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A rule base modification scheme in fuzzy controllers for time-delay systems
Expert Systems with Applications 36 (2009) 8476–8486
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A rule base modification scheme in fuzzy controllers for time-delay systems Hakkı Murat Genc a,b, Engin Yesil b, Ibrahim Eksin b,*, Mujde Guzelkaya b, Ozgur Aydın Tekin b a b
Marmara Research Center, Information Technologies Institute, 41470 Gebze, Kocaeli, Turkey Istanbul Technical University, Faculty of Electrical and Electronics Engineering, Control Engineering Department, Maslak TR-34469, Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Fuzzy controllers Rule base shifting Time-delay systems Microcontrollers Heat transfer system
a b s t r a c t In time-delay control systems, the observed information is naturally related to a past instant and this delayed information signal will usually cause unsatisfactory results. This study deals with how the time-delay information can be used in reorganizing the rule base of the fuzzy controller so as to improve system performance. It basically proposes a new scheme of appropriate shifting of the rule base to compensate the information lag caused by time delay in the system. The parameters affecting the shifting scheme are elaborated in detail and the new shifting scheme is proposed in a tabulated form that assumes the system time constant and the value of time delay as the main parameters. The effectiveness of the proposed methodology has firstly been tried to be illustrated on different simulation examples and, secondly, a real time application has been done on a heat transfer experimental setup. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Following the first fuzzy control application carried by Mamdani (1974), fuzzy logic is utilized quite often in control problems. During the past few decades, there are many successful applications with Fuzzy Logic Controllers (FLCs) in industry. It has been reported that they are successfully used in a number of complex and non-linear processes (Caner, Umurkan, Tokat, & Ustun, 2008; Elmas, Ustun, & Sayan, 2008; Sugeno, 1985; Yesil, Güzelkaya, & Eksin, 2004). Moreover, the experience has shown that fuzzy controls are often a favored method of designing controllers for dynamic systems even if traditional methods can be used (Mamdani, 1993). PI type FLCs are the most common and practical ones within the various types of FLCs just like the widely used conventional PI controllers in process control systems (Mudi & Pal, 1999). The research for improving FLC performance spreads over number of areas. The performance of FLC can be improved by adaptations over the design parameters categorized mainly in two groups (Hu, Mann, & Gasine, 1999): (a) structural parameters, and (b) tuning parameters. Basically, structural parameters include input/output (I/O) variables of fuzzy inference, fuzzy linguistic sets, membership functions, fuzzy rules, inference mechanism and defuzzification mechanism. Tuning parameters include I/O scaling factors (SFs) and parameters of membership functions (MFs). The structural * Corresponding author. Tel.: +90 212 2853500; fax: +90 212 2856700. E-mail address: [email protected] (I. Eksin). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.10.072
parameters are usually determined in an off-line manner while the tuning parameters are adjusted with an on-line mechanism. The scaling factors are simply the most effective parameters on system performance and, therefore, various on-line tuning algorithms are presented in literature (Guzelkaya, Eksin, & Yesil, 2003; Mudi & Pal, 1999; Qiao & Mizumoto, 1996). Most of the practical processes are high order non-linear systems and some of them may have considerable dead time (Mudi & Pal, 1999). The dead time is recognized as the most difficult dynamic element naturally occurring in physical systems (Shinskey, 1998). It is another known fact that high order systems can be modeled by a first order system with dead time counterpart (Skogestad, 2002). Therefore, any useful technique of designing a control system must be capable of dealing with dead time. Conventional PI (or PID) controllers are usually inefficient in managing systems with dominant dead time. Moreover, the controller output or process input should be a non-linear function of error and change of error to have a satisfactory performance. FLCs try to incorporate this nonlinearity by limited number ‘‘IF–THEN” rules (Mudi & Pal, 1999). As a consequence, FLCs are the commonly used controllers as the time delay or nonlinearity is the matter of concern. In a time-delay process, the present control action depends on the observed data that is related to a past instant. Therefore, when the observed data are directly used then the transportation lag would naturally produce a misleading or incorrect information for the rule base and, hence, an incorrect control action for the process (Li & Tso, 1999). A shifting scheme of the fuzzy rule base has been proposed by Li and Gatland (1995) to overcome this information lag. Later, Chang and Wang (1996) have used the same shifting strategy in their Cellular CDMA System. Li and Tso (1999) have
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argued that this method requires large rule bases and, therefore, it may not be accurate enough for low order rule base and it is hard to provide proper compensation for the controller. Finally, Bai, Zhuang, and Roth (2005) have proposed another method for their Laser Tracking System and found fairly good results for considerably small time delays. This paper is basically devoted to enlighten the use of timedelay information of the system in reorganizing the rule base of FLCs. There might be many aspects to be taken into account in reorganizing operation and obtaining a ‘‘better” rule base for a FLC. The reorganizing operation considered here depends upon the shifting of the rows of rule base of FLC. The main idea in the new approach developed in this study is that the shifting amount should be distinct for different time delays; it should even vary in between rows, where rows designate De. More precisely, the shifting amounts are tried to be tabulated using the parameter set of first order plus dead time (FOPDT) system models; namely, the time delay and time constant values. The effectiveness of the proposed methodology is shown both on different simulation examples and on a heat transfer experimental setup. The rest of the paper is divided into five sections. In Section 2, the most common architecture of a FLC is given. The proposed rule base shifting strategy is detailed in Section 3. In Section 4, simulations are presented to illustrate the effectiveness of the method. Moreover, in Section 5, experimental results on heat transfer system PT 326 (Feedback) are given. Possible extensions and conclusions are finally given in Section 6. 2. The architecture of fuzzy logic controller Although it is obvious that the proposed rule base modification scheme is valid for all kinds of FLCs, it has been preferred to make this implementation on PI type of FLC since it is common, adequate and efficient for various practical real time processes. The error (e) and the change of error (De) will be used as the inputs and the change of control signal (Du) as the output of the PI type FLC as illustrated in Fig. 1. The universe of discourse is chosen to be [1, 1] for the membership functions of input and output variables. The input and output parameters are scaled to fit this range via scaling factors. The input scaling factors (ISFs) normalize the real world inputs to a range in which membership functions are defined. The output scaling factor (OSF) is used to change the normalized control effort to its practical value. The relation between real and normalized values of the parameters can be simply given as
E ¼ eke;
DE ¼ Dekde;
Du ¼ DUko;
ð1Þ
where E and DE are the normalized inputs of the FLC controller, DU is the FLC output; e, De and Du are respective actual outputs, and ke, kde, ko are the error scaling factor, the change of error scaling factor and the control effort change scaling factor, respectively. The input variables are decomposed into at least seven fuzzy linguistic levels in order to make a considerable distinction between the fuzzy regions and, thus, to obtain fine tuned control action. These levels are chosen and named as NL (Negative Large),
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NM (Negative Medium), NS (Negative Small), Z (Zero), PS (Positive Small), PM (Positive Medium), PL (Positive Large) in this study. In Fig. 2 the MFs of input and output variables are illustrated. The output variables are chosen to be singletons for ease of calculations in the implementation phase. The symmetrical rule base given in Table 1 can be seen as one of the general and most used form. 3. The proposed rule base shifting scheme When a delay is introduced in a system either by the nature of the system or by the external devices used, the observed information is related to a past instant. Using this measured information for control purposes may cause unsatisfactory results especially in time dominant delay systems. However, if one can somehow guess the actual output information related to the measured instant from the process trend, then one can reorganize the rule base accordingly so that a ‘‘better” control performance might be achieved. A regular rule base might be divided into four regions not including e and De which are equal to ‘zero’ rows as illustrated in Fig. 3. The error is positive and decreases in magnitude in region R1 since the change of error is negative. Therefore, the actual (e, De) pair should lie somewhere in the left hand side of the measured one for a time-delay system. The rules corresponding to region R1 should be shifted to right to compensate this deviation. If the shifting amount is appropriate, then the controller would most probably perform ‘‘better” actions. The error and the change of error are both negative in R2 region and the error magnitude has the tendency to increase. The actual point is again in the left of the observed one and shifting the rules to right would provide again a ‘‘better” performance if this shifting operation is done in appropriate amounts for each row of the rule base. Similar arguments are valid for the upper side of rule base. In regions R3 and R4, the actual points are at the right of observed ones and rule base should be shifted towards left. When De is ‘zero’, then that row is unbiased and no shifting is needed. The time-delay effect is illustrated in Fig. 4a as a shift in the plane in the system step response of an arbitrarily chosen system. The arrow labeled 1 shows the difference of error magnitudes at the point t = 1 s. The rules at region R1 get activated for this point. The error for zero delay system is measured of about 0.6 at this point; whereas, the actual error of the delayed system is around 0.2 which implies a huge deviation from the actual value. Therefore, we can assume that the control effort for the delayed systems should have been decreased for that region. Then, the critical question is about the amount of that decrease. Li and Gatland (1995) shifted all the rules except the row in which De equals to ‘zero’ for one cell to obtain that decrease. This shifting scheme yields favorable performances for some systems but since it demands the same amount of shifting even for small error changes (around the set point) it may even cause oscillatory behavior and chattering. On the contrary, if the system delay value is too high, this one unit shifting amount might not be adequate enough for favorable system performance. This means that the shifting amount, as a matter of fact, should also depend upon both the time-delay
Fig. 1. The basic control block diagram illustration with PI type FLC.
