ON THE SAMPLING PERIOD IN STANDARD AND FUZZY

CONTROL ALGORITHMS FOR SERVODRIVES

A Multicriterial Design and a Timing Strategy for Constant Sampling

Dan Mihai

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

Keywords: Sampling period, PI algorithm, Fuzzy control, On-line timing, Servodrives.

Abstract: The paper deals with the best choice for the control sampling period in term of a multicriterial conditioning,

with the on-line timing and with a comparison between the conventional (like PI) control algorithms and the

fuzzy control. Several useful relations are followed by diagrams obtained in simulation and by different

real-time recordings both for the timing and for characteristic variables of the system. Implementations with

microcontroller and DSP are used for analyzing the design criterion and the timing strategy. The application

field concerns a servodrive, so the real-time constraints are quite strong. The author conceived a general

control strategy for the on-line timing based on imbricate interrupts, each pulse encoder acting on the

hardware input interrupt of the control processor.

1 INTRODUCTION

The sampling frequency plays an essential role in

implementing a digital control algorithm in real-

time, especially for the fast systems. Most of the

reference books give evaluations only for the upper

limit of the sampling period (T), as if the ideal value

would be as little as possible – only the capacity of

the control processor being the constraint. The

efficient choice of the sampling rate in closed-loop

system is based on its influence on the performance

of the control system. The absolute lower bound to

the sample rate is set by the system bandwidth. The

classical controllers (and their loops) are not robust

and their tuning (including the additional T

parameter) - although stated as well settled (Astrom,

1997), seems to be very difficult in complex

conditions. Not a few applications and studies

concern the sampling period design for different

kind of fuzzy control. An earlier idea (Coleman,

1994) about the robustness comparison between

fuzzy logic, PID control and sliding mode control,

will be extended now in the area of the control

sampling period.

During the last decades, the electrical drives

field has integrated more and more design

techniques, fast control processors and acquisition

modules for high performance platforms. A hybrid

approach is to have an inner current loop monitored

by a fuzzy controller (Mrozek, 2000) while the main

speed loop is monitored by a classical PI controller.

Another approach is to design a self tuning fuzzy

logic controller, based on some desired output

behaviour and hence, does not requiring a precise

model of the machine (Ibbini, 2002). Sometimes, the

results are quite close in term of performance (in

steady state or dynamic regime) but the

implementation effort is much lower for the fuzzy

solution (Silveira, 2002). Industrial equipment support

from few kHz to 20 kHz sample rate for the velocity

loop. This high rate of sampling combined with the

velocity observer, allows equipment to provide a very

fast control for industrial servo drives. Despite the

availability of several high performance DSP

controllers, many researchers are interested in

designing optimal control algorithms based on low-

cost solutions; such approach is typical for linear and

sliding-mode controllers designed for a DC servo

drive with a microcontroller and low sampling rate,

typical for embedded systems (Kosek, 2007).

The author developed a general on-line timing

for all digital control algorithms for the servodrives

with a hardware position loop and a software one for

the speed, proposing several relations for the

multicriterial correlation of the involved parameters,

T included. Several such relations require the T

value to be superior to some threshold limits.

Mihai D. (2008).

ON THE SAMPLING PERIOD IN STANDARD AND FUZZY CONTROL ALGORITHMS FOR SERVODRIVES - A Multicriterial Design and a Timing

Strategy for Constant Sampling.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 72-77

DOI: 10.5220/0001477600720077

 SciTePress

2 THE SERVODRIVE AND THE

ON-LINE CONTROL TIMING

Figure 1a presents the main parts of the system.

Each encoder pulse acts on the interrupt entry of the

processor. The next control strategy and the timing

remain the same for any motor (and its associated

power supply), for different kind of control

algorithms (standard, fuzzy). The general systemic

structure is given by figure 1b; figure 1c is for a

conventional control and the next one (1d) presents

the fuzzy control. The main notations: N

*,N

αk

position set-point and the real position, in encoder

pulses; ε

αk

, Δε

αk

: position error, its variation referred

to a sampling period; ΔN

αk

- pulses encoder during

T; c

, c

kout

- the computed control and its outputted

value; Norm

: normalization blocks for each Fuzzy

Logic Controller (FLC); CPB: Control Processing

Block; PS - Power supply with digital control input;

gen

- torque generator; M - motor; En - encoder.

