A Big Bang-Big Crunch Optimization Based Approach for Interval Type-2 Fuzzy PID Controller Design

Transcription

1 A Big Bang-Big Crunh Optimization Based Approah or Interval Type-2 Fuzzy PID Controller Design Tuan Kumbasar Control Engineering Department Istanbul Tehnial University Istanbul, Turkey Abstrat In this paper, we will present a big bang-big runh optimization (BB-BC) based approah or the design o an interval type-2 uzzy PID ontroller. The implemented global optimization algorithm has a low omputational ost and a high onvergene speed. As a onsequene, the BB-BC method is a very eiient searh algorithm when the number o the optimization parameters is relatively big. The optimized type-2 uzzy ontroller is ompared with PID and type- uzzy PID ontrollers whih were optimized with either the BB-BC optimization method or onventional design strategies. The paper will also show the eet the extra degrees o reedom provided by the anteedent interval type-2 uzzy sets on the losed loop system perormane. We will present a omparative study perormed on the highly nonlinear asaded tank proess to show the superiority o the optimized interval type-2 uzzy PID ontroller ompared to its optimized PID, type- ounterparts. Keywords interval type-2 uzzy sets; interval type-2 uzzy PID; nonlinear ontrol; Big Bang-Big Crunh Optimization. I. INTRODUCTION In the reent years, Interval Type-2 Fuzzy Logi Controllers (IT2-FLC) have been widely used to ontrol nonlinear systems where the IT2-FLCs demonstrated signiiant perormane improvements. The internal struture o the IT2-FLC is similar to the type- ounterpart. However, the major dierenes are that IT2-FLCs employ interval type-2 uzzy sets (rather than type- uzzy sets) and the IT2-FLCs proess Interval Type-2 Fuzzy Sets (IT2-FSs) and thus the IT2- FLC has the extra type-redution proess [], [2]. It has been shown in various works that IT2-FLCs ahieve better ontrol perormane beause o the additional degree o reedom provided by the Footprint o Unertainty (FOU) in their Membership Funtions (MFs) [3], [4]. Consequently, IT2- FLCs have attrated muh researh interest, espeially in ontrol appliations [3-9]. In [0], [], it has been reported that the IT2-FLCs are generally more robust than type- ounterparts. However, IT2-FLCs are diiult to analyze and design sine their internal struture is typially more omplex than their type- ounterpart [2]. In this ontext, several studies have been perormed in order to design the IT2-FLCs [3-5]. Nevertheless, a systemati design or type-2 uzzy ontrollers is still a hallenging problem due to the main diiulty in determining the IT2-FSs and rulebase. Reently, Hani Hagras The Computational Intelligene Centre University o Essex Colhester, United Kingdom hani@essex.a.uk several studies have employed evolutionary algorithms or the design o IT2-FLCs [5, 5]. In this paper, we will present the novel appliation o the evolutionary algorithm alled Big Bang- Big Crunh (BB-BC) optimization [6] to optimize the anteedent parameters o the Interval Type-2 Fuzzy PID (IT2-FPID) ontroller or a highly nonlinear asaded two tank proess. Sine the number o the design parameters o the IT2-FPID is relatively big, the BB-BC global optimization algorithm has been preerred as it has a low omputational ost and a high onvergene speed. The perormane o the BB-BC Optimized IT2-FPID (OIT2-FPID) ontroller will be ompared with a BB-BC Optimized Type- Fuzzy PID (OT-FPID) and PID (OPID) ontrollers. In order to make a air omparison, the rule base and saling ators are kept ixed or the uzzy ontroller strutures while only the anteedent uzzy sets parameters are optimized or ertain reerene trajetory. It will be illustrated that the extra degree o reedom o the anteedent IT2-FSs gives the BB-BC method an opportunity to ind a more optimal solution than those obtained or the optimized OPID and OT-FPID with respet to the Integral o the Absolute Error (IAE) perormane index. The optimized ontrollers are also ompared with an Internal Model Control based PID (IMC-PID) and an Internal Model Control based Type- Fuzzy PID (IMC-T-FPID) ontroller strutures. A detailed omparative study has been onduted to show that the ontrol perormane o the OIT2-FPID is better in dierent