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Fig. 2. Membership functions for: (a) e and De; (b) Du.
Table 1 A typical and symmetrical rule base for an FLC.
De/e
NL
NM
NS
Z
PS
PM
PL
PL PM PS Z NS NM NL
Z NS NM NL NL NL NL
PS Z NS NM NL NL NL
PM PS Z NS NM NL NL
PL PM PS Z NS NM NL
PL PL PM PS Z NS NM
PL PL PL PM PS Z NS
PL PL PL PL PM PS Z
Fig. 4. Arbitrary system step response without time delay and (a) a dead time of 0.5 s; (b) a dead time of 1 s.
Fig. 3. Fuzzy rule base regions.
magnitude, and the time constant of the (first order) system under control. In this work, the shifting amounts for the rows of the rule base are deduced and given in table form in a very general scheme that
depends on both the system parameters; namely, the time-delay value and the time constant of the plant. It is easily seen from Fig. 4a that the difference of the two responses in terms of error magnitude is not the same at each point if one examines the points illustrated by the three arrows. The error change rate is quite high and the response curve climbs up very rapidly to the set point at the arrow labeled as 1 for the delay time of 0.5 s. The system response is at the set-point value at arrow labeled as 2, but considering the time delay and interpreting change of error correctly, it is obvious that the actual system response value must be somewhere above that point. As the system response gets closer to the steady state region designated by the arrow labeled as 3, the difference between the two graphs diminishes. Thus, if the change in error is ‘‘high” then the difference between systems responses gets higher. Therefore, the amount of shifting must be kept ‘low’ as the system gets closer to the steady state region (when De is low) and it should be ‘high’ when the response rate of change of the system is high. In fact, no rule base shifting is appropriate for the small values of change of error to prevent oscillations and decrease chattering effect. The amount of time delay is an important factor in system responses as it is presented in Fig. 4. It is seen that the shifting amounts must be adjusted in accordance with the amount of
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H.M. Genc et al. / Expert Systems with Applications 36 (2009) 8476–8486 Table 3 Proposed shifting schemes.
Fig. 5. Ideal shifting scheme for infinite resolution.
time-delay values. The resolution of the membership functions plays another key role in rule base shifting. The effect of the time delay can be handled more precisely if more fuzzy regions are defined for each parameter. Fig. 5 illustrates the variation of the shifting in the infinite resolution case. This is impractical in real world applications since it can only be achieved by the use of unlimited number linguistic variables. It is obvious that adjusting the shifting amounts for each De row with higher resolutions would be more efficient. Naturally, on the other side, this causes computational complexity and this fact should be handled by the designer. In this work, seven fuzzy regions are defined for each variable as discussed in detail in Section 2. In this paper, combining and utilizing all the facts stated above, almost zero shifting for small De and increasing amount of shifting for larger values of De is suggested. The suggested shifting scheme
Normalized dead time (s)
Appropriate shift
s=0 0 < s < 0.05 0.05 6 s < 0.15 0.15 6 s < 0.7 sP7
000 001 011 023 033
seems to be efficient for minimum of seven linguistic terms for each input. As the resolution is increased zero shifting operation for the small De rows become more crucial. The reason for this is that the rules are rarely fired for high De values or at the extreme De rows. Therefore, adjusting the shifting amount at low and medium values of De rows becomes much more important for high resolution cases. Moreover, as the time-delay value gets higher the shifting amount should be increased as well while keeping it around zero for small De rows. As illustrated in Fig. 5, the shifting amounts are equal and opposite in direction in lower and upper sides of De = Z (zero) row for the infinite dimension case. Table 2 illustrates different shifting schemes on the standard rule base given in Table 1, which is a finite dimension application of the shifting scheme idea. The shifting amounts of the rows of the rule bases are coded as in Table 2. For instance, all the rows in Table 2b are not shifted equally; that is, the neighboring ‘Zero’ rows of De are not shifted while the others are shifted for one cell in appropriate directions. This shifting scheme is coded as 011 (shifted number of cells from low values to high values of De) and the corresponding controller is abbreviated as FLC_011. Table 2 includes four different shifting schemes for FLCs that will be used. Then, comparisons will be made of those four throughout the paper.
Table 2 FLC rule base for different shifting schemes: (a) FLC_001; (b) FLC_011; (c) FLC_023; and (d) FLC_033.
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Fig. 6. The effect of various rule base shifting schemes for a FOPDT system (T = 1 s) with a time-delay values of (a) 0.05 s; (b) 0.1 s; (c) 0.2 s; (d) 0.9 s; and (e) 3 s.
In this study, all of the systems are chosen to be first order plus dead time (FOPDT) systems since one can model most of the systems FOPDT counterpart Skogestad (2002). The shifting amounts for the rows of the rule base are tabulated using the system param-
eters; namely, the time-delay value and the time constant of the plant, and also on the magnitude of the error and the change of error. The transfer function of a FOPDT system can be given as follows:
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and rules. Moreover, when different performance indices are used, these transition points or borders might slightly change. Since the extreme rules are rarely fired, a change made in the shifting scheme related to the small and medium values of De will naturally make the most important effect on the system performance. Moreover, no shifting choice is convenient for NS or PS rows of De in order to prevent oscillatory behavior.
