Extreme value controller

With an extreme value controller, a concept from control engineering is implemented, according to which this controller should maximize or minimize a target variable, i.e. H. an optimum should be achieved. The associated control parameters are continuously adapted depending on the influence of the disturbance variables .

English term

The term extreme value control has been established for many years in German-speaking countries . In the English-speaking world, however, there does not seem to be a fixed definition of the term. However, the term extremum-seeking control is likely to be the one that most frequently names this regulatory principle today.

Classification in control engineering

Due to the optimum character of the characteristic field of the system, the extreme value controller is always a controller for non-linear systems. It differs from the commonly used controllers in that it has no reference variable and only a single sampling point on the measurement signal side does not provide any information about the direction in which the manipulated values are have to be changed in order to get in the direction of the optimum, and how great the distance to the optimum is. In order to get at least the direction information, two or more scans on the characteristic curve are always necessary. The extreme value controller therefore always needs a "searching component" in its structure.

Note the functional difference to adaptive control and optimal control . With extreme value control, it is not the parameters or the structure of a controller that are optimized in order to optimally guide the system according to certain characteristics, but rather the system is guided by the extreme value controller in such a way that it itself drives to an optimal point of its characteristic field.

Fig. 1: The most general representation of an extreme value control loop as a block diagram

Figure 1 shows the general mode of operation of an extreme value control loop as a block diagram . Therein mean:

R - extreme value controller
P - process (includes control device and system to be controlled)
M - measuring device (includes sensors, signal conversion and signal evaluation)
${\ displaystyle y}$ - control value
${\ displaystyle x}$ - process variable
${\ displaystyle r}$ - feedback variable

The sizes and can generally also appear as vectors and . ${\ displaystyle y}$ ${\ displaystyle x}$ ${\ displaystyle {\ vec {y}}}$ ${\ displaystyle {\ vec {x}}}$

Fig. 2: General representation of a control loop as a block diagram

This block diagram can be converted into the general representation of a control loop (Figure 2) if the reference variable is set to zero. In the case of an extreme value controller, the summation element and the resulting control deviation are replaced by a necessary structure which estimates the position of the extreme value of the feedback variable r more or less precisely. The control deviation corresponds to the distance and the direction from the current operating point to the extreme value. In comparison, with the general control loop, a restriction to a scalar value of the feedback variable is not absolutely necessary. ${\ displaystyle w}$ ${\ displaystyle e}$ ${\ displaystyle r}$

Target size

The target variable of the extreme value control corresponds to the scalar feedback variable in the general representation. The target variable must be determined by evaluating the measured variables in such a way that their extreme value corresponds to the optimal state of the system to be controlled desired by the user. This evaluation then results in the connection ${\ displaystyle r}$

${\ displaystyle r = f ({\ vec {x}})}$

as a characteristic curve or surface with an extreme value in the optimal system state.

So that the extreme value controller can find the extreme value from all operating points in the manipulated variable range, only one local extreme may be located in this area. In general, the extreme value controls also work when a saddle point is present, if the gradient next to the saddle point can be resolved to the extent of the resolution of the measured values and the "search width" in the working point.

functionality

The basic principle of all extreme value controllers is that the characteristic field is scanned at the current operating point. This can be done continuously, quasi-continuously or time-discrete. If a gradient is determined, the operating point is shifted "in the direction" of this gradient. Since the gradient is exactly zero in the optimum, the control loop remains at this point, but continues to scan it. If the position of the optimum point now shifts, the extreme value controller detects this and follows the optimum point. There are methods, such as the relay extreme value controller mentioned below, which do not make any explicit distinction between gradient sampling and operating point and / or also do not determine an intermediate value that can be interpreted as a gradient. Nevertheless, these methods are based on the gradient or on its sign .

When designing an extreme value controller, it should be noted that the systems to be controlled are usually dynamic systems. A distinction must also be made between one-dimensional and multi-dimensional systems. One-dimensional systems are used when the manipulated variable is a scalar value. In the case of multi-dimensional systems, however, the manipulated variable is a vector made up of a fixed number of manipulated values.