operating points even at those at whih the ontrollers are not optimized or tuned and is more robust against noise and disturbanes when ompared to its optimized T-FPID and traditional PID ounterparts. Setion II will present the BB-BC optimization algorithm. Setion III will briely desribe the nonlinear two asaded tank proess. Setion IV will present the ontrol design strategies or the onventional PID, T-FPID and IT2-FPID strutures. Setion V will present the simulation studies and results while Setion VI will present the onlusions and uture work. II. A BRIEF OVERVIEW ON BIG BANG BIG CRUNCH OPTIMIZATION In [6], a new evolutionary algorithm named Big Bang Big Crunh (BB-BC) was proposed. The leading advantage o BB-

2 BC is the high onvergene speed and the low omputation time. The working priniple o this method an be explained as the transormation o a onvergent solution to a haoti state and then bak to a single tentative solution point. The BB-BC algorithm has been implemented in various engineering appliations where the omputational time and onvergene time are important ators. The eiieny o this evolutionary algorithm has been demonstrated espeially where the optimization problem must be solved in relatively small sampling times suh as in inverse uzzy model ontrol and uzzy model adaptation [7]. Moreover, in the optimization o highly nonlinear engineering problems suh as ontroller design [8], design o spae trusses [9], and size redution o spae trusses [20], this algorithm has been preerred beause o its high onvergene speed. The Big Bang-Big Crunh (BB-BC) optimization method proposed in [6] onsists o two main steps. The irst step is the Big Bang phase where andidate solutions are randomly distributed over the searh spae and the next step is the Big Crunh phase where a ontration proedure alulates the enter o mass or the population. The initial Big Bang population is randomly generated over the entire searh spae just like the other evolutionary searh algorithms. All subsequent Big Bang phases are randomly distributed about the enter o mass or the best it individual in a similar ashion [6]. Ater the Big Bang phase, a ontration proedure is applied as the Big Crunh phase to orm a enter or a representative point or urther Big Bang operations. In this phase, the ontration operator takes the urrent positions o eah andidate solution in the population and its assoiated ost untion value and omputes a enter o mass. The enter o mass an be omputed as: N i i = x = () N i = where x is the position o the enter o mass, x i is the position o the andidate, i is the ost untion value o the i th andidate, and N is the population size. Instead o the enter o mass, the best it individual an also be hosen as the starting point in the Big Bang phase. The new generation or the next iteration Big Bang phase is normally distributed around x. The new andidates around the enter o mass are alulated by adding or subtrating a normal random number whose value dereases as the iterations elapse. This an be ormalized as x new i x rα ( xmax - xmin ) = x + k where r is random number; α is a parameter limiting the size o the searh spae, x max and x min are the upper and lower limits; and k is the iteration step. In [6] the working priniple o this evolutionary method is explained as to transorm a onvergent solution to a haoti state whih is a new set o solutions. III. THE TWO CASCADED TANK PROCESS The shemati diagram o the two-asaded tank proess is illustrated in Fig.. The irst tank ( Tank 2 ) is a straight tank i (2) with a ross setional area A 2. The seond tank ( Tank ) is an inlined tank. The ross-setional area is related to the level o the tank. The angle o its sidewall is β, and the width o Tank is L. The dierential equation related to the tank level system is given as: dh -γ 2gh + γ 2 2gh2 = dt 2( d + h tanβ ) L dh γ = - 2gh + Fin t dt A A The system output is deined as: [ 0] h h = h2 The proess parameters are: A 2 =85 m2, g = m 2, g 2 = m2, L=20 m, β=5., d=3.32 m, the aeleration o gravity g=9.8 ms -2 and the initial levels are h 0 = 5m, h 20 = 5m while the inlet low range is 0-00 m 3 /s [2]. Fig.. Illustration o the two-asaded tank proess IV. () THE CONTROLLERS DESIGN STRATEGIES In this setion, we will present