Table 4 Performance results for systems with various time-delay values. Delay time, L (s)
Normalized dead time (s)
Controller type
Performance index
0.05
0.046
0.1
0.09
0.2
0.17
FLC FLC_001 FLC FLC_011 FLC FLC_011 FLC_023 FLC FLC_023 FLC FLC_033
111.11 123.00 93.00 109.97 65.18 84.24 90.78 26.83 37.02 11.76 15.72
0.9
0.47
3
0.75
GðsÞ ¼
K esL ; Ts þ 1
4. Analysis of the proposed shifting scheme via simulations 4.1. Analysis with different time-delay systems
ð2Þ
where L is the delay time, K is the gain and T is the time constant for the process model. The normalized dead time or controllability ratio s can be defined in the interval [0, 1] as follows:
s¼
L : LþT
ð3Þ
The parameter s is a ratio that gives the relative magnitude of the time delay with respect to system time constant. It has generally and roughly been found out that processes with small s are easy to control; whereas, difficulty in controlling the system increases as s increases (Åström & Hägglund, 1995). Several shifting schemes are investigated for various time-delay values under the same performance index for the controller developed in Section 2. In the light of these results, the shifting amounts with respect to s are deduced and given in Table 3. The results are consistent with our presumptions. The transition points of Table 3 are determined through a search of scaling factors that are maximizing the following performance index
J¼
1000 ; 100mp þ 3t s þ 6tr þ 100ess
8481
ð4Þ
where mp is the maximum overshoot, ts is the settling time, tr is the rise time and ess is the steady state error. The transition from one shifting scheme to another is actually continuous. These borders are not crisp for all cases, but changes slightly for different configurations of the membership functions
In this section, the performance results are provided so as to designate the effect of different rule base shifting schemes for various time-delay values of a FOPDT system. The gain of this system is set to unity and the time constant to 1 s for simplicity. Therefore, the transfer function of the system becomes as follows:
GðsÞ ¼
1 sL e : sþ1
ð5Þ
The most commonly used controller given in Section 2 is used for all simulations. The input scaling factor ‘‘ke” is chosen to be unity in all cases since reference is unit step and error is already normalized to [1, 1]. The other two scaling factors shown in Fig. 1 are tuned for each case separately. There is unfortunately no commonly accepted method for choosing the optimum scaling factors and they are usually found out by trial error techniques. In this study, the scaling factors are searched via a basic genetic algorithm which maximizes the performance index given in Eq. (4). In the first simulation example, the delay time is chosen to be 0.05 s which is small in comparison to the time constant. Therefore, a small s value arises. The simulation result for this case is given in Fig. 6a. As it can be depicted from figure and Table 4, and 001 shifting scheme slightly improves the performance index value as expected since the extreme rules have little influence on the controller. Fig. 6b illustrates the case when 0.1 s delay is introduced to system. This corresponds to a normalized dead time s of 0.09 and the proposed shifting scheme is 011. The controller with the new rule base has provided a shorter rise time and as a result yielded about 20% performance improvement. If the time-delay value is 0.2 s, then s becomes 0.17 and Table 3 proposes 023 shifting scheme. This situation provides 40% improvement in the performance index. However, since these borders are not crisp and not discontinuous from one another, 011
Fig. 7. Illustration of the effect of various time constants on the system responses.
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Fig. 8. The effect of various rule base shifting schemes for a FOPDT system (L = 1 s) with a time constant values of (a) 20; (b) 10; (c) 5; (d) 1; and (e) 0.3 s.
shifting scheme also provides respectful results. This case is illustrated in Fig. 6c and the performance index values for the three rule bases are given in Table 4.
Fig. 6d shows the case when delay value and time constant are close to each other. Increasing the time delay far beyond the time constant of the plant case, results are illustrated in Fig. 6e.
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Fig. 9. Illustration of the effect of the rule base shifting for the same normalized dead time: (a) L = 1.5 s; T = 15.17 s; (b) L = 0.5 s, T = 5.06 s.
Table 5 Performance results for systems with various time constant values.
Table 7 Performance results for with a large normalized dead time (s = 0.75).
Time constant, T (s)
Normalized dead time (s)
Controller type
Performance index
Delay time, L (s)
Time constant, T (s)
Controller type
Performance index
20
0.476
0.66
0.0909
3
1
5
0.166
5
1.66
1
0.5
10
3.33
FLC FLC_033 FLC FLC_033 FLC FLC_033 FLC FLC_033
21.52 23.40 11.76 15.72 9.02 9.98 4.59 5.10
0.3
0.77
21.06 25.03 22.12 25.40 23.50 28.47 30.79 35.55 40.51 44.03
2
10
FLC FLC_001 FLC FLC_011 FLC FLC_023 FLC FLC_023 FLC FLC_033
Table 6 Performance results for systems with a small normalized dead time (s = 0.09). Delay time, L (s)
Time constant, T (s)
Controller type
Performance index
Improvement (%)
0.1
1 10
0.5
5.06
1.5
5.17
93.00 109.97 22.12 25.40 38.43 48.73 12.66 18.74
18.2
1
FLC FLC_011 FLC FLC_011 FLC FLC_011 FLC FLC_011
14.92 26.80
33.67 10.58 11.01
ness of the proposed method, the simulation results of the system given in Eq. (6) for five different time constants are presented in Fig. 8. Furthermore, the performance index values for the different simulation results are tabulated in Table 5. As the time constant T decreases, the normalized dead time s of the system increases so do shifting amounts (Table 3). Note that in the absence of time delay, there would be no need to shift. This result can also be derived from the normalized dead time s given in Eq. (3). 4.3. Analysis with different (L, T) combinations
In this part, time-delay value has been assumed to remain constant and time constant parameter of the FOPDT system has been changed gradually. Delay value is set to 1 s and the transfer function of the system then becomes as follows:
1 es : Ts þ 1
8.77
48.03
4.2. Analysis over systems with various time constants
GðsÞ ¼
Improvement (%)
ð6Þ
The set of other parameters of the system is configured as explained in Section 4.1. Changing the time constant of the system model practically means a change in the system response speed and Fig. 7a and b provides better perception of this effect. A smaller time constant provides a faster system response as seen in 7a and the difference between the actual and the observed system responses becomes larger. We eventually can say that the needed shifting amount should be ‘‘greater”. In order to show the effective-
The same normalized dead time value can be obtained by infinite number of (L, T) combinations. Many tests have been conducted to illustrate the robustness of proposed shifting scheme to these variations. More specifically, the performance results of the system with a very small normalized dead time (s = 0.09) are given in Table 6 and another system with a relatively large normalized dead time (s = 0.75) is presented in Table 7. Moreover, the simulation results for two different cases are presented in Fig. 9. 4.4. Analysis with different rule base sizes It should be an obvious fact that the rule base shifting scheme would be more effective as the number of variables has been increased which are defined on the fuzzy regions. This is because of the fact that non-linear structure of the shifting scheme can be fit more precisely with more resolution. All the previous illustrations in this study are based on the case where seven fuzzy regions are defined for each parameter and we will now consider the case with nine fuzzy regions. A comparison of the ‘‘shifted” and
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Fig. 10. Comparison of two fuzzy controllers with different number of fuzzy rules.
The control system realization is based on the Motorola MPC555 32-bit single-chip microcontroller whose main features are as follows: 448 kB on-chip flash memory; 26 kB on-chip RAM; two 10-bit analog-to-digital (A/D) converters with 16-input channels; modular input/output system (MIOS) (eight channels for PWM); 2 serial communication interfaces (SCIs); 64-bit floating-point unit with 0.1-ls floating-point multiply; and 0.25-ls floating-point divide (at 40-MHz clock frequency). The MPC555 hardware is compatible with the MATLAB/SIMULINK. Mentioned features of the MPC555 are available as SIMULINK blocks in the ‘Embedded Target for Motorola MPC555 Block Library’. The code generation process for this design is thus handled in SIMULINK using Embedded Target for Motorola MPC555, RealTime Embedded Coder and Real-Time Workshop which externally needs the C compiler software ‘Metrowerks CodeWarrior Development Studio for MPC5xx’. The experimental setup used in this study is given in Fig. 11. A step input is applied and then the well-known area method is used in order to obtain FOPDT model of the PT 326 (Åström & Hägglund, 1995). The following transfer function model is then obtained
PðsÞ ¼ Table 8 Performance results for controller with nine fuzzy regions. Controller type
Performance index
FLC FLC_023 FLC_0233
17.64 23.08 24.18
0:74 e0:3s : 0:58s þ 1
‘‘non-shifted” controllers for a FLC with nine fuzzy regions (81 rules) on a same system with time delay of 1.75 s and time constant 1 is illustrated in Fig. 10. A shifting scheme of ‘‘0233” yields the best results in terms of performance index given in Eq. (4). The improvements are illustrated in Table 8 and it is not so worthy in comparison with results attained in 7 region counterpart. It should also keep in mind that the improvement achieved by increasing the number of fuzzy regions may not compensate the computational complexity. 5. Experiment Fig. 11. Experimental setup.
