Robots Extension Control using Fuzzy Smoothing

Victor Vladareanu1, Paul Schiopu3

Faculty of Electronics, II, 313 SplaiulIndependenţei, 060042

“Politehnica” University of Bucharest,

Bucharest, ROMANIA

vladareanuv@gmail.com

Mingcong Deng2

Dept. of Electrical and Electronic Eng., Koganei,,

Tokyo University of Agriculture and Technology

Tokyo 184-8588, JAPAN

deng@cc.tuat.ac.jp ntry

Abstract— The paper presents a new control method for the

robot extension control, using the Extenics concepts and

Extension Theory techniques. The controller uses the Dependent

Function to measure the degree of compatibility of the process

variable and then takes the appropriate action to force the system

into convergence around a desired set point. The output of the

Dependent Function classifies the process variable value into one

of four categories, concurrent with the nested intervals used in

Extenics, which are then equated to fuzzy inputs sets of a

linguistic variable for a simple fuzzy controller. This acts as a

fuzzy smoothing of the controller output. The rationality and

validity of the proposed model are demonstrated through

simulation in the Matlab/Simulink environment. The results show

that the proposed new controller architecture obtains remarkable

results, while having the advantage of increased simplicity in

design and setting of parameters. Throughout the paper,

opportunities for further improvement and research are

highlighted and discussed.

Keywords—robot control; extension theory; extenics; fuzzy net

I. INTRODUCTION (Heading 1)

This paper proposes a new type of controller for robot

actuators using elements of Extenics, as well as fuzzy control.

There are a number of key differences from the usual

controller paradigm, which are discussed throughout the paper.

Extenics is a relatively new science which deals with the

condition and solving of contradictory problems. At its core lie

a new formalism for the description of problem elements and a

theory of transformations which seek to turn incompatible

problems into compatible problems. It researches to what

extent innovative ideas can be generated using algorithms and

computers. It is classified as being part of Artificial

Intelligence, but is frequently referred to as being a mixture of

Mathematics, Engineering and Philosophy [1].

Extenics has been used extensively over the past decade in

fields such as Data Mining, Marketing, Operations Research,

Control and Detection [2]. Applications involving Extension

control, however, have generally been limited to extending the

range of controllability of a given process [3].

As a first test, this new controller is then used to

regulate the speed of a DC motor. This is roughly equivalent

to simulating a one-dimensional robot actuator and is very

convenient to test the operation of the Extenics controller in a

relatively simple task. Extensions to 2- and n- dimensional

spaces can then be made using the advances [4] of Extension

Theory as a whole.

While Direct Current motors have been around for a long time,

they continue to enjoy broad usage in industry and everyday

applications alike. They are easy to use and simulate and

representations of DC motors are readily available for virtually

all programming languages. In spite of this, the performance

of a given controller in regulating a DC motor is a good

indication of how it will perform in increasingly complex tasks,

such as robot actuators. They therefore provide a good

benchmark for controller comparison and indeed have often

been used as such in academia.

Due to space constraints and the rather involved nature of

some of the concepts present in Extenics, this paper will

proceed to give a brief outline of such concepts, as are needed

for the subject matter at hand, in the next chapter. Chapter III

will then discuss the features and key concepts involved in the

development and design of the Extension controller. A

detailed description of the construction of the controller as

well as the concepts needed for the simulation is presented in

Chapter IV, while the final chapter shows the obtained results,

discusses their significance and draws conclusion and

inferences from these. Suggestions for further research are also

comprised in the last chapter and throughout the paper, as

appropriate.