the design strategies o the implemented uzzy and onventional PID ontrollers. We will start by presenting the two design strategies employed or the onventional PID ontroller struture. We will then present the two design strategies employed or T-FPID. Finally, we will present the design strategy employed or the design o the IT2- FPID ontroller. A. Conventional PID Controller Design Methods In this subsetion, we will present the employed design strategies or the onventional PID ontrollers. The irst design strategy employs an Internal Model Control based PID ontroller based on the First Order Plus-Time Delay (FOPDT) model approximation [22]. As it is well-known, high order or nonlinear proesses an be represented by FOPDT models that orrespond to [23]: Ls Ke Gs () = Ts + (3) (4) (5)

3 (a) () Fig. 2. (a) T-FPID/ IT2-FPID ontroller struture Rule base () Type- Fuzzy Membership Funtions (d) Interval Type-2 Fuzzy Membership Funtions (d) When the nonlinear tank system is modeled as FOPDT or the operating point (h =0m, F in =50m 3 /s), the model parameters are obtained based on the step response method as ollows [23]: K=0.22, T=50s, L=0s. In the IMC based design method [22], the PID struture inherits a low pass ilter and is deined as ollows: IMC PID() s = K + + τ Ds + τis τ s Where is the proportional gain is the integral gain, is the derivative gain and is the time onstant o ilter. These parameters are deined as: T + 0.5L K = ; τ I = T L; K( τ + L) 0.5Lτ 0.5Lτ τd = ; τ = T + 0.5L τ + L In this design proedure, the desired losed-loop time onstant is the only design parameter [22]. For the two asaded tank system, the time onstant is hosen as 20 seonds so as to obtain moderate losed loop system response with a low IAE value or the operating point. Then, the ontroller parameters are alulated as 8.29, 49.5, τ D =6.06 and 4. In the seond PID design strategy, the three parameters (,, ) o a onventional PID ontroller are optimized so as to minimize the IAE perormane index via the BB-BC optimization algorithm. The IAE is deined as: 0 (6) (7) IAE = r() t y() t dt (8) where y(t) is the proess output and r(t) is the reerene value. The transer untion o the optimized PID ontroller is given as: OPID() s = K + + τ Ds τ I s B. The Type- and Interval Type-2 Fuzzy PID Controller Design Methods In this subsetion, the design strategies or the implemented type- and the interval type-2 uzzy PID ontrollers are presented. Here, the standard two input uzzy PID ontrollers as illustrated in Fig.2a is preerred and used [24]. The input to the uzzy PID ontrollers are the error (e=r-y) and the hange o error (Δe), the output o the FLCs is the hange o the ontrol signal (u). The input saling ators K e or error and K d or the hange o error normalize the inputs to the range in whih the membership untions o the inputs are deined. While K 0 and K are the output saling ators. In the implemented T-FLCs and IT2-FLC strutures, a symmetrial 3x3 rule base is used as shown in Fig. 2b. Here, the outputs are deined with three risp singleton onsequents (Negative (N)=-, Zero (Z)=0, Positive (P)=0). The implemented T-FLCs use the produt impliation and the entroid deuzziiation method while the IT2-FLC uses the enter o sets type redution/ deuzziiation method [2]. For the sake o simpliity, the inputs o all FLC strutures are deined with three (J=3) triangular type membership untions and are denoted as N (Negative), Z (Zero) and P (Positive). The type- uzzy membership untion o the T- FLC is deined with three parameters (l ij, ij, r ij ; i=,2, j=,2,3), as shown in Fig.2, while the interval type-2 uzzy membership (9)