The heat transfer process trainer (PT 326) that has been widely used by many different researchers to check their new control strategies is also used here to present the advantages of the proposed methodology (Bandyopadhyay, Chakraborty, & Patranabis 2001; de la Pena, Ramirez, Camacho, & Alamo 2005; Dias & Dourado, 1999; Ng & Cook, 1998; Pereira, Henriques, & Dourado 2000; Yesil, Guzelkaya, Eksin, & Tekin, 2007). PT 326 has the basic characteristics of a large plant that involves a tube through which air is drawn from atmosphere by a centrifugal blower and the air is heated as it passes over a heater grid before being released into the atmosphere. Temperature control is achieved varying the electrical power supplied to the heater grid. The mass flow of air through the duct can be adjusted by setting the opening of the throttle. The pure time delay depends on the position of the temperature sensor element that can be inserted into the air stream at any one of the three points spaced at 28, 140 and 280 mm far from the heating point along the tube. In this study, the sensor element is placed at the third place and therefore the allowable longest dead time is achieved. The damper position is set to 40°. The system input u(k) is the voltage applied to the power electronic circuit feeding the heating resistance and the output y(k) is the outlet air temperature, which is expressed as a voltage between 0 and 10 V (Yesil, Guzelkaya, Eksin, & Tekin, 2008).
Fig. 12. Simulation results for the system outputs.
ð7Þ
H.M. Genc et al. / Expert Systems with Applications 36 (2009) 8476–8486
spect to the normalized dead time. The simulation results show that proposed rule base shifting scheme improves system performance in considerable amounts. Also it provides considerably acceptable results even for the large variations in structural parameters. Shifting all rows of the fuzzy rule base by one cell is proposed in previous studies and it rarely yields improved results. Therefore, in addition to one cell shifting of all rows, two sets of scaling factors (for transient and steady state responses) and a change in some entries of the rule base has been proposed. Under these circumstances, some performance ameliorations have been observed. However, the new approach proposed in this study provides better results even in very generic form. Also, in case of different design purposes or performance indices, the proposed shifting scheme can be redefined. In the simulations done on various systems, it has been observed that the proposed methodology ameliorate the overall performance of the system with respect to set-point following. Therefore, the proposed fuzzy controller that has been embedded into a microcontroller is then used to control a heat transfer process (PT 326 – Feedback). Finally, the effectiveness of the proposed methodology with respect to the transient response has been demonstrated and compared with fuzzy controller that has classical diagonal rule base.
Fig. 13. Experiment results for the system outputs.
Table 9 Comparison of performance results for the experimental study. Controller type
Maximum overshoot (%)
Rise time (s)
Settling time (s)
Performance index
FLC FLC_023
3.67 0.67
3.5 2.1
6.3 4.5
22.95 34.76
References
As it is seen from Eq. (7), the apparent time delay of the system is small with respect to its dominant time constant. A time delay of 0.9 s has been added via software at the input of the process so as to obtain a time dominant system and to emphasize the benefit of the rule base shifting idea. Then, the overall transfer function of the model is obtained as
PðsÞ ¼
0:74 e1:2s 0:58s þ 1
8485
ð8Þ
and the normalized dead time s is calculated as 0.674 for this timedelay dominant system. As it can easily be deduced from the proposed scheme given in Table 3, the best rule base for the FLC is the one obtained via shifting mechanism coded as FLC_023. As a first step, simulations with two controllers are performed on the system: the first controller is a standard FLC that has the symmetrical rule base given in Table 1 and the other FLC that has shifted rule base. In both the cases, the optimized input and output scaling factors are used and the reference value is set to 35 °C. The step response of both cases is illustrated in Fig. 12. As the second step, these two FLCs are implemented on MPC 555 hardware and the real time step responses given in Fig. 13 are obtained. To prove the effectiveness of the devised technique other performance criteria are used as maximum overshoot, rise time, and settling time. The performance comparison of the experimental results is presented in Table 9. When the shifted FLC_023 rule base is used the overall improvement in terms of the performance index is obtained at about 51%. The results of the experiments show that maximum overshoot, rise time and settling time are reduced about 27%, 40% and 28%, respectively. 6. Conclusion In this study, a fuzzy rule base shifting scheme for systems with time delay is proposed. The shifting scheme is tabulated with re-
Åström, K. J., & Hägglund, T. (1995). PID controllers: Theory, design, and tuning. Research Triangle Park, NC: Instrument Society of America. Bai, Y., Zhuang, H., & Roth, Z. S. (2005). Fuzzy logic control to suppress noises and coupling effects in a laser tracking system. IEEE Transactions on Control Systems Technology, 13(1), 113–121. Bandyopadhyay, R., Chakraborty, U. K., & Patranabis, D. (2001). Autotuning a PID controller: A fuzzy-genetic approach. Journal of Systems Architecture, 47(7), 663–673. Caner, M., Umurkan, N., Tokat, S., & Ustun, S. V. (2008). Determination of optimal hierarchical fuzzy controller parameters according to load condition with ANN. Expert Systems with Applications, 34, 2650–2655. Chang, P.-R., & Wang, B.-C. (1996). Adaptive fuzzy proportional integral power control for a cellular CDMA system with time delay. IEEE Journal on Selected Areas in Communications, 14(9), 1818–1829. de la Pena, D. M., Ramirez, D. R., Camacho, E. F., & Alamo, T. (2005). Application of an explicit min–max MPC to a scaled laboratory process. Control Engineering Practice, 13(12), 1463–1471. Dias, J. M., & Dourado, A. (1999). A self-organizing fuzzy controller with a fixed maximum number of rules and an adaptive similarity factor. Fuzzy Sets Systems, 103(1), 27–48. Elmas, C., Ustun, O., & Sayan, H. H. (2008). A neuro-fuzzy controller for speed control of a permanent magnet synchronous motor drive. Expert Systems with Applications, 34, 657–664. Guzelkaya, M., Eksin, I., & Yesil, E. (2003). Self-tuning of PID-type fuzzy logic controller coefficients via relative rate observer. Engineering Applications of Artificial Intelligence, 16, 227–236. Hu, B., Mann, G. K. I., & Gasine, R. G. (1999). New methodology for analytical and optimal design of fuzzy PID controllers. IEEE Transactions on Fuzzy Systems, 7(5), 521–539. Li, H. X., & Gatland, H. B. (1995). A new methodology for designing a fuzzy logic controller. IEEE Transactions on Systems, Man and Cybernetics, 25(3), 505–512. Li, H.-X., & Tso, S. K. (1999). Higher order fuzzy control structure for higher order or time-delay systems. IEEE Transactions on Fuzzy Systems, 7(5), 814–829. Mamdani, E. H. (1974). Application of fuzzy algorithm for simple dynamic plant. In Proceedings of the IEEE (pp. 1585–1588). Mamdani, E. H. (1993). Twenty years of fuzzy control: Experiences gained and lessons learnt. In Proceedings of the 2nd IEEE International Conference on Fuzzy Systems (pp. 339–344). San Francisco, CA. Mudi, R. K., & Pal, N. R. (1999). A robust self-tuning scheme for PI and PD type fuzzy controllers. IEEE Transactions on Fuzzy Systems, 7(1), 2–6. Ng, G. W., & Cook, P. A. (1998). Real-time control of systems with unknown and varying time-delays, using neural networks. Engineering Applications of Artificial Intelligence, 11(3), 401–409. Pereira, C., Henriques, J., & Dourado, A. (2000). Adaptive RBFNN versus conventional self-tuning: Comparison of two parametric model approaches for non-linear control. Control Engineering Practice, 8(1), 3–12. Qiao, W. Z., & Mizumoto, M. (1996). PID type fuzzy controller and parameters adaptive method. Fuzzy Sets and Systems(78), 23–35. Shinskey, F. G. (1998). Process control systems—application, design, and tuning. New York: McGraw-Hill.
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Skogestad, S. (2002). Simple analytic rules for model reduction and PID controller tuning. Journal of Process Control, 13(4), 291–309. Sugeno, M. (1985). Industrial applications of fuzzy control. Amsterdam, The Netherlands: Elsevier. Yesil, E., Güzelkaya, M., & Eksin, I. (2004). Self-tuning fuzzy PID-type load– frequency controller. Energy Conversion and Management, 45(3), 377–390.
Yesil, E., Guzelkaya, M., Eksin, I., & Tekin, O. A. (2007). Cross-coupled ratio control structure. Instrumentation Science & Technology, 35(5), 523–536. Yesil, E., Guzelkaya, M., Eksin, I., & Tekin, O. A. (2008). Set-point regulator with blending mechanism. Instrumentation Science & Technology, 36(1), 1–17.