The encode has N

p/r

pulses per revolution and the

speed is monitored through a software image:

ksp

r/p

kdiv1kk

Nk2

Δ⋅=

⋅

Δ⋅⋅π

α−α

≈ω

−

(1)

Figure 1: The drive with different controllers.

div

is a division / multiplication factor for encoder

pulses. The real-time control strategy is based on 2

imbricate interrupts – figure 2. INT0 is a high level

priority interrupt generated by encoder pulses (the

falling edge). The low level priority interrupt is

software generated, marking the sampling period -

T. PULSE means the encoder signals. For standard

or non-conventional control algorithms, T must be

correlated with: a. the specific system dynamic; b.

min

- the minimum accepted value of the real speed

for which ω

k soft

is detectable; c. n

- the size of the

data word (register) for ω

; d. the accepted

resolution for the speed; e. the program length of the

on-line processing; f. The amount of memory

available for on-line recordings. b., c and d. give the

next restrictions for T:

[RPM]n

p/r

div

(

[RPM]n

p/r

div

max

6012

min

⋅

⋅⋅−

≤≤

⋅

(2)

When the range speed covers a full data register, for

having 1 LSB at minimum speed, T must be:

(2 -1) 60

[]

max

reg

div

nRPM

××

≥

(3)

The e. condition generates another constraint for T

(Mihai, 1999). INT0 requested by the falling edge

encoder pulses computes ε

αk

, ΔN

and needs the

time-Δt

INT0

. Δt

av.instr.

is the average time for a

processor instruction. Having the code program

length for the software interrupt routine-N

max.ALG. instr.

imposed by the algorithm, T must be:

div

INTrp

instravinstrALG

tNn

⋅

Δ⋅⋅

−

Δ⋅

≥

0/max

...max

(4)

As for f. condition, the data memory space for all

on-line records is given by the number of data bytes

Figure 2: The proposed timing for on-line control based on

interupts.

ON THE SAMPLING PERIOD IN STANDARD AND FUZZY CONTROL ALGORITHMS FOR SERVODRIVES - A

Multicriterial Design and a Timing Strategy for Constant Sampling

Bytes

saved for each T and the regime duration Δt

pos

A too much little T can lead to an outrunning of the

available memory space V

data mem.

. Then:

max.

memdata

Bytespos

⋅Δ

≥

(5)

The basic software structure for the on-line control

is presented in figure 3. The main program performs

an initial preparation of the interrupt system and T2

timer. Then, the interruptible loop makes some

auxiliary processing concerning data displaying and

a test for the flag F which signals a null position

error. Main tasks are those monitored by the

hardware interrupt INT0 and the software interrupt

ALG / T2. The first is a very fast one and is

absolutely the same for all type algorithms, which

control the position and speed. A more complex

interrupt routine is ALG / T2. The specific tasks

concern some additional computing procedures for

the control c

and a characteristic up-to-date for

addresses content allocated to regressive variables of

standard algorithms. The processing of the obtained

control means the extraction of one effective control

k out

in 8 bit format which acts on a power supply.

2.1 Standard Digital (Micro)

Controller

The author combined the computations for both

loops into a single relation for the control:

Figure 3: Flow chart of the on-line control program.

= c

k-1

+ k

·(

− ε

k-1

)+ k

·T·

/ T

(6)

pα

, k

pω

and T

are the tuning parameters of position

and speed loops. Figure 4 has a more detailed

systemic structure of the figure 2a. The next form

was obtained as an on-line optimal one:

= c

k-1

+ A·

k-1

+ B·

Nk + C·

(7)

A = k

· c

; B = - k

· c

·( k

T / T

)

(8)

C = - k

· k

· T / T

A, B and C are the new tuning parameters and all

other variables are delivered by INT0. The real-time

implementation was prepared by various simulations

on a model having the structure and the

characteristics very close to the real operation and

capabilities of the system and of the microcontroller.

The algorithm was applied for simulation and on-

line control for a drive system with: a 12 V DC

brushed low-inertia motor; an 8 bits microcontroller;

an encoder with 2500 lines / rev. A multicriterial

optimal conditioning (Mihai, 1999, 2004) of the

involved data meant T = 2.456 ms and c

= 1.02 ≅1

(an ideal value). The figure 5 presents the simulation

results by the main macroscopic variables. The on-

line results are those from figure 6, the behaviour

being like the expected one by simulation. Some

differences are in connection with a particular

strategy for the pre-final time segment (Mihai,

2004). Figure 7 is a witness of what is happening in

real-time, the analyser recordings giving details for

the timing (including the T value, interrupt events)

and for precise evaluations for each activated task.

