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Dead Time Compensation for Interpolative-type Control Systems

Toma L. Dragomir*, Alexandru Codrean*, Vlad Ceregan*, Sanda Dale**

* “Politehnica” University of Timi oara, 2 Vasile Pârvan Bld., 300223 Timi oara, România (e-mail: [email protected], [email protected], [email protected])

**University of Oradea, 1 Universit ii Street, 410087 Oradea, România (e-mail: [email protected])

Abstract- Signal transmission in control structures is usually

characterized by propagation processes, i.e. existence of time delays. For the interpolative-type control structures the problem is often disregarded, especially in the design step. Undoubtedly, the effect consists in worse real performances that those designed. In this context, the present paper proposes and presents a solution able to compensate, via a discrete-time PD compensator, the effect of the dead times appeared on the feedback channels for both a fuzzy and an interpolative control structure. For systems with variable time delays an adaptive-interpolative type PD compensator is suited. Experimental results, obtained through simulation on a second order positioning system, validate the proposed solution. Finally, the paper discusses the issue of the accuracy of simulation. It is emphasized the manner in which the simulation can be done using initial segment generators for the elementary dead time transfer elements.

I. INTRODUCTION Frequently, the control of second order non-linear plants,

can be successfully implemented using conventional fuzzy controllers based on error (e) and its derivative ( e ). Due to simplicity in implementation and design, the two-input fuzzy controllers are recommended even for complex, higher order plants, as a sub optimal solution. For conventional fuzzy controllers using the error and the change-of-error as input variables, the established rule table may be represented in a 2-dimensional space of the phase plane.

As a rule, in such table the values of output variables are skew-symmetric and the absolute magnitude of the control input is proportional to the distance of the characteristic point from its main diagonal line, which acts like a switching line in the normalized input space, Choi et al. [2], Palm [8], Dale et. al. [4]. Based on this observation, the use of single-input fuzzy controller (denoted as RG_1F) with an input variable called signed distance (sd), suggested in Palm [8], was intensive developed in Dale et. al. [4] and Dale [3]. The variable sd is the distance of the actual state from the input space to the main diagonal line (or hyper-plane) of the rule table and it is positive or negative according to the position of the actual state related to this switching line (hyper-plane). To obtain a better and more elastic control structure the RG_1F can be replaced with a single input interpolative controller (denoted as RG-1I) Drechsel [6] and Drechsel [7].

It is well known that during signal circulation in control

processes different constant or variable dead times appear. Indubitable, their presence, in situations in which they weren’t taken into account in the design, affects the system’s performances. However, in design procedures for the fuzzy and, more generally, for interpolative type controllers, dead times are not considered. The present paper has the aim to underline this issue considering situations in which the dead times can appear on the feedback channels of the above referred systems.

In this purpose two control structures are presented (structures designed in Dale et. al. [4] and Dale [4]), together with their performances (section II). If dead times appear on the feedback channels of the two structures, their performances diminish (section III). In section IV is presented the manner chosen to correct this inconvenience and also some simulation refinements.

II. TWO POSITIONING CONTROL SYSTEMS

The objective of the study is to control a second order linear positioning system with the transfer function:

)1s(s1)s(H (1)

The fuzzy control scheme in Fig. 1 with a synthetic input sd for the fuzzy bloc FB is the first solution proposed and used in Dale et. al. [4] and Dale [3].

The single-input fuzzy controller RG_1F based on the simplified rule base given in Table I calculates in terms of signed-distance, associated with the signed-distance calculus block

22

1

exye

F)u(Fsd (2)

the control signal c. Here, corresponds to the slope of the switching line.

TABLE I – RULE BASE FOR THE RG_1F

sd NB NM ZE PM PB c NB NM ZE PM PB

ISCIII 2009 • 4th International Symposium on Computational Intelligence and Intelligent Informatics • 21–25 October 2009 Egypt

119

Fig.1. Fuzzy control scheme with RG_1F for the positioning system

The domains considered for implementation of the fuzzy

block FB are Dsd = [-2.85, 2.85] and Dc = [-1.4, 1.4], with Dsd determined by geometric means in Dale [3] and Dc validated by considering the necessity of obtaining zero steady state error. The associated fuzzy variables for sd and c, described by the 5 linguistic values from Table I have the membership functions given in Fig. 2a and in Fig. 2b.