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
A rule base modification scheme in fuzzy controllers for time-delay systems Hakkı Murat Genc a,b, Engin Yesil b, Ibrahim Eksin b,*, Mujde Guzelkaya b, Ozgur Aydın Tekin b a b
Marmara Research Center, Information Technologies Institute, 41470 Gebze, Kocaeli, Turkey Istanbul Technical University, Faculty of Electrical and Electronics Engineering, Control Engineering Department, Maslak TR-34469, Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Fuzzy controllers Rule base shifting Time-delay systems Microcontrollers Heat transfer system
a b s t r a c t In time-delay control systems, the observed information is naturally related to a past instant and this delayed information signal will usually cause unsatisfactory results. This study deals with how the time-delay information can be used in reorganizing the rule base of the fuzzy controller so as to improve system performance. It basically proposes a new scheme of appropriate shifting of the rule base to compensate the information lag caused by time delay in the system. The parameters affecting the shifting scheme are elaborated in detail and the new shifting scheme is proposed in a tabulated form that assumes the system time constant and the value of time delay as the main parameters. The effectiveness of the proposed methodology has firstly been tried to be illustrated on different simulation examples and, secondly, a real time application has been done on a heat transfer experimental setup. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Following the first fuzzy control application carried by Mamdani (1974), fuzzy logic is utilized quite often in control problems. During the past few decades, there are many successful applications with Fuzzy Logic Controllers (FLCs) in industry. It has been reported that they are successfully used in a number of complex and non-linear processes (Caner, Umurkan, Tokat, & Ustun, 2008; Elmas, Ustun, & Sayan, 2008; Sugeno, 1985; Yesil, Güzelkaya, & Eksin, 2004). Moreover, the experience has shown that fuzzy controls are often a favored method of designing controllers for dynamic systems even if traditional methods can be used (Mamdani, 1993). PI type FLCs are the most common and practical ones within the various types of FLCs just like the widely used conventional PI controllers in process control systems (Mudi & Pal, 1999). The research for improving FLC performance spreads over number of areas. The performance of FLC can be improved by adaptations over the design parameters categorized mainly in two groups (Hu, Mann, & Gasine, 1999): (a) structural parameters, and (b) tuning parameters. Basically, structural parameters include input/output (I/O) variables of fuzzy inference, fuzzy linguistic sets, membership functions, fuzzy rules, inference mechanism and defuzzification mechanism. Tuning parameters include I/O scaling factors (SFs) and parameters of membership functions (MFs). The structural * Corresponding author. Tel.: +90 212 2853500; fax: +90 212 2856700. E-mail address: [email protected] (I. Eksin). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.10.072
parameters are usually determined in an off-line manner while the tuning parameters are adjusted with an on-line mechanism. The scaling factors are simply the most effective parameters on system performance and, therefore, various on-line tuning algorithms are presented in literature (Guzelkaya, Eksin, & Yesil, 2003; Mudi & Pal, 1999; Qiao & Mizumoto, 1996). Most of the practical processes are high order non-linear systems and some of them may have considerable dead time (Mudi & Pal, 1999). The dead time is recognized as the most difficult dynamic element naturally occurring in physical systems (Shinskey, 1998). It is another known fact that high order systems can be modeled by a first order system with dead time counterpart (Skogestad, 2002). Therefore, any useful technique of designing a control system must be capable of dealing with dead time. Conventional PI (or PID) controllers are usually inefficient in managing systems with dominant dead time. Moreover, the controller output or process input should be a non-linear function of error and change of error to have a satisfactory performance. FLCs try to incorporate this nonlinearity by limited number ‘‘IF–THEN” rules (Mudi & Pal, 1999). As a consequence, FLCs are the commonly used controllers as the time delay or nonlinearity is the matter of concern. In a time-delay process, the present control action depends on the observed data that is related to a past instant. Therefore, when the observed data are directly used then the transportation lag would naturally produce a misleading or incorrect information for the rule base and, hence, an incorrect control action for the process (Li & Tso, 1999). A shifting scheme of the fuzzy rule base has been proposed by Li and Gatland (1995) to overcome this information lag. Later, Chang and Wang (1996) have used the same shifting strategy in their Cellular CDMA System. Li and Tso (1999) have
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argued that this method requires large rule bases and, therefore, it may not be accurate enough for low order rule base and it is hard to provide proper compensation for the controller. Finally, Bai, Zhuang, and Roth (2005) have proposed another method for their Laser Tracking System and found fairly good results for considerably small time delays. This paper is basically devoted to enlighten the use of timedelay information of the system in reorganizing the rule base of FLCs. There might be many aspects to be taken into account in reorganizing operation and obtaining a ‘‘better” rule base for a FLC. The reorganizing operation considered here depends upon the shifting of the rows of rule base of FLC. The main idea in the new approach developed in this study is that the shifting amount should be distinct for different time delays; it should even vary in between rows, where rows designate De. More precisely, the shifting amounts are tried to be tabulated using the parameter set of first order plus dead time (FOPDT) system models; namely, the time delay and time constant values. The effectiveness of the proposed methodology is shown both on different simulation examples and on a heat transfer experimental setup. The rest of the paper is divided into five sections. In Section 2, the most common architecture of a FLC is given. The proposed rule base shifting strategy is detailed in Section 3. In Section 4, simulations are presented to illustrate the effectiveness of the method. Moreover, in Section 5, experimental results on heat transfer system PT 326 (Feedback) are given. Possible extensions and conclusions are finally given in Section 6. 2. The architecture of fuzzy logic controller Although it is obvious that the proposed rule base modification scheme is valid for all kinds of FLCs, it has been preferred to make this implementation on PI type of FLC since it is common, adequate and efficient for various practical real time processes. The error (e) and the change of error (De) will be used as the inputs and the change of control signal (Du) as the output of the PI type FLC as illustrated in Fig. 1. The universe of discourse is chosen to be [1, 1] for the membership functions of input and output variables. The input and output parameters are scaled to fit this range via scaling factors. The input scaling factors (ISFs) normalize the real world inputs to a range in which membership functions are defined. The output scaling factor (OSF) is used to change the normalized control effort to its practical value. The relation between real and normalized values of the parameters can be simply given as
E ¼ eke;
DE ¼ Dekde;
Du ¼ DUko;
ð1Þ
where E and DE are the normalized inputs of the FLC controller, DU is the FLC output; e, De and Du are respective actual outputs, and ke, kde, ko are the error scaling factor, the change of error scaling factor and the control effort change scaling factor, respectively. The input variables are decomposed into at least seven fuzzy linguistic levels in order to make a considerable distinction between the fuzzy regions and, thus, to obtain fine tuned control action. These levels are chosen and named as NL (Negative Large),
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NM (Negative Medium), NS (Negative Small), Z (Zero), PS (Positive Small), PM (Positive Medium), PL (Positive Large) in this study. In Fig. 2 the MFs of input and output variables are illustrated. The output variables are chosen to be singletons for ease of calculations in the implementation phase. The symmetrical rule base given in Table 1 can be seen as one of the general and most used form. 3. The proposed rule base shifting scheme When a delay is introduced in a system either by the nature of the system or by the external devices used, the observed information is related to a past instant. Using this measured information for control purposes may cause unsatisfactory results especially in time dominant delay systems. However, if one can somehow guess the actual output information related to the measured instant from the process trend, then one can reorganize the rule base accordingly so that a ‘‘better” control performance might be achieved. A regular rule base might be divided into four regions not including e and De which are equal to ‘zero’ rows as illustrated in Fig. 3. The error is positive and decreases in magnitude in region R1 since the change of error is negative. Therefore, the actual (e, De) pair should lie somewhere in the left hand side of the measured one for a time-delay system. The rules corresponding to region R1 should be shifted to right to compensate this deviation. If the shifting amount is appropriate, then the controller would most probably perform ‘‘better” actions. The error and the change of error are both negative in R2 region and the error magnitude has the tendency to increase. The actual point is again in the left of the observed one and shifting the rules to right would provide again a ‘‘better” performance if this shifting operation is done in appropriate amounts for each row of the rule base. Similar arguments are valid for the upper side of rule base. In regions R3 and R4, the actual points are at the right of observed ones and rule base should be shifted towards left. When De is ‘zero’, then that row is unbiased and no shifting is needed. The time-delay effect is illustrated in Fig. 4a as a shift in the plane in the system step response of an arbitrarily chosen system. The arrow labeled 1 shows the difference of error magnitudes at the point t = 1 s. The rules at region R1 get activated for this point. The error for zero delay system is measured of about 0.6 at this point; whereas, the actual error of the delayed system is around 0.2 which implies a huge deviation from the actual value. Therefore, we can assume that the control effort for the delayed systems should have been decreased for that region. Then, the critical question is about the amount of that decrease. Li and Gatland (1995) shifted all the rules except the row in which De equals to ‘zero’ for one cell to obtain that decrease. This shifting scheme yields favorable performances for some systems but since it demands the same amount of shifting even for small error changes (around the set point) it may even cause oscillatory behavior and chattering. On the contrary, if the system delay value is too high, this one unit shifting amount might not be adequate enough for favorable system performance. This means that the shifting amount, as a matter of fact, should also depend upon both the time-delay
Fig. 1. The basic control block diagram illustration with PI type FLC.