4 untion is deined with our parameters(l ij, ij, r ij, m ij ; i=,2, j=,2,3), as shown in Fig.2d. In the irst T-FPID design strategy, an IMC based tuning method is used to determine the saling ators or a type- uzzy PID based on the FOPDT model approximation [25]. In this uzzy struture, the anteedent triangular membership untions must stritly partition the input domain, i.e., 50% overlapping. The parameters o the anteedent membership untions o the IMC-T-FPID are tabulated in Table I. In this ase, the input and output saling ators an then be alulated via the ollowing equations: where K0 = ; K = β K0; Kd = α K L KKe( τ + ) 2 β = max( T, L/ 2), α = min( T, L/ 2) Ke = rt ( ) yt ( ) e (0) () where r(t ) and y(t ) are the values o the reerene and system output at the time o the reerene variation (t=t ), respetively. Then, the saling ators are alulated as: K = 5 K, K = 5.525, K = (2) d e 0 I In the seond uzzy PID design strategy, the parameters o the anteedent membership untions are optimized via the BB- BC method suh that to minimize the IAE perormane index given in (8) or both the type- and interval type-2 uzzy PID ontrollers. In the optimization o the uzzy PID ontrollers, the BB-BC will only tune the parameters o the anteedent MFs, while the saling ators, given in (2), and the rule base, given in Fig.2b, are kept ixed. The anteedent membership untions o OT-FPID ontroller that are labeled as the N and P are deined or eah input with two parameters eah whih are ( i, r i ) and (l i3, i3, i=,2), respetively while the one labeled as Z is deined with three parameters (l i2, i2, r i2, i=, 2). Hene or two inputs, the total number o the parameters to be optimized or the OT- FPID design is then 2x7=4. Whereas, the anteedent IT2-FSs o OT-FPID ontroller that are labeled as the N and P or eah input are deined with three parameters eah whih are ( i, r i, m i ) and (l i3, i3, m i3, i=, 2), respetively while the IT2- FS labeled as Z is deined with our parameters (l i2, i2, r i2, m i2, i=, 2). Consequently, or the two inputs the total numbers to be optimized or the OIT2-FPID design is 2x0=20. It is obvious that, the OIT2-FPID has 6 more parameters, i.e. extra degrees o reedom. C. IAE perormane evaluation o the optimized ontroller strutures In this subsetion, the optimization results o the OPID, OT-FPID and OIT2-FPID ontroller strutures or the asaded nonlinear tank system are examined and ompared. Sine the aim o this subsetion is to make a air omparison, the population size and the number o iterations or all three BB-BC based design strategies have been hosen as 00 and 000, respetively. In the optimization studies, sine the nonlinearity is related to the level o the both tanks, a reerene trajetory with the values o 2, 20 and 29m, in suessive order, have been applied to all three ontroller strutures. The design parameters o the ontrollers are optimized suh that to minimize total IAE perormane value or the reerene trajetory. For the desired reerene trajetory, the best values o the OPID ontroller parameters are ound as 23.96, 49.92, 7.2. The best obtained parameter set or the OT-FPID and OIT2-FPID ontrollers are tabulated in Table I. The obtained best IAE values o the OPID, OT-FPID and OIT2-FPID are , and respetively.in Fig.3, the variations o the IAE values are illustrated. It an be learly seen that the BB-BC optimizing the OPID and OT-FPID onverge with a muh higher IAE values than the OIT2-FPID whih ends up with a lower IAE value. This shows that the extra degrees o reedom enable the BB- BC to ind a more optimal solution than those obtained by OPID and OT-FPID or the desired reerene trajetory with the values o 2, 20 and 29m. Fig. 3. The variations o the IAE values

5 TABLE I. IMC-T-FPID OT-FPID OIT2-FPID E E E E E E PARAMETERS OF THE ANTECEDENT MFS Parameters o the MFs l r m N Z P N Z P N Z P N Z P N Z P N Z P Sine the BB-BC optimization method is a stohasti global searh method, a statistial evaluation is perormed. The optimization routine or eah o the ontroller strutures is repeated 5 times and the best IAE value or eah run are reorded. In Table II, the best, the worst and the average IAE values or the ive runs are tabulated. Although IT2-FPID has the largest number o parameters to be optimized, it has a lower IAE value in all three ases. TABLE II. BEST, AVERAGE AND WORST IAE VALUES FOR THE FIVE TRIALS IAE perormane Best Average Worst OPID OT-FPID OIT2-FPID V. CLOSED LOOP CONTROL PERFORMANCE ANALYSES In this setion, dierent analyses are presented to investigate the losed loop ontrol perormanes o the IMC- PID, OPID, IMC-T-FPID, OT-FPID and OIT2-FPID. In order to make a air omparison, three perormane measures are onsidered. Two o these perormane measures are seleted rom the lassial transient system response riteria; namely, the settling time (Ts), and the overshoot (OS %) while the other