Notations: SPER-sampling period; PULSE- encoder

pulses; INT0-external interrupt service routine;

PROC-all speed loop tasks; ARITH- arithmetic

routines; SCON-control tasks. Figure 7a reveals the

timing for a low speed and the total processing time

for having a control: 452 μs. The next diagram is for

the rated speed and the real T - 2.47 ms. 7c makes a

precise evaluation of the final position error: 2.5

pulses, that meaning 1 / 1000 rev. during 11.37 ms.

Figure 4: For the standard control algorithm.

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

Fig. 8 is for T= 2.456 ms (the ideal value) and T =

1ms. It can be seen the worsening of the

performance; the control is saturated and a big

position overshoot is present. So, a lower T value

does not bring a higher performance.

Figure 5: Simulation results / standard algorithm.

Figure 6: Real-time results / standard algorithm.

Figure 7: The on-line tasks / standard algorithm.

2.2 Standard Algorithm by DSP

Controller

The first idea for improving the bad results from fig.

8b is to use a much more performing hardware. The

next experiment was made with a brushless DC

motor and a DSP controller (Technosoft, 1997). The

figure 9a presents the system. The results (in real-

time) from figure 9b are for no-load conditions. The

figure 9c proves a good general tuning when the

motor has a load. This first two result sets were for

T=1 ms (speed loop) and T=0.1 ms (current loop).

With a faster control sampling (twice), the results

are loosing the quality-fig. 9d. A good choice for T

is better than an expensive hardware solution.

2.3 A Fuzzy Digital Controller

The input variables for the FLC are:

= (N

*- N

αk

) ·

nk max

/ N

* ≥ 0 (9)

n k

nk-1

Nk·Δ

nk max

·c

max

≤ 0 (10)

The most reduced on-line computational effort is

obtained using a look-up table (LUT) filled off-line.

Fig. 10 gives the image of the equivalent systemic

structure of FLC. The control values are stored by

the columns concatenation of a matrix with the final

control values, so the additional software block CPB

is no more necessary. The fuzzy LUT strategy was

Figure 8: T = 2.456 ms (a) and T = 1 ms (b).

ON THE SAMPLING PERIOD IN STANDARD AND FUZZY CONTROL ALGORITHMS FOR SERVODRIVES - A

Multicriterial Design and a Timing Strategy for Constant Sampling

applied for the same servodrive as for 2.1. The main

characteristic elements are presented by fig. 11. The

notations: az-almost zero; vs-very small; s-small; rs-

relative small; m-medium; b-big; vb-very big. The

off-line tuning for the fuzzy rule base was made by

Figure 9: The real-time results for a standard algoritm and

a DSP controller.

Figure 10: A LUT based FLC.

simulation on model, with the results from the figure

12. The on-line results are depicted in figure 13 and

the on-line timing is given by the figure 14. It can be

notice a good concordance between the simulated

and the real-time results. The meaning of the

additional notations used by the figure 14: NORM-

normalization task; LUT-searching in look-up table.

The apparent discrepancy between the position

smooth evolution and the speed diagram is explained

by the fact that the software image ΔN

is a

truncated value for speed, while the real speed and

the position support a filtering effect of mechanical

inertia. The ringing of the control is, basically, a

result of fuzzy rules commutation. The control

variable evolution (motor voltage) reveals a very

sensitive behaviour of the controller, better than for

the standard controller–the figures 5 and 6. The

quality of the whole evolution is also presented in

the (ε

, ω) state-space coordinates, with a good

(smooth) behaviour and an entry with very low

speed in the proximity of the final point. The same

sampling control period as for the standard control

algorithm was used: T = 2.456 ms, according to the

same multicriterial optimum. It is confirmed the

right operation of the 2 interrupts and the existence

of quite large time reserve during each T for

additional tasks. The diagram from the figure 14a

concerns a quasi steady state regime, revealing the

typical task distribution. Several values for the tasks

time are precisely measured. Δt SPER = 2.456 ms

<< T

= 50 ms (the electromechanical time constant

of the motor); Δt PULSE = 125.5 μs - for a speed of

192 RPM. The last diagram caught the final

algorithm time segment. The final detectable speed

is 3.39 RPM. After the T2 timer (that marks the

sampling rate and the cyclic processing for the

control determination) is disabled, it can be seen a

single pulse coming from the encoder. It is the ideal

position error and the real too. Other durations are:

Δt INT0 = 10 μs; Δt PROC = 558 μs < 25 % x T; Δt

NORM = 310.5 μs; Δt LUT = 20.5 μs; Δt SAVES =

26 μs (globally). The most of time is necessary for

the normalization arithmetic operations.