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Fig. 2. Membership functions for: (a) e and De; (b) Du.
Table 1 A typical and symmetrical rule base for an FLC.
De/e
NL
NM
NS
Z
PS
PM
PL
PL PM PS Z NS NM NL
Z NS NM NL NL NL NL
PS Z NS NM NL NL NL
PM PS Z NS NM NL NL
PL PM PS Z NS NM NL
PL PL PM PS Z NS NM
PL PL PL PM PS Z NS
PL PL PL PL PM PS Z
Fig. 4. Arbitrary system step response without time delay and (a) a dead time of 0.5 s; (b) a dead time of 1 s.
Fig. 3. Fuzzy rule base regions.
magnitude, and the time constant of the (first order) system under control. In this work, the shifting amounts for the rows of the rule base are deduced and given in table form in a very general scheme that
depends on both the system parameters; namely, the time-delay value and the time constant of the plant. It is easily seen from Fig. 4a that the difference of the two responses in terms of error magnitude is not the same at each point if one examines the points illustrated by the three arrows. The error change rate is quite high and the response curve climbs up very rapidly to the set point at the arrow labeled as 1 for the delay time of 0.5 s. The system response is at the set-point value at arrow labeled as 2, but considering the time delay and interpreting change of error correctly, it is obvious that the actual system response value must be somewhere above that point. As the system response gets closer to the steady state region designated by the arrow labeled as 3, the difference between the two graphs diminishes. Thus, if the change in error is ‘‘high” then the difference between systems responses gets higher. Therefore, the amount of shifting must be kept ‘low’ as the system gets closer to the steady state region (when De is low) and it should be ‘high’ when the response rate of change of the system is high. In fact, no rule base shifting is appropriate for the small values of change of error to prevent oscillations and decrease chattering effect. The amount of time delay is an important factor in system responses as it is presented in Fig. 4. It is seen that the shifting amounts must be adjusted in accordance with the amount of
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H.M. Genc et al. / Expert Systems with Applications 36 (2009) 8476–8486 Table 3 Proposed shifting schemes.
Fig. 5. Ideal shifting scheme for infinite resolution.
time-delay values. The resolution of the membership functions plays another key role in rule base shifting. The effect of the time delay can be handled more precisely if more fuzzy regions are defined for each parameter. Fig. 5 illustrates the variation of the shifting in the infinite resolution case. This is impractical in real world applications since it can only be achieved by the use of unlimited number linguistic variables. It is obvious that adjusting the shifting amounts for each De row with higher resolutions would be more efficient. Naturally, on the other side, this causes computational complexity and this fact should be handled by the designer. In this work, seven fuzzy regions are defined for each variable as discussed in detail in Section 2. In this paper, combining and utilizing all the facts stated above, almost zero shifting for small De and increasing amount of shifting for larger values of De is suggested. The suggested shifting scheme
Normalized dead time (s)
Appropriate shift
s=0 0 < s < 0.05 0.05 6 s < 0.15 0.15 6 s < 0.7 sP7
000 001 011 023 033
seems to be efficient for minimum of seven linguistic terms for each input. As the resolution is increased zero shifting operation for the small De rows become more crucial. The reason for this is that the rules are rarely fired for high De values or at the extreme De rows. Therefore, adjusting the shifting amount at low and medium values of De rows becomes much more important for high resolution cases. Moreover, as the time-delay value gets higher the shifting amount should be increased as well while keeping it around zero for small De rows. As illustrated in Fig. 5, the shifting amounts are equal and opposite in direction in lower and upper sides of De = Z (zero) row for the infinite dimension case. Table 2 illustrates different shifting schemes on the standard rule base given in Table 1, which is a finite dimension application of the shifting scheme idea. The shifting amounts of the rows of the rule bases are coded as in Table 2. For instance, all the rows in Table 2b are not shifted equally; that is, the neighboring ‘Zero’ rows of De are not shifted while the others are shifted for one cell in appropriate directions. This shifting scheme is coded as 011 (shifted number of cells from low values to high values of De) and the corresponding controller is abbreviated as FLC_011. Table 2 includes four different shifting schemes for FLCs that will be used. Then, comparisons will be made of those four throughout the paper.
Table 2 FLC rule base for different shifting schemes: (a) FLC_001; (b) FLC_011; (c) FLC_023; and (d) FLC_033.
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Fig. 6. The effect of various rule base shifting schemes for a FOPDT system (T = 1 s) with a time-delay values of (a) 0.05 s; (b) 0.1 s; (c) 0.2 s; (d) 0.9 s; and (e) 3 s.
In this study, all of the systems are chosen to be first order plus dead time (FOPDT) systems since one can model most of the systems FOPDT counterpart Skogestad (2002). The shifting amounts for the rows of the rule base are tabulated using the system param-
eters; namely, the time-delay value and the time constant of the plant, and also on the magnitude of the error and the change of error. The transfer function of a FOPDT system can be given as follows:
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and rules. Moreover, when different performance indices are used, these transition points or borders might slightly change. Since the extreme rules are rarely fired, a change made in the shifting scheme related to the small and medium values of De will naturally make the most important effect on the system performance. Moreover, no shifting choice is convenient for NS or PS rows of De in order to prevent oscillatory behavior.
Table 4 Performance results for systems with various time-delay values. Delay time, L (s)
Normalized dead time (s)
Controller type
Performance index
0.05
0.046
0.1
0.09
0.2
0.17
FLC FLC_001 FLC FLC_011 FLC FLC_011 FLC_023 FLC FLC_023 FLC FLC_033
111.11 123.00 93.00 109.97 65.18 84.24 90.78 26.83 37.02 11.76 15.72
0.9
0.47
3
0.75
GðsÞ ¼
K esL ; Ts þ 1
4. Analysis of the proposed shifting scheme via simulations 4.1. Analysis with different time-delay systems
ð2Þ
where L is the delay time, K is the gain and T is the time constant for the process model. The normalized dead time or controllability ratio s can be defined in the interval [0, 1] as follows:
s¼
L : LþT
ð3Þ
The parameter s is a ratio that gives the relative magnitude of the time delay with respect to system time constant. It has generally and roughly been found out that processes with small s are easy to control; whereas, difficulty in controlling the system increases as s increases (Åström & Hägglund, 1995). Several shifting schemes are investigated for various time-delay values under the same performance index for the controller developed in Section 2. In the light of these results, the shifting amounts with respect to s are deduced and given in Table 3. The results are consistent with our presumptions. The transition points of Table 3 are determined through a search of scaling factors that are maximizing the following performance index
J¼
1000 ; 100mp þ 3t s þ 6tr þ 100ess
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ð4Þ
where mp is the maximum overshoot, ts is the settling time, tr is the rise time and ess is the steady state error. The transition from one shifting scheme to another is actually continuous. These borders are not crisp for all cases, but changes slightly for different configurations of the membership functions
In this section, the performance results are provided so as to designate the effect of different rule base shifting schemes for various time-delay values of a FOPDT system. The gain of this system is set to unity and the time constant to 1 s for simplicity. Therefore, the transfer function of the system becomes as follows:
GðsÞ ¼
1 sL e : sþ1
ð5Þ
The most commonly used controller given in Section 2 is used for all simulations. The input scaling factor ‘‘ke” is chosen to be unity in all cases since reference is unit step and error is already normalized to [1, 1]. The other two scaling factors shown in Fig. 1 are tuned for each case separately. There is unfortunately no commonly accepted method for choosing the optimum scaling factors and they are usually found out by trial error techniques. In this study, the scaling factors are searched via a basic genetic algorithm which maximizes the performance index given in Eq. (4). In the first simulation example, the delay time is chosen to be 0.05 s which is small in comparison to the time constant. Therefore, a small s value arises. The simulation result for this case is given in Fig. 6a. As it can be depicted from figure and Table 4, and 001 shifting scheme slightly improves the performance index value as expected since the extreme rules have little influence on the controller. Fig. 6b illustrates the case when 0.1 s delay is introduced to system. This corresponds to a normalized dead time s of 0.09 and the proposed shifting scheme is 011. The controller with the new rule base has provided a shorter rise time and as a result yielded about 20% performance improvement. If the time-delay value is 0.2 s, then s becomes 0.17 and Table 3 proposes 023 shifting scheme. This situation provides 40% improvement in the performance index. However, since these borders are not crisp and not discontinuous from one another, 011
Fig. 7. Illustration of the effect of various time constants on the system responses.