II. EXTENICS

Extension Set Theory is a new set theory which aims to

describe the change of the nature of matters, thus taking both

qualitative, as well as quantitative aspects into account. The

theoretical definition for an extension set is as follows:

supposingU to be an universe of discourse, u is any one

element in U, k is a mapping of U to the realfield I, T=(TU, Tk,

Tu) is given transformation, we call:

an extension set on the universe of discourse U, y=k(u) the

Dependent Function of E (T ), and y’= Tkk(Tu u) the extension

Manuscript received August 10th, 2013. This work was supported in part

by the Romanian Academy, the FP7IRSES RABOT project no. 318902/2012-

2016 and the Romanian Scientific Research National Authority under Grant

PN-II-PT-PCCA-2011-3.1-0190 Contract 149/2012(sponsor and financial

support acknowledgment goes here).

Proceedings of the 2013 International Conference on Advanced Mechatronic Systems, Luoyang, China, September 25-27, 2013

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function of E(T), wherein, TU, Tk and Tu are transformations of

the respective universe of discourse U. Dependent Function k

and element u. If T≠e, that is to say the transformation is not

identical, four more concepts can be outlined, as follows:

- positive extensible field (or positive qualitative change

field) of E (T):

- negative extensible field (or negative qualitative change

field) of E (T ):

- positive stable field (or positive quantitative change

field) of E (T ):

- negative stable field (or negative quantitative change

field)of E (T ):

- extension boundary of E (T ):

This is further illustrated in Figure 1 [2].

Fig. 1. Universe of Discourse in an Extenics Transformation

The tool for solving contradictory problems is extension

transformation. Through certain extension transformations,

unknowable problems can be transformed to knowable

problems and unfeasible problems can be transformed to

feasible problems. In any real environment, change is a

constant, which is to say at any given moment any number of

transformations is taking place. It is therefore convenient, in

attempting to solve a contradictory problem, or indeed any

problem, to consider which transformation may effect a

change in the qualitative aspects of the problem formulation,

so as to arrive at the desired result. One of the main research

areas of Extenics is investigating and formalizing the effects of

transformation upon the objects, elements, conditions and

universe of discourse of a given problem.

With the aim of measuring the degree of compatibility or

incompatibility in a given problem set, Extension Theory has

introduced the notion of Extension Distance. New concepts of

“distance” and “side distance” which describe distance are

established, to break the classical mathematics rule that the

distance between points and intervals is zero if the point is

within the interval. The Dependent Function established on the

basis of this can quantitatively describe the objective reality of

“differentiation among the same classification” and further

describe the process of qualitative change and quantitative

change [2].

Suppose x is any point in real axis, and X=<a,b>is any

interval in real field, then

is the Extension Distance between point x and interval<a,b>,

where <a,b>can be an open interval, a closed interval, or a

half-open and a half-closed interval X. This is, in effect, the

distance between the point considered and the closest border

of the interval. It can be noticed that when the point is on the

border of the interval (i.e. x=a or x=b), the Extension Distance

will be null, while the minimum possible value for the

Extension Distance is the negative of the half of the interval

length. This of course applies to the definition of Extension

Distance as considered here, taking into account the centre of

the interval as the point of interest and in keeping with the

one-dimensional aspect of the formulation. Further

generalizations to „side distance” and „n-dimensional

distance” are put forth in papers such as [4, 5], which merit

consideration but are beyond the scope of this application.

With the aid of this new take on the distance between

a point and an interval, a new concept can be introduced.

Place value is an indicator of the relative position of a point

with relation to two nested intervals.

Suppose , then the

specified place value of point x about the nest of intervals

composed of intervals X0and X is

This describes the locational relation between point x and the

nest of intervals composed of X0 and X [2].

Further use is made of this new definition of distance in

order to define a new indicator for the measurement of

compatibility within an Extension Set. This indicator is called

Dependent Function and is defined as follows.

Constitute a nest of three intervals by standard positive field

X0, positive field X and interval, i.e.

,

then for any , the elementary Dependent Function k(x) of optimal point at the midpoint of interval X0 is

This provides an indicator for the degree of compatibility of

a given problem which has been expressed numerically, much in the same way as a membership function determines the degree of membership in a fuzzy set. In Extenics, however, the Dependent Function is generalized to the entire real domain, so that it also takes into account qualitative changes, as well as

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quantitative ones. It should be noted that this is the simplest, earliest definition of the Dependent Function. Subsequent papers both by the original authors [2, 6] as well as new researchers [5,7] have led to generalized and, in some cases, very involved expressions for dependents functions. For the purposes of this paper, however, the elementary Dependent Function will suffice.