perormane measure is the IAE. At irst, we will analyze the ontrol perormanes or the reerene trajetory at whih the ontrollers are designed. Next, the ability o the ontrollers to handle with output noise is investigated or the same reerene trajetory. Moreover, the ability o the ontrollers to ope with nonlinear dynamis is investigated by deining a reerene trajetory at whih the ontrollers are neither optimized nor tuned. Finally, the input and output disturbane rejetion perormanes o the implemented ontrollers are presented and ompared. A. Trajetory Control Perormane Analyses In the irst perormane study, the reerene trajetory is examined at whih the ontrollers are optimized. The ontrol signals and proess outputs o the implemented ontroller strutures are illustrated in Fig.4. As it an be learly seen in Table III, the OIT2-FPID struture improves the overall perormane o the proess muh better in every sense ompared to the other ontrollers. For instane, or the reerene value variation rom 20m to 29m (without output noise), the OT-FPID has the perormane measures Ts=222s, OS=24% and a total IAE value o as tabulated in Table III. For this operating point, the OIT2-FPID redues the overshoot to 7%; it also dereases the settling time to about 50s and the total IAE value to when it is ompared to the OT-FPID. Similar omments an be made or the other two reerene variations. Moreover, in order to understand how muh the ontrollers are robust against noise, white noise is added at the output o the proess. The simulation is then repeated or the same reerene trajetory. The proess outputs are illustrated in Fig.5. As it an be learly seen, the OIT2- PID is more robust against noise ompared to the uzzy and onventional strutures. The perormane measures are tabulated in in Table III. (a) Fig. 4. Illustration o the (a) proess outputs and ontrol signals or irst trajetory ontrol perormane without output noise.

6 TABLE III. PERFORMANCE COMPARISONS FOR THE FIRST TRAJECTORY CONTROL PERFORMANCE STUDY IMC-PID OPID IMC-T-FPID OT-FPID OIT2-FPID TRANSIENT PERFORMANCE 5-2m 2-20m 20-29m IAE Ts OS Ts OS Ts OS w/o Noise 90s 00% 296s 48% 432s 58% w/ Noise 92s 00% 30s 50% 62s 65% w/o Noise 80s 23% 46s 52% 248s 58% w/ Noise 88s 23% 60s 63% 436s 65% w/o Noise 68s 8% 74s 45% 300s 57% w/ Noise 20s 48% 86s 55% 330s 64% w/o Noise 90s 00% 086s 5% 222s 24% w/ Noise 02s 45% 226s 25% 422s 38% w/o Noise 36s 00% 074s 07% 30s 07% w/ Noise 74s 22% 0s % 50s 0% that the OIT2-FPID struture provides better transient perormanes than the type- uzzy and onventional ontroller strutures or this varying reerene values. For instane, or the reerene variation rom 26m to 6m, the OIT2-FPID produes superior ontrol perormane than the other ontrollers as its proess response has the smallest overshoot and settling time. The perormane measures are tabulated in in Table IV. It an be onluded that, the OIT2- FPID struture has the ability to ope better with the nonlinear dynamis ompared to the other ontrollers. TABLE IV. PERFORMANCE COMPARISONS FOR THE SECOND TRAJECTORY CONTROL PERFORMANCE STUDY Fig. 5. Illustration o the proess outputs or the irst trajetory ontrol perormane with output noise. The perormanes o the designed ontrollers have been also tested or another reerene trajetory. Sine the nonlinearity is related to the level o the both tanks, the ontrol perormanes o the implemented ontrollers is investigated or two operating points at whih they are not optimized or tuned. Thus, a reerene trajetory with the values o 26 and 6 m, in suessive order, has been hosen in order to examine how the ontrollers handle nonlinearity. As it has been shown in Fig.6 TRANSIENT PERFORMANCE 5-26m 26-6m IAE Ts OS Ts OS IMC-PID 456s 34% 90s 42% OPID 306s 4% 48s 40% IMC-T-FPID 358s 33% 60s 40% OT-FPID 20s 4% 26s 73% OIT2-FPID 62s 09% 8s 28% B. Disturbane Rejetion Perormane Analyses This subsetion examines the input and output disturbane rejetion perormanes o the implemented ontrollers. In this ontext, it is assumed that the proess is steady state at the operating point (h =20m, F in =70m 3 /s) at whih the ontrollers are neither optimized nor tuned. (a) Fig. 6. Illustration o the (a) proess outputs and ontrol signals or the seond trajetory ontrol perormane.