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Fig. 8. The effect of various rule base shifting schemes for a FOPDT system (L = 1 s) with a time constant values of (a) 20; (b) 10; (c) 5; (d) 1; and (e) 0.3 s.
shifting scheme also provides respectful results. This case is illustrated in Fig. 6c and the performance index values for the three rule bases are given in Table 4.
Fig. 6d shows the case when delay value and time constant are close to each other. Increasing the time delay far beyond the time constant of the plant case, results are illustrated in Fig. 6e.
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Fig. 9. Illustration of the effect of the rule base shifting for the same normalized dead time: (a) L = 1.5 s; T = 15.17 s; (b) L = 0.5 s, T = 5.06 s.
Table 5 Performance results for systems with various time constant values.
Table 7 Performance results for with a large normalized dead time (s = 0.75).
Time constant, T (s)
Normalized dead time (s)
Controller type
Performance index
Delay time, L (s)
Time constant, T (s)
Controller type
Performance index
20
0.476
0.66
0.0909
3
1
5
0.166
5
1.66
1
0.5
10
3.33
FLC FLC_033 FLC FLC_033 FLC FLC_033 FLC FLC_033
21.52 23.40 11.76 15.72 9.02 9.98 4.59 5.10
0.3
0.77
21.06 25.03 22.12 25.40 23.50 28.47 30.79 35.55 40.51 44.03
2
10
FLC FLC_001 FLC FLC_011 FLC FLC_023 FLC FLC_023 FLC FLC_033
Table 6 Performance results for systems with a small normalized dead time (s = 0.09). Delay time, L (s)
Time constant, T (s)
Controller type
Performance index
Improvement (%)
0.1
1 10
0.5
5.06
1.5
5.17
93.00 109.97 22.12 25.40 38.43 48.73 12.66 18.74
18.2
1
FLC FLC_011 FLC FLC_011 FLC FLC_011 FLC FLC_011
14.92 26.80
33.67 10.58 11.01
ness of the proposed method, the simulation results of the system given in Eq. (6) for five different time constants are presented in Fig. 8. Furthermore, the performance index values for the different simulation results are tabulated in Table 5. As the time constant T decreases, the normalized dead time s of the system increases so do shifting amounts (Table 3). Note that in the absence of time delay, there would be no need to shift. This result can also be derived from the normalized dead time s given in Eq. (3). 4.3. Analysis with different (L, T) combinations
In this part, time-delay value has been assumed to remain constant and time constant parameter of the FOPDT system has been changed gradually. Delay value is set to 1 s and the transfer function of the system then becomes as follows:
1 es : Ts þ 1
8.77
48.03
4.2. Analysis over systems with various time constants
GðsÞ ¼
Improvement (%)
ð6Þ
The set of other parameters of the system is configured as explained in Section 4.1. Changing the time constant of the system model practically means a change in the system response speed and Fig. 7a and b provides better perception of this effect. A smaller time constant provides a faster system response as seen in 7a and the difference between the actual and the observed system responses becomes larger. We eventually can say that the needed shifting amount should be ‘‘greater”. In order to show the effective-
The same normalized dead time value can be obtained by infinite number of (L, T) combinations. Many tests have been conducted to illustrate the robustness of proposed shifting scheme to these variations. More specifically, the performance results of the system with a very small normalized dead time (s = 0.09) are given in Table 6 and another system with a relatively large normalized dead time (s = 0.75) is presented in Table 7. Moreover, the simulation results for two different cases are presented in Fig. 9. 4.4. Analysis with different rule base sizes It should be an obvious fact that the rule base shifting scheme would be more effective as the number of variables has been increased which are defined on the fuzzy regions. This is because of the fact that non-linear structure of the shifting scheme can be fit more precisely with more resolution. All the previous illustrations in this study are based on the case where seven fuzzy regions are defined for each parameter and we will now consider the case with nine fuzzy regions. A comparison of the ‘‘shifted” and
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Fig. 10. Comparison of two fuzzy controllers with different number of fuzzy rules.
The control system realization is based on the Motorola MPC555 32-bit single-chip microcontroller whose main features are as follows: 448 kB on-chip flash memory; 26 kB on-chip RAM; two 10-bit analog-to-digital (A/D) converters with 16-input channels; modular input/output system (MIOS) (eight channels for PWM); 2 serial communication interfaces (SCIs); 64-bit floating-point unit with 0.1-ls floating-point multiply; and 0.25-ls floating-point divide (at 40-MHz clock frequency). The MPC555 hardware is compatible with the MATLAB/SIMULINK. Mentioned features of the MPC555 are available as SIMULINK blocks in the ‘Embedded Target for Motorola MPC555 Block Library’. The code generation process for this design is thus handled in SIMULINK using Embedded Target for Motorola MPC555, RealTime Embedded Coder and Real-Time Workshop which externally needs the C compiler software ‘Metrowerks CodeWarrior Development Studio for MPC5xx’. The experimental setup used in this study is given in Fig. 11. A step input is applied and then the well-known area method is used in order to obtain FOPDT model of the PT 326 (Åström & Hägglund, 1995). The following transfer function model is then obtained
PðsÞ ¼ Table 8 Performance results for controller with nine fuzzy regions. Controller type
Performance index
FLC FLC_023 FLC_0233
17.64 23.08 24.18
0:74 e0:3s : 0:58s þ 1
‘‘non-shifted” controllers for a FLC with nine fuzzy regions (81 rules) on a same system with time delay of 1.75 s and time constant 1 is illustrated in Fig. 10. A shifting scheme of ‘‘0233” yields the best results in terms of performance index given in Eq. (4). The improvements are illustrated in Table 8 and it is not so worthy in comparison with results attained in 7 region counterpart. It should also keep in mind that the improvement achieved by increasing the number of fuzzy regions may not compensate the computational complexity. 5. Experiment Fig. 11. Experimental setup.
The heat transfer process trainer (PT 326) that has been widely used by many different researchers to check their new control strategies is also used here to present the advantages of the proposed methodology (Bandyopadhyay, Chakraborty, & Patranabis 2001; de la Pena, Ramirez, Camacho, & Alamo 2005; Dias & Dourado, 1999; Ng & Cook, 1998; Pereira, Henriques, & Dourado 2000; Yesil, Guzelkaya, Eksin, & Tekin, 2007). PT 326 has the basic characteristics of a large plant that involves a tube through which air is drawn from atmosphere by a centrifugal blower and the air is heated as it passes over a heater grid before being released into the atmosphere. Temperature control is achieved varying the electrical power supplied to the heater grid. The mass flow of air through the duct can be adjusted by setting the opening of the throttle. The pure time delay depends on the position of the temperature sensor element that can be inserted into the air stream at any one of the three points spaced at 28, 140 and 280 mm far from the heating point along the tube. In this study, the sensor element is placed at the third place and therefore the allowable longest dead time is achieved. The damper position is set to 40°. The system input u(k) is the voltage applied to the power electronic circuit feeding the heating resistance and the output y(k) is the outlet air temperature, which is expressed as a voltage between 0 and 10 V (Yesil, Guzelkaya, Eksin, & Tekin, 2008).