III. EXTENSION CONTROL

The dominant view with respect to the use of Extenics for

control applications relates to improving the range of

controllability, rather than the quality of control parameters. In

this regard, papers such as [3, 8] merit consideration. Excellent

results have also been obtained by [9].

This paper presents a new method, which aims to improve

control quality by designing a controller that uses some of the

innovative concepts and the mathematical apparatus of

Extenics. The basic idea is using the Dependent Function as a

means of judging whether the controlled parameter is within

an acceptable range and altering the output of the controller

commensurately with the degree of incompatibility shown by

the Dependent Function. Thus it provides a powerful indicator

of reduced complexity for the controller status of the system.

This means that the controller output should be higher in

cases where compatibility is low to non – existent and will

decrease as the system nears convergence. This, in Extenics

terms, is the transformation occurring in the universe of

discourse which should transform the problem from an

incompatible form (lack of convergence) to compatibility. In

this sense, the aim is simply to equate the property of converge

with the positive transitive field pertaining to this

transformation.

It should be noted that, as transformations go, increasing or

decreasing the controller output is one of the most rudimentary

options available. Therefore, as seen from the point of view of

Extenics, ample opportunities for further research should be

present in investigating the effect of different, more complex

transformations on the universe of discourse.

The aim of the Extension controller is bring the desired

controlled parameter (the speed of the DC motor) to converge

to the reference value set by the operator. The nested intervals

in the Extenics representation of the problem are symmetrical

ranges around the optimum (i.e. the reference value). These

can be set at will, provided the configuration of the nested

intervals remains the same (as in, they still include each other

as in the definition of the Dependent Function). In fact, further

research may experiment with unsymmetrical intervals for one

or more of the extension sets, different and more complex

formulations for the Dependent Function in order to achieve

different behaviour.

As such, a classification of sorts can be made, with each

nested interval acting as a class. These can be equated with

fuzzy sets or linguistic variables for a fuzzy smoothing of the

output (as is seen later) or can be useful in a number of

different implementations. It is even possible to provide

different controllers to act on the process, with the selection

being made depending on the class the current output is

assigned to (sort of like a switch of controllers).

The input variable for the fuzzy controller falls within one

of four classes, as discussed in the theoretical approach: “Not –

Controlled” (NC) for values of the Dependent Function (k)

smaller than -1; “Transformable” (Tr) for k between -1 and 0;

“Acceptable” (Acc) for k between 0 and 1 and “Compatible”

(OK) for k greater than 1. These are the Dependent Function

values of the limits of the nested intervals and are the same in

any Extenics application, being a property of how the

Dependent Function is calculated rather than the problem itself.

In our case, “Not – Controlled” corresponds to the process

variable value being outside the last nested interval, where an

Extenics problem would be considered completely non –

compatible. This makes the controller act vigorously, in order

to force the system into compatibility. “Transformable” is the

label assigned to the largest nested interval. Since it is past the

rise time mark (0.7 of the reference value), the controller is

clearly having an appropriate effect, however controller action

must still be maintained in order to reach the goal. The name

of the label comes from this being the region, in Extension

Theory, where a generic transformation can most likely lead to

compatibility. For a Dependent Function value that is positive,

but less the one, the process value is classified as “Acceptable”.

This is very close to the desired settling band (±2% of the

reference value), so the controller action is minimal. Once the

signal passes into the settling band, controller output is zero

and the system is classified as being “Compatible”.

Figure 2 shows these categories for the Dependent Function

value in relation to the process variable, assuming a reference

value of 1.

