59326-8 , pp. 151-154 .
4. ↑ ^{a b} Manfred Reuter, Serge Zacher: Control technology for engineers: analysis, simulation and design of control loops . 10th edition. Vieweg, Braunschweig 2002, ISBN 3-528-94004-2 , pp. 236-237 .
5. ↑ ^{a b c d} Otto Föllinger, Ulrich Konigorski (edit.): Control engineering : Introduction to the methods and their application . 11th edition. VDE-Verlag, Berlin 2013, ISBN 978-3-8007-3231-9 , p. 201-203 .
6. ^ ^{A b c} Thomas Beier, Petra Wurl: Control engineering : basic knowledge, fundamentals, examples . 2nd Edition. Fachbuchverlag Leipzig, Munich 2015, ISBN 978-3-446-44210-8 , p. 177-185 .
7. ↑ ^{a b} Jörg Kahlert: Crash Course Control Engineering: A practice-oriented introduction with accompanying software . 2nd Edition. VDE-Verlag, Berlin 2015, ISBN 978-3-8007-3642-3 , p. 150-153 .
8. ↑ A. Weigl-Seitz: Control engineering . Darmstadt 2015, p. 23-35 .
9. ↑ Hans-Werner Philippsen: Entry into control engineering: procedural model for practical controller design . Fachbuchverlag Leipzig, Munich 2004, ISBN 3-446-22377-0 , p. 141-143 .
10. ^ Rolf Schönfeld: Regulations and controls in electrical engineering . Verlag Technik, Berlin, Munich 1993, ISBN 3-341-01027-0 , p. 78-96, 151-154 .