ON THE SAMPLING PERIOD IN FUZZY CONTROL

ALGORITHMS FOR SERVODRIVES

A Strategy for Variable Sampling

Dan Mihai

University of Craiova, Decebal Blvd, 107, Craiova, Romania

Keywords: Fuzzy control, Adaptive sampling period, On-line timing, Microcontroller, Servodrives.

Abstract: The paper deals with a variable control sampling period for the fuzzy control algorithms implemented on

low inertia servodrives. The robustness of the fuzzy control strategy is extended on the sampling period

values and hence an adaptive sampling algorithm is proposed. The author analyzes the possibility to vary

continuously the sampling frequency upon a basic process variable. Principles, models and simulation

results inserted here give reliance in this technique and an enhancement of the fuzzy control

implementation. The distribution of the sampling moments in different adaptive conditions and the

behaviour of the servodrive are obtained by means of some models and simulations in accordance with the

real-time target hardware system.

1 INTRODUCTION

The author found (Mihai, 2001) that the fuzzy logic

has the ability to drive the system in good conditions

for very different control sampling period - T values

over more than a magnitude order. In such a context,

the idea to vary continuously the T value, in

accordance with a dynamic parameter of the system,

finds a suitable application area. The standard digital

models become non-linear because the variable

coefficients and the classical algorithms are very

sensitive to T. A variable T means, in almost all the

approaches, acquisitions with a variable frequency.

Less studies and experience concern the real-time

control with a variable cycle. Most of the involved

authors and equipment use several pre-computed

constant values T. Computer graphics applications

refer to an adaptive sampling in term of an

adjustment of the sampling resolution in exploiting

the image (Adamson, 2005). The adapting sampling

in the fuzzy control could also provide means to

reduce noise in computer graphics, like for global

illumination algorithms (Xu, 2006). Also some other

special or non-conventional application fields

implement an adaptive sampling (radio telemetric

system for missiles, drying processes in food

industry). Although some papers still prove the

natural idea that a sampled-data fuzzy controller

recovers the performance of the continuous-time

fuzzy controller as the sampling period approaches

zero (Do Wan, 2007), several authors have noticed

that the fuzzy control is flexible and reliable for a

low rate control sampling (Popescu, 1997; Mihai,

2001). Using an adaptive sampling frequency for the

control of a servodrive is a complex task because of

the fast reaction speed of such a system and its high

associated performance.

2 THE FUZZY CONTROLLER

AND T VALUES

Although T seems, apparently, not being an essential

variable for the main characteristics of a FLC (fuzzy

sets and the rule base), this parameter is involved in

a fuzzy loop in two ways:

- as a real - time “integration step” of the system,

(acquisition–processing–control cycle;

- as an input FLC variables generator by:

= V(kT); ΔVb

= V

–V

k-1

; Vck= (V

–V

k-1

) / T (1)

The author considered a low inertia servodrive with

DC motors. The figure 1 gives the essential structure

for the drive with disk rotor motor and an encoder.

The FLC entries are the normalized position error

and the normalized variation of the position error:

221

Mihai D. (2008).

ON THE SAMPLING PERIOD IN FUZZY CONTROL ALGORITHMS FOR SERVODRIVES - A Strategy for Variable Sampling.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 221-224

DOI: 10.5220/0001477702210224

 SciTePress

αn k =

αn k

· (α

) /

(2)

Δε

αnk

=ε

αnk

−ε

αnk-1

= [Τ ·(−ω

)]

=ω

·Δε

αnk max

/Ω

max

(3)

α*/α

- the position set point/ the actual position;

*/N

αk

– same, in encoder pulses; ΔN

αk

- pulses

encoder during T; c

, c

kout

- the computed control

and its outputted value; Norm

: normalization

blocks; CPB: Control Processing Block; PS - Power

supply; T

gen

- torque generator; M - motor; En -

encoder. The encoder has N

p/r

pulses per revolution

and the speed is computed with:

ksp

r/p

kdiv1kk

Nk2

Δ⋅=

⋅

Δ⋅⋅π

α−α

≈ω

−

(4)

The simulation results from figure 2 are obtained

using a FLC from Fuzzy Toolbox (Guley, 1995),

with fuzzy sets and rules presented in (Mihai, 2008).

The quality of the results is proved by the final

position error (null) and the fuzzy state-space

trajectory, between the initial point (10, 0) and the

final point (0,0)–last window. When T increases, the

FLC task becomes more difficult. Although for the

whole range the controller succeeds in bringing the

system to the final point, some internal ringing or

steps appear. It is obviously also that for low

sampling frequencies, the speed (quite well filtered

by the mechanical system) is far from the position

error variation. Another model is designed as a fuzzy

position / speed loop for the same system but with a

Look-Up-Table (LUT) FLC-figure 3. An additional

argument for the adaptation of T is given by the real-

time recordings presented in figure 4, for an on-line

inference fuzzy control (Mihai, 2006). During every

T (SPER), each falling edge of the encoder pulses

(PULSE) leads to a fast hardware interrupt routine

(INTO) that up-dates the FLC entries. FUZZ is the

fuzzyfier task, INFER-the on-line inference task,

DEFUZ-defuzzification task and AUX concerns

other processing tasks, like savings. The 2 diagrams

were recorded for different conditions, revealing the

ability of the FLC to manage the microcontroller

resources even at maximum speed, when the

processing algorithm is interrupted at maximum rate.