- a -

- b -

Fig.2. Form and distribution of membership functions for RG_1F controller: sd (a) and c (b)

In order to illustrate how the system is stabilized in a

certain state, different from the origin, the time response of the system with RG_1F for the reference signal

r(t) = (t) - (t-10) (3)

is considered. The response of the system to this pulse signal applied in zero initial conditions is depicted in Figure 3 by the curve RG-1F.

The second solution discussed and used in Dale et. al. [4] and Dale [3] differs from the previous one by replacing the fuzzy controller RG_1F with an interpolative controller RG_1I like in Fig.4. It uses the same sd-generator f(u) as in the previous case and an interpolative block IB with the rule base from Table I, the linguistic variables described in Table II and the support points from Table III. IB implements this table through a 5 points “look-up table”, through the method

described in Dale et. al. [4] and Dale [3]. The five support points represent the middles of the intervals describing the linguistic terms.

Fig.3. Comparative responses for the systems with RG_1F and RG_1I at the

reference signal (3)

Fig.4. Interpolative-based control scheme with RG_1I for the positioning

system

TABLE II. LINGUISTIC TERMS OF RG_1I l.t. NB NM ZE PM PB sd [-2.5,-2] [-1.4,0] [-0.2,0.2] [0,1.4] [2 2.5] c [-1.4,-0.6] [-0.7,0] [-0.1,0.1] [0,0.7] [0.6,1.4]

TABLE III. SUPPORT POINTS FOR THE IMPROVED RG_1I

l.t. NB NM Nμ ZE Pμ PM PB sd -2.25 -0.9 -0.48 0 0.48 0.9 2.25 c -1 -0.62 -0.03 0 0.03 0.62 1

The time behavior of this second control system for the same scenario as for the first structure is given in Fig.3 also (curve RG-1I).

The main advantages of the interpolative structure with RG_1I are: the reduced complexity of the controller, reduced calculus time and slightly better performances comparatively to those obtained with the fuzzy controller RG_1F.

III. THE INFLUENCE OF DEAD TIME PRESENCE ON THE

DYNAMICAL BEHAVIOUR Both structures from the previous section have as common

feature a signal transmission unaffected by propagation processes. While the control structures are based on interpolative reasoning, i.e. on a field in which the studies have been orientated on controlled plants without dead time, it doesn’t surprise, Zadeh [10]. Researches in this area are relatively rare, Sajidman [9]. Consequently, it is for major interest to investigate the conservation of performances of the

T. L. Dragomir, A. Codrean, V. Ceregan, S. Dale • Dead Time Compensation for Interpolative-type Control Systems

120

two control structures in the hypothess that dead times appear on the feedback loops.

For simplicity, the transmission of feedback signals from Fig. 1 and 4 is considered to take place in channels affected by propagation processes, i.e. by dead times. Thus the second control scheme takes the aspect from Fig. 5. The first control scheme will take, as well, a similar form.

Fig.5. Interpolative-based control scheme with RG_1I for the positioning

system with dead time

In Figures 6 and 7 the responses of both systems are represented for two different dead times versus the reference signal (3).

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

r, y

r

y

tau1 = tau 2 =0.1 second

- a -

0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

1.5

t

r,y

r

y

tau 1 = tau 2= 1 second

- b -

Fig.6. The behaviour of the Fuzzy control system with dead time for: (a) sec1.021 , (b) sec121

Comparing the responses of the structures without dead time (Fig. 3), respectively with dead time (Fig. 6 - 7), it is obvious that the appearance of delays determine a significant depreciation of the control system’s performances. This is

visible by a stationary error of approximately 5% in the case with the dead times sec1.021 , respectively of 50% in the case when sec121 .

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

r,y

r

y

tau1 = tau2= 0.1 second

- a -

0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

1.5

2

t

r, y

r

y

tau1 = tau2 =1 second

- b -

Fig.7. The behaviour of the interpolative type control system with dead time: (a) sec1.021 , (b) sec121

More, it can be observed that the interpolative system

enters in an oscillation trying to return to the prescribed value, while the fuzzy system, being more rigid, conserves its stationary value of the output, which differs greatly from the prescribed one (Fig. 7 and 6).

IV. DEAD-TIME COMPENSATION

A. Using of PD-compensators for systems with constant delay

According to the examples in section III, dead time presence makes necessary the redesign of the control loops. In the absence of systematic methods for this operation, the best alternative seems to be the usage of correction blocks.