7 At irst, a step input disturbane with a magnitude o +50 has been applied the proess at 50 th seond. The ontrol signals and proess outputs or the input rejetion perormanes are given in Fig.7. As it an be observed rom Fig.7 that, OIT2- FPID ontroller notieably showed superior perormanes ompared with type- and onventional PID ounterparts. Seondly, a step output disturbane with a magnitude o -0 is applied to the proess at 000 th seond. The ontrol signals and proess outputs or the output disturbane rejetion perormanes are given in Fig.8. As it an be seen, the proposed OIT2-FPID ontrol struture ompensated very eetively the disturbane in a short period o time ompared to the other implemented ontroller strutures. The alulated perormane values or the input and output disturbane rejetion perormanes are tabulated in Table V. The perormane measures demonstrate that the OIT2-FPID ontroller is more apable to handle input and output disturbanes. TABLE V. DISTURBANCE REJECTION PERFORMANCE COMPARISONS IAE Input Disturbane Output Disturbane IMC-PID OPID IMC-T-FPID OT-FLCPID OIT2-FLCPID VI. CONCLUISON In this paper, the BB-BC optimization algorithm was used to optimize anteedent membership untions o an interval type- 2 uzzy PID ontroller or a nonlinear asaded two tank proess. The optimization perormanes o the IT2-FPID ontroller (20 parameters) was ompared with an optimized T-FPID ontroller (4 parameters) and an optimized PID ontroller (3 parameters) based on the IAE perormane index. Sine the number o optimization parameters o type-2 uzzy ontroller is relatively big, we preerred the BB-BC optimization method whih has a low omputational ost and a high onvergene speed. In order to make a air omparison, the rule base and saling ators are kept ixed or the uzzy ontroller strutures while only the anteedent parameters are optimized or a deined reerene trajetory. The optimization results demonstrated that the OIT2-FPID, whih has more design parameters, outperorms the optimized type- uzzy and onventional PID ontroller strutures with respet to IAE perormane index. Sine the FOU provides the anteedent type-2 uzzy sets with extra degrees o reedom, it an be onluded that an IT2-FPID is more apable o providing lower IAE perormane measure ompared to a type- ounterpart that has the same rules and saling ators. In other words, this extra degree o reedom gives the BB-BC global searh method an opportunity to ind a more optimal solution than those obtained or the optimized OPID and OT- FPID. The losed loop system perormanes o the optimized ontrollers have been also analyzed in detail. In this ontext, a detailed omparative study is presented on the highly nonlinear asaded two tank proess. The ontrol perormane results o the optimized ontrollers are also ompared with IMC-PID and IMC-T-FPID ontrollers based on perormane measures whih are settling time, overshoot and IAE. At irst, it has been illustrated that the OIT2-FPID improves the overall system perormane o the proess signiiantly or the reerene trajetory at whih the ontrollers are optimized or tuned in every sense. Then, the ability o the ontrollers to ope with noise, nonlinear dynamis and disturbanes is investigated. The perormane results o the omparative study show that the OIT2-FPID struture attenuates the disturbanes and noises as well as improves transient state o the losed loop response in dierent operating points even those at whih the ontroller was not optimized. It an be onluded that the ontrol perormane o an IT2- FPID an be improved by tuning the FOUs o its anteedent IT2-FSs. Moreover, the optimized IT2-FPID appears to be more robust against nonlinear dynamis, output noise, and input & output disturbanes in omparison with the type- uzzy and onventional PID ontroller strutures. When IT2- FPID is tuned oline, this beneit is espeially useul in order to redue the eet o unexpeted nonlinear dynamis, noises and disturbanes. Future work will ous on implementing the proposed type- 2 uzzy ontrol struture on a real-time proess and extending the optimization based design strategy or generalized type-2 uzzy logi ontroller strutures. ACKNOWLEDGMENT This researh is supported by the projet (059B ) o Sientii and Tehnologial Researh Counil o Turkey (TUBITAK). All o these supports are appreiated. REFERENCES [] N.N. Karnik, J.M. Mendel and Q. Liang, Type-2 Fuzzy Logi Systems, IEEE Transation on Fuzzy Systems, vol. 7 (6), pp , 999. [2] Q. Liang and J.M. Mendel, Interval Type-2 Fuzzy Logi Systems: Theory and Design, IEEE Transations on Fuzzy Systems, vol. 8(5), pp , [3] T. 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