However, the available time is very depending on

the motor speed. A higher speed could lead to the

situation when the control processor is no more able

to fulfil the real-time task inside T. Its adaptation to

the speed would be the solution. For adding

robustness related with the load variation, a special

strategy was proposed by the author keeping the

same reference LUT. Additional procedures were

implemented for on-line adaptation of the control to

the load value, both by an estimated current and

some external computations and decisions blocs.

Figure 1: The servodrive with FLC.

Figure 2: Results: T=2.456 ms (a) and T=50 ms (b).

Figure 3: Results for T = 2.456 ms (a) and T= 50 ms (b)

with a LUT based FLC.

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

222

Figure 4: An on-line T in fuzzy control.

3 A FUZZY CONTROLLER

WITH ADAPTIVE T VALUES

The idea is to relate T with the variation rate of the

main variable of the system. During the intervals

with small variations (or in steady state regimes), T

is greater and during the high rate dynamic regimes,

T decreases. The variation for the (generic) variable

v from the step t induces an adequate adaptation of T

at t+δt. The figure 5 gives an image of the principle

and helps for obtaining some relations. A first

possibility is to evaluate the amplitude variation for

the main variable during a constant time interval

(easily in real-time). Another idea is to use an

amplitude quantization of the v variable using a

constant step Δ and to evaluate then the time

intervals associated with this variation. They can be

directly assimilated with the adapted T. Next

relations make connects the derivative value and Δ.

v = f(t); Δ = f(t

i+1

) - f(t

) ; tg α

= Δ / T

(5)

= Δ / tg α

≅ Δ / ( ⏐|df / dt|

⏐+ ε) (6)

Δ could be chosen by practical considerations. If

f is known, (5) is useful for evaluate T. If not, t

result by detecting the amplitude thresholds and by

that the next T value is available. ε is for a

limitation of max.T. A limitation is also necessary

for minT, (systemic, on-line processing constraints):

min

≤ Τ ≤ Δ / ε (7)

For a servodrive where the main variable is the

position, let be the speed ω the variable v (the

variation of the position). It is more suitable to use a

T adaptation in accordance with the variation rate

not after the amplitude of the v variable. Indeed, for

that last case, even in a steady-state regime, the

sampling rate is high and for a low speed during a

strong dynamic regime the sampling has a slow

rate. If the main characteristic variable of the

system is the speed, v could be the acceleration.

The next idea is to adapt also the step value Δ

upon another characteristic variable of the

controlled system. The results for two Δ (constant)

values are depicted by figure 6, with the distribution

of the sampling moment. Position 1 is the sampled

position with an adaptive period. “State space FLC

path” is the trajectory of the system. The global

behaviour is good for both variants, the final

position error being null and the system response in

speed and position being a smooth one. Figures 7a

and 7b give the variations of T for Δ=2.456 and

Δ=20, during the whole regime. The max/min rate

values are almost 100/Δ=10 and 35/Δ=50. Figures

7c, 7d make visible the large variation range of the

max/min T along 3 magnitude ranges (logarithmic

scale).

The next idea is to use a variable quantization step

for adapting T as a double adaptive sampling

strategy. Another adaptation parameter is involved.

The fig. 8 gives the elements for that, considering

the speed as an additional modulator (by its change

rate) for the adaptive sampler of the position. In this

way, it is no more necessary to make different

experiments in order to adopt the best step value

Δ (Q). The distribution of the sampling moments is

different (fig. 8a) and the step value is variable (fig.

8b). The image of the new sampled position is given

by fig. 8c. The results from 8d concern another

values range for the modulator of the adaptive

sampler (a larger one – see Q_ad). It is depicted

also the quantified speed – Speed 1, as the source

of the modulator for the quantization step necessary

for the sampler with double adaptive T. The

overshoot for the position is related with its

quantized final values.

Figure 5: For adaptation of T values.

ON THE SAMPLING PERIOD IN FUZZY CONTROL ALGORITHMS FOR SERVODRIVES - A Strategy for Variable

Sampling

223

4 CONCLUSIONS

A variable sampling frequency could give a better

control. This approach leads to some serious

robustness problems for the classical algorithms but

not for the fuzzy control. A good robustness

regarding the sampling period for the fuzzy control

induced the idea to try a control with adaptive

sampling period. This idea is applied for a

servodrive - a fast and precise system. Several

variants were considered: adaptive sampler with a

constant quantized step, with a multi-step

modulation and with a continuous variation.

Figure 6: Main variables of the system for a step Δ = 2.5

(a), Δ = 20 (b) and T evolution.

Figure 7: T evolution in time and upon Δ.

Figure 8: Adaptive FLC /variable step.

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