Differentiation from optimization procedures

Due to the necessary behavior of the extreme value controllers to find and maintain an optimal point of a system, they not only have similarities with the optimization methods generally known from mathematics and other scientific fields (see Operations Research ), but also represent a subset of such optimization methods.

In principle, an extreme value controller can be implemented using many known optimization methods . (see for example) In general, however, the requirements from the point of view of control engineering usually set limits that do not allow the use of such processes or lead to modifications that greatly impair the effectiveness of these processes.

These control requirements are above all:

continuous behavior
Stability in the working point
limited adjustment range
dynamic behavior of the system to be controlled
Influence of interfering signals on the target variable.

Optimization methods that work with the derivation of the target function cannot be used in principle, since in the case of extreme value control only an approximate determination of the gradient at an operating point is possible, but not the entire function of the family of characteristics is known as an equation or system of equations.

Good optimization methods are characterized by the fact that they find the optimum point as quickly as possible or with the least possible computing effort. For this reason, most of the methods work with an adaptation of the step size with which the characteristic curve field is scanned. At the beginning of the search, the family of characteristics is scanned with rough steps and this step size is successively reduced when the optimum point is approached. As a rule, an undershoot of the step size due to this step size control is used as a termination criterion for "optimum point found".

This behavior is in contradiction to the above claims 1 to 4. The sampled values are obtained discontinuously over a large area of the family of characteristics at the points that the respective method precalculates as a sensible sampling point for quickly finding the optimum. This means that the scanning does not take place continuously, not in the area of an operating point, does not take into account the limits of the setting range without special measures and, if it is dynamic, which is usually the case in technical systems, can stimulate the system to turn it into critical Operating states (vibrations, limitations). Although this can be counteracted by massive limitation of the step size, this method, however, loses precisely the advantages that make it possible to quickly find the optimum point.

Many mathematical methods are also intended for "smooth" function curves. In technical systems, however, one always has to deal with an interference level in the measurement signal (point 5). Especially in the area of the optimum, where the gradient of the family of characteristics approaches zero, these disturbances become noticeable with small sampling steps, since they "trick" the algorithm into thinking that it is not in the optimum. Filtering the measurement data brings a remedy here, but this further reduces the search speed compared to the search speed theoretically possible with the respective method. However, this does not represent a difference to the extreme value control methods usually used, since these also perform smoothing, mostly by integration.

A fundamental difference in the use of extreme value controllers compared to conventional optimization methods is ultimately that it is not just a question of finding the optimum point (ideally only when switching on the control loop), but also of keeping it stable over a long period of time.

stability

Even with the extreme value rules needs the stability will be, as in all other control systems also ensured. The conditions are comparable to non-linear regulations . Special attention must be paid to the fact that the working point not only leads to the typical "swinging up" through an effective loop gain> 1, but that the working point itself does not move stably to the extreme point or remains there. If the design is incorrect, this behavior can also occur without the typical vibration behavior of control systems. The reason for this can be phase shifts between the signal of the control vector and that of the feedback if their mutual behavior is used to determine the direction (gradient estimation) in the extreme value controller.

Known extreme value control methods

As with the optimization process , it is not possible to create a comprehensive image of the various forms of extreme value controllers, since each process can ultimately be further developed in some form depending on the application. In the following overview, the basic procedures should therefore be named, to which corresponding further developments can be traced back.

One-dimensional extreme value controllers

The dimension relates to the control vector, which must be a scalar variable in this method. All of the methods described below can only be applied to such one-dimensional problems. How these can be extended for multi-dimensional problems: s. Multi-dimensional extreme value controllers

Relay extreme value controller

Multi-dimensional extreme value controllers

If the controller has to set several manipulated variables at the same time or quasi simultaneously so that a target variable is taken to its extreme or should be kept there, it is a multi-dimensional process.