Fig. 2. Dependent Function Classification

This is the same graph that is being shown in all subplots, each of them has been zoomed and panned so as to better illustrate each of the four categories. As can be seen from last subplot, the range for the “Compatible” classification is quite small. For an added measure of accuracy, it has been reduced to ±1% of the reference value.

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IV. EXTENION CONTROL METHOD WITH FUZZY SMOOTHING

After the text edit has been completed, the paper is ready for

the template. As mentioned in the previous chapter, the degree

of incompatibility, measured using the Dependent Function,

will provide a scale of the output of the controller.

The controller simulations and tests are performed using a

simple DC motor configuration in the Matlab / Simulink

programming environment. An overall representation of the

process, as seen in a Simulink simulation, is given in Figure 3.

The motor model, while not at all involved, takes into account

the existence of load (although this is not used in this

application) and provides a scope to the armature current,

which needs to be monitored, as very high currents may cause

permanent damage to a motor in a real-life setting.

The Dependent Function is calculated with respect to the

process variable and not the error, as is the most common case

with controllers. It shows the degree of compatibility between

the current value of the process variable and the desired state

(convergence on the reference value). It cannot, however, offer

information on whether the error is positive or negative.

Therefore, the simulation must account for the sign of the error,

which is extracted and multiplied to the end result of the

controller output, as can be seen in the overall diagram in

Figure 3.

Fig. 3. Overall Model of the Simulation

Considering the three nested interval to be bands around the

desired set point value of the process variable, the value of the

Dependent Function provides information as to where the

current process value is located.

Fig. 4. Nested Intervals of the Extension Controller

Again, the width and position of the bands around the set

point value may be altered at will, but for this first

implementation they were chosen to be concentric,

symmetrical and roughly coinciding with established

indicators such as the marks for response time, rise time and

settling time. This is explained visually in Figure 4.

The Dependent Function value is obtained in the simulation

by using an Interpreted Matlab Function block, after having

defined the elementary Dependent Function in a Matlab m-file.

This provides a simple, instantaneous calculation for the

degree of compatibility of the process variable.

Within the simulation, this classification is done using the

fuzzy controller. Each of the four classes is equated with a set

of the input fuzzy linguistic variable. Minor adjustments can

be made to account for the way a fuzzy controller processes

inputs: the sets do not need to be mutually exclusive and the

process is usually smoother when there is some overlapping.

The exception to this is in the third set, which does not pass its

right outer boundary, so as to prevent the controller from

having a residual output once the “Compatible” stage is

reached. Figure 5 below shows the arrangements of the input

variable and output variable sets.

Fig. 5. Fuzzy Controller Sets

While it is certainly possible to assign direct mathematical

meaning to the degree of compatibility, it seems more feasible

to pass the controller output through a process of so-called

“fuzzy smoothing”. This simply means that an additional

fuzzy controller is attached to the end of the output path and is

indeed an integral part of the controller as a whole. The output

of the Dependent Function is passed to the fuzzy controller,

which classifies it with a degree a membership and

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reconstructs an output through defuzzification. While this is in

no way necessary, fuzzy smoothing has shown markedly

improved results in reducing jamming in an array of

applications [10, 11] and is also convenient for the task at

hand, since establishing a direct mathematical equivalence

would require considerably more experimentation.

Fig. 6. Fuzzy Controller Rule Base

The obvious interpretation of the nested Extenics intervals

as fuzzy linguistic variables also makes the use of a fuzzy

smoother an easy choice for implementation.

Fig. 7. Extenics and Regular FLC RuleViewers (comparison)

As is quite the norm in implementations of fuzzy smoothing,

neither the rule base, nor the structure of the fuzzy controller,

are very complex. The controller has a 1 input to 1 output

structure, making use of the equivalence between the nested

interval ranges and fuzzy linguistic variables as described

earlier. The representation of the fuzzy rule base is given

below in Figure 6.