In order to compensate the signal’s delay created by the dead time on the transmission channel, a derivative type component added to the current signal can be used. This means a cascade connection on the channels with dead time, positioned on the up-stream of the calculus block of the c signal, with PD type blocks.

A first possibility represents the catenation of a PD block for every dead time element. A second one consists in the catenation of a single PD block on the channel of the signal


121

sd. The study of these variants concluded that the better one is the second method, the method being also more advantageous from the computational point of view.

The schemes derived from this method are presented in Figures 8a and 8b. It can be observed that the implementation of the control blocks is made in the discrete-time variant. This compensation blocks are integrated in the controllers RG-1F and RG-1I and have the same transfer functions:

z

1zKK)z(H dp (4)

i.e. ])1t[s]t[s(K]t[sK]t[s ddddpd . (5)

SE and ZOH means sampling element and zero order holder. The determination of the gain values of Kp and Kd of the

proportional and derivative channels can be done simply by “trial and error” method, according to the manner in which the response y(t) deviates from de reference signal (3), respectively from the responses from Figure 3 of the systems without dead time. The values

1300K,985.0K dp (6)

were found for the scheme from Fig. 8a and the values

295K,77.0K dp (7)

for the scheme from Fig. 8b.

In Fig. 9a the first system’s response is shown, with the PD block parameters set according to (6). In Fig. 9b an intermediate situation appears, between the one from Fig. 6b and the one from Fig. 9a. It can be easily observed the effect of the Kd parameter. By comparing this result with the one form Fig. 3 it can be stated that the compensation is complete.

In Fig. 10 the same idea is presented for the control system from Fig. 8b with the parameters tuned according to (7). Also in this case, the system benefits of full compensation.

Considering the responses in Fig. 3 (for initial structures without dead time), Fig. 6 and 7 (for structures with dead times) and in Fig. 9 and 10 (for structures with dead times and PD-blocks), respectively it’s obvious that the proposed solution can successfully compensate the influence of delays appeared on the feedback channels for both fuzzy and interpolative structure. In fact, the responses of the compensated systems perfectly match with ones of the initial, unaffected systems.

- a -

- b -

Fig.8. Control systems structures with PD compensator for the effect of dead times

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

r, y

r

y

tau1 = tau2 = 1 second

Kp = 0.985, Kd = 1300

- a -

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

r, y

r

y

tau1 = tau2 = 1 second

Kp = 0.985, Kd = 1000

- b -

Fig.9. The behaviour of the Fuzzy control system with dead times and PD compensator

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

r, y

r y

tau 1 = tau 2 = 1 sec

Kp = 0.835, Kv = 284

- a -


122

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r y

tau1 = tau 2 = 1second

Kp = 0.77, Kd = 205

- b -

Fig.10. The behaviour of the interpolative type control system with dead times sec121 and PD compensator

B. Using of PD-compensators for systems with step-wise variable delay

In wireless remote control systems the transmission of command signals as well as the reception of feedback signals is achieved with variable transfer rates, therefore with variable emission times, processes which can be modeled through variable dead times. Qualitatively, the control problem for such a system dealt in the same manner as in Åstrom [1].

In this context the systems from Fig. 1 and 4 have been investigated when the dead time is variable, remaining constant on time subintervals, for example according to the scenario from Fig. 11a. For the system from Fig. 4 the result is shown in Fig. 11b.

For correction, this time an adaptive compesation will be used, based on interpolative adaptation of the parameters Kp and Kd from the correction block. The control structure associated to the one from figure 4, through the extension of the one from Fig. 8, has the appearance from Fig. 12. The dead time is measured through the means specific for internet transmissions, by a software module DTC (dead time converter). The measured value m is used to generate through interpolation the gains Kp and Kd. The interpolators I-Kp and I-Kd use the data from Table IV.

TABLE IV. THE LINEAR INTERPOLATION TABLE FOR THE BLOCKS I-KP

AND I-KD.

m 0.05 0.3 0.55 0.8 1.05 Kp 1.474 1.19 1.015 0.931 0.878 Kd 860 820 780 740 700

m 1.3 1.55 1.8 2.05 2.3 Kp 0.837 0.805 0.777 0.753 0.734 Kd 660 620 580 540 500

In Fig. 13 the results for the same scenario as the one in

Fig. 11 are presented. It can be observed that the deviations from Fig. 11b are compensated significantly. In order to obtain a better compensation, the structure from Fig. 12 can be filled with discriminating blocks Drechsel [6].