Fig. 3: Multi-dimensional extreme value controller with subordinate one-dimensional extreme value controller

As with the general optimization methods , there are two fundamentally different solutions for this task. The first way is that the problem is transformed into a one-dimensional problem that is then solved with one of the above-mentioned one-dimensional extreme value controllers. Figure 3 shows such a principle in a block diagram.

The subordinate one-dimensional extreme value controller does not act directly on the process, but is multiplied by a direction vector. The superimposed controller R specifies this direction vector, which evaluates variables from the subordinate circle and, if necessary, also uses a priori knowledge of the process. Depending on the structural design, the higher-level controller R can work continuously or discontinuously. Logical links and step sequences are also possible. In the actual sense of control technology, it is not always to be referred to as a controller in the respective application, but only the overall structure. ${\ displaystyle {\ textrm {R}} _ {u}}$ ${\ displaystyle {\ vec {y_ {d}}}}$

In the simplest application, block R as a pure control element in an open chain of effects does nothing more than gradually and systematically switch the controller output from one manipulated variable to the next. ${\ displaystyle y_ {r}}$

The second solution is that the structure of the extreme value controller is already designed in such a way that it influences several manipulated variables at the same time. So far, only one principle is known from the literature that can be classified here. This procedure is referred to as an extreme value controller with synchronous detector. This procedure is basically described in.

Applications

Combustion processes
Flow processes
Electrical engineering
Accelerator technology
chemistry
Refrigeration

Compared to the total amount of literature on control engineering, the field of extreme value control only accounts for a negligibly small proportion. It should be concluded from this that the proportion of applications is correspondingly low. However, it is more likely that, on the one hand, some of the potential applications were solved with instruments of mathematical optimization or, on the other hand, processes of extreme value control were unconsciously implemented without explicitly assigning this to this subject, which in turn had a negative effect on the number of publications.

literature

Unfortunately, at least in German and English-speaking countries, there is no literature that covers the subject in detail. However, in the last few years the procedure based on a correlation of test and target signals has evidently become established. This is described from the theoretical side and provided with numerous application cases and references in.

Ariyur, Kartik B .; Krsti, Miroslav: Real-Time Optimization by Extremum-Seeking Control. 1st edition. Wiley, 2003, ISBN 978-0-471-46859-2

Morossanow, IS: Relay Extreme Value Control Systems . Verlag Technik, Berlin, 1967

Solodovnikov, WW: Nonlinear and Self-Adjusting Systems. Verlag Technik, Berlin, 1975

Ivachnenko, AG: Technical Cybernetics. Introduction to the basics of automatic, adaptive systems. 2nd edition. Verlag Technik, Berlin, 1964

Individual evidence

↑ ^a ^b ^c Ariyur, Kartik B .; Krsti, Miroslav: Real-Time Optimization by Extremum-Seeking Control. 1st edition. Wiley, 2003, ISBN 978-0-471-46859-2 .

↑ Chunlei, Z .; Ordonez R .: Numerical optimization-based extremum seeking control with application to ABS design . fully published in IEEE transactions on automatic control, 2007, vol. 52, no3, pp. 454-467, ISSN 0018-9286 .

↑ ^a ^b Herbrand, F .: Extreme value controller for automatic beam guidance at linear accelerators for ions. Atp magazine 8/2001.

↑ Schulze, K.-P. and Rehberg, K.-J .: Design of adaptive systems. A representation for engineers. Verlag Technik, Berlin 1988.

↑ Multi-scale asymptotic method for controlling thermoacoustic processes in combustion chambers. TU Berlin

↑ AIR-FUEL MIXTURE REGULATING SYSTEM IN AN INTERNAL COMBUSTION ENGINE Patent application PCT / DE1998 / 003708

↑ Development of a real-time capable indexing system. ( Memento of the original from February 22, 2004 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. TU Darmstadt @1@ 2

↑ Schulz, Jan: Active noise reduction of the torsional sound of axial turbo machines by influencing the flow. ( Memento of the original from June 23, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Dissertation 2004, TU Berlin (PDF, 9MB). @1@ 2

↑ Flow regulation of blunt bodies. ( Memento of the original from June 18, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. in doctoral colloquium, program & book of abstracts. TU Berlin, p. 28ff (PDF 4MB). @1@ 2

↑ Extreme value controller for the optimal setting of the operating point of regenerative energy converters (MPP controllers). Patent application DE9217595U1.

