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BO - Magnitude Optimum (Betragsoptimum) by Kessler specified for
undelayed inputs (classical Betragsoptimum) and
delayed inputs (Symmetrical Optimum) |
The basic ideas of the Magnitude Optimum (Betragsoptimum) were already
formulated in detail by C. Kessler in 1955 for undelayed signals and were
also comprehensively extended to delayed signals by C. Kessler in 1958
under the name Symmetric Optimum. Sources of inspiration were fundamental
considerations by A. L. Whiteley and a non-generalised example according to
R. C. Oldenbourg / H. Sartorius. Substantial contributions to the
application were later made by R. Schönfeld, D. Schröder and G. Brandenburg
. Since exactly one system of equations provides the basis for all
calculations, it is recommended from today's point of view to use the terms
Magnitude Optimum for undelayed input signals (conventional Magnitude
Optimum) and Magnitude Optimum for delayed input signals
(Symmetrical Optimum).
A - Freeware Toolbox BO with m-files and
demos is available for MATLAB to optimize
continuous controllers based on the Magnitude Optimum: current >>>
version 1.1<<<
The features of this toolbox for the optimization of continuous
controllers are based on the generalised application of the optimization
equation system resulting from the requirements of the Magnitude Optimum,
i.e. neither a priori pole compensation is specified, nor are
approximations or optimization equations tailored to special plant
structures used. The following properties apply to the Magnitude Optimum
in general:
- Optimization for undelayed input signals
- Basically only possible for one integrator in the open loop
- Yields a linear equation system, matrix form makes sense -
structural similarity to the
Digital Magnitude Optimum (BOD)
- Simplest cases are P- and I-controllers
- Optimization for delayed input signals
- Does not imply a restriction on the number of integrators in
the open loop - Optimization for delayed input signals
(Symmetric Optimum) is thus more universally relevant than
optimization for undelayed input signals (classical Magnitude
Optimum)
- A pre-filter is definitely an integral part of the controller
design - contrary to what is sometimes stated otherwise in the
references
- Yields a non-linear equation system - structural similarity to
the Digital Magnitude Optimum (BOD);
matrix form may increase clarity
- Simplest cases are PI- and PD-controllers based on a quadratic
equation system - structural similarity to the
Digital Magnitude Optimum (BOD)
- Can be transformed into a special case of an RST structure
according to
I.D. Landau / G. Zito
(of formal interest, since the foresight of C. Kessler can be
seen, but pure numerator polynomials are of practical interest
only for digital control)
B - Historical background and issues of misunderstanding in the
application of the Magnitude Optimum
In the early period of control engineering, rather little computational
support was typical and application examples for the control of electric
drives later caused an incomprehensible misjudgement of the potential of
this optimization method, see e.g.
J.W. Umland / M. Safiuddin or
K.J. Åström / T. Hägglund.
The close connection of the Magnitude Optimum according to C. Kessler to the
Naslin Polynomial Method (Doppelverhältnisse) according to P. Naslin,
is emphasized e.g. by
R. Schönfeld and explained by
B. Ufnalski
with the aid of an instructive example.
- Selected conditions / tools in the early days of control
technology and consequences thereof:
- Bodediagrams (logarithmic amplitude / phase responses),
controllers using multiplicative representation (time constants
<==> zeros, poles), simple analytical optimization equations,
nomograms, approximations for plants.
- Bodediagrams and controllers using multiplicative representation
require time constants, i.e. real numerical results;
Bodediagrams, simple equations and nomograms benefit from pole
compensation; approximations produce ranges of validity
- Some of the untenable counter-arguments / misunderstandings,
e.g. discussed at length by K.G. Papadopoulos:
- Pole compensation is only a possible option (and also only with
undelayed inputs), but by no means a condition of the Magnitude
Optimum
- Requirement for no more than one integrator in the open loop
only applies to optimization for undelayed inputs
- Optimization for undelayed inputs cannot result in optimum
behaviour in the case of disturbances at the input of the plant
- Optimization for delayed inputs includes by definition the
specification of a pre-filter as design default
- The Magnitude Optimum excludes a priori neither more
complicated plants nor controllers
C - Exploiting the potential opportunities of the Magnitude Optimum
through the BO Toolbox
Recently, there has been a renewed interest in the Magnitude Optimum and
various authors have published generalised results. The equations on which
this BO Toolbox is based avoid the remaining
limitations, and a publication is in preparation.
- The optimization equation system
- Consideration of multiple integrators in the design for delayed
inputs is possible without approximations in the application of the
system of equations for optimization, i.e. without setting one side to
zero. Similarly, higher-order controllers can optionally be designed
for undelayed inputs or controllers for delayed inputs with only one
integrator in the open loop - all in contrast to
K.G. Papadopoulos. Here the weak point of
K.G. Papadopoulos' type classification becomes apparent, since in fact two
type I optimizations are possible, but K.G. Papadopoulos only treats
optimization for undelayed inputs.
- If required, the design of higher order controllers, i.e.
weighting of higher derivatives, is possible for both undelayed inputs
and delayed inputs and in each case without the need to calculate
characteristic areas. The weighting factor for power zero of the
Laplace variable in the denominator polynomial of the plant is
usefully not fixed at unity - all in contrast to
D. Vrancic.
- The properties of the Magnitude Optimum in case of PI controllers were
analysed in detail by J. Cvejn and it
was stated that the "... criterion
usually results in very good control quality and can be applied directly
for high-order linear models with dead time, without need of any model
reduction" and that although "... the stability
margin requirements are not explicitly included in the design objective
... proper open-loop behavior in the middle and high frequency ranges
... is ensured automatically ..." under certain circumstances.
- Selected special features included
- Complex controller zeros through additive representation of the
controller (gain, reset time, rate times)
- Non-minimum phase plants, complex poles, multiple poles etc. by
polynomial representation
- Oscillation elements, optional dead time, also with PID
controllers - in contrast to J. Cvejn
who has studied PI controllers
- Optional integrated approximation of dead times by Strejc, Pade
or Polynomial
- Optional consideration of the smoothing time constant for
controller D-terms as part of the design
- Optional optimization of the pre-filter to reduce the overshoot