Fig. 12. Simulation results for the system outputs.
ð7Þ
H.M. Genc et al. / Expert Systems with Applications 36 (2009) 8476–8486
spect to the normalized dead time. The simulation results show that proposed rule base shifting scheme improves system performance in considerable amounts. Also it provides considerably acceptable results even for the large variations in structural parameters. Shifting all rows of the fuzzy rule base by one cell is proposed in previous studies and it rarely yields improved results. Therefore, in addition to one cell shifting of all rows, two sets of scaling factors (for transient and steady state responses) and a change in some entries of the rule base has been proposed. Under these circumstances, some performance ameliorations have been observed. However, the new approach proposed in this study provides better results even in very generic form. Also, in case of different design purposes or performance indices, the proposed shifting scheme can be redefined. In the simulations done on various systems, it has been observed that the proposed methodology ameliorate the overall performance of the system with respect to set-point following. Therefore, the proposed fuzzy controller that has been embedded into a microcontroller is then used to control a heat transfer process (PT 326 – Feedback). Finally, the effectiveness of the proposed methodology with respect to the transient response has been demonstrated and compared with fuzzy controller that has classical diagonal rule base.
Fig. 13. Experiment results for the system outputs.
Table 9 Comparison of performance results for the experimental study. Controller type
Maximum overshoot (%)
Rise time (s)
Settling time (s)
Performance index
FLC FLC_023
3.67 0.67
3.5 2.1
6.3 4.5
22.95 34.76
References
As it is seen from Eq. (7), the apparent time delay of the system is small with respect to its dominant time constant. A time delay of 0.9 s has been added via software at the input of the process so as to obtain a time dominant system and to emphasize the benefit of the rule base shifting idea. Then, the overall transfer function of the model is obtained as
PðsÞ ¼
0:74 e1:2s 0:58s þ 1
8485
ð8Þ
and the normalized dead time s is calculated as 0.674 for this timedelay dominant system. As it can easily be deduced from the proposed scheme given in Table 3, the best rule base for the FLC is the one obtained via shifting mechanism coded as FLC_023. As a first step, simulations with two controllers are performed on the system: the first controller is a standard FLC that has the symmetrical rule base given in Table 1 and the other FLC that has shifted rule base. In both the cases, the optimized input and output scaling factors are used and the reference value is set to 35 °C. The step response of both cases is illustrated in Fig. 12. As the second step, these two FLCs are implemented on MPC 555 hardware and the real time step responses given in Fig. 13 are obtained. To prove the effectiveness of the devised technique other performance criteria are used as maximum overshoot, rise time, and settling time. The performance comparison of the experimental results is presented in Table 9. When the shifted FLC_023 rule base is used the overall improvement in terms of the performance index is obtained at about 51%. The results of the experiments show that maximum overshoot, rise time and settling time are reduced about 27%, 40% and 28%, respectively. 6. Conclusion In this study, a fuzzy rule base shifting scheme for systems with time delay is proposed. The shifting scheme is tabulated with re-
Åström, K. J., & Hägglund, T. (1995). PID controllers: Theory, design, and tuning. Research Triangle Park, NC: Instrument Society of America. Bai, Y., Zhuang, H., & Roth, Z. S. (2005). Fuzzy logic control to suppress noises and coupling effects in a laser tracking system. IEEE Transactions on Control Systems Technology, 13(1), 113–121. Bandyopadhyay, R., Chakraborty, U. K., & Patranabis, D. (2001). Autotuning a PID controller: A fuzzy-genetic approach. Journal of Systems Architecture, 47(7), 663–673. Caner, M., Umurkan, N., Tokat, S., & Ustun, S. V. (2008). Determination of optimal hierarchical fuzzy controller parameters according to load condition with ANN. Expert Systems with Applications, 34, 2650–2655. Chang, P.-R., & Wang, B.-C. (1996). Adaptive fuzzy proportional integral power control for a cellular CDMA system with time delay. IEEE Journal on Selected Areas in Communications, 14(9), 1818–1829. de la Pena, D. M., Ramirez, D. R., Camacho, E. F., & Alamo, T. (2005). Application of an explicit min–max MPC to a scaled laboratory process. Control Engineering Practice, 13(12), 1463–1471. Dias, J. M., & Dourado, A. (1999). A self-organizing fuzzy controller with a fixed maximum number of rules and an adaptive similarity factor. Fuzzy Sets Systems, 103(1), 27–48. Elmas, C., Ustun, O., & Sayan, H. H. (2008). A neuro-fuzzy controller for speed control of a permanent magnet synchronous motor drive. Expert Systems with Applications, 34, 657–664. Guzelkaya, M., Eksin, I., & Yesil, E. (2003). Self-tuning of PID-type fuzzy logic controller coefficients via relative rate observer. Engineering Applications of Artificial Intelligence, 16, 227–236. Hu, B., Mann, G. K. I., & Gasine, R. G. (1999). New methodology for analytical and optimal design of fuzzy PID controllers. IEEE Transactions on Fuzzy Systems, 7(5), 521–539. Li, H. X., & Gatland, H. B. (1995). A new methodology for designing a fuzzy logic controller. IEEE Transactions on Systems, Man and Cybernetics, 25(3), 505–512. Li, H.-X., & Tso, S. K. (1999). Higher order fuzzy control structure for higher order or time-delay systems. IEEE Transactions on Fuzzy Systems, 7(5), 814–829. Mamdani, E. H. (1974). Application of fuzzy algorithm for simple dynamic plant. In Proceedings of the IEEE (pp. 1585–1588). Mamdani, E. H. (1993). Twenty years of fuzzy control: Experiences gained and lessons learnt. In Proceedings of the 2nd IEEE International Conference on Fuzzy Systems (pp. 339–344). San Francisco, CA. Mudi, R. K., & Pal, N. R. (1999). A robust self-tuning scheme for PI and PD type fuzzy controllers. IEEE Transactions on Fuzzy Systems, 7(1), 2–6. Ng, G. W., & Cook, P. A. (1998). Real-time control of systems with unknown and varying time-delays, using neural networks. Engineering Applications of Artificial Intelligence, 11(3), 401–409. Pereira, C., Henriques, J., & Dourado, A. (2000). Adaptive RBFNN versus conventional self-tuning: Comparison of two parametric model approaches for non-linear control. Control Engineering Practice, 8(1), 3–12. Qiao, W. Z., & Mizumoto, M. (1996). PID type fuzzy controller and parameters adaptive method. Fuzzy Sets and Systems(78), 23–35. Shinskey, F. G. (1998). Process control systems—application, design, and tuning. New York: McGraw-Hill.
8486
H.M. Genc et al. / Expert Systems with Applications 36 (2009) 8476–8486
Skogestad, S. (2002). Simple analytic rules for model reduction and PID controller tuning. Journal of Process Control, 13(4), 291–309. Sugeno, M. (1985). Industrial applications of fuzzy control. Amsterdam, The Netherlands: Elsevier. Yesil, E., Güzelkaya, M., & Eksin, I. (2004). Self-tuning fuzzy PID-type load– frequency controller. Energy Conversion and Management, 45(3), 377–390.
Yesil, E., Guzelkaya, M., Eksin, I., & Tekin, O. A. (2007). Cross-coupled ratio control structure. Instrumentation Science & Technology, 35(5), 523–536. Yesil, E., Guzelkaya, M., Eksin, I., & Tekin, O. A. (2008). Set-point regulator with blending mechanism. Instrumentation Science & Technology, 36(1), 1–17.