To illustrate the simplicity in designing the fuzzy controller,

Figure 7 shows a comparison between the Rule Viewer used

here and one used in an implementation of a simple fuzzy PD

controller [11]. For this application, the effects of load, disturbance and

noise were ignored within the simulation. The overall controller output was also limited, which would protect the motor in a real-life situation against damage to its components.

RESULTS AND CONCLUSION

Figure 8 below shows the result of the Extension controller

action on the process variable. It has very small overshoot

(8%) with good values for rise time (2.4s) and settling time

(4.7s). The steady state error is negligible at 0.03% of the

reference value.

Fig. 8. Extension Controller Performance (Scope)

Observing the effect on the armature current values (Figure

9), it is easy to see that the controller performance is more than

satisfactory, with maximum currents in the region of 4A not

posing a danger to the motor.

Fig. 9. Armature Current (Scope)

The advantage of the Extension controller is that

these results were obtained with no need for added complexity,

either in the design or the implementation of the simulation.

The controller architecture is very straightforward, once the

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function interpretation block is in place, and even the

additional fuzzy controller could have been omitted and

replaced with a PID – like structure, for example. This,

however, is not necessary, as even the design of the fuzzy

smoother is not at all involved, with a 1:1 equivalence between

the classification sets and the fuzzy variable sets, and then

later between the fuzzy input and output variables. As opposed

to a normal, standalone fuzzy controller implementation, there

are no complex rule bases to sift through, and there are no

schemas or complex algorithms needed to determine the

simulation parameters (such as the gains in a PID

implementation). While the locations and ranges of the nested

extended sets needs to be specified and there is some tweaking

involved in this, their optimization is not vital and perfectly

viable results can be obtained with simple, intuitive values (for

example setting the “Acceptable” range the same as the

settling band of ±2% of the reference value).

The novelty presented in this paper is implementing

Extension Theory methods in a controller structure, with the

aim of improving control quality as regards classical control

indicators (overshoot, settling time, expended current, etc.), as

well as robustness and ease of implementation. These aims

have been followed and highlighted throughout the paper. It is

also important, in the authors’ opinion, to propose a controller

structure using the innovations brought about by Extension.

Theory, which can then be improved and perfected by

subsequent research.

With that in mind, using the Extension controller for the

purpose of controlling the speed of a simple DC motor with no

load, disturbance or noise present is only the first step. Further

simulation in this regard is needed to test the controller

behaviour for robot actuators with load, noise, and disturbance.

The final goal to be reached is a physical implementation on

an actual mechatronics system. Good results have also been

achieved with virtual projection methods [12, 13, 14].

Throughout the paper, possibilities for further research have

been outlined and discussed. Extension control, as discussed in

this paper, benefits greatly from being a novelty approach to

controller design. While this paper proves a working model

can be established with basic parameters, the possibilities for

tweaking and optimizing in the hopes of obtaining improved

performance are virtually limitless. Changes, both subtle and

large, will be brought about by further experimentation and the

development of Extension Theory as a whole.

There are of course parallels to be drawn to Fuzzy control,

as well as other types of Artificial Intelligence control

algorithms. However, Extension control is unique in a number

of aspects. Perhaps most importantly, it represents a shift in

the paradigm of controller structure. While the controllers

themselves have evolved greatly over the years, changes in the

way one looks at controllers and controller structures have not

been frequent, save perhaps for the first implementation of

fuzzy controllers and their acceptance in industry. By way of

being an implementation of a more generalized theory, whose

aim is precisely to formalize the process of innovation, there is

virtually no end to the possibilities for further research. Also,

as Extension Theory continues to grow and mature as a

discipline in itself, the theoretical advances made are sure to

have a favourable impact on this type of implementation.

ACKNOWLEDGMENT

This work was supported in part by the Romanian Academy, the FP7 IRSES RABOT project no. 318902/2012-2016 and the Romanian Scientific Research National Authority under PN-II-PT-PCCA-2011-3.1-0190 Contract 149/2012 (sponsor and financial support acknowledgment goes here).

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