↑ Martínez, E .: Extremum-seeking control of redox processes in wastewater chemical treatment plants . Contribution to 17th European Symposium on Computer Aided Process Engineering (PDF 300kB).

↑ Yaoyu Li et al .: Extremum Seeking Control of a Tunable Thermoacoustic Cooler (PDF, 1.3MB) also published in IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 4, JULY 2005, pp. 527-538.

Remarks

↑ ^a ^b This fact is very confusing due to the conceptual situation in the English-speaking world. There the term 'adaptive extremum-seeking control' appears, which alone does not make it clear whether it is a pure extreme value controller or an extreme value controller whose properties (parameters) are influenced by adaptive processes, or whether an extreme value controller within an adaptive controller In the sense of a search method for parameter estimation for the process model is used (with Model Identification Adaptive Controllers ) or the parameters of the controller block in the adaptive controller are changed directly (with Model Reference Adaptive Controllers ).
↑ Theoretically, a vector can also be thought of here, comparable to vector optimization problems . However, this can lead to problems such that either several local extrema arise unnoticed or the individual vector elements compete with one another. However, since it is not the process that is responsible for specifying the optimal operating point, but rather the user, he must ultimately be able to create a corresponding relationship from the measured variables, which from the user's point of view reflects the optimal state of the system to be controlled.
↑ This condition presented here also applies to all optimization methods that work with the determination of the gradient of the objective function. In the literature on optimization methods, this target variable is often referred to as the quality value. In some cases, the term costs is also used there, based on optimizations in the economic area.

[ArKrRTO-1] Ariyur, Kartik B .; Krsti, Miroslav: Real-Time Optimization by Extremum-Seeking Control. 1st edition. Wiley, 2003, ISBN 978-0-471-46859-2 .

[5] Chunlei, Z .; Ordonez R .: Numerical optimization-based extremum seeking control with application to ABS design . fully published in IEEE transactions on automatic control, 2007, vol. 52, no3, pp. 454-467, ISSN 0018-9286 .

[HerATP-6] Herbrand, F .: Extreme value controller for automatic beam guidance at linear accelerators for ions. Atp magazine 8/2001.

[SchReh-7] Schulze, K.-P. and Rehberg, K.-J .: Design of adaptive systems. A representation for engineers. Verlag Technik, Berlin 1988.

[8] Multi-scale asymptotic method for controlling thermoacoustic processes in combustion chambers. TU Berlin

[9] AIR-FUEL MIXTURE REGULATING SYSTEM IN AN INTERNAL COMBUSTION ENGINE Patent application PCT / DE1998 / 003708

[AdaptR-2] This fact is very confusing due to the conceptual situation in the English-speaking world. There the term 'adaptive extremum-seeking control' appears, which alone does not make it clear whether it is a pure extreme value controller or an extreme value controller whose properties (parameters) are influenced by adaptive processes, or whether an extreme value controller within an adaptive controller In the sense of a search method for parameter estimation for the process model is used (with Model Identification Adaptive Controllers ) or the parameters of the controller block in the adaptive controller are changed directly (with Model Reference Adaptive Controllers ).

[3] Theoretically, a vector can also be thought of here, comparable to vector optimization problems . However, this can lead to problems such that either several local extrema arise unnoticed or the individual vector elements compete with one another. However, since it is not the process that is responsible for specifying the optimal operating point, but rather the user, he must ultimately be able to create a corresponding relationship from the measured variables, which from the user's point of view reflects the optimal state of the system to be controlled.

[4] This condition presented here also applies to all optimization methods that work with the determination of the gradient of the objective function. In the literature on optimization methods, this target variable is often referred to as the quality value. In some cases, the term costs is also used there, based on optimizations in the economic area.