3 CONCLUSIONS

The sampling control period is more than additional

tuning parameters for the digital control. The author

conceived a multicriterial conditioning relation set

with the limitation both for the upper and the lower

values for the sampling period. The general timing

proposed for the servodrive control is based on

imbricate interrupts. A strong real-time constraint is

accomplished by a hardware interrupt that makes an

acquisition rate different than the control rate. The

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

Figure 11: The basic elements for a LUT - FLC.

Figure 12: Results for a FLC in simulation.

Figure 13: On-line results for a LUT based FLC.

Figure 14: The on-line recordings for a LUT FLC.

on-line results, both for the standard algorithm and

the fuzzy control, are very good in terms of the

macroscopic variables and for the timing revealed by

on-line recordings with a logic analyzer. Some

experiments proved that a good choice for the

sampling control period is not necessarily related

with a high end control processor but is the result for

all correlations previously mentioned.

REFERENCES

Astrom, K.J., Wittenmark, B., 1997. Computer-controlled

systems: theory and design, Prentice Hall, USA.

Coleman C. P., Godbole D., 1994. A Comparison of

Robustness: Fuzzy Logic, PID and Sliding Mode

Control, Proc. of the American Control Conference,

pp. 1654-1659.

Do Wan, K., Jin Bae, P., Young Hoon, J., 2007. Effective

digital implementation of fuzzy control systems based

on approximate discrete-time models, Automatica

(IFAC Jour.), Volume 43 , Issue 10, pp. 1671-1683.

Ibbini, M.S., Jafar, A.S., 2002. Self-Tuning Fuzzy Logic

Controller for a Series DC Motor, Proc. (369) Power

and Energy Systems, Acta Press.

Kozek, M., Lorenz, A., Kampas, Ph., 2007. Modeling and

control of an electric servo drive with strong

restrictions in the control variable, Int. Journal of

Applied Electromagnetics and Mechanics, Vol. 25,

Number 1-4, pp. 521 – 527.

Mihai, D., Constantinescu C., 1999. Fuzzy Versus

Standard Digital Control for a Precise Positioning

System with Low-Cost Microcontroller, PCIM99,

Nurnberg, Proceedings, pp. 249-255.

Mihai, D., 2004. Systèmes d’entraînements électriques I.

Problèmes fondamentaux. Systèmes avec moteurs à

courant continu, Ed. Universitaria, Craiova.

Mihai, D., 2006a. Additional Mathematical Pre-processing

for the Fuzzy Control of a Servodrive, WSEAS Trans.

on Circuits and Systems, Is. 11, Vol. 5, pp. 1575-1580.

Mihai, D., 2006b. An Optimized Fuzzy Control Algorithm

for Servodrives. Some Real-Time Experiments.

Proceedings, IS ’06, London, paper 1-4244-0195-

X/06-CD ROM, pp. 192-197.

Mrozek, B, Mrozek, Z., 2000. Modelling and Fuzzy

Control of DC Drive, ESM 2000, Ghent, pp186-190.

Silveira, P. E., Souza, J.R., Biazotto R. de, V. M., 2002.

Speed Control of an Autonomous Mobile Robot:

Comparison between a PID Control and a Control

Using Fuzzy Logic, J. Braz. Soc. Mech. Sci., vol.24,

no.2, pp.127-129.

Vas, P., 1999. Artificial-Intelligence-Based Electrical

Machines and Drives, Oxford University Press.

*** Technosoft S. A., 1997, MCK 240 DSP Motion

Control Kit. User Manual, Switzerland.

ON THE SAMPLING PERIOD IN STANDARD AND FUZZY CONTROL ALGORITHMS FOR SERVODRIVES - A

Multicriterial Design and a Timing Strategy for Constant Sampling