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

tau1

= ta

u2

- a -

0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

1.5

2

t

r, y

r

y

- b -

Fig.11. The performances of the system from Fig. 4 alter considerably when

the dead time varies: a) the variation scenario of the dead time, b) the reference signals and the output signals, of the system

Fig.12. The interpolative control structure with adaptive PD compensation of

interpolative type


123

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

t

r, y

r y

* Variables tau 1 and tau 2 changed as in Figure 11a

** Kp and Kd obtained by interpolation based on Table 1

Fig.13. The results obtained with the structure from figure 12

C. Using of PD-compensators for systems with constant delay

In order to assure accuracy in simulations made during the study form section IV.A and IV.B, initial segment generators (isg type blocks) were used. The problem of using these modules in order to correctly simulate processes with dead time is presented in detail in Dragomir et. al. [5]. The idea is presented in Fig. 14 for the interpolative structure case. It was also used for the fuzzy control structure. The initial segment generators, isg- 1 and isg- 2, are connected at the output of the dead time blocks denoted by their transfer functions s1e and s2e . The purpose of the isg is to store the signal from the input of the dead time elements on the time intervals:

),[ 010 tt , respectively ),[ 020 tt .

Fig.14. Interpolative type-control system with compensator from Fig. 4

completed with isg for simulation

As an example, Figure 15 shows the segment corresponding to the system’s response with the fuzzy controller from figure 8a on the time interval [3, 10] seconds. It can be observed that the global response from figure 9a is very well reproduced.

3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

r, y

y

r

r

tau1 = tau2 = 1second

Kp=0.985, Kd = 1300

x1(0) =0.1742 , x2(0) = 0.8236

Fig.15 Detailed response obtained using isg

V. CONCLUSION

The control systems of interpolative type, with fuzzy blocks or with proper interpolation blocks prove to be very sensitive to the presence of dead time. In order to reduce this disadvantage the paper proposes a method of compensation based on the employment of PD compensators. The problem is approached through a case study referring to a second order positioning system. The obtained results indicate the possibility of a very good compensation. The paper approaches some important aspects for practical applications:

i) The control structures of interpolative type use a synthetic control signal, which simplifies the base of rules and the real time implementation; ii) The PD type compensator is used on the channel of the synthetic control signal; iii) In the case in which the dead time varies, the interpolative structures can be extended in an adaptive variant also of interpolative type. iv) For the refining of the simulation initial segment generators have been used associated with the dead time elements.

REFERENCES

[1] Åstrom K.J., Murray R.M., “Feedback systems”, Princeton University Press, 2008.

[2] Choi B.-J., Kwak S.-W., Kim B.K., “Design and stability analysis of single-input fuzzy controller”, IEEE Transactions on Systems, Man and Cybernetics, part B: Cybernetics, vol.30/2, pp. 303-309, 2000.

[3] Dale S., “Contributions to the study of control systems with interpo-lative type controllers”, Editura Politehnica, Timi oara, 2006.

[4] Dale S. and Dragomir T.L., “Interpolative-type control solutions”, in V. E. Balas, J. C. Fodor, and AM. R. Varkonyi-Koczy, Soft Computing Based Modeling in Intelligent Systems, Serie Studies in Computational Intelligence, Vol. 196, Springer Berlin, Heildelberg, pp. 169-203, 2009.

[5] Dragomir T.L., Codrean A., Ceregan V., “On an elementary problem of modelling dead time systems”, Journal of Electrical Engineering, Vol. 9, No. 2, 2009.

[6] Drechsel D., “Regelbasierte Interpolation und fuzzy control,” Vieweg, Wiesbaden, 1996.

[7] Drechsel D., “RIP control”, CEAI, 1, 7-17, 1999. [8] Palm R., “On the compatibility of fuzzy control and conventional control

techniques, in M.J. Patira and D.M. Mlynek, Fuzzy logic – Implementation and Applications”, 63-105, John Wiley & Sons, New York, 1996.

[9] Sajidman M. and Kuntze H.-B., “Fuzzy-Regelung stark gestörter, verfahrenstechnischer Prozesse mit großer Meßtotzeit. Fuzzy Control“, 118-123, 5.Workshop des GMA., Dormunth, 1995.

[10] Zadeh L.A., “Interpolative reasoning as a common basis for inference in fuzzy logic, neural network theory and the calculus of fuzzy If/Then rules”, in Opening talk of 2nd International conference on fuzzy logic and neural networks, pp. XIII-XIV, Iizuka, 1992.


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